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+{"text":"\\section{Introduction}\nThe systematic study of the higher finiteness properties of groups\nwas initiated forty years ago by Wall \\cite{wall} and Serre \\cite{serre}.\nIn 1963, Stallings \\cite{stall} constructed the first example\nof a finitely presented group $\\Gamma$ with $H_3(\\Gamma;\\mathbb Q)$ infinite\ndimensional; his example was a subgroup of a direct product of three\nfree groups. \nThis was  the first indication of the\ngreat diversity to be found amongst the finitely presented subgroups\nof direct products of free groups, a theme developed in \\cite{bieri}.\n\nIn contrast, Baumslag and Roseblade \\cite{BR}\n proved that in a direct product of two\nfree groups the only finitely presented subgroups are the \n{\\em{obvious ones}}: \nsuch a subgroup is either free or has a subgroup of finite index\nthat is a direct product of free groups. \nIn \\cite{bhms} the present\nauthors explained this contrast  by proving that\n the exotic behaviour among the finitely presented\nsubgroups of direct products of free groups is accounted for entirely \nby the failure of higher\nhomological-finiteness conditions. In particular, we\nproved that the only subgroups $S$ of type ${\\rm FP}_n$ in a \ndirect product of $n$ free\ngroups are the obvious ones: if $S$ intersects each of the direct factors\nnon-trivially, it virtually splits as the direct product of these intersections.\nWe also proved that this splitting phenomenon persists when one replaces\nfree groups by the fundamental groups of compact surfaces \\cite{bhms};\nin the light of the work of Delzant and Gromov \\cite{DG}, \nthis has significant implications for  the structure of \nK\\\"ahler groups.\n\nExamples show that the splitting phenomenon for ${\\rm FP}_\\infty$\nsubgroups does not extend to products of more general\n2-dimensional hyperbolic groups or\nhigher-dimensional Kleinian groups \\cite{mb:haef}. But recent work\nat the confluence of logic, group theory and topology has brought to\nthe fore a class of groups that is more profoundly tied to surface\nand free groups than either of the above classes, namely {\\em{limit groups}}.\n\nLimit groups arise naturally from several points of view. They are\nperhaps most easily described as the finitely generated\ngroups $L$ that are {\\em{fully residually free}} (or \n{\\em{$\\omega$--residually free}}):\n for any finite subset $T\\subset L$ there exists a homomorphism\nfrom $L$ to a free group that is injective on $T$. \nIt is in this guise that limit groups were studied extensively by \nKharlampovich and Myasnikov \\cite{KM1,KM2,KM3}.\nThey are also known as {\\em{$\\exists$-free groups}} \\cite{res}, reflecting the\nfact that these are precisely the finitely generated groups that have\nthe same existential theory as a free group.\n\nMore geometrically,  limit groups are the\nfinitely generated groups that have a Cayley graph in which each\nball of finite radius is isometric to a ball of the same radius in\nsome Cayley graph of a free group of fixed rank.\n\nThe name {\\em{limit\ngroup}} was introduced by Sela. His original definition\ninvolved a certain limiting action on an $\\mathbb R$-tree, but \nhe also emphasized that these are\nprecisely the groups that arise when one takes limits of \nstable sequences of homomorphisms $\\phi_n:G\\to F$, where $G$ is an arbitrary\nfinitely generated group and $F$ is a free group;  {\\em{stable}} means that\nfor each $g\\in G$ either $I_g=\\{n\\in\\mathbb{N} : \\phi_n(g)=1\\}$ or \n $J_g=\\{n\\in\\mathbb{N} : \\phi_n(g)\\neq 1\\}$ is finite, and the {\\em{limit}} of $(\\phi_n)$\nis the quotient of $G$ by $\\{g\\mid |I_g|=\\infty\\}$.\n\n\nIn his account \\cite{S9}\nof the outstanding problems concerning limit groups, Sela\nasked whether the main theorem of \\cite{bhms} could be extended to cover\nlimit groups. The present article represents the culmination of a project\nto prove that it can. Building on ideas and results from \\cite{bhms,\nbh1, bh2, bh3, BM1} we prove:\n\n\\begin{thmA}\\label{main} If $\\Gamma_1,\\dots,\\Gamma_n$ are limit\ngroups and $S\\subset \\Gamma_1\\times\\cdots\\times\\Gamma_n$ is a subgroup\nof type ${\\rm FP}_n(\\mathbb{Q})$, then $S$ is virtually a direct product of  $n$\nor fewer limit groups.\n\\end{thmA}\n\nCombining this result with the fact that every finitely generated residually\nfree group can be embedded into a direct product of finitely many limit groups\n(\\cite[Corollary 2]{KM2}, \\cite[Claim 7.5]{S1}), we obtain:\n\n\\begin{corollary}\\label{FPinfty} Every residually free group of type\n${\\rm FP}_\\infty(\\mathbb{Q})$ is virtually a direct product of a finite number of limit groups.\n\\end{corollary}\n\nB.~Baumslag \\cite{benjy} proved that a \nfinitely generated,\nresidually free group is fully residually free (i.e.~a limit group)\nunless it contains a subgroup isomorphic to $F\\times\\mathbb{Z}$, where $F$\nis a free group of rank 2. Corollary\n\\ref{FPinfty} together with the methods used to prove Theorem \\ref{main}\nyield the following generalization of Baumslag's result:\n\n\\begin{corollary} Let $\\Gamma$ be a residually free group of type ${\\rm FP}_n(\\mathbb{Q})$\nwhere $n\\ge 1$, let $F$ be a free group of rank 2 and let\n$F^n$ denote the direct product of $n$ copies of $F$. Either $\\Gamma$\ncontains a subgroup isomorphic to $F^n\\times\\mathbb{Z}$ or else $\\Gamma$ is\nvirtually a direct product of $n$ or fewer limit groups.\n\\end{corollary}\n\nWe also prove that if a subgroup of a direct product \nof $n$ limit groups \nfails to be of type ${\\rm FP}_n(\\mathbb{Q})$, then\none can detect this failure in the homology of a subgroup of finite\nindex. \n\n\\begin{thmA}\\label{split}\nLet $\\Gamma_1,\\dots,\\Gamma_n$ be limit groups \nand  let $S\\subset \\Gamma_1\\times\\cdots\\times\\Gamma_n$ be a \nfinitely generated subgroup\nwith \n$L_i=\\Gamma_i\\cap S$ non-abelian for $i=1,\\dots,n$.\n\nIf $L_i$ is finitely generated for $1\\le i\\le r$ and not finitely\ngenerated for $i>r$, then there is a subgroup of finite index $S_0\\subset S$\nsuch that $S_0=S_1\\times S_2$, where $S_1$ is the direct\nproduct of the limit groups $S_0\\cap\\Gamma_i,\\ i\\le r$ and (if $r<n$) \n$S_2=S_0\\cap(\\Gamma_{r+1}\\times\\cdots\\times\\Gamma_n)$ has\n$H_k(S_2;\\mathbb Q)$  infinite dimensional for some $k\\le n-r$.\n\\end{thmA}\n\nIn Section \\ref{s:last} we shall prove a more technical version\nof Theorem \\ref{split} and account for abelian intersections.\n  \n Theorems \\ref{main} and \\ref{split} are \nthe exact analogues  of  Theorems A and B of \\cite{bhms}. \nIn Section  \\ref{reductions} \nwe introduce a sequence of reductions that will allow\nus to deduce both theorems from the following result \n(which, conversely, is \nan easy consequence of Theorem \\ref{split}). We remind the reader\nthat a subgroup of a direct\nproduct is called a {\\em{subdirect product}} if its projection to each \nfactor is surjective. \n\n\\begin{thmA}\\label{main3} Let $\\Gamma_1,\\dots,\\Gamma_n$ be non--abelian limit groups \nand  let $S\\subset \\Gamma_1\\times\\cdots\\times\\Gamma_n$ be a \nfinitely generated subdirect product which intersects each factor non-trivially. \nThen either :\n\\begin{enumerate}\n\\item\n$S$ is of finite index; {\\em{or}}\n\n\\item $S$ is of infinite index and has a finite-index subgroup \n$S_0 < S$ such that $H_j(S_0;\\mathbb{Q})$ has infinite $\\mathbb{Q}$-dimension\nfor some $j\\leq n$.\n\\end{enumerate}\n\\end{thmA} \n  \n \nFor simplicity of exposition, the homology of a group $G$ in this\npaper will almost always be with coefficients in a $\\mathbb{Q}\\,G$-module -- typically the\ntrivial module $\\mathbb{Q}$.  But with minor modifications, \nour arguments also apply\nwith other coefficient modules, giving corresponding results under\nthe finiteness conditions ${\\rm FP}_n(R)$ for other suitable rings $R$. \n\nA notable aspect of the proof of the above theorems is that following\na raft of reductions based on geometric methods, the proof takes\nan unexpected twist in the direction of nilpotent groups.\nThe turn of events that leads us in\nthis direction is explained in Section \\ref{s:elements}\n-- it begins with a\nsimple observation about higher commutators from \\cite{BM1}\nand proceeds via a spectral sequence argument.\n\n\n\nSeveral of our results shed light on the nature of arbitrary finitely\npresented subgroups of direct products of limit groups, most notably\nTheorem 4.2. These results suggest that there is\na real prospect of understanding all such subgroups, i.e.~all\nfinitely presented residually free groups. We  take up this\nchallenge in  \\cite{bhms3}, where we describe precisely which subdirect\nproducts of limit groups are finitely presented and present a solution\nto the conjugacy and membership problems for these subgroups\n(cf.~\\cite{BM1}, \\cite{BW}). But\nthe isomorphism problem for finitely presented residually free groups remains\nopen, and beyond that lie many further challenges.\nFor example, with applications of the surface-group case to K\\\"ahler \ngeometry in mind \\cite{DG},\none would like to know if all finitely presented subdirect\nproducts of limit groups satisfy a polynomial isoperimetric inequality. \n\n\\iffalse\nSuch an\nunderstanding certainly entails a calculation of the \nBieri-Neumann-Strebel invariants\nof direct products of limit groups -- see for example \\cite{meinert,MMvW,gehrke}.\nThe BNS invariants, however, apply only to subgroups that contain\nthe commutator subgroup, so further methods will need to be developed\nto cope with the general case, and many challenges remain. \n\\fi \n\n\n\\smallskip\n\nWe are grateful to the many colleagues with whom we have had useful\ndiscussions at various times about aspects of this work, particularly\nEmina Alibegovi\\'c, Mladen Bestvina, Karl Gruenberg, Peter Kropholler, Zlil Sela\nand Henry Wilton. We are also grateful to the anonymous referee\nfor his\/her helpful comments.\n\n\\section{Limit groups and their decomposition}\n\nSince this is the fourth in a series of papers on limit groups\n(following \\cite{bh1,bh2,bh3}), we shall only recall the minimal\nnecessary amount of information about them. The reader unfamiliar\nwith this fascinating class of groups should consult \nthe introductions in \\cite{AB,BF}, the original papers of\nSela \\cite{S1,S2,S6},  or those of\n Kharlampovich and Myasnikov \\cite{KM1,KM2,KM3} where\nthe subject is approached from a perspective more in keeping with\ntraditional combinatorial group theory; a further\n perspective is developed in \\cite{CG}.\n \n\\subsection{Limit groups}\n\nOur results rely  on  the fact that limit groups\nare the finitely generated subgroups of \n$\\omega$-residually free tower ($\\omega$-rft)\ngroups \n\\cite[Definition 6.1]{S1} (also known as NTQ-groups\n\\cite{KM2}).  A useful summary of Sela's proof\nof this result was given by Alibegovi\\'c and Bestvina in\nthe appendix to \\cite{AB} (cf.~\\cite[(1.11), (1.12)]{S2}).\nThe equivalent result of Kharlampovich and Myasnikov\n \\cite[Theorem 4]{KM2} is presented in a more algebraic manner. \n\nAn $\\omega$-rft group is the fundamental group of a\ntower space  assembled from graphs, tori\nand surfaces in a hierarchical manner. The number of\nstages in the process of assembly is the {\\em{height}}\nof the tower. Each stage in the construction involves\nthe attachment of an orientable surface along its boundary, or\nthe attachment of an $n$-torus $T$  along an\nembedded circle representing a primitive element\nof $\\pi_1T$. (There are additional constraints in each\ncase.)\n\nThe {\\em{height}} \nof a limit group $\\Gamma$ is the minimal height of an \n$\\omega$-rft group\nthat has a subgroup isomorphic to $\\Gamma$. Limit\ngroups of height $0$ are free products of \nfinitely many free abelian groups (each of finite rank)\nand surface\ngroups of Euler characteristic at most $-2$.\n\n\nThe splitting described in the following proposition is obtained as follows:\nembed $\\Gamma$ in an $\\omega$-rft group $G$ of minimal height, take  the\ngraph of groups decomposition of $G$  that   the Seifert-van Kampen\nTheorem associates to the addition of the final block in the tower, \nthen apply Bass-Serre theory to get an induced graph of \ngroups decomposition  of $\\Gamma$.\n\n\nRecall that a graph-of-groups decomposition is termed\n{\\em{$k$-acylindrical}} if in the action on the associated\nBass-Serre tree, the stabilizer of each geodesic edge-path of length\ngreater than $k$ is trivial; if the value of $k$ is unimportant,\none says simply that the decomposition is {\\em{acylindrical}}.\n\n\\begin{prop}\\label{graph} If $\\Gamma$ is a freely-indecomposable\nlimit group of height $h\\ge 1$, then it is the fundamental group of a\nfinite graph of groups that has infinite cyclic edge groups and has a\nvertex group that is a non-abelian limit group of height $\\le h-1$.\nThis decomposition \nmay be chosen to be 2-acylindrical.\n\\end{prop}\n \nNote also that any non-abelian limit group of height $0$  splits\nas $A\\ast_C B$ with $C$ infinite-cyclic or trivial, and this splitting\nis 1-acylindrical for surface groups, and 0-acylindrical for free products.\n \n\\subsection{The class of groups ${\\mathcal C}$}\n\nWe define a class of finitely presented groups\n${\\mathcal C}$ in a hierarchical manner; it is the union of the\nclasses ${\\mathcal C}_n$ defined as follows.\n\n\nAt {\\em{level $0$}} we have the class ${\\mathcal C}_0$ consisting\nof free products $A\\ast B$ of non-trivial,\nfinitely presented groups, where at least one of $A$ and $B$\nhas cardinality at least $3$ -- in other words, all finitely\npresented non-trivial free products with the exception of $\\mathbb{Z}_2*\\mathbb{Z}_2$. \n \nA group lies in ${\\mathcal C}_n$ if and only if it is the fundamental group\nof a finite, acylindrical graph of finitely presented groups, where all of the\nedge groups are cyclic, and at least one of the vertex groups\nlies in ${\\mathcal C}_{n-1}$.\n\n\\bigskip\n\nThe following is an immediate consequence of Proposition \\ref{graph}.\n\n\\begin{cor}\\label{LinC}\nAll non-abelian limit groups lie in ${\\mathcal C}$.\n\\end{cor}\n\n\\subsection{Other salient properties}\\label{ss:props}\n\nIn the proof of Theorems \\ref{main} and \\ref{split}, the\nonly properties of limit groups $\\Gamma$ that will be needed are the\nfollowing.\n\\begin{enumerate}\n\\item Limit groups are finitely presented, \nand their finitely generated subgroups\nare limit groups.\n\\item \nIf $\\Gamma$ is non-abelian, it lies in ${\\mathcal C}$ (Corollary \\ref{LinC}).\n\\item Cyclic subgroups are\nclosed in the profinite topology on $\\Gamma$.\n (This is true for\nall finitely generated subgroups \\cite{wilt}.)\n\\item If a subgroup $S$ of $\\Gamma$ has finite dimensional $H_1(S;\\mathbb Q)$,\nthen  $S$ is finitely generated (and hence is a \nlimit group) \\cite[Theorem 2]{bh1}.\n\\item Limit groups are of type ${\\rm FP}_\\infty$ (in fact $\\mathcal F_\\infty$).\nThis follows directly from the fact that the class of limit groups\ncoincides with the class of constructible limit groups, \\cite[Definition 1.14]{BF}.\n\\end{enumerate} \n\n\n\n\\subsection{Notation}\n Throughout this paper, we consistently use the notational convention that\n$S$ is a subgroup of the\ndirect product of the limit groups $\\Gamma_i$ ($1\\le i\\le n$),  that\n$L_i$ denotes the intersection $S\\cap\\Gamma_i$, and that\n$p_i : \\Gamma_1\\times\\dots\\times \\Gamma_n\\to\\Gamma_i$ is the coordinate\nprojection.\n\n\n\\subsection{Subgroups of finite index} \\label{ss:fi}\nThroughout the proof Theorems \\ref{main}, \\ref{split}\nand \\ref{main3}\nwe shall repeatedly pass to subgroups of finite\nindex $H_i\\subset\\Gamma_i$. When we do so, we shall assume\nthat our original subgroup\n$S$\nis replaced by $p_i^{-1}(H)\\cap S$ and \nthat each $\\Gamma_j$ ($j\\neq i$) is replaced\nby $p_jp_i^{-1}(H_i)$. This does not affect the intersections\n$L_j= S \\cap \\Gamma_j$ ($j\\neq i$). \n\n\nRecall \\cite[VIII.5.1]{ksbrown} that the property ${\\rm FP}_n$ \nis inherited by finite-index subgroups and persists in finite\nextensions. In the proof of Theorem \\ref{main3}\nwe detect the failure of property\n${\\rm FP}_n$ by considering the homology of subgroups of finite\nindex: if $H_k(S_1;\\mathbb{Q})$ is infinite dimensional for some $S_1<S$\nof finite index, then neither $S$ nor $S_1$ is of type ${\\rm FP}_k$.\n\nSome care is required here because one cannot conclude\nin the previous sentence that $S$ has an infinite-dimensional\nhomology group: the finite-dimensionality\nof homology groups is a property that persists in finite extensions \nbut is not, in general, inherited by finite-index subgroups. \nIn  the context of the proof\nof Theorem \\ref{main3}, care has been taken to ensure\nthat each passage to a finite-index subgroup\nrespects this logic.\n\n\n\n\\section{Reductions of the main theorem}\\label{reductions}\n\nThe following proposition reduces Theorem \\ref{main} \nto Theorem \\ref{main3}.\n\n\\begin{prop}\\label{assume}\nTheorem \\ref{main} is true if and only if it holds under the following\nadditional assumptions.\n\\begin{enumerate}\n\\item $n\\ge 2$.\n\\item Each projection $p_i:S\\to\\Gamma_i$ is surjective.\n\\item Each intersection $L_i=S\\cap\\Gamma_i$ is non-trivial.\n\\item Each $\\Gamma_i$ is a non-abelian limit group.\n\\item Each $\\Gamma_i$ splits as an HNN-extension over a cyclic subgroup $C_i$\nwith stable letter $t_i\\in L_i$.\n\\end{enumerate}\n\\end{prop}\n\n\\begin{pf}\n\\noindent (1) The case $n=0$ of Theorem \\ref{main} is trivial.  \n\nIn the case $n=1$, $S<\\Gamma_1$ has type ${\\rm FP}_1(\\mathbb{Q})$, so is finitely generated.\nBut a finitely generated subgroup of a limit group is again a limit group,\n and there is nothing more to prove.\n(The case $n=2$ was proved in \\cite{bh3} but an independent proof is\ngiven below.)\n \n\\smallskip\\noindent (2)  Since $S$ has type ${\\rm FP}_n(\\mathbb{Q})$ it is finitely generated,\nhence so is $p_i(S)$  and we can  replace  each $\\Gamma_i$ by $p_i(S)$.\n \n\\smallskip\\noindent (3) If, say, $L_n$ is trivial, then the projection map\n$q_n:S\\to\\Gamma_1\\times\\cdots\\times\\Gamma_{n-1}$ is injective, and $S$\nis isomorphic to a subgroup $q_n(S)$ of $\\Gamma_1\\times\\cdots\n\\times\\Gamma_{n-1}$.\nAfter iterating  this argument,  we may assume that each\n$L_i$ is non-trivial.\n\n\\smallskip\\noindent (4) Suppose that one or more of the $\\Gamma_i$ is abelian. \nA group is an abelian limit group if and only if it is free abelian of finite\nrank.  Hence a direct product of finitely many abelian limit groups is again\nan abelian limit group.  This reduces us to the case where precisely\none of the $\\Gamma_i$ -- say $\\Gamma_n$ -- is abelian.\n\nNow, replacing $\\Gamma_n$ by a finite index subgroup if necessary, we may\nassume that $L_n\\subset\\Gamma_n$ is a direct factor of $\\Gamma_n$:\nsay $\\Gamma_n=L_n\\oplus M$.  Since $M\\cap S$ is trivial, the projection\n$\\Gamma_1\\times\\cdots\\times\\Gamma_n\\to\\Gamma_1\\times\\cdots\n\\times\\Gamma_{n-1}\\times L_n$ \nwith kernel $M$ maps $S$ isomorphically onto a subgroup $T$\nof $\\Gamma_1\\times\\cdots\\times\\Gamma_{n-1}\\times L_n$.  Since \n$L_n\\subset T$, it follows that $S\\cong T=U\\times L_n$ for some\nsubgroup $U$ of $\\Gamma_1\\times\\cdots\\times\\Gamma_{n-1}$.  \nBut then $U$\nhas type ${\\rm FP}_n(\\mathbb{Q})$, since $S$ does, and if Theorem \\ref{main} holds\nin the case where all the $\\Gamma_i$ are non-abelian, then $U$ is virtually\na direct product of $n-1$ or fewer limit groups.  But then $S\\cong U\\times L_n$\nis virtually a direct product of $n$ or fewer limit groups, so Theorem \\ref{main}\nholds in full generality.\n\n\\smallskip\\noindent (5)  \nThe subgroup $L_i$ of $\\Gamma_i$ is normal by (2) and non-trivial by (3).\nHence it contains an element\n$t_i$ that acts hyperbolically on the tree of the splitting described in \nProposition \\ref{graph}\n(see \\cite[Section 2]{bh2}).  \nThen by\n\\cite[Theorem 3.1]{bh2}, $t_i$ is the stable letter in some HNN decomposition\n(with cyclic edge-stabilizer)  of a finite-index subgroup $\\Delta_i\\subset\\Gamma_i$.\n\nReplacing each $\\Gamma_i$ by the corresponding subgroup $\\Delta_i$, and\n$S$ by $S\\cap (\\Delta_1\\times\\cdots\\times\\Delta_n)$, gives us the desired\nconclusion. \n\n(The above argument extends to all groups in ${\\mathcal C}$ under the additional hypothesis that\nthe edge groups in the splittings defining ${\\mathcal C}$  are all closed in the profinite topology.)\n \\end{pf}\n\n\\section{The elements of the proof of Theorem \\ref{main3}}\\label{s:elements}\n\nWe have seen that Theorem \\ref{main} follows  from\nTheorem \\ref{main3}.\nThe proof of Theorem \\ref{main3} extends from Section   \\ref{fgnormal} to \nSection \\ref{finalstep}. \nIn the present section we give an overview of the contents of these sections\nand indicate how they will be assembled to complete the proof.\n\nIn Section \\ref{fgnormal} we prove the following extension of the basic result that\nnon-trivial, finitely-generated normal subgroups of non-abelian limit groups have\nfinite index \\cite{bh1}.\n\n\\begin{theorem}\\label{index}\nLet $\\Gamma$ be a group in ${\\mathcal C}$, and $1\\ne N<G<\\Gamma$ with $N$ normal \nin $\\Gamma$ and $G$ finitely generated. \n Then $|\\Gamma:G|<\\infty$. \n\\end{theorem}\n\nUsing this result, together with the HNN decompositions of the\n$\\Gamma_i$ described in Proposition \\ref{assume}, \nwe deduce (Section \\ref{vna}):\n\n\\begin{theorem}\\label{propvna}\nLet $\\Gamma_1,\\dots,\\Gamma_n$ be non-abelian limit groups.\nIf $S\\subset \\Gamma_1\\times\\cdots\\times\\Gamma_n$ \nis a finitely generated subgroup with $H_2(S_1;\\mathbb{Q})$ finite dimensional\nfor all finite-index subgroups $S_1<S$,\nand if $S$\nsatisfies  conditions (1) to (5) of Proposition \\ref{assume},\nthen:\n\n\\begin{itemize}\n\\item the image of each projection $S\\to\\Gamma_i\\times\\Gamma_j$ \nis of finite index in $\\Gamma_i\\times\\Gamma_j$;\n\\item the quotient groups\n$\\Gamma_i\/L_i$ are virtually nilpotent\nof  class at most $n-2$.\n\\end{itemize}\n\\end{theorem}\n\nWe highlight the case $n=2$. \n\n\n\\begin{cor}\\label{cor2factor} \nIf $\\Gamma_1$ and $\\Gamma_2$ are non-abelian limit groups, \nand $S<\\Gamma_1\\times\\Gamma_2$\nis a subdirect product  intersecting each factor non-trivially, \nwith $H_2(S_1;\\mathbb{Q})$ finite dimensional\nfor all finite-index subgroups $S_1<S$,\n then $S$ has finite index in $\\Gamma_1\\times\\Gamma_2$.\n\\end{cor}\n\nAn important special case of Theorem \\ref{main3},\nconsidered in Section \\ref{kernelZ},\narises where $S$ is the kernel of an epimorphism\n$\\Gamma_1\\times\\cdots\\times\\Gamma_n\\to\\mathbb{Z}$.\n\n\\begin{theorem}\\label{theoremkernelZ}\nLet $\\Gamma_1,\\dots,\\Gamma_n$ be non-abelian limit groups, and\n$N$ the kernel of an epimorphism\n$\\Gamma_1\\times\\cdots\\times\\Gamma_n\\to\\mathbb{Z}$.  \nThen there is a subgroup of finite index $N_0\\subset N$\nsuch that \nat least one of the homology groups $H_k(N_0;\\mathbb{Q})$ ($0\\le k\\le n$)\nhas infinite $\\mathbb{Q}$-dimension. \n\\end{theorem}\n\n\nWe complete the proof of Theorem \\ref{main3} in Section \\ref{finalstep}.\nWe have seen that each of the $\\Gamma_i\/L_i$ is virtually nilpotent.\nSetting $\\Gamma=\\Gamma_1\\times\\cdots\\times\\Gamma_n$ and \nnoting that $S$ contains the product $L=L_1\\times\\cdots\\times L_n$,\nwe argue by induction on the difference in Hirsch lengths $d=h(\\Gamma\/L)-h(S\/L)$\nto prove that $H_k(S;\\mathbb{Q})$ has infinite $\\mathbb{Q}$-dimension for some $k\\le n$\nif $d>0$. \nThe initial step of the induction is provided by Theorem \\ref{theoremkernelZ},\nand the inductive step is established using the LHS spectral sequence.\nSection \\ref{s:last} contains a proof of Theorem \\ref{split}.\n\n\n\\section{Subgroups containing normal subgroups}\\label{fgnormal}\nIn this section we prove Theorem \\ref{index}.\nWe assume that the reader is familiar with Bass-Serre theory\n\\cite{Serre2},\nwhich we shall use freely. \nAll our actions on trees are without inversions.\n\n\\begin{lemma}\\label{lemmacocompact}\nLet $\\Delta$ be a group acting $k$-acylindrically, \ncocompactly and minimally on a tree $X$.\nLet $H$ be a finitely generated subgroup of $\\Delta$.\nSuppose that $M<H$ is a non-trivial subgroup which is normal in $\\Delta$.\nThen the action of $H$ on $X$ is cocompact.\n\\end{lemma}\n\n\\begin{proof}\nIf $X$ is a point there is nothing to prove, so we may assume that\n$X$ has at least one edge.\nBy hypothesis, $\\Delta$ has no global fixed point in its action on $X$.\nBy \\cite[Corollary 2.2]{bh2}, the non-trivial normal subgroup $M<\\Delta$ contains\nelements which act hyperbolically on $X$, and the union of the axes of all such elements\nis the unique minimal $M$-invariant subtree $X_0$ of $X$.\nSince $M$ is normal in $\\Delta$, the $M$-invariant subtree $X_0$ is also\ninvariant under the action of $\\Delta$.  But $X$ is minimal as a $\\Delta$-tree,\nso $X_0=X$.\n\nWe have shown that $M$ acts minimally on $X$. Since $M<H$, it\nfollows that $H$ acts minimally on $X$, so the quotient graph of\ngroups $\\mathcal G$ has no proper sub-(graph of groups) such that the inclusion induces an\nisomorphism on $\\pi_1$. A standard argument in Bass-Serre theory shows that\nsince $H$ is finitely\ngenerated, the topological graph underlying $\\mathcal G$ is compact, as claimed.\n\\end{proof}\n\n\\begin{prop}\\label{propindex}\nLet $\\Gamma\\in{\\mathcal C}$, and let $C,G$ be \nsubgroups of $\\Gamma$ with $C$ cyclic and $G$ finitely\ngenerated.  If $|G\\backslash\\Gamma\/C|<\\infty$,\nthen $|\\Gamma:G|<\\infty$.\n\\end{prop}\n\n\\begin{pf}\nLet $\\Gamma$ be a group in ${\\mathcal C}$.\nWe argue by induction on the level $\\ell=\\ell(\\Gamma)$ in the hierarchy \n$\\mathcal C=\\cup_n{\\mathcal C}_n$ where $\\Gamma$ first appears. \nBy definition,  $\\Gamma$ has a non-trivial,\n$k$-acylindrical, cocompact\naction on a tree $T$, with cyclic edge stabilizers.\nWithout loss of generality we can suppose that this action is minimal.\n \nIf $\\ell=0$ there is a single orbit of edges, the edge stabilizers are trivial and the vertex stabilizers are non-trivial.\nIf $\\ell>0$ the edge-stabilizers are non-trivial and the stabilizer of some vertex $w$ is in ${\\mathcal C}_{\\ell-1}$.\n\nLet $c$ be a generator for $C$.\nWe treat the initial and inductive stages of the argument simultaneously, \nbut distinguish two cases according to the action\nof $c$.\n\n\\smallskip\\noindent{\\bf Case 1.} Suppose that $c$ fixes a vertex $v$ of $T$.\n\nThen, by our double-coset hypothesis, the $\\Gamma$-orbit of $v$ consists of only\nfinitely many $G$-orbits $Gv_i$.  Since the action of $\\Gamma$ on $T$\nis cocompact, there is a constant $m>0$ such that $T$ is the $m$-neighbourhood\nof $\\Gamma v$, and hence the quotient graph $X=G\\backslash T$ is the \n$m$-neighbourhood of the finitely many vertices $Gv_i$. In other\nwords,  $X$ has finite diameter.  \n\nNote also that $\\pi_1X$ has finite rank, because it is a retract\nof $G$ which is finitely\ngenerated.\n\nFinally, note that $X=G\\backslash T$ has only finitely many valency 1 vertices.  \nFor otherwise, we can deduce a contradiction\nas follows.  Since $G$ is finitely generated, if there are infinitely\nmany vertices of valency 1, then the induced graph-of-groups\ndecomposition of $G$ is degenerate, in the sense that there\nis a valency 1 vertex $\\bar u$ with $G_{\\bar u}=G_{\\bar e}$,\nwhere $\\bar e$ is the unique edge of $G\\backslash T$\nincident at $\\bar u$.\n\nNow $\\bar u=Gu$ for some vertex $u$ in $T$,\n and $\\bar e=Ge$ for an edge $e$ incident at $u$.\nThe group $G_{\\bar u}$\nis the stabilizer of $u$ in $G$, and $G_{\\bar e}$ is the stabilizer\nof $e$ in $G$.  The fact that $\\bar u=Gu$ has valency $1$ in \n$G\\backslash T$ means that $G_{\\bar u}$ acts transitively on the link\n{\\rm{Lk}} of $u$ in $T$.  Hence $|{\\rm{Lk}}|=|G_{\\-u}:G_{\\-e}|=1$, so\n$u$ is a valency\n$1$ vertex of $T$.  But this contradicts the fact that $T$ is minimal\nas a $\\Gamma$-tree.\n\n\nWe have shown that $X=G\\backslash T$ has finite diameter, finite rank,\nand only finitely many vertices of valency 1.  It follows that $X$\nis a finite graph. \n\nIn the case where $\\Gamma$ has level $\\ell=0$, the stabilizer\n$\\Gamma_e$ of any edge $e$ of $T$ is trivial. The number of edges\nin $X=G\\backslash T$ that are images of edges $\\gamma e\\in\\Gamma e$\ncan therefore be counted as $|G\\backslash\\Gamma\/\\Gamma_e|=|G\\backslash\\Gamma|=|\\Gamma:G|$.\nHence, in this case, $|\\Gamma:G|<\\infty$, as required.\n\nIn the case where $\\ell>0$, there is a vertex $w$ of $T$ whose\nstabilizer $\\Gamma_w$ in $\\Gamma$ is a group in ${\\mathcal C}_{\\ell-1}$.\nLet $\\Gamma_e$ denote the stabilizer of some edge\n$e$ incident at $w$.  Then\n$|(G\\cap\\Gamma_w)\\backslash\\Gamma_w\/\\Gamma_e|$ is bounded above\nby the finite number of edges of\n$X=G\\backslash T$ incident at $Gw\\in G\\backslash T$\nthat are images of edges $\\gamma e\\in \\Gamma e$. \nBy inductive hypothesis, $G\\cap\\Gamma_w$ has finite index in\n$\\Gamma_w$.  Similarly, for each $\\gamma\\in\\Gamma$,\n$G\\cap\\gamma\\Gamma_w\\gamma^{-1}$ has finite index in\n$\\gamma\\Gamma_w\\gamma^{-1}$.  \nConsider the action of $\\Gamma_w$ by right multiplication on $G\\backslash\\Gamma$: the orbits\nare\nthe double cosets $G\\backslash\\Gamma\/ \\Gamma_w$ and hence are finite in number\nbecause they index a subset of the vertices of $X=G\\backslash T$; moreover the\nstabilizer of $G\\gamma$ is $\\gamma^{-1} G \\gamma \\cap \\Gamma_w$, which we have just\nseen has finite index in $\\Gamma_w$.  Thus $G\\backslash\\Gamma$ is finite.\n\n\n\\smallskip\\noindent{\\bf Case 2.}\nSuppose that $c$ acts hyperbolically on $T$, with\naxis $A$ say. \n\nThen the double coset hypothesis implies that the axes\n$\\gamma(A)$, for $\\gamma\\in\\Gamma$, belong to only finitely many \n$G$-orbits.   On the other hand, the convex hull of \n$\\bigcup_{\\gamma\\in\\Gamma}\\gamma(A)$ is a $\\Gamma$-invariant subtree of $T$, \nand hence by minimality is the whole of $T$.  \n\n\nLet $T_0$ be the minimal $G$-invariant subtree of $T$.  If $T_0=T$\nthen $X=G\\backslash T$ is finite since $G$ is finitely generated, and so\n$|G\\backslash\\Gamma\/\\Gamma_e|<\\infty$ for any edge-stabilizer \n$\\Gamma_e$ in $\\Gamma$.  \nIf $\\ell=0$, then $\\Gamma_e$ is trivial, so $|\\Gamma:G|<\\infty$.  \nOtherwise, choose  \n $e$ incident at a vertex $w$ whose stabilizer $\\Gamma$\n is in ${\\mathcal C}_{\\ell-1}$ and apply the inductive hypothesis as above\n to deduce that $|\\Gamma:G|<\\infty$.\n\nIt remains to consider the case $T_0\\ne T$.\n\nNow, for any subgraph $Y$ of $T$, and any $g\\in G$, \nwe have $$d(g(Y),T_0)=d(g(Y),g(T_0))=d(Y,T_0).$$\nSince the $\\Gamma$-orbit of $A$ contains only finitely many\n$G$-orbits, there is a global upper bound $K$, say, on\n$d(\\gamma(A),T_0)$ as $\\gamma$ varies over $\\Gamma$.\n\nSince $T\\neq T_0$ and $T$ is spanned by the $\\Gamma$-orbit of $A$, \nthere is a \ntranslate $\\gamma(A)$ of $A$ that is not contained in $T_0$.  \nRecall that the action is $k$--acylindrical.\nChoose a vertex $u$ on $\\gamma(A)$ with $d(u,T_0)>K+k+2$\nand let $\\Gamma_u$ denote its stabiliser in $\\Gamma$. Let $p$ be the\nvertex a distance $K$ from $T_0$ on the unique shortest \npath from $T_0$ to $u$. Since $d(\\gamma(A),T_0)\\le K$,\nthe geodesic $[p,u]$ is contained in $\\gamma(A)$. Similarly,\n$[p,u]$ is contained\nin any translate of $A$ that passes through $u$.\nIn particular, if $\\delta\\in\\Gamma_u$ \nthen $[p,u]\\subset\\delta\\gamma(A)$, and since $\\delta$\nfixes $u$ we have $\\delta(p)=p$ or $\\delta(p')=p$,\nwhere $p'$ is the unique point of $\\gamma(A)$ other than\n$p$ with $d(u,p)=d(u,p')$. \n\nIf $\\delta$ fixes the edge of $[p,u]$\nincident at $u$, then $\\delta(p)=p$ hence $\\delta$ fixes\n$[p,u]$ pointwise, which  contradicts the $k$-acylindricality of\nthe action unless $\\delta=1$. Thus the stabiliser of this edge\nis trivial, which is a contradiction  unless $\\ell=0$.\n\nIf $\\ell=0$ then, replacing $u$ by an adjacent vertex if\nnecessary, we may assume that $|\\Gamma_u|>2$.\nChoose distinct non-trivial elements $\\delta_1,\\delta_2\\in\\Gamma_u$.\nIt cannot be that all three of $\\delta_1,\\delta_2,\\delta_1\\delta_2^{-1}$\nsend $p'$ to $p$. Thus one of them fixes $p$, hence\n$[p,u]$, which again contradicts the $k$-acylindricality of\nthe action.\n\n\n\\end{pf}\n\nWe are now able to complete the proof of Theorem \\ref{index}.\n\n\\begin{pfof}{Theorem \\ref{index}}\nSuppose that $\\Gamma\\in{\\mathcal C}$, $G<\\Gamma$ is finitely generated, and $N$ is a non-trivial\nnormal subgroup of $\\Gamma$ that is contained in $G$. \n Then by definition of ${\\mathcal C}$, $\\Gamma$ acts non-trivially,\n cocompactly and $k$-acylindrically on\na tree $T$ with cyclic edge stabilizers.  There is no loss of generality\nin assuming the action is minimal, so we may apply \nLemma \\ref{lemmacocompact} to see that the \naction of $G$ is cocompact.  \nThe stabilizer $\\Gamma_e$ in $\\Gamma$\nof an edge $e$ is cyclic, and the finite number of edges in\n$G\\backslash T$ is an upper bound on $|G\\backslash\\Gamma\/\\Gamma_e|$.\nIt follows from Proposition \\ref{propindex} that $|\\Gamma:G|<\\infty$, as claimed.\n\\end{pfof}\n\n\\section{Nilpotent quotients}\\label{vna}\n\nIn this section we prove Theorem \\ref{propvna}, which steers us\naway from the study of groups acting on trees and into the realm\nof nilpotent groups.\n\nWe first prove a general lemma (from \\cite{BM1})\nabout a subdirect product $S$ of $n$ arbitrary (not necessarily limit) groups $\\Gamma_1,\\dots,\\Gamma_n$.  As before, we write $L_i$ for\nthe normal subgroup $S\\cap\\Gamma_i$ of $\\Gamma_i$.\nWe also introduce the following notation.  We write $K_i$ for the\nkernel of the $i$-th projection map $p_i:S\\to\\Gamma_i$, \nand $N_{i,j}$ for the image of $K_i$ under the $j$-th projection \n$p_j:S\\to\\Gamma_j$.  \nThus $N_{i,j}$ is a normal subgroup of $\\Gamma_j$.\n\n\nWe shall denote by  $[x_1,x_2,\\dots,x_n]$ the  left-normed $n$-fold commutator   \n$[[..[x_1,x_2],x_3],\\dots],x_n]$.\n \n\n\\begin{lemma}\\label{multicomm} \n$[N_{1,j},N_{2,j},\\dots,N_{j-1,j},N_{j+1,j},\\dots,N_{n,j}]\\subset L_j$.\n\\end{lemma}\n\n\\begin{pf} Suppose that $\\nu_{i,j}\\in N_{i,j}$ for a fixed $j$ and for all $i\\ne j$.\nThen\n there exist $\\sigma_i\\in S$ with $p_i(\\sigma_i)=1$ and $p_j(\\sigma_i)=\\nu_{i,j}$.  \nLet $\\sigma$ denote the $(n-1)$-fold commutator\n$[\\sigma_1,\\dots,\\sigma_{j-1},\\sigma_{j+1},\\dots,\\sigma_n]\\in S$.  Then\n$p_j(\\sigma)$ is the $(n-1)$-fold commutator\n$$[\\nu_{1,j},\\dots,\\nu_{j-1,j},\\nu_{j+1,j},\\dots,\\nu_{n,j}]\\in\\Gamma_j.$$\n\nOn the other hand, for $i\\ne j$, we have $p_i(\\sigma)=1$ since $p_i(\\sigma_i)=1$.\nHence $\\sigma\\in L_j$, and $p_j(\\sigma)=\\sigma\\in L_j$.\n\nSince the choice of $\\nu_{i,j}\\in N_{i,j}$ was arbitrary, we have \n$$[N_{1,j},N_{2,j},\\dots,N_{j-1,j},N_{j+1,j},\\dots,N_{n,j}]\\subset L_j$$\nas claimed.\n\\end{pf}\n\n\nWe now consider a finitely generated\nsubdirect product $S$ of non-abelian\nlimit groups $\\Gamma_1,\\dots,\\Gamma_n$\nsuch that  $H_2(S_1;\\mathbb{Q})$ is finite dimensional for every\nfinite-index subgroup $S_1<S$.\n  \nLet $L_i,C_i$ and $t_i$ be as in Proposition \\ref{assume}.\nWe consider the image $A_{i,j}:=p_j(p_i^{-1}(C_i))$ \nunder the projection\n$p_j$ of the preimage under $p_i$ of the cyclic group $C_i$.  \nClearly $N_{i,j}<A_{i,j}<\\Gamma_j$.\n\nIn the remainder of this section we shall prove that $N_{i,j}\\subset \\Gamma_j$\nis of finite index for all $i$ and $j$. Lemma \\ref{multicomm}\n then implies that\n$\\Gamma_i\/L_i$ is virtually nilpotent of class at most $n-2$,\nas is claimed in Theorem \\ref{propvna}. \n\nAs a first step towards showing that $N_{i,j}\\subset \\Gamma_j$\nis of finite index, we prove the following lemma.\n\n\\begin{lemma}\\label{comms}\nLet $\\Gamma_1,\\dots,\\Gamma_n$  be non-abelian limit groups.\nIf $S < \\Gamma_1\\times\\cdots\\times\\Gamma_n$ \nis a finitely generated subgroup\nwith $H_2(S;\\mathbb{Q})$ finite dimensional, and if $S$\nsatisfies  conditions (1) to (5) of Proposition \\ref{assume},\nthen for all $i,j$:\n\\begin{enumerate}\n\\item $|\\Gamma_j:A_{i,j}|<\\infty$;\n\\item $A_{i,j}\/N_{i,j}$ is cyclic.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{pf}\n(1) It suffices to consider the case $i=1$.\nThe HNN decomposition $\\Gamma_1= B_1*_{C_1}$ described in\nProposition \\ref{assume} (5) pulls back to an HNN\ndecomposition of $S$ with stable letter $\\widehat t_1=(t_1,1,\\dots,1)$, \nbase group $\\widehat {B_1}={p_1}^{-1}(B_1)$, and\n amalgamating subgroup $\\widehat C_1={p_1}^{-1}(C_1)$.\nAs $C_1$ is cyclic, $\\widehat C_1=K_1\\rtimes\\<\\widehat c_1\\>$\nwhere $\\widehat c_1$ is a choice of a lift of a generator of $C_1$.\nConsider the Mayer-Vietoris sequence for the HNN decomposition \nof $S$.\n$$\\cdots\\to H_2(S;\\mathbb{Q})\\to H_1(\\widehat C_1;\\mathbb{Q}) \\overset{\\phi}\\to \nH_1(\\widehat{B_1};\\mathbb{Q}) \\to H_1(S;\\mathbb{Q})\\to\\cdots$$\nThe map $\\phi$ is the difference between the map \ninduced by inclusion and the map  induced by the \ninclusion twisted by the action of $\\widehat t_1$ by conjugation.\nNotice that $\\widehat t_1$ commutes with $K_1$ \nand so acts trivially on $H_*(K_1;\\mathbb{Q})$. Thus $\\phi$\nfactors through the map $H_1(\\widehat C_1;\\mathbb{Q})\\to H_1(\\langle\\widehat c_1\\rangle;\\mathbb{Q})$,\nin particular the image of $\\phi$ has dimension at most 1. \nSince $H_2(S;\\mathbb{Q})$ is finite dimensional by hypothesis, \nit follows that $H_1(\\widehat C_1;\\mathbb{Q})$ is finite dimensional.\nFor each $j$, $A_{1,j}=p_j(\\widehat C_1)$ is a homomorphic image\nof $\\widehat{C_1}$, so $H_1(A_{1,j};\\mathbb{Q})$ is finite-dimensional.\nSince $A_{1,j}$ is a subgroup of the non-abelian limit group $\\Gamma_j$,\nit follows that it is finitely generated.  Since it\ncontains the non-trivial normal subgroup $L_j$,\nTheorem \\ref{index} now implies that $A_{1,j}$\nhas finite index in $\\Gamma_j$, as claimed.\n\n(2) As $p_j$ is surjective,\n  $A_{i,j}\/N_{i,j}=p_j(\\widehat C_i)\/p_j(K_i)$ is a homomorphic\nimage of $\\widehat C_i\/K_i$, so it is also cyclic, as claimed.\n\\end{pf}\n\nThe other crucial ingredient in the proof of Theorem \\ref{propvna}\nis the following proposition.\n \n\\begin{prop}\\label{surprising}\n Let $G$ be an HNN extension of the form\n$B\\ast_C$ with stable letter $t$, finitely generated base-group\n$B$ and infinite-cyclic edge group $C$. Suppose that $G$\nhas normal subgroups $L$ and $N$ such that $t\\in L$,\n$C\\cap N=\\{1\\}$ and $G\/N$ is infinite-cyclic. Suppose further\nthat $H_1(N;\\mathbb{Q})$ is infinite dimensional. Let $\\Delta\\subset G$ be the\nunique subgroup of index 2 that contains $B$. Then, there\nexists an element $x\\in L\\cap B\\cap N$ such that \n$R\\overline x\\subset H_1(N\\cap \\Delta;\\mathbb{Q})$ is a free $R$-module\nof rank 1, where $R=\\mathbb{Q}[\\Delta\/(N\\cap \\Delta)]$ and $\\overline x$\nis the homology class determined by $x$.\n\\end{prop}\n\n\\begin{proof}  Let $T$ be the Bass-Serre tree of the splitting\n$G=B\\ast_C$ and consider the graph of groups decomposition\nof $N_2:=N\\cap \\Delta$ with underlying graph $X=N_2\\backslash T$; since $N_2C$\nhas finite index in $G$, this is a finite graph. Each vertex\ngroup in this decomposition is a conjugate of $B\\cap N_2$,\nand the edge groups are trivial since $C\\cap N_2=\\{1\\}$.\n\nThus, as an abelian group, $H_1(N_2;\\mathbb{Q})$ is the direct sum of\n$H_1(X;\\mathbb{Q})$ and $p$ copies of $H_1(B\\cap N_2;\\mathbb{Q})$,\nwhere $p$ is the \nindex of $BN_2$ in $G$. The first of these summands is finite-dimensional, and hence $H_1(B\\cap N_2;\\mathbb{Q})$ is\ninfinite-dimensional (since $H_1(N;\\mathbb{Q})$ is infinite-dimensional,\nimplying that $H_1(N_2;\\mathbb{Q})$ is too). \n\nLet $\\tau$ be a generator of $G\/N$. Then $M:=H_1(B\\cap N_2;\\mathbb{Q})$ is a\n$\\mathbb{Q}[\\tau^{\\pm p!}]$-module, which is finitely generated\nbecause $B$ is finitely generated and $B\/(B\\cap N_2)$\nis finitely presented. \nSince $\\mathbb{Q}[\\tau^{\\pm p!}]$ is a principal ideal domain, the\nmodule $M$ has a free direct summand. We fix $z\\in B\\cap N_2$\nso that $\\overline z\\in M$ generates this free summand. \nIt follows that $R\\overline z$ has infinite $\\mathbb{Q}$-dimension, and so\nis a free submodule of \nthe $R$-module $H_1(N_2;\\mathbb{Q})$.\n \n Since $t\\notin \\Delta$,\n $z_1:=z$ and $z_2:=tzt^{-1}$ belong to distinct vertex groups\n in $X$.  Hence $x:=[z,t]=z_1z_2^{-1}\\in L\\cap N\\cap \\Delta$\n is such that $\\overline x=\\overline z_1-\\overline z_2$\n generates a free $\\mathbb{Q}[\\tau^{\\pm p!}]$-submodule of $H_1(N_2;\\mathbb{Q})$, and hence also a free $R$-submodule.\n \\end{proof}\n\nThe following proposition completes the proof of Theorem \\ref{propvna}.\n\n\\begin{prop} \nLet $\\Gamma_1,\\dots,\\Gamma_n$  be non-abelian limit groups.\nIf $S < \\Gamma_1\\times\\cdots\\times\\Gamma_n$ \nis a finitely generated subgroup\nwith $H_2(S_1;\\mathbb{Q})$ finite dimensional for each subgroup\n$S_1$ of finite index in $S$, and if $S$\nsatisfies  conditions (1) to (5) of Proposition \\ref{assume},\nthen (in the notation of Lemma\n\\ref{comms})\n$N_{i,j}\\subset \\Gamma_j$\nis of finite index for all $i$ and $j$.\n\\end{prop}\n\n\\begin{pf} It suffices to consider the case $(i,j)=(2,1)$.\nLet $T$ be the projection of $S$ to $\\Gamma_1\\times\\Gamma_2$, and define\n$M_i=T\\cap\\Gamma_i$ for $i=1,2$.  Notice that $M_1 = N_{2,1}$, the projection to $\\Gamma_1$\nof the kernel of the projection $p_2:S\\to\\Gamma_2$,\nand similarly $M_2=N_{1,2}$.  \n\nSince $S$ projects onto each of $\\Gamma_1$ and $\\Gamma_2$, the same is\ntrue of $T$.  Hence we have isomorphisms\n$$\\frac{\\Gamma_1}{M_1}\\cong\\frac{T}{M_1\\times M_2}\\cong\\frac{\\Gamma_2}{M_2}.$$\nWe will assume that these groups are infinite, and obtain a contradiction.\n\nBy Lemma \\ref{comms}, $T\/(M_1\\times M_2)$ is virtually cyclic,\nso we may choose a finite index subgroup $T_0<T$ containing\n$M_1\\times M_2$ such that $T_0\/(M_1\\times M_2)$ is infinite\ncyclic.  Hence $G_i:=p_i(T_0)$ is a finite-index subgroup\ncontaining $M_i$ for $i=1,2$, such that $G_i\/M_i$ is infinite\ncyclic.  Choose $\\tau\\in T_0$ such that $\\tau.(M_1\\times\nM_2)$ generates $T_0\/(M_1\\times M_2)$, and let \n$\\tau_i=p_i(\\tau)\\in G_i$\nfor $i=1,2$.\n\nThe HNN-decomposition of $\\Gamma_i$ from Proposition \\ref{assume} (5)\ninduces\n an HNN decomposition $G_i=B'_i\\ast_{C'_i}$\nwith stable letter $t'_i\\in L_i$, where $C'_i=C_i\\cap G_i$\nand $t'_i$ an appropriate power of the stable letter $t_i$ of\n$\\Gamma_i$.\nNotice that, by Lemma \\ref{comms}, $C'_i\\cap M_i=\\{1\\}$.\nFor each $i=1,2$, Proposition \\ref{surprising} \n(with $G=G_i$, $N=M_i$, $L=L_i$, $t=t'_i$, $B=B'_i$,\n$C=C'_i$) provides an index $2$ subgroup $\\Delta_i$\nin $G_i$ and an element $ x_i\\in M_i\\cap\\Delta_i\\cap L_i$ such\nthat $\\overline x_i$ generates a free $\\mathbb{Q}[\\tau_i^{\\pm 1}]$-submodule\nof $H_1(M_i\\cap\\Delta_i;\\mathbb{Q})$.\n\nNow define $M'_i:=M_i\\cap\\Delta_i$.\nIt follows that  $\\overline x_1\\otimes \\overline x_2$ generates a free $\\mathbb{Q}[\\tau_1^{\\pm 1},\\tau_2^{\\pm 1}]$-submodule of \n$$H_1(M'_1;\\mathbb{Q})\\otimes_\\mathbb{Q} H_1(M'_2;\\mathbb{Q})\\subset H_2(M'_1\\times M'_2;\\mathbb{Q}).$$\n\nLet $T_1$ be the finite-index subgroup of $T_0$ defined by\n$T_1:=(M'_1\\times M'_2)\\rtimes\\<\\tau\\>$, and let $S_1<S$ be the \npreimage of $T_1$ under the projection $S\\to T$.\nUsing the LHS spectral sequence for the short exact sequence\n$M'_1\\times M'_2\\to T_1\\to\\<\\tau\\>$, we see that \n$$H_0(\\langle \\tau\\rangle;H_2(M'_1\\times M'_2;\\mathbb{Q}))\\subset H_2(T_1;\\mathbb{Q})$$\nhas an infinite dimensional $\\mathbb{Q}$-subspace generated by the images of\n$$\\{(\\tau_1^mx_1\\tau_1^{-m})\\otimes (\\tau_2^nx_2\\tau_2^{-n});~m,n\\in\\mathbb{Z}\\}.$$\nIn particular, the image of the map\n$H_2(L_1\\times L_2;\\mathbb{Q})\\to H_2(T_1;\\mathbb{Q})$ induced by inclusion is infinite-dimensional.\nBut this contradicts the hypothesis that $H_2(S_1;\\mathbb{Q})$ is finite dimensional,\n since the inclusion $(L_1\\times L_2)\\to T_1$ factors\nthrough $S_1$. This is the desired contradiction which completes the proof.\n\\end{pf}\n\n \n\n\\section{Normal subgroups with cyclic quotient}\\label{kernelZ}\n\n\n\n\\begin{prop} If $\\Gamma_1,\\dots, \\Gamma_n$ are groups of\ntype ${\\rm FP}_n(\\mathbb{Z})$ and $\\phi:\\Gamma_1\\times\\cdots\\times\n\\Gamma_n\\to\\mathbb{Z}$ has non-trivial restriction to each\nfactor, then $H_{j}(\\ker\\phi;\\mathbb{Z})$\nis finitely generated for $j\\le n-1$. \n\\end{prop}\n\n\\begin{proof} We first prove the result in the special\ncase where the restriction of\n$\\phi$ to each factor is epic.\nThus we may write $\\Gamma_i = L_i\\rtimes\\langle t_i\\rangle$ where\n$S=\\ker\\phi$, $L_i=S\\cap\\Gamma_i$ is the kernel of $\\phi|_{\\Gamma_i}$ and $\\phi(t_i)$ is a fixed\ngenerator of $\\mathbb{Z}$.\n\nIf $n\\ge 2$\nand we fix a finite set $A_i\\subset L_i$ such that $\\Gamma_i=\\langle A_i,t_i\\rangle$,\nthen $S$ is generated by $A_1\\cup\\dots\\cup A_n\\cup\n\\{ t_1t_2^{-1},\\dots,t_1t_n^{-1}\\}$. \n\n We proceed by induction on $n$\n(the initial case $n=1$ being trivial), considering\nthe LHS spectral sequence in homology for\nthe projection of $S$ to $\\Gamma_n$,\n$$\n1\\to S_{n-1}\\to S \\overset{p_n}\\to \\Gamma_n\\to 1,\n$$\nwhere $S_{n-1}$ is the kernel of the restriction of\n$\\phi$ to $\\Gamma_1\\times\\cdots\\times \\Gamma_{n-1}$.  In particular,\nthe inductive hypothesis applies to $S_{n-1}$.\n\nSince $\\Gamma_n$ is of type ${\\rm FP}_n(\\mathbb{Z})$ and $H_q(S_{n-1};\\mathbb{Z})$ is\nfinitely generated for $q\\le n-2$, by induction, \non the $E^2$ page of the spectral sequence\nthere are only finitely generated groups in the rectangle \n $0\\le p\\le n$ and $0\\le q\\le n-2$. It follows that all of the\ngroups on the $E^\\infty$ page that contribute to $H_j(S;\\mathbb{Z})$\nwith $j\\le n-1$ are finitely generated, with the possible exception\nof that in position $(0,n-1)$.\n\nOn the $E^2$ page, the group in position $(0,n-1)$ is\n$H_0(\\Gamma_n;H_{n-1}(S_{n-1};\\mathbb{Z}))$, which is the quotient of \n$H_{n-1}(S_{n-1};\\mathbb{Z})$ by the action of $\\Gamma_n$. \nThis action\nis determined by taking a section of $p_n:S \\to \\Gamma_n$\nand using the conjugation action of $S$. The section\nwe choose is that with image $L_n\\rtimes\\langle t_1t_n^{-1}\\rangle$.\nSince $L_n$ and $t_n$ commute with $S_{n-1}$, we have\n$$H_0(\\Gamma_n;H_{n-1}(S_{n-1};\\mathbb{Z})) = H_0(\\langle t_1\\rangle;H_{n-1}(S_{n-1};\\mathbb{Z})) \\ .$$\nThe latter group is the $(0,n-1)$ term on the $E^2$ page\nof the spectral sequence for the extension \n$$1\\to S_{n-1}\\to \\Gamma_1\\times\\cdots\\times\\Gamma_{n-1}\\overset{\\phi}\\to \\mathbb{Z}\\to 1.$$\nThis is a 2-column spectral sequence, so the $E^2$ page\ncoincides with the $E^\\infty$ page. \nSince $\\Gamma_1\\times\\cdots\\times \\Gamma_{n-1}$\nis of type ${\\rm FP}_{n-1}$ (indeed of type ${\\rm FP}_{n}$), it follows that   \n$H_0(\\langle t_1\\rangle;H_{n-1}(S_{n-1};\\mathbb{Z}))$ is finitely generated, and\nthe induction is complete.\n\nFor the general case, replace $\\mathbb{Z}$ by the finite index subgroup\n$\\phi(\\Gamma_1)\\cap\\cdots\\cap\\phi(\\Gamma_n)$ ($=m\\mathbb{Z}$, say); replace each $\\Gamma_i$\nby the finite-index subgroup $\\Delta_i=\\Gamma_i\\cap\\phi^{-1}(m\\mathbb{Z})$,\nand replace $S$ by the finite-index subgroup $T=S\\cap(\\Delta_1\\times\\cdots\\times\\Delta_n)$.  Since $\\phi(\\Delta_i)=m\\mathbb{Z}$ for each $i$,\nthe above special-case argument applies to $T$, to show that\n$H_j(T;\\mathbb{Z})$ is finitely generated for each $0\\le j\\le n-1$.  Moreover,\n$T$ is normal in $S$, and we may consider the LHS spectral sequence of the short exact sequence\n$$1\\to T\\to S\\to S\/T\\to 1.$$\nOn the $E^2$ page of this spectral sequence, the terms $E^2_{pq}$\nin the region $0\\le q\\le n-1$ are homology groups of the finite\ngroup $T\/S$ with coefficients in the finitely generated modules\n$H_q(T;\\mathbb{Z})$, and so they are finitely generated abelian groups.\nBut all the terms that contribute to $H_j(S;\\mathbb{Z})$ for $0\\le j\\le n-1$\nlie in this region, so $H_j(S;\\mathbb{Z})$ is finitely generated for $j\\le n-1$,\nas required.\n\\end{proof}\n\n\n\n\\begin{theorem}\\label{thm14}\nLet $\\Gamma_1,\\dots,\\Gamma_n$ be non-abelian limit groups and\nlet $S$ be the kernel of an epimorphism $\\phi:\\Gamma_1\\times\\cdots\\times\n\\Gamma_n\\to\\mathbb{Z}$. \nIf the restriction of $\\phi$ to each of the $\\Gamma_i$\nis epic, then  $H_n(S;\\mathbb{Q})$ has infinite $\\mathbb{Q}$-dimension.\n\\end{theorem}\n\n\\begin{proof}\nThe proof is by induction on $n$. \nThe case $n=1$ was established in \\cite{bh1}: the group $S=\\ker\\phi$ is a normal subgroup of the \nnon-abelian limit group $\\Gamma_1$, and if $H_1(S,\\mathbb{Q})$ were finite dimensional\nthen $S$ would be finitely generated, and hence would have finite\nindex in $\\Gamma_1$. \n\nThe preceding proposition shows that $H_j(S;\\mathbb{Z})$ is finitely generated, and hence $H_j(S;\\mathbb{Q})$  is\nfinite dimensional for $j<n$. As in the proof of that proposition, we\nconsider  the LHS spectral\nsequence for\n$$\n1\\to S_{n-1}\\to S\\overset{p_n}\\to \\Gamma_n\\to 1,\n$$\nnow with $\\mathbb{Q}$-coefficients. There are now \nonly finitely generated $\\mathbb{Q}$-modules in the region $0\\le q\\le n-2$. \n(Recall that $\\Gamma_i$ is of type ${\\rm FP}_\\infty$.)\nIn particular, the terms on the $E^2$ page\ninvolved in the calculation of $H_n(S;\\mathbb{Q})$ are all finitely generated\nexcept for \n$$H_0(\\Gamma_n;H_{n}(S_{n-1};\\mathbb{Q})) = H_0(\\langle t_1\\rangle;H_{n}(S_{n-1};\\mathbb{Q}))$$\nand\n$$H_1(\\Gamma_n;H_{n-1}(S_{n-1};\\mathbb{Q})).$$\n\nIt suffices to prove that the latter is infinite dimensional\nover $\\mathbb{Q}$. (The former is actually finite dimensional, but this\nis irrelevant.) \n   \nThe module $M=H_{n-1}(S_{n-1};\\mathbb{Q})$ is a homology group of \nthe kernel of a map from an ${\\rm FP}_\\infty$ group to $\\mathbb{Z}$.\nIt is thus a homology group of a chain complex of\nfree $R=\\mathbb{Q}[t,t^{-1}]$ modules of finite rank. \nThe ring $R$ is Noetherian, so such a homology\ngroup is finitely generated as an $R$-module.\nBy the inductive hypothesis, $M$ has infinite $\\mathbb{Q}$-dimension. \nSo by the classification of finitely generated modules over a principal\nideal domain, \n$M$ has a free direct summand, that is $M=M_0\\oplus R$.\n\nThe $\\Gamma_n$-action on $M$ factors through \n the quotient $\\Gamma_n\\to\\Gamma_n\/L_n = \\<t_n\\>$, since\n$L_n$ acts trivially, so the direct sum decomposition \n passes to $M$ considered as a $\\mathbb{Q}\\Gamma_n$ module.\nHence $H_1(\\Gamma_n;M)=H_1(\\Gamma_n;M_0) \\oplus H_1(\\Gamma;R)$.\n\nFinally, as a $\\mathbb{Q}\\Gamma_n$ module, \n$R=\\mathbb{Q}\\Gamma_n \\otimes_{\\mathbb{Q} L_n} \\mathbb{Q}$, so by Shapiro's Lemma \n$H_1(\\Gamma_n;R) \\cong H_1(L_n;\\mathbb{Q})$ \n(see for instance \\cite[III.6.2. and III.5]{ksbrown}) . \n  \n As $L_n$ is an infinite index normal subgroup of a non-abelian\n limit group, it is not finitely generated, and therefore neither is\nthe $\\mathbb{Q}$-module  $H_1(L_n;\\mathbb{Q})$ \\cite{bh1}.\n \\end{proof} \n\nTheorem \\ref{theoremkernelZ} follows immediately from Theorem \\ref{thm14}\nin the\nlight of the K\\\"unneth formula, after one has passed to a subgroup \nof finite index to ensure that whenever $\\Gamma_i\\to \\mathbb{Z}$ is non-trivial it\nis onto.\n\n\\section{Completion of the proof of the Main Theorem}\\label{finalstep}\n\nThe following lemma and its corollary provide an extension to the \nvirtual context of known results about finitely generated nilpotent\ngroups.\nWe shall apply them to direct products of the  \nvirtually nilpotent quotients of $\\Gamma_i\/L_i$ resulting from \n Theorem \\ref{propvna}.\n\n\\begin{lemma}\\label{nilp}\nLet $G$ be a finitely generated virtually nilpotent group and \nlet $\\overline S$ be\na subgroup of infinite index.  Then there exists a subgroup $K$\nof finite index in $G$ and an epimorphism $f:K\\to\\mathbb{Z}$ such that \n$(\\overline S\\cap K)\\subset {\\mathrm{ker}}(f)$.\n\\end{lemma}\n\n\\begin{pf}\nWe argue by induction on the Hirsch length\n$h(G)$, which is strictly positive, since $G$ is infinite. \n\nIn the initial case, $h(G)=1$ means that $G$ has an infinite cyclic\nsubgroup $K$ of finite index.  \nSince $\\overline S$ has infinite index in $G$,\n$\\overline S$ is finite, so $(\\overline S\\cap K)$ is trivial, \nand we can take $f:K\\to\\mathbb{Z}$ to be an isomorphism.\n\n\\medskip\nFor the inductive step, let $H$ be a finite index torsion-free\nsubgroup of $G$, and $C$ an infinite cyclic central subgroup of\n$H$.  \nIf $C\\-S$ has infinite index in $G$, then the inductive hypothesis\napplies to $H\/C$ and we are done.  \nOtherwise, $\\overline S$ has infinite index in $C\\-S$, \nso $C\\cap \\overline S$ has infinite index in $C\\cong\\mathbb{Z}$.  \nBut then $C\\cap \\overline S=\\{1\\}$, and since $C<H$, it follows that \n$C\\overline S\\cap H=C\\times (\\overline S\\cap H)$.  \nPut $K=C\\overline S\\cap H$ and let $f$ be the projection\n$K\\to C$ with kernel $\\overline S\\cap H$. \n\\end{pf}\n\nWe note that Lemma \\ref{nilp}\nwould not remain true if one assumed only that $G$ were polycyclic. For example, it fails for\nlattices $G=\\mathbb{Z}^2\\rtimes\\langle t \\rangle$ \nin the 3-dimensional Lie group $\\rm{Sol}$\nif one takes $S=\\langle t\\rangle$.\n\n\\smallskip\nRepeated applications of Lemma \\ref{nilp} yield the following.\n \n\n\n\\begin{cor}\\label{chain}\nLet $G$ be a finitely generated, virtually nilpotent group and\nlet $\\overline S$ be a subgroup of $G$.  \nThen there is a subnormal chain\n$\\-S_0<\\-S_1<\\cdots <\\-S_r=G$, \nwhere $\\-S_0$ is a subgroup of finite index\nin $\\-S$ and for each $i$ the quotient group \n$\\-S_{i+1}\/\\-S_i$ is either finite or cyclic.\n\\end{cor}\n\n \n For the benefit of topologists, we should note that the following algebraic argument is\nmodelled on the geometric proof of the Double Coset Lemma in \\cite{bh2}.\n\n\\begin{pfof}{Theorem \\ref{main3}}\n\nLet $\\Gamma=\\Gamma_1\\times\\cdots\\times\\Gamma_n$.\nRecall that \nthe $\\Gamma_i$ are non-abelian, the projections $p_i:S\\to\\Gamma_i$\nare surjective, and the intersections $L_i=S\\cap\\Gamma_i$\nare non-trivial. \nBy passing to a subgroup of finite index we may assume that each $\\Gamma_i$\nsplits as in Proposition \\ref{assume}(5).\nLet $L=L_1\\times\\dots\\times L_n$.\n\nWe only need consider the \ncase when $S$ has infinite index in $\\Gamma$.\nWe shall \nderive a contradiction from the assumption that for\nall subgroups\n$S_0<S$ of finite index  and for all $0\\le j\\le n$,\n$H_j(S_0,\\mathbb{Q})$ is finite-dimensional.\n \nFrom Theorem \\ref{propvna} we know\nthat each of the quotient groups $\\Gamma_i\/L_i$ is virtually\nnilpotent, and hence so is $\\Gamma\/L$.\n\nSince $L\\subset S$ and $S$ has infinite index in $\\Gamma$,\n the image $\\-S$ of $S$ in $\\Gamma\/L$ is of infinite index\nand we may apply\nLemma \\ref{nilp} with $\\Gamma\/L$ in the role of $G$.\nLet $\\Lambda<\\Gamma$ be the preimage of the subgroup $K$\nprovided by the lemma. \nNote that $\\Lambda$ has finite index in $\\Gamma$,\ncontains $L$, and admits an epimorphism $f:\\Lambda\\to\\mathbb{Z}$\nsuch that $S\\cap \\Lambda\\subset\\mathrm{ker}(f)$.\nAs in (\\ref{ss:fi}), we may replace the groups\n$\\Gamma_i$ and $S$ by  finite-index \nsubgroups so as to ensure that $L\\subset S\\subset N$, \nwhere $N$  is the\nkernel of an epimorphism $\\Gamma\\to\\mathbb{Z}$.\nBy Theorem  \\ref{theoremkernelZ}, there is a finite index subgroup\n$N_0<N$ and an integer $j\\le n$ such that  $H_j(N_0;\\mathbb{Q})$ is infinite\ndimensional.  \n\nBy Corollary \\ref{chain} (applied to the image of $S\\cap N_0$ in\n$\\Gamma\/L$) there is a \nsubgroup $S_0$ contained in $S\\cap N_0$, \nwhich has finite index in $S$, \nand a subnormal chain of subgroups\n$S_0\\triangleleft S_1\\triangleleft\\cdots\\triangleleft S_k=N_0$\nwith $S_{i+1}\/S_i$ either finite or cyclic for each $i$.  \nWe now use the following lemma to contradict  the\nassumption that $H_j(S_0;\\mathbb{Q})$ is finite-dimensional. \n\n\\begin{lemma}\\label{chain2}\nLet $S_0\\triangleleft S_1$ be groups with\n $S_1\/S_0$ \nfinite or cyclic.\nIf $H_j(S_0;\\mathbb{Q})$ is finite dimensional for $0\\le j\\le n$,\nthen $H_j(S_1;\\mathbb{Q})$ is finite dimensional for  $0\\le j\\le n$.\n\\end{lemma}\n\n\\begin{proof}\nIn the LHS spectral sequence for the\ngroup extension $S_0\\to S_1\\to (S_1\/S_0)$ we have\n$E^2_{p,q}=H_p(S_1\/S_0;H_q(S_0;\\mathbb{Q}))$.\nBy hypothesis,\n$E^2_{p,q}$ has finite $\\mathbb{Q}$-dimension for $q\\le n$.  \nMoreover, $E^2_{p,q}=0$ for $p>1$, since $S_1\/S_0$ \nhas homological dimension at most $1$ over $\\mathbb{Q}$.\nThus the derivatives on the $E^2$ page all vanish and the spectral sequence stabilizes at the $E^2$ page.  \nHence, for $0\\le j\\le n$, we have\n$$\\mathrm{dim}_\\mathbb{Q}(H_j(S_1;\\mathbb{Q}))=\\mathrm{dim}_\\mathbb{Q}(E^2_{0,j})+\\mathrm{dim}_\\mathbb{Q}(E^2_{1,j-1})<\\infty,$$\nas required.\n\\end{proof}\n\nRepeatedly applying this lemma to the subnormal sequence \n$S_0\\triangleleft S_1\\triangleleft\\cdots\\triangleleft S_k=N_0$\nimplies that $H_j(N_0;\\mathbb{Q})$ is finite dimensional for all $j\\le n$, contradicting Theorem \\ref{theoremkernelZ}.\n\n\\end{pfof}\n\n\nThis completes the proof of Theorem \\ref{main3}, \nfrom which Theorem \\ref{main} follows immediately.\n\n \\section{From Theorem \\ref{main3} to Theorem \\ref{split}} \\label{s:last}\n\nLet $\\Gamma_i,L_i$ and $S$ be as in the statement of Theorem \\ref{split},\nbut without necessarily assuming that the $L_i$ are non-abelian for all $i$.  We first discuss how this situation differs from the special case\nstated in Theorem \\ref{split}.\n\nIf some $L_i$ is trivial, then $S$ is isomorphic to a subgroup\nof the direct product of the $\\Gamma_j$ with $j\\ne i$, as in Proposition\n\\ref{assume} (3).  We now assume that $L_i\\ne \\{1\\}$ for each $i$.\n\nAs in Proposition \\ref{assume} (2), we may replace each $\\Gamma_i$ by\n$p_i(S)$, where $p_i:S\\to\\Gamma_i$ is the projection, and\nhence assume that $p_i$ is surjective, and so each $L_i$ is normal in\n$\\Gamma_i$.\n\nIf some $L_i$ is non-trivial and abelian, then it is free abelian of finite\nrank, by \\cite[Corollary 1.23]{BF}.  Since $L_i$ is normal, it has finite\nindex in $\\Gamma_i$, and it follows immediately from the $\\omega$-residually\nfree property that $\\Gamma_i$ is itself abelian.\n\nArguing as in Proposition \\ref{assume} (4), we may assume that only one of\nthe $\\Gamma_i$ is abelian, say $\\Gamma_1$,  and that $L_1$ is the only\nnon-trivial abelian $L_i$.   We may also assume that $L_1$ is a direct \nfactor of $\\Gamma_1$; say $\\Gamma_1=L_1\\times M_1$.  But then $S$ virtually\nsplits as a direct product $L_1\\times S'$, where $S'=S\\cap (\\Gamma_2\\times\\cdots \\Gamma_n)$.\n\nNote that the above reduction involved only one passage to a finite index subgroup,\nand that was within the abelian factor $\\Gamma_1$.  The other $\\Gamma_i$ and $L_i$\nare left unchanged.  In particular, the $L_i$ remain non-abelian.\n\nWe have now reduced to the situation of the statement of Theorem \n\\ref{split}, with the additional hypothesis that each $p_i:S\\to\\Gamma_i$\nis surjective.\n\nIn particular, each  $L_i$ is normal in   $\\Gamma_i$, and hence\nis of finite index   for $i=1,\\dots,r$. \n\nLet\n $\\Pi_r:\\Gamma_1\\times\\cdots\\times\\Gamma_n\\to \\Gamma_1\\times\\cdots\\times\\Gamma_r$\n be the natural projection,\n let $\\Lambda =  L_1\\times\\cdots\\times L_r$\n and let $\\hat S_0=S\\cap\\Pi_r^{-1}(\\Lambda)$.\n Then $\\hat S_0$ has finite index \n in $S$ and $\\hat S_0=\\Lambda\\times \\hat S_2$, where\n  $\\hat S_2 = \\hat S_0\\cap(\\Gamma_{r+1}\\times\\cdots\\times\\Gamma_n)$.\nTheorem \\ref{main3} now says that  that $\\hat S_2$ has\n a subgroup of finite index $S_2$ with\n$H_k(S_2;\\mathbb Q)$  infinite dimensional for some $k\\le n-r$.   \n  \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nA recent class of (mostly medical) imaging modalities, called hybrid, coupled-physics, or multi-wave modalities offers the possibility to reconstruct high-contrast parameters of interest with high resolution. High contrast is important to discriminate between, say, healthy and non-healthy tissues. Resolution is important to detect anomalies at an early stage. \n\nSuch hybrid modalities typically involve two steps. In the first step, not considered in this paper, a high resolution modality takes as an input measurements performed at the boundary of a domain of interest and provides as an output internal functionals of the parameters of interest and of specific solutions of underlying partial differential equations describing the probing (medical imaging) mechanism. This paper is concerned with the second step, involving the quantitative reconstruction of the parameters from knowledge of said internal functionals. For recent books and reviews on hybrid inverse problems, we refer the reader to, e.g., \\cite{A-Sp-08,AS-IP-12,B-IO-12,KK-EJAM-08,S-SP-2011,WW-W-07}.\n\nMost practically used hybrid inverse problems involve internal functionals that are polynomials in the parameters of interest and the specific solutions mentioned above. Combined with the equations describing the latter solutions, we observe that all available information represents a coupled, often redundant, system of nonlinear partial differential equations.\n\nIn some instances, local algebraic manipulations allow us to solve such a system explicitly. In the framework of functionals of solutions to second-order equations (and not of their derivatives), we refer the reader to, e.g., \\cite{BR-IP-11,BRUZ-IP-11,BU-CPAM-13,BU-IP-10,BU-AML-12}. Such theories find applications in the quantitative step of the imaging modalities Photo-acoustic tomography, Thermo-acoustic tomography, Transient Elastography, and Magnetic Resonance Elastography; see also \\cite{CAB-IP-07,CLB-SPIE-09,MY-IP-04,MZM-IP-10,PS-IP-07,SU-IO-12}. In the framework of functionals of the gradients of solutions, which find applications in Ultrasound Modulated tomography and in Current Density Imaging, we refer the reader to, e.g., \\cite{ABCTF-SIAP-08,B-APDE-13,BBMT-13,BGM-IP-13,BGM-IPI-13,BM-LINUMOT-13,BS-PRL-10,CFGK-SJIS-09,GS-SIAP-09,KK-AET-11,MB-aniso-13,MB-IP-12,MB-IPI-12}.\n\nIn many cases, explicit algebraic inversions may not be known or may not be applicable because not enough information is available. This paper proposes a framework to address several such problems when the linearization of the coupled system is {\\em elliptic}. Hybrid inverse problems need not be elliptic; see the example of the $0$-Laplacian in \\cite{B-APDE-13,CFGK-SJIS-09} (also recalled below in section \\ref{sec:zerolap}) or the Photo-acoustic problem as treated in, e.g., \\cite{BR-IP-11,BU-IP-10}. However, when the number of internal functionals increases, the resulting hybrid system becomes more redundant and hence more likely to be elliptic. We consider such a setting in section \\ref{sec:ellipticity}. We recall that elliptic systems augmented with boundary conditions that satisfy the Lopatinskii conditions admit left-parametrices. This follows from the theory of Agmon-Douglis-Nirenberg \\cite{ADN-CPAM-59,ADN-CPAM-64} and the extensions to redundant systems by Solonnikov \\cite{S-JSM-73}. The existence of parametrices allows us to solve the linear problem up to possibly a finite dimensional space. Along with the construction of a parametrix, elliptic regularity theory provides optimal stability results for the linearization of the hybrid inverse problem. \n\nThe analysis of elliptic hybrid inverse problems was first addressed in \\cite{KS-IP-12} by means of systems of pseudo-differential operators that were shown to be elliptic in the sense of Douglis and Nirenberg. The differential systems considered in this paper simplify the analysis of boundary conditions and hence of injectivity for the linearized and nonlinear hybrid inverse problems as we now describe.\n\n\n\nThe possible existence of a finite dimensional kernel for the linearized hybrid inverse problem prevents us from determining whether the available internal functionals uniquely determine the coefficients of interest. Moreover, the dimension of the finite dimensional kernel is not stable with respect to small perturbations, which prevents us from analyzing the uniqueness and stability properties of the nonlinear hybrid problem. A powerful methodology to obtain uniqueness results in the framework of elliptic systems of equations is the notion of {\\em unique continuation}. In section \\ref{sec:ucp}, we revisit two classical notions of unique continuation. One is based on the Holmgren theorem, which we generalize to the setting of redundant systems considered in this paper. The second one is based on the use of Carleman estimates as they are formulated in Calder\\'on's uniqueness theorem. See \\cite{C-AJM-58,C-PSFD-62,H-SP-63,H-I-SP-83,H-III-SP-94,L-AM-57,N-AMS-73,Z-Birk-83} for references on these unique continuation results. Several extensions of these results, following the presentation in \\cite{N-AMS-73}, are given in the setting of redundant systems in section \\ref{sec:ucp} with proofs postponed to the appendix.\n\nOnce a reasonable uniqueness result has been obtained for the linearization of the nonlinear hybrid inverse problem, several statements about uniqueness and iterative reconstruction procedures can be formulated for the nonlinear hybrid inverse problem. A constructive fixed point iteration method and a non-constructive local uniqueness result for the nonlinear problem are presented in section \\ref{sec:nonlinear}.\n\nAs an application of the conditions of ellipticity including boundary conditions and the conditions for unique continuation, we consider the case of power density internal functionals $H_j(x)=\\gamma(x)|\\nabla u_j|^2(x)$, where $u_j$ is the solution to $\\nabla\\cdot\\gamma\\nabla u_j=0$ on an open domain $X\\subset\\Rm^n$ with boundary conditions $u_j=f_j$ on $\\partial X$ for $1\\leq j\\leq J$. For such a problem, we characterize the conditions under which the redundant problem is elliptic (for $J=2$ in dimension $n=2$ and $J=3$ in higher dimension) and analyze cases in which a unique continuation principle (UCP) applies.\n\n\n\n\\section{Inverse Problems with local internal functionals.}\n\\label{sec:ellipticity}\n\n\\subsection{Systems of nonlinear partial differential equations}\n\nLet $\\gamma$ be a set of constitutive parameters in (linear or nonlinear, scalar or systems of) partial differential equations of the form\n\\begin{equation}\n\\label{eq:pde} \n \\mL(\\gamma,u_j) =0  \\quad \\mbox{ in } X,\\qquad \\tilde \\mB u_j = f_j\\quad\\mbox{ on } \\partial X,\n\\end{equation}\nwhere $\\mL$ is a polynomial in the derivatives of the solution $u_j$ and those of $\\gamma$ on the open domain $X\\subset\\Rm^n$, with $u_j$ augmented with boundary conditions on $\\partial X$ for $1\\leq j\\leq J$. \n\nLet us now assume knowledge of the functionals\n\\begin{equation}\n\\label{eq:fct}\n\\mM(\\gamma,u_j) = H_j \\quad\\mbox{ in } X, \\qquad 1\\leq j\\leq J.\n\\end{equation}\nwhere $\\mM$ is a polynomial in the derivatives of the solution $u_j$ and those of $\\gamma$.\n\nSeveral hybrid inverse problems may be recast in this general framework. More generally, we could have knowledge of functionals of the form $\\mM(\\gamma,u_i,u_j) = H_{ij}$, or functionals $\\mM$ depending on more than two solutions $u_j$. We restrict ourselves to \\eqref{eq:fct} to simplify notation.\n\nThe above problem may thus be recast as a system of nonlinear partial differential equations for $(\\gamma,\\{u_j\\})$:\n\\begin{equation}\n\\label{eq:syst}\n\\begin{array}{rcl}\n\\mL(\\gamma,u_j) &=&0  \\quad \\mbox{ in } X,\\qquad \\tilde \\mB u_j = f_j\\quad\\mbox{ on } \\partial X, \\qquad 1\\leq j\\leq J \\\\\n\\mM(\\gamma,u_j) &=& H_j \\quad\\mbox{ in } X, \\qquad 1\\leq j\\leq J.\n\\end{array}\n\\end{equation}\n\nThe first relevant question for the inverse problem is whether the above system admits a {\\em unique solution}. Since the solutions $u_j$ are uniquely determined by knowledge of $\\gamma$, we are primarily interested in finding a unique solution to the parameters $\\gamma$. The strategy followed in \\cite{KS-IP-12} consists of writing a system of pseudo-differential equations for $\\gamma$. Considering the higher-dimensional coupled system of equations for $(\\gamma,\\{u_j\\})$ allows us to simplify the analysis of the uniqueness question for \\eqref{eq:syst}.\n\nThe second question pertains to the stability properties of the reconstruction. Provided that the solution to the inverse problem is unique, we wish to understand how perturbations in the information $\\{H_j\\}$ propagates to the reconstruction of $\\gamma$.\n\nThe uniqueness and stability properties of the system depend on the number of acquired internal functionals $J$ and on the way the medium was probed via the boundary conditions $\\{f_j\\}$. Understanding how the uniqueness and stability properties are affected by changes in $J$ and the boundary conditions $\\{f_j\\}$ is the third question we wish to (very partially) answer.\n\n\\subsection{Linearization}\n\nSome problems of the form \\eqref{eq:syst} can directly be solved as non-linear systems. For instance, when $\\mL$ is a linear second-order equation in $u_j$ and $\\mM(\\gamma,u_j)=u_j$ the solution itself, the full non-linear problem is analyzed in \\cite{BU-CPAM-13,BU-AML-12}.\n\nFor many problems in which direct reconstruction procedures may not readily be available, it is fruitful to analyze the linearization of \\eqref{eq:syst}. Neglecting boundary conditions at first, this yields\n\\begin{equation}\n\\label{eq:linsyst}\n\\begin{array}{rcl}\n\\partial_\\gamma \\mL(\\gamma,u_j) \\delta\\gamma + \\partial_u \\mL(\\gamma,u_j) \\delta u_j&=&0  \\quad \\mbox{ in } X, \\qquad 1\\leq j\\leq J\\\\\n\\partial_\\gamma \\mM(\\gamma,u_j) \\delta\\gamma + \\partial_u \\mM(\\gamma,u_j) \\delta u_j &=& \\delta H_j \\quad\\mbox{ in } X, \\qquad 1\\leq j\\leq J.\n\\end{array}\n\\end{equation}\nNote that the above differential operators may all be of different orders. With $v = (\\delta\\gamma,\\{\\delta u_j\\})$, we may recast the above system as\n\\begin{equation}\n\\label{eq:linear}\n\\mA v = \\mS,\n\\end{equation}\nfor an implicitly defined source $\\mS$. Let us assume that each $u_j$ is a scalar solution and that each $H_j$ is also a scalar information. Then $\\mA$ is a system of differential operators of size $2J\\times(J+M)$, where $M$ is the number of scalar functions describing $\\gamma$. Note that for $J<M$, the above system is under-determined. We consider here the case $J\\geq m$. In several practical problems, $J=M$ gives rise to a determined system $\\mA$ that is either not invertible, or invertible with non-optimal stability properties. It is therefore also fruitful to consider the setting with $J>M$.\n\n\n\\subsection{Ellipticity}\n\nIn applications, $\\mL$ is often a linear, elliptic, operator in the variables $\\{u_j\\}$. Adding the constraints \\eqref{eq:fct}, however, may render the coupled system \\eqref{eq:syst} or its linear version \\eqref{eq:linsyst} non-elliptic. One fruitful strategy to solve \\eqref{eq:linear} on the whole domain $X$ (with appropriate boundary conditions) is therefore to ``ellipticize\" $\\mA$, i.e., to find a number of constraints $J$ sufficiently large so that $\\mA$ is elliptic, provided that such a $J$ exists.\n\nWhat we mean by elliptic is defined as follows. For each $x\\in X$, $\\mA(x,D)_{ij}$ is a polynomial in $D=(\\partial_{x_1},\\ldots,\\partial_{x_n})$ for $1\\leq i\\leq 2J$ and $1\\leq j\\leq J+M$. We define the {\\em principal part} $\\mA_0$ of $\\mA$ in the sense of Douglis and Nirenberg \\cite{DN-CPAM-55}. For each row $1\\leq i\\leq 2J$ of the system, we associate an integer $s_i$ and for each column $1\\leq j\\leq J+M$ of the system an integer $t_j$. We normalize these integers by assuming that ${\\rm max}(s_i)=0$.\n\nWe assume that $\\mA_{ij}(x,D)$ is a polynomial in $D$ of degree not greater than $s_i+t_j$.  Then $\\mA_{0,ij}(x, D)$ is the part of the polynomial in $\\mA_{ij}(x,D)$  of degree exactly equal to $s_i+t_j$. \n\nWhen all differential operators in \\eqref{eq:linsyst} are of the same order $t$, then we may choose $s_i=0$ and $t_j=t$, in which case $\\mA_0(x,D)$ is composed of entries that are homogeneous polynomials of degree $t$ in $D$. Many practical problems arise in forms in which the differential operators in \\eqref{eq:linsyst} have different orders.\n\nWe say that $\\mA$ is {\\em elliptic} when the matrix $\\mA_0(x,\\xi)=\\{\\mA_{0,ij}(x,\\xi)\\}$, the symbol of the operator $\\mA_0$, is {\\em full-rank} (i.e., of rank $J+M$ here) for all $\\xi\\in\\Sm^{n-1}$ the unit sphere and all $x\\in\\bar X$.\n\nBeing full-rank is ``more likely\" when $J$ is large, i.e., when $\\mA$ is over-determined. It is then useful to acquire redundant information $H_j$ until \\eqref{eq:linear} above is elliptic because elliptic systems enjoy more favorable (and in fact optimal) stability estimates than non-elliptic systems.\n\n\\subsection{Lopatinskii boundary conditions}\n\\label{sec:Lop}\n\nLet us assume that we have been able to prove that $\\mA$ was a redundant elliptic system of equations. Then the system can be solved, up to possibly a finite dimensional subspace, when it is augmented by boundary conditions that satisfy the Lopatinskii criterion. This is defined as follows; see \\cite{S-JSM-73}.\n\nWe consider the system\n\\begin{equation}\n\\label{eq:linbc}\n\\mA v = \\mS\\quad\\mbox{ in } X,\\qquad \\mB v = \\phi\\quad\\mbox{ on } \\partial X,\n\\end{equation}\nwhere $\\mB(x,D)$ is a $Q\\times (J+M)$ matrix with entries $\\mB_{ij}(x,D)$ that are polynomial in $D$ for each $x\\in\\partial X$.  We denote by $b_{ij}$ the order of $\\mB_{ij}$ and by $\\sigma_i=\\max_j (b_{ij}-t_j)$. Then $\\mB_0(x,D)$ is the principal part of $\\mB$ and consists of entries $\\mB_{0,ij}(x,D)$ defined as the polynomials of $\\mB_{ij}(x,D)$ of degree exactly equal to $\\sigma_i+t_j$. \n\nThe Lopatinskii conditions are defined as follows. For each $x\\in\\partial X$, we denote by $\\nu(x)$ the outward unit normal to $X$ at $x\\in\\partial X$. We then think of $z$ as the parameterization of the half line $x-z\\nu(x)$ for $z\\geq0$. Let $\\zeta\\in\\Sm^{n-1}$ with $\\zeta\\cdot\\nu(x)=0$ and consider the system of ordinary differential equations\n\\begin{equation}\n\\label{eq:lopat}\n\\begin{array}{rcl}\n \\mA_0 (x,i\\zeta + \\nu(x) \\dr{}z) u(z) &=& 0 \\quad \\mbox{ in } z>0 \\\\[2mm]\n \\mB_0 (x,i\\zeta + \\nu(x) \\dr{}z) u(z) &=& 0 \\quad \\mbox{ at } z=0.\n\\end{array}\n\\end{equation}\nWe assume that for each $x\\in\\partial X$, the only solution to the above system such that $u(z)\\to 0$ as $z\\to\\infty$ is $u\\equiv0$. This is the Lopatinskii condition for $(\\mA,\\mB)$. We then also say that $\\mB$ covers $\\mA$.\n\nThe above conditions need to be verified for the specific problems being considered. In some situations, the boundary conditions provided by \\eqref{eq:pde} (or their linearization) generate a cover of $\\mA$. In other situations, they need to be augmented with additional boundary conditions for $\\delta u_j$ as well as for $\\delta\\gamma$ as we shall see.\n\nWhen $\\mA$ is elliptic and $\\mB$ covers $\\mA$, we say that $(\\mA,\\mB)$ is an elliptic system.\n\n\\subsection{Parametrices and stability estimates.}\n\\label{sec:param}\n\nFollowing work in \\cite{ADN-CPAM-59,ADN-CPAM-64,DN-CPAM-55} on determined systems, the case of overdetermined elliptic systems was treated in \\cite{S-JSM-73}. The salient feature of these works is that the operator $A=(\\mA,\\mB)$ admits a left-parametrix (a left-regularizer) in the following sense. Let $(\\mS,\\phi)$ in \\eqref{eq:linbc} be in the space\n\\begin{displaymath}\n \\mR(p,l) = W_p^{l-s_1}(X)\\times\\ldots\\times  W_p^{l-s_{2J}}(X) \\times W_p^{l-\\sigma_1-\\frac1p}(\\partial X) \\times \\ldots \\times W_p^{l-\\sigma_Q-\\frac1p}(\\partial X),\n\\end{displaymath}\nfor some $l\\geq0$ and $p>1$ and let us assume that $(\\mA,\\mB)$ is a bounded operator from \n\\begin{displaymath}\n v\\in \\mU(p,l) = W_p^{l+t_1}(X)\\times\\ldots\\times W_p^{l+t_{J+M}}(X)\n\\end{displaymath}\nto $(\\mA,\\mB)v=(\\mS,\\phi)\\in\\mR(p,l)$. Such is the case when the coefficients of $\\mA$ and $\\mB$ are sufficiently regular. More precisely, with $l$ sufficiently large so that $p(l-s_i)>n$ for all $1\\leq i\\leq 2J$ (to simplify; see \\cite{S-JSM-73} for slight generalizations), we assume that $\\mA_{ij}$ is a sum of homogeneous operators of degree $s_i+t_j-\\kappa$ for $0\\leq \\kappa\\leq s_i+t_j$ and that the coefficients of these operators are of class $W^{l-s_i}_p(X)$. Moreover, assuming $l$ large enough so that $p(l-\\sigma_q)>n$ as well for $1\\leq q\\leq Q$, we assume that $\\mB_{qj}$ is a sum of homogeneous operators of degree $\\sigma_q+t_j-\\kappa$ for $0\\leq \\kappa\\leq \\sigma_q+t_j$ and that the coefficients of these operators are of class $W^{l-\\sigma_q-\\frac 1p}(\\partial X)$.\n Here $W_p^s(X)$ is the standard Sobolev space of functions with $s$ derivatives that are $p$-integrable in $X$ with standard extensions for $s$ not an integer \\cite{adams}.\n\nThe main result in \\cite{S-JSM-73} is the existence of a bounded operator $R$ from $\\mR(p,l)$ to $\\mU(p,l)$ such that \n\\begin{equation}\n\\label{eq:leftparam}\nRA=I-T,\n\\end{equation}\nwhere $I$ is the indentity operator and $T$ is compact in $\\mU(p,l)$. \nWhen $1$ is not in the spectrum of $T$ so that $I-T$ is invertible, then $A$ is invertible with bounded inverse $(I-T)^{-1}R$.\nHowever, $1$ could very well be in the spectrum of $T$, in which case ${\\rm dim Ker}A$ is finite but positive.\n\nMoreover, we have the following stability estimate\n\\begin{equation}\n\\label{eq:stabell}\n\\dsum_{j=1}^{J+M} \\|v_j\\|_{W_p^{l+t_j}(X)} \\leq C \\Big( \\dsum_{i=1}^{2J} \\|\\mS_i\\|_{W_p^{l-s_i}(X)} + \\dsum_{i=1}^Q \\| \\phi_i\\|_{W_p^{l-\\sigma_i-\\frac 1p}(\\partial X)} \\Big) + C_2 \\dsum_{t_j>0} \\|v_j\\|_{L^p(X)},\n\\end{equation}\nfor some constants $C>0$ and $C_2>0$. \n\nThe presence of $C_2>0$ indicates the possibility that $A$ may not be invertible. The presence of finite dimensional kernels is a serious difficulty in the analysis of the nonlinear problem \\eqref{eq:syst} because such a dimension is not stable with respect to perturbations. What we can ensure is that for $A_1$ sufficiently small, then ${\\rm dim Ker}(A+A_1)\\leq{\\rm dim Ker}A$; see, e.g., \\cite{H-III-SP-94}.\n\nWhether we can choose $C_2=0$ above, i.e., whether $A$ is invertible, depends on lower-order terms that are not captured by the principal part $(\\mA_0,\\mB_0)$. Their analysis can prove quite complicated in practical settings and we do not follow that route here. Instead, our aim is to modify $A$ so that a unique continuation principle may be applied. In section \\ref{sec:ucp}, we augment the properly modified system $(\\mA,\\mB)v=(\\mS,\\phi)$ with additional boundary conditions, which in some cases allow us to obtain injectivity results. \n\nNote that the parametrix $R$ is clearly not unique. It is theoretically constructive, as can be seen by following the proof in \\cite{ADN-CPAM-64,S-JSM-73}. However, its practical, for instance numerical, implementation is not straightforward. The modified, higher-order, systems proposed later in the section offer a more direct numerical inversion procedure.\n\n\\subsection{Example of power-density measurements}\n\\label{sec:pdm}\nTo illustrate the theoretical result of this paper, we consider the example of the reconstruction of a scalar coefficient from knowledge of the so-called power density measurements. Consider the scalar elliptic equation\n\\begin{equation}\n\\label{eq:ellj}\n\\mL(\\gamma, u_j) := \\nabla\\cdot \\gamma\\nabla u_j =0 \\quad \\mbox{ in } \\quad X,\\qquad u_j = f_j \\quad \\mbox{ on } \\partial X,\\qquad 1\\leq j\\leq J.\n\\end{equation}\nHere, $X$ is an open domain in $\\Rm^n$ for $n\\geq2$ with smooth boundary $\\partial X$.\nThe objective is to reconstruct the scalar coefficient $\\gamma$, uniformly bounded above and below by positive constants, from knowledge of the power densities\n\\begin{equation}\n\\label{eq:pdj}\nH_j(x) = \\mM(\\gamma,u_j) := \\gamma(x) |\\nabla u_j|^2 ,\\qquad x\\in X,\\qquad 1\\leq j\\leq J,\n\\end{equation}\nwhere $u_j$ is the solution to \\eqref{eq:ellj}.\n\nThis problem and some variations have received significant theoretical and numerical analyses in recent years; see, e.g., \\cite{ABCTF-SIAP-08,B-APDE-13,BBMT-13,BS-PRL-10,CFGK-SJIS-09,GS-SIAP-09,KK-AET-11,KS-IP-12,MB-IPI-12}; generalizations for anisotropic coefficients $\\gamma$ can be found in \\cite{BGM-IP-13,BGM-IPI-13,MB-IP-12,MB-aniso-13}. Explicit reconstruction procedures exist when the number of internal functionals $J$ is sufficiently large; see \\cite{BBMT-13,BGM-IP-13,CFGK-SJIS-09,MB-IPI-12,MB-aniso-13}. The case $J=1$, which does not correspond to an elliptic system, was analyzed in \\cite{B-APDE-13}. The main features of this analysis are recalled in section \\ref{sec:zerolap}. For intermediate values of $J$, the above hybrid inverse problem may not have an explicit reconstruction but may still be modeled by a redundant elliptic system. Such a problem was also analyzed in \\cite{KK-AET-11,KS-IP-12}. The conditions of ellipticity of the system $(\\mA,\\mB)$ are described in detail in section \\ref{sec:linell}. A modified system is presented in \\ref{sec:elimell}, whereas optimal stability estimates of the form \\eqref{eq:stabell} are presented in section \\ref{sec:stabestim} for the power density measurement problem.\n\n\\subsubsection{The $0-$Laplacian when $J=1$}\n\\label{sec:zerolap}\n\nWhen $J=1$, the $2\\times2$ system of nonlinear partial differential equations is formally determined with two unknown coefficients $(\\gamma,u_1)$. The elimination of $\\gamma$ from such a system is in fact straightforward and we obtain the equation for $u:=u_1$ (with $H:=H_1$) given by\n\\begin{equation}\n\\label{eq:0Lap}\n \\nabla\\cdot \\dfrac{H(x)}{|\\nabla u|^2}\\nabla u =0 \\quad \\mbox{ in } \\quad X,\\qquad u = f \\quad \\mbox{ on } \\partial X.\n\\end{equation} \nThe above equation may be transformed as \n\\begin{equation}\n  \\label{eq:Cauchy2}\n  (I-2\\widehat{\\nabla u}\\otimes\\widehat{\\nabla u}) : \\nabla^2 u + \\nabla \\ln H\\cdot\\nabla u =0\\,\\mbox{ in } X,\\qquad u=f \\,\\,\\mbox{ and } \\,\\, \\pdr{u}{\\nu} = j \\,\\, \\mbox{ on } \\partial X.\n\\end{equation}\nHere $\\widehat{\\nabla u}=\\frac{\\nabla u}{|\\nabla u|}$ and we introduced Cauchy data on $\\partial X$ anticipating the fact that \\eqref{eq:Cauchy2} is a quasilinear strictly hyperbolic equation, at least provided that $\\widehat{\\nabla u}$ is defined. Indeed, we observe that the operator $(I-2\\widehat{\\nabla u}\\otimes\\widehat{\\nabla u}) : \\nabla^2$ is hyperbolic with respect to the (unknown) direction $\\widehat{\\nabla u}$.\n\nThe above problem is analyzed in \\cite{B-APDE-13}. The two salient features of that analysis are: (i) unique reconstructions of $u$, and hence $\\gamma$, are guaranteed only on part of the domain $X$; see \\cite{B-APDE-13}; and (ii) the stability estimates are {\\em sub-elliptic}: first-order derivatives of $u$ are controlled by the gradient of $H$ rather than second-order derivatives as would be the case if $(I-2\\widehat{\\nabla u}\\otimes\\widehat{\\nabla u}) : \\nabla^2$ was replaced by an elliptic operator. For $\\gamma$, this translates into an inequality of the following form. Let $H$ and $\\tilde H$ be two measurements corresponding to the pairs $(u,\\gamma)$ and $(\\tilde u,\\tilde\\gamma)$, respectively. Assume that the Cauchy data of $u$ and $\\tilde u$ agree on $\\partial X$. Then we find that on an appropriate (see \\cite{B-APDE-13}) subdomain $\\mO\\subset X$, we have the following stability estimate:\n\\begin{equation}\n\\label{eq:subelliptic}\n \\|\\gamma-\\tilde\\gamma\\|_{L^2(\\mO)} \\leq C \\| \\nabla H-\\nabla \\tilde H\\|_{L^2(\\mO)}.\n\\end{equation}\nAs we shall see below, this estimate is sub-optimal (with a loss of one derivative) when compared to elliptic estimates of the form \\eqref{eq:stabell}. It is however optimal for (principally normal) operators of principal type \\cite{H-SP-63}.\n\n\\subsubsection{Linearization and ellipticity}\n\\label{sec:linell}\n\nThe linearization of the above problem with $J=1$ is a hyperbolic equation. We now wish to show that redundancy in the data ($J\\geq2$) allows us to render the system elliptic under some conditions. We consider two ways to obtain elliptic systems of equations.\n\nWe first linearize the coupled system \\eqref{eq:ellj}-\\eqref{eq:pdj} about solutions $(\\gamma,u_j)$ and obtain\n\\begin{equation}\n\\label{eq:pdlinsyst}\n\\begin{array}{rcll}\n\\nabla\\cdot \\delta\\gamma\\nabla u_j + \\nabla\\cdot \\gamma\\nabla \\delta u_j &=& 0 &\\quad \\mbox{ in } \\, X\\\\\n\\delta\\gamma|\\nabla u_j|^2 + 2 \\gamma \\nabla u_j\\cdot\\nabla\\delta u_j &=& \\delta H_j &\\quad \\mbox{ in } \\, X\\\\\n\\delta u_j &=& 0 & \\quad \\mbox{ on } \\partial X.\n\\end{array}\n\\end{equation}\nWe define \n\\begin{equation}\n\\label{eq:Fj}\nF_j=\\nabla u_j\n\\end{equation}\nand {\\em assume} that $|F_j|\\geq c_0>0$ is bounded from below by a positive constant uniformly. Such an assumption is valid for and appropriate open set of boundary conditions $f_j$; see, e.g., \\cite{B-IO-12,BBMT-13,BU-CPAM-13,BU-IP-10} for details of constructions based on complex geometric optics solutions or unique continuation principles, which we do not reproduce here.\n\nLet $\\mA_J$ be the operator applied to $\\delta v=(\\delta \\gamma,\\{\\delta u_j\\})$ in the above system. Its principal part $\\mP_J$ has for (principal) symbol $\\mp_j$ a $2J\\times(J+1)$ matrix given by\n\\begin{equation}\n\\label{eq:pdpJ}\n\\mp_J(x,\\xi) =  \\left(\\begin{matrix}\n |F_1|^2  & 2 \\gamma F_1\\cdot i\\xi & \\ldots & 0 \\\\\nF_1\\cdot i\\xi  & -\\gamma|\\xi|^2 & \\ldots & 0 \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\n |F_J|^2 & 0 & \\ldots & 2 \\gamma F_J\\cdot i\\xi\\\\\nF_J\\cdot i\\xi  & 0 & \\ldots & -\\gamma|\\xi|^2\n\\end{matrix}\\right).\n\\end{equation}\nWhen $J=1$, the determinant of $\\mp_J$ is given by $\\gamma|F_1|^2(2(\\hat{F_1}\\cdot\\xi)^2-|\\xi|^2)$, which is hyperbolic with respect to $F_1$ as seen in section \\ref{sec:zerolap} above. The above system is in Douglis-Nirenberg form for $s_{2k}=1$, $s_{2k+1}=0$, $t_1=0$, $t_j=1$ for $j\\geq2$.\n\nWhen $J\\geq1$, we observe that the sub-determinants with the largest number of powers of $|\\xi|^2$ are of the form\n\\begin{displaymath}\n   \\gamma^{J}|\\xi|^{2(J-1)} |F_j|^2 q_j(x,\\xi),\\qquad q_j(x,\\xi) := 2(\\hat{F_j}(x)\\cdot\\xi)^2-|\\xi|^2.\n\\end{displaymath}\nWe thus obtain that $\\mp_J(x,\\xi)$ is injective if\nthe quadratic forms $q_j(x,\\xi)=0$ for all $1\\leq j\\leq J$ imply that $\\xi=0$. \n\\begin{definition}\n\\label{def:ellcondpj}\nDefine the quadratic forms and operators\n\\begin{equation}\n\\label{eq:qjPj}\nq_j(x,\\xi) = 2 \\big(\\hat F_j\\cdot \\xi)^2 - |\\xi|^2 ,\\qquad P_j(x,D) = \\Delta - 2 \\hat F_j\\otimes\\hat F_j : \\nabla\\otimes \\nabla.\n\\end{equation}\nHere $\\hat F_j(x)$ are vector fields of unit vectors defined on $\\bar X$. We say that the family $\\{q_j\\}$ or $\\{P_j\\}$ is elliptic at $x$ if\n\\begin{equation}\n\\label{eq:condellip}\n q_j(x,\\xi) =0 \\mbox{ for all }1\\leq j\\leq J \\quad\\mbox{ implies } \\quad \\xi=0.\n\\end{equation}\nWe say that such families are elliptic in $X$ if they are elliptic in at all points $x\\in X$. \n\\end{definition}\nWe can then prove the \n\\begin{lemma}\n\\label{lem:ellipAJ}\nWe assume that $F_j:=\\nabla u_j$ is such that $|F_j|$ is bounded from below by a positive constant on $\\bar X$ for all $1\\leq j\\leq J$.\n\nThe operator $\\mA_J$ defined in \\eqref{eq:pdlinsyst} with principal symbol given in \\eqref{eq:pdpJ} is elliptic in $\\bar X$ if and only if the above family of quadratic forms $\\{q_j\\}$ is elliptic in $\\bar X$.\n\\end{lemma}\n\\begin{proof}\n  We have already seen the sufficiency of the condition. Let us prove its necessity and assume that $\\mp_J(x,\\xi)$ is maximal rank for $x\\in X$ and $\\xi\\not=0$. This means that one determinant of $(J+1)\\times(J+1)$ sub-matrices of $\\mp_J$ is non-vanishing. Each column of $\\mp_J$ beyond the first one has two non-vanishing entries. For all $1\\leq j\\leq J$ except for one entry $j_0$, then either $-\\gamma(x)|\\xi|^2$ or $2\\gamma F_j\\cdot i\\xi$ appears as a multiplicative factor in the determinant of the sub-matrix. Since $\\gamma(x)|\\xi|^2$ never vanishes, we may discard the determinant involving $2\\gamma F_j\\cdot i\\xi$. We thus obtain that if one determinant of a sub-matrix does not vanish, then that determinant may be chosen as $\\gamma^{J}|\\xi|^{2(J-1)} |F_{j_0}|^2 q_{j_0}(x,\\xi)$. Since by assumption $\\gamma^{J}|\\xi|^{2(J-1)} |F_{j_0}|^2$ is bounded away from $0$, we observe that the injectivity of $\\mp_J$ implies that (at least) one of the quadratic forms $q_{j}(x,\\xi)$ does not vanish. Thus, $\\mA_J$ being elliptic implies that $\\{q_j\\}$ is elliptic in the sense of definition \\ref{def:ellcondpj}.\n\\end{proof}\n\nThe ellipticity of $\\mA_J$ is thus a consequence of the fact that the null cones of quadratic forms intersect only at $0$. \nWe have the following properties:\n\\begin{proposition}\n\\label{prop:ell}\n (i) Let $x\\in\\Rm^2$ and assume that $\\hat F_1(x)$ and $\\hat F_2(x)$ are neither parallel nor orthogonal. Then the corresponding $(q_1,q_2)$ in \\eqref{eq:qjPj} form an elliptic family at $x$. \n \n (ii) In dimension $n\\geq2$, let $\\hat F_1(x)$ and $\\hat F_2(x)$ be two different directions and define $\\hat F_3(x)=\\alpha\\hat F_1(x)+\\beta\\hat F_2(x)$ with $\\alpha\\beta\\not=0$ such that $|\\hat F_3(x)|=1$. Then the corresponding $(q_1, q_2, q_3)$ in \\eqref{eq:qjPj} form an elliptic family at $x$. \n \n (iii) In dimension $n\\geq3$, a family $(q_1,q_2)$ is never elliptic at a given point $x$ independent of the choice of $\\hat F_1,\\hat F_2$.\n\\end{proposition}\n\\begin{proof}\n  (i) In dimension $n=2$, it is clear that the null cones (where $q_j$ vanishes, two lines of vectors $\\xi$ in $\\Rm^2$) coincide if and only if $\\hat F_1$ and $\\hat F_2$ are either parallel or orthogonal.\n  \n(iii) The result is obvious if $\\hat F_1=\\pm \\hat F_2$. Assume otherwise and let $F_3$ be a unit vector orthogonal to $\\hat F_1$ and $\\hat F_2$. Assume $\\hat F_1\\cdot\\hat F_2\\geq0$ for otherwise change the sign of $\\hat F_2$ (which does not modify the quadratic forms $q_j$). $\\xi$ belongs to the intersection of the null cones $\\{q_j(x,\\xi)=0\\}$ for $j=1,2$ if $2(\\hat F_1\\cdot\\xi)^2=2(\\hat F_2\\cdot\\xi)^2=|\\xi|^2$. Define $\\xi=\\hat F_1+\\hat F_2+\\lambda F_3$. The first constraint is satisfied when $2(1+\\hat F_1\\cdot\\hat F_2)^2 = |\\hat F^1+\\hat F^2|^2+\\lambda^2$, i.e., when  $\\lambda=\\pm \\big(2\\hat F_1\\cdot\\hat F_2+ 2(\\hat F_1\\cdot\\hat F_2)^2)^{\\frac12}$. But then it is clear that this $\\xi\\not=0$ also belongs to the second cone so that $(q_1,q_2)$ is not elliptic.\n\n(ii) Let us assume that $\\hat F_3=\\alpha \\hat F_1+\\beta \\hat F_2$ with $\\alpha\\beta\\not=0$. Then $\\xi$ belongs to the three null cones if \n\\begin{displaymath}\n \\Big(\\dfrac{\\alpha \\hat F_1+\\beta \\hat F_2}{|\\alpha \\hat F_1+\\beta \\hat F_2|}\\cdot\\xi\\Big)^2 = \\dfrac12 |\\xi|^2 = (\\hat F_1\\cdot \\xi) ^2 = (\\hat F_2\\cdot\\xi)^2.\n\\end{displaymath}\nExpanding the first constraint, we get\n\\begin{displaymath}\n \\alpha^2(\\hat F_1\\cdot\\xi)^2 + \\beta^2(\\hat F_2\\cdot\\xi)^2 + 2\\alpha\\beta \\hat F_1\\cdot\\xi\\hat F_2\\cdot\\xi = \\frac12|\\xi|^2(\\alpha^2+\\beta^2+2\\alpha\\beta \\hat F_1\\cdot\\hat F_2).\n\\end{displaymath}\nThe last two constraints imply that $2\\hat F_1\\cdot\\xi\\hat F_2\\cdot\\xi=\\eps|\\xi|^2$ with $\\eps=\\pm1$. Combined with the latter, they also imply that\n\\begin{displaymath}\n\\alpha\\beta \\eps = \\alpha\\beta \\hat F_1\\cdot\\hat F_2.\n\\end{displaymath}\nSince $\\alpha\\beta\\not=0$, this can only occur if $\\hat F_1=\\pm\\hat F_2$, which is a contradiction.\n\\end{proof}\n\nFrom the practical point of view, this result says that if $F_1=\\nabla u_1$ and $F_2=\\nabla u_2$ are not parallel, then the internal functionals $H_j$ for $u_1$, $u_2$, and $u_1+u_2$ (with $F_3=\\nabla (u_1+u_2)=\\nabla u_1+\\nabla u_2$) generate three quadratic forms $q_j$, $1\\leq j\\leq 3$ that form an elliptic family. Such internal functionals are obtained by choosing three boundary conditions of the form $f_1$, $f_2$, and $f_3=f_1+f_2$. This result holds for all $n\\geq2$.\n\n\\subsubsection{Sufficient conditions for ellipticity} We have thus obtained the following result. In dimension $n=2$, $J\\geq2$ is necessary for $\\mA_J$ to be elliptic. Moreover, $J=2$ is sufficient when $n=2$ if $\\nabla u_1$ and $\\nabla u_2$ are nowhere parallel or orthogonal. In dimension $n\\geq3$, $J\\geq3$ is necessary for $\\mA_J$ to be elliptic. Moreover, $J=3$ is sufficient for $\\mA_J$ to be elliptic in all dimensions $n\\geq2$ by choosing as boundary conditions, e.g.,  $(f_1,f_2,f_1+f_2)$ provided that $\\nabla u_1$ and $\\nabla u_2$ are nowhere parallel.\n\n\\subsubsection{Boundary conditions and Lopatinskii condition.} In order to obtain an optimal theory of stability estimates, the system needs to be augmented with boundary conditions that satisfy the Lopatinskii condition. Dirichlet conditions on $\\delta u_j$ and no condition on $\\delta\\gamma$ satisfy such conditions. Indeed, we need to show that $v(z)=(\\delta\\gamma(z),\\ldots,\\delta u_J(z))\\equiv 0$ is the only solution to\n \\begin{equation}\\label{eq:pdLP}\n \\begin{array}{l}\n\\delta u_j(0)=0,\\qquad    (iF_j\\cdot\\zeta+F_j\\cdot N \\partial_z) \\delta\\gamma + \\gamma (i\\zeta+\\partial_z)^2 \\delta u_j =0 ,\\,\\\\|F_j|^2 \\delta\\gamma +2\\gamma(iF_j\\cdot\\zeta+ F_j\\cdot N \\partial_z) \\delta u_j=0,\\,\\, z>0\n\\end{array}\n\\end{equation}\nwith $v(z)$ vanishing as $z\\to\\infty$ for $N=\\nu(x)$ at $x\\in\\partial X$ and $z$ coordinate along $-N$. Eliminating $\\delta\\gamma$ as earlier, we deduce that \n\\begin{displaymath}\n \\Big( |F_j|^2 (i\\zeta+\\partial_z)^2 - 2 (iF_j\\cdot\\zeta+F_j\\cdot N\\partial_z)^2 \\Big) \\delta u_j=0\n\\end{displaymath}\nThe leading term of the above second order equation with constant coefficients is $|F_j|^2q_j(x,N)\\partial_z^2$.  If $q_j(x,N)\\not=0$ for some $j=j_0$, which is the condition for joint ellipticity described in definition \\ref{def:ellcondpj}, then the same proof showing that Dirichlet conditions cover the Laplace operator show that $\\delta u_{j_0}=0$. We then deduce that $\\delta\\gamma=0$ from the second line in \\eqref{eq:pdLP} and by ellipticity on the first line in \\eqref{eq:pdLP} that all $\\delta u_j=0$ and hence $v\\equiv0$.\n\n\nLet us define the spaces $\\mX=W^l_{p}(X)\\times W^{l+1}_p(X;\\Rm^J)$ and $\\mY^i=W^l_p(X;\\Rm^{2J})$ with $l$ large enough so that the latter spaces are all algebras; i.e., $pl>n$. We also define $\\mY^\\partial = W^{l+1-\\frac 1p}_p(X;\\Rm^{J})$ for the traces of $\\delta u_j$ on $\\partial X$, which vanish by construction. Then we observe that $(\\mA,\\mB)$ in \\eqref{eq:pdlinsyst} maps $\\mU(p,l)=\\mX$ to $\\mR(p,l)=\\mY^i\\times\\mY^\\partial$. Moreover, the coefficients $\\gamma$ and $u_j$ appearing in the definition of $\\mA$ belong to $W^l_{p}(X)$ and $W^{l+1}_p(X;\\Rm^J)$, respectively, by assumption for $\\gamma$ and by elliptic regularity for $u_j$ solution of the elliptic problem \\eqref{eq:ellj}.\n\n\\subsubsection{Elimination and ellipticity}\n\\label{sec:elimell}\n\nThe above system involves $J+1$ unknowns. Strategies to lower the dimension of the system include: (i) eliminating $\\gamma$ as we did in obtaining \\eqref{eq:0Lap}; or (ii) eliminating all $u_j$. The second strategy follows from the observation\n\\begin{displaymath}\n   \\delta u_j = L_\\gamma^{-1} (\\nabla\\cdot \\delta\\gamma \\nabla u_j),\\qquad L_\\gamma = -\\nabla\\cdot\\gamma \\nabla,\n\\end{displaymath}\nwith $L_\\gamma^{-1}$ defined by solving $L_\\gamma$ with vanishing Dirichlet conditions.\n\nThe result is a redundant system of the form $P_j\\delta\\gamma=\\delta H_j$ where $P_j$ is a pseudo-differential operator with principal symbol given by $|F_j|^2q_j(x,\\xi)$. The redundant system for $\\delta\\gamma$ is therefore elliptic under the same conditions as those for \\eqref{eq:pdpJ} above. The main difficulty is that $P_j$ is no longer local (no longer a (system of) partial differential equation). It is not clear how one may approach the question of uniqueness for such a system. The unique continuation principles presented in section \\ref{sec:ucp} do not apply directly. See \\cite{KS-IP-12} for an analysis of such a method.\n\nThe first strategy based on the elimination of $\\delta\\gamma$ preserves the differential structure of the original system since $\\delta\\gamma$ appears undifferentiated in the second equation in \\eqref{eq:pdlinsyst}. We find that\n\\begin{equation}\n\\label{eq:deltauj}\n\\nabla\\cdot\\gamma\\big(-\\nabla \\delta u_j + 2 \\hat F_j \\hat F_j\\cdot\\nabla \\delta u_j\\big) = \\nabla\\cdot \\dfrac{\\delta H_j}{|F_j|^2} F_j.\n\\end{equation}\nThe symbol for such an equation is then given by $\\gamma q_j(x,\\xi)$ so that the above equation is not elliptic, as we already know. However, the elimination of $\\delta\\gamma$ also provides the constraint\n\\begin{equation}\n\\label{eq:constraintjk}\n2\\gamma\\dfrac{1}{|F_j|^2} F_j\\cdot\\nabla \\delta u_j - \\dfrac{\\delta H_j}{|F_j|^2}  =  2\\gamma\\dfrac{1}{|F_k|^2} F_k\\cdot\\nabla \\delta u_k - \\dfrac{\\delta H_k}{|F_k|^2} \\quad 1\\leq j<k\\leq J.\n\\end{equation}\nIt turns out that the combination of \\eqref{eq:deltauj} with \\eqref{eq:constraintjk} makes the redundant system for the $\\{u_j\\}$ elliptic provided $q_j(x,\\xi)=0$ for all $j$ implies that $\\xi=0$ as above. To see this, assume that \n\\begin{displaymath}\n  q_j(x,\\xi) \\delta u_j =0,\\qquad |F_k|^2F_j\\cdot\\xi \\delta u_j = |F_j|^2 F_k\\cdot\\xi \\delta u_k.\n\\end{displaymath}\nLet $\\xi\\not=0$. Then, not all $q_j(x,\\xi)$ vanish. Assume that $q_1(x,\\xi)\\not=0$. Then $\\delta u_1=0$ so that $F_j\\cdot\\xi \\delta u_j=0$ for all $j\\geq2$. However, $F_j\\cdot\\xi$ and $q_j(x,\\xi)$ cannot vanish at the same time so that $\\delta u_j=0$. This shows the injectivity of the symbol of the redundant system, which is easily found to be in Douglis-Nirenberg form. \n\nMoreover, unlike the redundant system of strategy (ii), the above system \\eqref{eq:deltauj}-\\eqref{eq:constraintjk} is differential. It may therefore be augmented with boundary conditions that satisfy the Lopatisnkii conditions. We leave the details to the reader to verify, as we did in \\eqref{eq:pdLP}, that such conditions are satisfied for Dirichlet conditions $\\delta u_j=0$ on $\\partial X$ under the same conditions guaranteeing ellipticity in $X$. The above system for the $(\\{\\delta u_j\\})$ is therefore elliptic when the family of quadratic forms $\\{q_j\\}$ is elliptic.\n\n\\subsubsection{Stability estimates}\n\\label{sec:stabestim}\n\nBoth systems of differential equations for $(\\delta\\gamma,\\{\\delta u_j\\})$ and for $(\\{\\delta u_j\\})$ after elimination of $\\delta\\gamma$ are elliptic with Dirichlet conditions for $\\delta u_j$ under the conditions stated, e.g., in Proposition \\ref{prop:ell}. We may therefore apply the general theory of elliptic redundant systems described above and obtain a parametrix for both systems. Moreover, in both cases, we obtain the following stability estimates\n\\begin{equation}\n\\label{eq:pdstab}\n  \\|\\delta\\gamma-\\delta\\tilde\\gamma\\|_{W_p^l(X)} + \\dsum_{j=1}^J \\|\\delta u_j-\\delta\\tilde u_j\\|_{W_p^{l+1}(X)} \\leq C \\dsum_{j=1}^J \\|\\delta H_j-\\delta \\tilde H_j\\|_{W_p^l(X)} +C_2 \\dsum_{j=1}^J \\|\\delta u_j-\\delta\\tilde u_j\\|_{L_p(X)}.\n\\end{equation}\nIn other words, $\\delta\\gamma$ and $\\nabla \\delta u_j$ are reconstructed from $\\delta H_j$ with {\\em no loss} of derivative, unlike the construction in \\eqref{eq:subelliptic}. However, we are not guaranteed that any of the linear systems is indeed invertible, and hence the presence of the last term on the right-hand-size in \\eqref{eq:pdstab}. Proving the injectivity of the above systems is much more delicate than proving their ellipticity. The following section proposes some generic strategies to do so. \n\n\\subsubsection{Generalization to similar models}\n\nThe results presented above generalize to the setting where the internal functionals are of the form\n\\begin{equation}\n\\label{eq:alphaH}\nH_j(x) = \\gamma^{\\alpha} |\\nabla u_j|^2,\n\\end{equation}\nwhere $\\alpha\\geq0$. The case $\\alpha=1$ was treated above. The case $\\alpha=2$ corresponds to the setting of CDII and MREIT \\cite{NTT-Rev-11,SW-SR-11}. As shown, e.g., in \\cite{KS-IP-12,MB-IPI-12}, the coupled problem is elliptic with $J=1$ for $\\alpha>2$, is degenerate elliptic for $\\alpha=2$, and is hyperbolic for $\\alpha<2$. The cases $\\alpha\\geq2$ become elliptic for $J$ chosen sufficiently large. We do not consider these extensions further here.\n\n\\subsection{Sufficient conditions for ellipticity}\n\\label{sec:ellipticitycond}\n\nTo summarize the above derivation, we have modeled the nonlinear hybrid inverse problem as a coupled system of nonlinear partial differential equations \\eqref{eq:syst}. Its linearization is then given by \\eqref{eq:linsyst}. These systems have $J+M$ unknowns for $2J\\geq J+M$ equations. When $J$ is large compared to $M$, we expect the principal part $\\mA_0$ of $\\mA$ in the Douglis-Nirenberg sense to be elliptic since a redundant matrix is ``more likely\" to be full-rank than a less elongated matrix. \n\n\nThe matrix $\\mA_0$ is full rank if we can prove that, for well chosen boundary conditions $f_j$, the internal functionals $H_j$ are sufficiently independent. Such a result is problem-dependent. In the setting of power-density measurements, we have obtained that the boundary conditions $f_j$ were well-chosen if the quadratic form $q_j(x,\\xi)$ were jointly elliptic at each point $x\\in \\bar X$. A sufficient condition to do so is to choose two boundary conditions $f_1$ and $f_2$ such that $F_1=\\nabla u_1$ and $F_2=\\nabla u_2$ are nowhere co-linear (for the quadratic forms corresponding to the three boundary conditions $f_1$, $f_2$, and $f_1+f_2$ are then jointly elliptic).\n\nStrategies to ensure that, e.g., $\\nabla u_1$ and $\\nabla u_2$ are nowhere co-linear have been presented in, e.g., \\cite{B-APDE-13,BBMT-13,BU-IP-10,BU-CPAM-13}, see also the review \\cite{B-IO-12}. Such strategies are based on the use of Complex Geometrical Optics when the latter are available, or on the use on local construction and unique continuation properties of the operators in \\eqref{eq:pde} as described in \\cite{BU-CPAM-13}. In both settings, it is proved that qualitative properties of solutions to elliptic equations, such as for instance the independence of gradients of solutions, hold for an open set of boundary conditions $f_j$. More explicit constructions of such boundary conditions are also proposed in \\cite{BC-JDE-13}.\n\n\\section{Injectivity results for the linearized problem}\n\\label{sec:ucp}\n\nIn this section, we consider two methods to obtain injectivity of an elliptic operator $\\mA$ augmented with appropriate boundary conditions $\\mB$.\nBoth are based on replacing the redundant system $\\mA$ by its normal, determined, form $\\mA^t\\mA$, and augmenting it with appropriate Dirichlet boundary conditions that ensure its injectivity under certain assumptions. The first method invokes a Holmgren unique continuation principle while the second method is based on unique continuation principles that are consequences of Carleman estimates.\n\n\nHere, the operator $\\mA^t$ is defined such that $(\\mA^t)_{ki}=(\\mA_{ik})^t=:\\mA^t_{ik}$, where $(\\mA_{ik})^t$ is the formal adjoint to $\\mA_{ik}$ for the usual inner product $(\\cdot,\\cdot)$ on $L^2(X)$.\n\nOne difficulty with operators $\\mA$ that are elliptic in the Douglis-Nirenberg (DN) sense is that the normal operator $\\mA^t\\mA$ need not be elliptic, even in the DN sense. Consider for instance the $2\\times2$ system in one independent variable defined by $\\mA_{11}=a$, $\\mA_{12}=\\mA_{21}=\\partial_x$ and $\\mA_{22}=\\partial^2_x$, which is DN elliptic with $s_1=t_1=0$ and $s_2=t_2=1$ when $a\\not=1$. Defining $\\mC=\\mA^T\\mA$, the principal term in $\\mC$ is $\\mC_{11}=\\partial_x^2$, $\\mC_{12}=\\mC_{21}=\\partial_x^3$, $\\mC_{22}=\\partial_x^4$, which is independent of $a$ and not elliptic.\n\nOne way to ensure that $\\mA^t\\mA$ is elliptic when $\\mA$ is elliptic is to assume that $\\tau=t_j$ independent of $j$ and $s_i=0$ independent of $i$.  We then verify that the leading term in $(\\mA^t\\mA)_{kl}$ is a differential operator of degree equal to $2\\tau$ and that  $\\mA^t\\mA$ is a strongly elliptic system of size $(J+M)\\times(J+M)$; see, e.g., \\cite[Proposition 4.1.16]{T-AV-95}. The principal part of ${\\rm Det}(\\mA^t\\mA)(x,\\xi)$ is equal to ${\\rm Det}(\\mA_0^t\\mA_0)(x,\\xi)$ and is a polynomial of degree $2(J+M)\\tau$ that is uniformly bounded from below for all $x\\in \\bar X$ and $\\xi\\in\\Sm^{n-1}$ by assumption of ellipticity.\n\nThe results in \\cite{C-JMAA-91} show that an elliptic system in DN form can always be transformed into an elliptic overdetermined system of first-order equations. The procedure increases the order of the system and differentiates any row involving a term with $s_i+t_j=0$. This forces us to impose boundary conditions on the parameters $\\delta\\gamma$ and the solutions $\\delta u_j$ that may not be necessary in the definition of $(\\mA,\\mB)$ above. \n\nFor the rest of the section, we {\\em assume} that $\\mA$ has been recast into a form where $\\tau=t_j$ is independent of $j$ and $s_i=0$ is independent of $i$; for instance with $\\tau=1$ or $\\tau=2$ as described in \\cite{C-JMAA-91}. We still keep the notation $2J$ and $J+M$ for the size of the system $\\mA$ so that with this notation, $\\mA^t\\mA$ is a system of size $(J+M)\\times(J+M)$.\n\n\n\n\n\n\n\n\nLet us augment $\\mA^t\\mA$ with the Dirichlet boundary conditions\n\\begin{equation}\n\\label{eq:Dirichlet} \n\\big(\\pdr{}\\nu\\big)^q v_j =\\phi_{qj}\\quad\\mbox{ on } \\partial X, \\qquad 0\\leq q\\leq \\tau-1,\\quad 1\\leq j\\leq J+M.\n\\end{equation}\nWe recast the above constraints as $\\mD v=\\phi$ on $\\partial X$. It is proved in \\cite[p.43-44]{ADN-CPAM-64} that such boundary conditions cover $\\mA^t\\mA$, i.e., that the Lopatinskii conditions are satisfied. We thus consider the problem\n\\begin{equation}\n\\label{eq:normal}\n\\mA^t\\mA v = \\mA^t\\mS\\quad\\mbox{ in } X,\\qquad \\mD v = \\phi\\quad\\mbox{ on } \\partial X.\n\\end{equation}\nSince $N:=(\\mA^t\\mA,\\mD)$ is elliptic, the above system admits a left parametrix $G$ such that $GN=I-T$ with $T$ compact  in $\\mU(p,l)$. Moreover, we have the stability estimate\n\\begin{equation}\n\\label{eq:stabnormal}\n\\dsum_{j=1}^{J+M} \\|v_j\\|_{W_p^{l+\\tau}(X)} \\leq C \\Big( \\dsum_{j=1}^{J+M} \\|(\\mA^t\\mS)_j\\|_{W_p^{l-\\tau}(X)} + \\dsum_{j,q} \\| \\phi_{qj}\\|_{W_p^{l-\\tau+q-\\frac 1p}(\\partial X)} \\Big) + C_2 \\dsum_{j=1}^{J+M} \\|v_j\\|_{L^p(X)},\n\\end{equation}\nWe verify that \n\\begin{displaymath}\n  \\|\\sum_{i=1}^{2J} \\mA^t_{ji} S_i \\|_{W_p^{l-\\tau}(X)} \\leq C \\sum_{i=1}^{2J} \\|S_i\\|_{W_p^{l}(X)},\n\\end{displaymath}\nso that \\eqref{eq:stabnormal} may also be seen as an analog of \\eqref{eq:stabell}.\n\nOur objective is to find sufficient conditions under which the above system is injective, and hence invertible, so that $C_2=0$ in the above estimates. We start with the following simple lemma:\n\\begin{lemma}\n\\label{lem:Normal}\nLet us assume that $v$ is a solution of \n\\begin{equation}\n\\label{eq:normal0}\n\\mA^t\\mA v = 0 \\quad\\mbox{ in } X,\\qquad \\mD v = 0\\quad\\mbox{ on } \\partial X.\n\\end{equation}\nThen $v$ is a solution of \n\\begin{equation}\n\\label{eq:syst0}\n\\mA v = 0 \\quad\\mbox{ in } X,\\qquad \\mD v = 0\\quad\\mbox{ on } \\partial X.\n\\end{equation}\n\\end{lemma}\nNote that the Dirichlet conditions associated to $\\mA^t\\mA$ are now redundant for \\eqref{eq:syst0}. For example, consider $\\mA=\\Delta+c(x)$ a scalar operator. For some choices of $c(x)$ (for instance $c(x)=\\lambda$ an eigenvalue of $-\\Delta$ on $X$ with Dirichlet conditions), $\\mA$ is not invertible. However, \\eqref{eq:syst0} corresponds to $(\\Delta+c(x))v=0$ with both $v=0$ and $\\partial_\\nu v=0$ on $\\partial X$. It is then known that the unique solution to the above constraints is $v=0$ and hence $(\\mA^t\\mA,\\mD)$ is injective.\n\\begin{proof}\n We observe that the differential operator $(\\mA^t)_{ki}=(\\mA_{ik})^t=:\\mA^t_{ik}$ is of the same order $\\tau$ as $\\mA_{ik}$. Since $\\partial_\\nu^q v_k=0$ for $0\\leq q\\leq \\tau-1$, we obtain by integrations by parts that\n \\begin{displaymath}\n 0=  \\sum_{ijk}(\\mA^t_{ik}\\mA_{ij} v_j,v_k) = \\sum_i (\\sum_j \\mA_{ij} v_j, \\sum_k \\mA_{ik} v_k) =\\sum_i \\|(\\mA v)_i\\|^2.\n\\end{displaymath}\nThis implies the result.\n\\end{proof}\n\\subsection{Holmgren unique continuation and generic results}\n\\label{sec:ucph}\n\nThe above result shows that the injectivity of \\eqref{eq:normal} is a consequence of the injectivity of \\eqref{eq:syst0} with redundant boundary conditions. The operator $\\mA$ is therefore augmented with more boundary conditions than is necessary to render it elliptic. This is a possible price to pay to guarantee that the solution to the normal system \\eqref{eq:normal} is uniquely defined.\n\nIn general, however, even with redundant boundary conditions, it is not entirely straightforward to ensure that $v=0$ is the only solution to \\eqref{eq:syst0}. In this section, we consider the case where $\\mA$ is well-approximated by an operator with analytic coefficients.  We have the following result adapted from \\cite{H-I-SP-83}.\n\\begin{proposition}\\label{prop:WFanalytic}\n Let $\\mA_A(x,D)$ be a $2J\\times (J+M)$ system of differential equations with analytic coefficients in $X$ such  that $\\mA_A^t \\mA_A(x,D)$ is an elliptic operator for all $x\\in X$. Assume that $\\mA_A v=0$ in $X$. Then $v$ is analytic in $X$.\n\\end{proposition}\n\\begin{proof}\n  Since $\\mA_Av=0$, we also have that\n  \\begin{displaymath}\n   ^{co}(\\mA_A^t\\mA_A) (\\mA_A^t\\mA_A) v = {\\rm Diag}\\big({\\rm det} (\\mA_A^t \\mA_A),\\ldots,{\\rm det} (\\mA_A^t \\mA_A) \\big) v =0,\n\\end{displaymath}\nwhere $^{co}(\\mA_A^t\\mA_A)$ is the matrix of co-factors of $\\mA_A^t\\mA_A$. Since $\\mA_A^t \\mA_A$ is elliptic (a consequence of the fact that the leading term in $\\mA_A(x,\\xi)$ is of full rank $J+M$ for all $(x,\\xi)\\in\\bar X\\times\\Sm^{n-1}$), then $P={\\rm det} (\\mA_A^t \\mA_A)$ is an elliptic operator such that ${\\rm Char}P=\\emptyset$, where the characteristic set of $P$ is $\\{(x,\\xi)\\in \\bar X\\times \\Sm^{n-1},\\, P_m(x,\\xi)=0\\}$ and $P_m$ is the principal part of $P$. We deduce  from \\cite[Theorem 6.1]{H-I-SP-83} that $WF_A(v_j)\\subset {\\rm Char}P \\cup WF_A(Pv_j)=\\emptyset$ so that each $v_j$ is analytic in $X$.\n\\end{proof}\n\nFrom this, we deduce the following unique continuation result:\n\\begin{theorem}[Holmgren]\n\\label{thm:holmgren}\nLet $\\mA_A$ be as in Proposition \\ref{prop:WFanalytic} and let us assume that $\\mA_A v=0$ in $X$. Then we have:\n\n(i) Assume that $v=0$ in an open set $\\Omega\\subset\\subset X$. Then $v=0$ in $X$.\n\n(ii) Assume that $\\mD v=0$ on an open set $\\Sigma$ of $\\partial X$. Then $v=0$ in $X$.\n\\end{theorem}\n\\begin{proof}\n  Let us first prove (i). We know from Proposition \\ref{prop:WFanalytic} that the functions $v_j$ are analytic. Since they vanish on an open set, they have to vanish everywhere.\n  \n  Now (ii) is a standard consequence of (i), which we write in detail for systems. Let $x_0\\in \\Sigma$ and $V$ a sufficiently small open ball around $x_0$ where the coefficients of $\\mA_A$ are analytic and where $\\mD v=0$ on $\\Sigma\\cap V\\subset\\Sigma$. Since $\\mA_A$ is injective, we deduce that $(\\partial_n)^{\\tau_j} v_j=0$ on $\\Sigma\\cap V$ as well, as for any higher-order derivative in fact. Let us extend $v$ by $0$ on $V$. Then we verify that $\\mA_A v=0$ in $V$ (all that needed verification was that the equation was satisfied point-wise on $\\Sigma\\cap V$). But now (i) implies that $v=0$ on $V\\cup X$.\n\\end{proof}\n\nThe above result shows that $A_A=\\mA_A^t\\mA$ is injective as well. Since $R_AA_A=I-T_A$ for some left-parametrix $R_A$ and compact operator $T_A$, we deduce that $1$ is not in the spectrum of $T_A$ so that $A_A^{-1}=(I-T_A)^{-1}R_A$. As a consequence, any operator $A$ sufficiently close to $A_A$ is also invertible with a bounded inverse. We  summarize this result as:\n\\begin{corollary}\n\\label{cor:generic} Let $\\mA=\\mA_A+\\mA_1$ with $\\mA_1$ sufficiently small in operator norm from $\\prod_i W_p^{l}(X)$ to $\\prod_j W_p^{l+\\tau}(X)$. Then $\\mA$ augmented with $\\mD v=0$ is injective and $\\mA^t\\mA$ augmented with the same boundary conditions is invertible with bounded inverse. In other words, \\eqref{eq:stabnormal} holds with $C_2=0$.\n\\end{corollary}\nThe invertibility of $\\mA^t\\mA$ is therefore {\\em generic}, i.e., holds for an open and dense set of coefficients in the definition of the elliptic operator $\\mA^t\\mA$; see \\cite{SU-JFA-09}. Note, however, that the size of norm of $\\mA_1$ for which $\\mA$ is invertible depends on $\\mA_A$.\n\n\\medskip\n\nAnother similar result states that the invertibility of $\\mA^t\\mA$ is guaranteed when $X$ is a sufficiently small domain.\n\\begin{theorem}\n\\label{thm:small} Let us assume that $\\mA(0,D)$ is an elliptic operator. Then for $X$ sufficiently small, we have that $\\mA$ is injective and that $\\mA^t\\mA$ is invertible when augmented with Dirichlet conditions $\\mD v=0$. As a consequence, \\eqref{eq:stabnormal} holds with $C_2=0$.\n\\end{theorem}\n As in the previous corollary, the size of the domain $X$ depends on the bound for $(\\mA(0,D)^t\\mA(0,D))^{-1}$ with Dirichlet conditions on $\\partial X$, which is independent of $X$ as we shall see. It is therefore possible to estimate the size of the domain $X$ for which the above theory applies; see also \\cite{BM-LINUMOT-13} for a similar result with a different method of proof.\n\\begin{proof}\n We assume that all coefficients of $\\mA$ are sufficiently smooth and that $0\\in X$. Let $P=\\mA(0,D)$ be the operator of order $m=\\tau$ with coefficients frozen at $x_0=0$. Then $P^tP$ with Dirichlet conditions on $\\partial X$  is invertible as an application of Theorem \\ref{thm:holmgren}. For constant coefficients, a more precise theory applies. Let $u\\in H^m(X)$ (we restrict ourselves to Hilbert spaces to simplify notation) be such that $u=\\partial^j_\\nu u=0$ for $1\\leq j\\leq m-1$, i.e., $\\mD u=0$. We assume $\\partial X$ sufficiently smooth that we can extend $u$ by $0$ outside of $X$ and obtain a function in $H^m(\\Rm^n)$ with the same norm. Let $Y$ be an open set such that $\\bar X\\subset Y\\subset\\Rm^n$. \n \n The elliptic theory for constant coefficient operators \\cite{H-I-SP-83} provides the existence of a fundamental solution $E$ such that\n\\begin{displaymath}\n  E * P^*Pu = u .\n\\end{displaymath}\nThis implies that \n\\begin{displaymath}\n   \\|u\\|_{H^{m}(X)} = \\|u\\|_{H^{m}(Y)} = \\|E * P^*Pu\\|_{H^{m}(Y)} \\leq\\|E*P^*\\|_{{\\mathcal L}(L^2(X),H^m(Y))} \\|Pu\\|_{L^2(X)}.\n\\end{displaymath}\nThis proves the existence of a constant $C$ independent of $X$ such that \n\\begin{displaymath}\n   \\|Pu\\|_{L^2(X)} \\geq C  \\|u\\|_{H^{m}(X)}.\n\\end{displaymath}\nThis holds for any operator with constant coefficients. \n\nNow let $Q$ be an operator of order at most $m$ and of small norm $\\eps$ from $L^2(X)$ to $H^m(X)$. Then\n\\begin{displaymath}\n \\|(P+Q)u\\|_{L^2(X)} \\geq (C -\\eps) \\|u\\|_{H^m(X)}\\geq C_1 \\|u\\|_{H^m(X)},\n\\end{displaymath}\nwhich implies that $P+Q$ is injective for $\\eps$ sufficiently small.\n\nThis is applied as follows. Let $P=\\mA(x,D)$ be an elliptic operator with not-necessarily constant coefficients and define $P_0=\\mA(0,D)$ the operator with coefficients frozen at $x_0=0$. Then $P-P_0$ is an operator bounded from $H^m(X_\\eps)$ to $L^2(X_\\eps)$ with bound $C_1\\eps$ for $C_1$ independent of $\\eps$ and $X_\\eps\\subset B(x_0,\\eps)$. From the above, we deduce that\n\\begin{displaymath}\n  \\|P_0 u\\|_{L^2(X_\\eps)}\\geq C \\|u\\|_{H^m(X)}\n\\end{displaymath}\nfor all function $u\\in H^m(X)$ such that $\\mD u=0$. For $C_1\\eps<C$, we find that\n\\begin{displaymath}\n  \\|P u\\|_{L^2(X_\\eps)}\\geq C_2 \\|u\\|_{H^m(X)}\n\\end{displaymath}\nfor some $C_2>0$. This proves that $P=\\mA$ is injective on sufficiently small domains.\n\\end{proof}  \n  \n  \n\\subsection{Unique continuation principle}\n\\label{sec:ucpc}\nWhen the domain $X$ is large or when the operator $\\mA$ is not sufficiently close to an operator $\\mA_A$ with analytic coefficients, then the unique continuation results must rely on an other principle than analyticity. \n\nUnique continuation in the absence of analyticity in the coefficients is not always guaranteed, even for scalar elliptic operators, although the construction of counter-examples is not straightforward. For general references on unique continuation principles (UCP) for mostly scalar, not necessarily elliptic, operators, as well as counter-examples, see \\cite{H-III-SP-94,N-CPAM-57,N-AMS-73,Z-Birk-83} and their references.\n\nIn this section, we revisit Calder\\'on's \\cite{C-AJM-58,C-PSFD-62} result of uniqueness from Cauchy boundary data closely following the presentation in \\cite{N-AMS-73} and extend it to redundant systems. Our objective is to adapt \\cite[Theorem 5]{N-AMS-73} to specific settings of redundant systems. We first present local uniqueness results, whose proofs are postponed to the appendix, and then use classical arguments to extend them to global uniqueness results. \n\n\\subsubsection{Local uniqueness result}\n\\label{sec:lur}\n\nLet $x$ be a point in $\\Rm^{n+1}$ and $N$ be a unit vector equal to $(0,\\ldots,0,1)$ in an appropriate system of coordinates. We want to address the local uniqueness of the Cauchy problem. Assume that $L_q$ for $1\\leq q\\leq Q$ are differential operators of order $m$ and that\n\\begin{equation}\n\\label{eq:redundsyst}\n L_q u =0 \\mbox{ in } V\\cap \\{x_{n+1}>0\\},\\qquad \\partial^j_{n+1} u=0 \\mbox{ on } V\\cap \\{x_{n+1}=0\\},\n\\end{equation}\nfor $1\\leq q\\leq Q$ and $1\\leq j\\leq m-1$, \nwhere $V$ is a neighborhood of $0$. We assume that $N$ is non characteristic (at $x$) for all $L_q$, i.e., $p_q(x,N)\\not=0$ with $p_q$ the principal symbol of $L_q$. This and \\eqref{eq:redundsyst} imply that all derivatives of $u$ vanish on $\\{x_{n+1}=0\\}$ and that $u$ can be extended by $0$ on $V\\cap \\{x_{n+1}<0\\}$. The uniqueness problem may therefore be recast as: if $u$ satisfies\n\\begin{equation}\n\\label{eq:UCPsyst}\n L_q u =0 \\mbox{ in } V \\quad \\mbox{ for } 1\\leq q\\leq Q\\qquad \\mbox{ and } \\qquad u=0 \\mbox{ in } \\{x_{n+1}<0\\},\n\\end{equation}\nthen $u\\equiv0$ in a full neighborhood of $0$. \n\nWhen $Q=1$, it is known that $L=L_1$ needs to satisfy several restrictive assumptions in order for the result to hold; see \\cite{N-AMS-73}. The main advantage of the redundancy in the above system is that such assumptions need to be valid only locally (in the Fourier variable $\\xi$) for each operator, and globally collectively.\n\nChanging notation, we define $t=x_{n+1}$ and still call $x=(x_1,\\ldots,x_n)$. Then $p_q=p_q(x,t,\\xi,\\tau)$ is the principal symbol of $L_q$ for this choice of coordinates. Sufficient conditions for the uniqueness to the above Cauchy problem involve the properties of the roots $\\tau$ of the above polynomials $\\tau\\mapsto p_q(\\cdot,\\tau)$ as a function of $\\xi$. In the setting of redundant measurements, these conditions may hold for different values of $q$ for different values of $\\xi$. This justifies the following definition of our assumptions. \n\nWe assume the existence of a finite covering $\\{\\Omega_\\nu\\}$  of the unit sphere $\\Sm^{n-1}$ (corresponding to $|\\xi|=1$) such that the following holds. For each $\\nu$, there exists $q=q(\\nu)$ and $\\eps>0$ such that for each $(x,t)$ close to $0$ and each $\\xi\\in\\Omega_\\nu$, we have:\n\\begin{equation*}\n\\label{eq:condCalderon}\n\\begin{array}{rl}\n(i) & p_q(x,t,\\xi,\\tau) \\mbox{  has at most simple real roots $\\tau$ and at most double complex roots} \\\\\n(ii) & \\mbox{distinct roots $\\tau_1$ and $\\tau_2$ satisfy } |\\tau_1-\\tau_2|\\geq\\eps>0\\\\\n(iii) & \\mbox{non-real roots $\\tau$ satisfy } |\\Im \\tau|\\geq\\eps.\n\\end{array}\n\\end{equation*}\nThen we have the following result:\n\\begin{theorem}\n\\label{thm:calderonsystem}[Calder\\'on's result for redundant systems.]\nAssume that $N$ is non characteristic for the operators $L_q$ at the origin and that for a finite covering $\\{\\Omega_\\nu\\}$ of the unit sphere, (i)-(ii)-(iii) above are satisfied.   Then \\eqref{eq:UCPsyst} implies that $u=0$ in a full neighborhood of $0$. \n\\end{theorem}\nThe proof, which  closely follows that of Theorem 5 in \\cite{N-AMS-73}, is presented in the appendix.\n\nIn the above theorem, $u$ is scalar. The above proof extends to vector-valued functions $u$ when the system for $u$ is diagonally dominant. For a determined system given by a matrix $L_{ij}$ of operators of order $m\\geq1$, this means that the operators $L_{ii}$ are of order $m$ and satisfy the hypotheses of the above theorem and the operators $L_{ij}$ for $i\\not=j$ are of order at most $m-1$.  This generalizes to the setting of redundant systems $L_{ij}u_j=S_i$ where for each $i$, only one operator $L_{ij}$ is of order $m$ and for each $j$, the operators $L_{kj}$ of order $m$ collectively satisfy the hypotheses of the above theorem; see Theorem \\ref{thm:calderonsystemred:app} in the appendix, which we do not reproduce here.\n\n\nThe above theorem may not apply directly in applications. However, it serves as a component to obtain more general results. We consider one such result that finds applications in the framework of power density measurements. \n\nWe consider a setting where the leading term in the system may not be diagonal but rather upper-triangular. Unique continuation properties may still be valid provided that (a sufficiently large number of) the diagonal operators are elliptic. However, the corresponding complex roots may no longer be double. Instead, we need the stronger condition for some $\\eps>0$:\n\\begin{equation*}\n\\label{eq:condCalderon2}\n\\begin{array}{rl}\n(iv) & p_q(x,t,\\xi,\\tau) \\mbox{ has at most simple roots $\\tau$} \\mbox{ that satisfy } |\\Im \\tau|\\geq\\eps.\n\\end{array}\n\\end{equation*}\n\nThen we have the following result.\n\\begin{theorem}\n\\label{thm:2by2system}\nConsider the redundant system of equations\n\\begin{equation}\n\\label{eq:2by2syst}\n\\left(\\begin{matrix} L_1 & L_0 \\\\ L_3 & L_2 \\end{matrix} \\right) \n\\left(\\begin{matrix} u_1 \\\\ u_2\\end{matrix} \\right) = 0\\mbox{ in } V\\cap \\{x_{n+1}>0\\},\\qquad \\partial^j_{n+1} u_k=0 \\mbox{ on } V\\cap \\{x_{n+1}=0\\},\n\\end{equation}\nfor $1\\leq j\\leq m-1$ and $1\\leq k\\leq 2$, with the following assumptions. The operators $L_1$ and $L_0$ are (vector-valued) $Q_1\\times1$ operators of order $m$,  where $L_1$ satisfies the hypotheses of Theorem \\ref{thm:calderonsystem} with $Q=Q_1$. The operators $L_2$ and $L_3$ are $Q_2\\times1$ operators of order $m$ and at most $m-1$, respectively. Moreover, $L_2$ satisfies the ellipticity hypothesis (i)-(ii)-(iv) with $Q=Q_2$.\nThen $(u_1,u_2)=0$ in a full neighborhood of $0$.\n\nThe same result extends to systems of the form\n\\begin{equation}\n\\label{eq:RbyRsyst}\n\\left(L_{ij} \\right)_{1\\leq i,j\\leq R} u = 0\\mbox{ in } V\\cap \\{x_{n+1}>0\\},\\qquad \\partial^j_{n+1} u_k=0 \\mbox{ on } V\\cap \\{x_{n+1}=0\\},\n\\end{equation}\nfor $1\\leq k\\leq R$, where $L_{ij}$ is (a vector-valued operator) of order $m-1$ for $i>j$, $L_{ii}$  is (a vector-valued operator) of order $m$ that satisfies the hypotheses of Theorem \\ref{thm:calderonsystem} with an appropriate value of $Q$ when all $L_{ik}$ are of order $m-1$ for $k\\not=i$ and the ellipticity hypothesis (i)-(ii)-(iv)  with an appropriate value of $Q$ when at least one operator $L_{ik}$ for $k<i$ is of order $m$.\n\\end{theorem}\nThe proof of this theorem is also given in the appendix. Note that {\\em (i)-(ii)-(iv)}, as opposed to {\\em (i)-(ii)-(iii)}, may be seen as an ellipticity condition since the symbol $p_{q(\\nu)}$ does not vanish on $\\Omega_\\nu$. It is known that Carleman estimates are sharper in such a setting; see, e.g., \\cite{LR-JPCS-11,N-AMS-73}.\n\n\\begin{remark}\n\\label{rem:LC}\\rm Let $\\mQ$ be an invertible $Q_1\\times Q_1$ matrix and define the linear operators $\\tilde L_1=\\mQ L_1$ and $\\tilde L_0=\\mQ L_0$. Then the conclusions of Theorem \\ref{thm:2by2system} are the same as those obtained when $L_1$ and $L_0$ are replaced by $\\tilde L_1$ and $\\tilde L_0$. In other words, the results of Theorem \\ref{thm:calderonsystem} hold if $L_q$ is replaced by an equivalent linear combination. The condition that $N$ be non characteristic for $L_q$ may then be replaced by the condition that it be non characteristic for the  linear combinations $\\tilde L_q=\\mQ_{q,q'} L_{q'}$.\n\\end{remark}\n\n\\begin{remark}\\label{rem:extensions} \\rm\n If $L_0$ is an operator of order $m-1$ in the latter theorem so that the above system is diagonally dominant, then it is sufficient to assume that  $L_1$ and $L_2$ satisfy the less constraining hypotheses of Theorem \\ref{thm:calderonsystem}; see Theorem \\ref{thm:calderonsystemred:app} and the discussion below Theorem \\ref{thm:calderonsystem}.\n\\end{remark}\n\n\\subsubsection{Global uniqueness result}\n\nThe above local results extend to global unique continuation results such as \\cite[Sections 8.5-8.6]{H-I-SP-83}:\n\\begin{theorem}\n\\label{thm:globalUCP}\n Let $\\mA$ be a (redundant) system of operators of order $m$, let $R_{\\mA}\\subset\\{(x,N)\\in \\bar X\\times \\Sm^{n-1}\\}$ be the set of points where the hypotheses of either Theorem \\ref{thm:calderonsystem} or Theorem \\ref{thm:2by2system}, (and possibly their extensions in Remark \\ref{rem:extensions}) are satisfied and let us define $\\Sigma_{\\mA}= \\bar X\\times\\Sm^{n-1} \\backslash R_{\\mA}$. Let\n \\begin{displaymath}\n \\mA u =0 \\quad \\mbox{ in } X,\\qquad \\partial^j_\\nu u =0 \\quad \\mbox{ on } \\partial X, \\,\\, 0\\leq j\\leq m-1.\n\\end{displaymath}\nLet $\\bar N({\\rm supp}(u))$ be the subset of  $(\\bar x,\\xi)\\in X\\times\\Sm^{n-1}$ composed of the closure of the normal set of the support of $\\{u_j\\}$; see \\cite{H-I-SP-83}. Heuristically, that set may be seen as the points $(x,\\xi)$ where $x$ belongs to the boundary of ${\\rm supp (u)}$ and $\\xi$ is a normal (inward or outward) unit vector to that boundary.\n\nThen $\\bar N({\\rm supp}(u))\\subset \\Sigma_{\\mA}$. When the latter set is empty, then $u\\equiv0$ in $X$.\n\\end{theorem}\nThe proof follows from the local results noting that the boundary to the support of $u$ cannot occur at a point and for a direction where a local UCP principle holds;  see \\cite[Sections 8.5-8.6]{H-I-SP-83} for the definitions and derivations. \n\n\n\\begin{remark}\\rm\n Let $\\mA$ be a $p\\times q$ system with $p\\geq q$ and let us assume that $\\mA u=0$ in the above theorem is replaced by \n\\begin{equation}\n\\label{eq:globalucpconst}\n|(\\mA u)_k|(x) \\leq C \\dsum_{|\\alpha|\\leq m-1} \\dsum_{l=1}^q |D^\\alpha u_l|(x),\n\\end{equation}\nfor a constant $C$ independent of $x\\in \\bar X$ and $1\\leq k\\leq p$. Let, as above,  $\\Sigma_\\mA$ be the set of points where the hypotheses of Theorem \\ref{thm:calderonsystem} do not hold (with obvious modifications for  Theorems \\ref{thm:2by2system}).  Then the conclusions of the above theorem, namely that $\\bar N({\\rm supp}(u))\\subset \\Sigma_{\\mA}$, still hold; see, e.g., \\cite{H-III-SP-94} and the proofs in the appendix.\n\\end{remark}\n\n\n\\subsection{Application to power-density measurements}\n\\label{sec:pduniq}\n\nWe now apply the unique continuation results of sections \\ref{sec:ucph} and \\ref{sec:ucpc} to the setting of power density measurements described in section \\ref{sec:pdm}.\n\n\\subsubsection{Second-order linear systems of equations}\n\nBoth unique continuation results require that the system $\\mA$ be elliptic in the regular sense, i.e., with $\\tau=t_j$ independent of $j$ and $s_i=0$ for all $i$. The systems \\eqref{eq:pdlinsyst} and \\eqref{eq:deltauj}-\\eqref{eq:constraintjk} are not in this form. We propose two modifications of the latter systems for which uniqueness results can be proven.\n\n\\subsubsection{System after elimination.}\nLet us first consider the system \\eqref{eq:deltauj}-\\eqref{eq:constraintjk}. The latter constraints are first-order. However, they easily become second-order by differentiation. We thus consider the system\n\\begin{equation}\n\\label{eq:elim2nd}\n\\begin{array}{rcl}\n  \\nabla\\cdot\\gamma\\big(-\\nabla \\delta u_j + 2 \\hat F_j \\hat F_j\\cdot\\nabla \\delta u_j\\big) &=& \\nabla\\cdot \\dfrac{\\delta H_j}{|F_j|^2} F_j.\\\\[3mm]\n\\nabla \\big( 2\\gamma\\dfrac{1}{|F_j|^2} F_j\\cdot\\nabla \\delta u_j - 2\\gamma\\dfrac{1}{|F_k|^2} F_k\\cdot\\nabla \\delta u_k\\big)  &=&   \\nabla \\big( \\dfrac{\\delta H_j}{|F_j|^2} - \\dfrac{\\delta H_k}{|F_k|^2}\\big) \\quad 1\\leq j<k\\leq J,\n\\end{array}\n\\end{equation}\nin $X$ with boundary conditions $\\delta u_j=\\partial_\\nu \\delta u_j=0$ on $\\dX$. The above system is clearly elliptic under the same hypotheses as  \\eqref{eq:deltauj}-\\eqref{eq:constraintjk}. Moreover, it satisfies the above ellipticity condition with $\\tau=2$.\n\n\\subsubsection{System in triangular form.}\nLet us come back to \\eqref{eq:pdlinsyst}. We apply the operator $K_j=2\\gamma |F_j|^{-2}F_j\\cdot\\nabla$ to the first line and $L_\\gamma |F_j|^{-2}$ to the second line with $L_\\gamma:=\\nabla\\cdot \\gamma\\nabla$ to obtain after subtraction\n\\begin{equation}\n\\label{eq:pdlinsyst2}\n\\begin{array}{rcl}\n\\nabla\\cdot \\delta\\gamma F_j + \\nabla\\cdot \\gamma\\nabla \\delta u_j &=& 0 \\\\\nL_\\gamma \\delta\\gamma - K_j \\nabla\\cdot \\delta\\gamma F_j +[L_\\gamma K_j,K_j L_\\gamma] \\delta u_j &=& L_\\gamma \\dfrac{\\delta H_j}{|F_j|^2}.\n\\end{array}\n\\end{equation}\nSince $L_\\gamma$ is second order and $K_j$ is first-order, the commutator $[L_\\gamma K_j,K_j L_\\gamma]$ is a second-order differential operator. We thus again obtain a system that is elliptic when \\eqref{eq:pdlinsyst} is (we leave this proof to the reader; see also \\eqref{eq:triangsyst} below) and that is in the appropriate form with $\\tau=2$.\n\n\n\\subsubsection{Holmgren unique continuation result}\n\nThe results of section \\ref{sec:ucph} apply to the power density measurement systems \\eqref{eq:elim2nd} and \\eqref{eq:pdlinsyst2}. Let $\\mA$ denote the linear operator in one of the aforementioned systems. Then $\\mA^t\\mA$ augmented with Dirichlet conditions is injective provided that $\\gamma$ and $u_j$ are sufficiently close to analytic coefficients or provided that the problem is posed on a sufficiently small domain, and provided of course that the quadratic forms $\\{q_j\\}$ are collectively elliptic on $\\bar X$ in the sense of Definition \\ref{def:ellcondpj}.\n\nThe main difference between \\eqref{eq:elim2nd}  and \\eqref{eq:pdlinsyst2} pertains to the boundary conditions. In the case of \\eqref{eq:pdlinsyst2}, vanishing Dirichlet and Neumann boundary conditions are imposed on $\\delta\\gamma$ and $\\delta u_j$. This means that the value of $(\\gamma,\\{u_j\\})$ and of its (normal) derivatives is known on $\\partial X$. In the case of \\eqref{eq:elim2nd}, only the values of $(\\{u_j\\})$ and of its (normal) derivatives need to be known. \n\nIn fact, the boundary conditions for $\\delta\\gamma$ at $x\\in\\partial X$ may be deduced from those for $\\delta u_1$, say, provided that $\\nu(x)$ is non characteristic for $P_1$. Indeed, we observe that \n\\begin{displaymath}\n  \\delta H_1 = \\delta\\gamma |\\nabla u_1|^2 + 2 \\gamma \\nabla u_j\\cdot\\nabla \\delta u_1.\n\\end{displaymath}\nThis provides boundary conditions for $\\delta\\gamma$ when $\\delta u_1$ and $\\partial_\\nu  \\delta  u_1$ are known on $\\partial X$. The derivation of $\\partial_\\nu \\delta\\gamma$ is obtained as follows. Using the equation\n\\begin{displaymath}\n   \\nabla\\cdot \\delta\\gamma \\nabla u_j + \\nabla\\cdot \\gamma\\nabla \\delta u_j =0,\n\\end{displaymath}\nand replacing $\\delta\\gamma$ above using the expression of $\\delta H_1$, we obtain after straightforward eliminations that\n\\begin{displaymath}\n   (1-2 (\\widehat{\\nabla u_1} \\cdot \\nu(x))^2) \\partial^2_\\nu u_1 =f_1,\n\\end{displaymath}\nwhere $f_1$ is a known expression involving $\\partial_\\nu u_1$ and tangential derivatives of $u_1$ of degree up to $2$ at $x$, which are known since $u_1$ is known on $\\partial X$. Since $\\nu$ is non characteristic for $P_1$, this provides an expression for $\\partial^2_\\nu u_1$, and hence for $\\partial_\\nu\\delta\\gamma$ using the above expression for $\\delta H_1$.\n\nSo knowledge of Cauchy data for $\\delta\\gamma$ is in fact a consequence of knowledge of Cauchy data for the solutions $\\delta u_j$. Thus, the major requirement in solving $\\mA^t\\mA v=\\mA^t\\mS$ is that we impose that $\\partial_\\nu\\delta u_j=g_j$ for a known function $g_j$, which implies that $\\partial_\\nu u_j$ is known on $\\dX$.\n\n\n\n\\subsubsection{Calder\\'on unique continuation result}\nThe symbol of the operator \\eqref{eq:pdlinsyst} is elliptic in the Douglis-Nirenberg sense. Recall that  the results of unique continuation principle (UCP) of the preceding sections apply to operators that are elliptic in a classical sense. We thus consider the  modification of the system \\eqref{eq:pdlinsyst2}. We were not able to obtain a satisfactory global UCP using Carleman-type estimates for the system \\eqref{eq:elim2nd}.\n   \nUCP for such  the modified system \\eqref{eq:pdlinsyst2} is based on proving a collective UCP for quadratic forms, which we now address in detail.\n\\begin{lemma}\n\\label{lem:ucphyper}\nLet $P_j(x,D)$ be second-order operators with principal symbols given by $q_j(x,\\xi)=2(\\hat F_j\\cdot\\xi)^2-|\\xi|^2$, where $\\hat F_j(x)\\in\\Sm^{n-1}$ for $1\\leq j\\leq J$. \nLet $(x,N)\\in\\bar X\\times\\Sm^{n-1}$. For $\\xi'\\in\\Rm^n$ such that $\\xi'\\cdot N=0$, we define\n\\begin{displaymath}\n  q_{j;N}(x,\\xi') = 2(\\hat F_j\\cdot\\xi')^2 - \\big(1-2(\\hat F_j\\cdot N)^2\\big) |\\xi'|^2.\n\\end{displaymath}\nLet us assume that the quadratic forms are independent in the following UCP sense:\n\\begin{equation}\\label{eq:UCPN}\n  q_{j;N}(x,\\xi') =0 \\quad \\mbox{ for all } 1\\leq j\\leq J\\mbox{ s.t. } q_j(x,N)\\not=0\\qquad \\mbox{ implies that } \\qquad \\xi'=0.\n\\end{equation}\n\nThen $\\{P_j\\}$ collectively satisfies a UCP for $(x,N)$ in the sense that we can find a partition $\\{\\Omega_\\nu\\}$ such that  (i)-(ii)-(iii) holds.\n\\end{lemma}\n\n\\begin{proof}\nThe proof in dimension $n=2$ is trivial: $P_j$ satisfies UCP except for four directions on the null cone, i.e., for $\\xi$ such that $q_j(x,\\xi)=0$. Since \\eqref{eq:UCPN} implies that $N$ is not in the intersection of the null cones, then UCP holds for one of the operators $P_j$ and we can choose $\\Omega_1=\\Sm^{1}$ in the partition.\n\n Let $(x,N)\\in\\bar X\\times\\Sm^{n-1}$ and $n\\geq3$. Let $\\xi'\\in\\Sm^{n-1}$ such that $\\xi'\\cdot N=0$. Then\n \\begin{displaymath}\n  q_j(x,\\xi'+\\tau N) = 2(\\hat F_j\\cdot (\\xi'+\\tau N))^2- |\\xi'+\\tau N|^2 = \\big(2(\\hat F_j\\cdot N)^2-1)\\tau^2 + 4 \\hat F_j\\cdot N\\hat F_j\\cdot \\xi' \\tau + 2 (\\hat F_j\\cdot \\xi')^2-1.\n\\end{displaymath}\nThis quadratic polynomial (because $q_j(x,N)=2(\\hat F_j\\cdot N)^2-1\\not=0$) with real-valued coefficients has a double root when the determinant\n\\begin{equation}\\label{eq:Deltaprime}\n \\Delta'_j = -1+2 \\big((\\hat F_j\\cdot N)^2 + (\\hat F_j\\cdot \\xi')^2\\big) = q_{j;N}(x,\\xi')=0.\n\\end{equation}\nFor $\\xi'$ away from such points, then the UCP condition holds for $P_j$ since the roots cannot be simple and complex roots (which come in conjugate pairs) cannot be close to each-other. By assumption, $\\Delta'_j=0$ for all $j$ is not possible and by continuity, $\\max_{j}|\\Delta'_j|$ is bounded away from $0$ so that different roots $\\tau$ are separated by some $\\eps>0$ (as well as the imaginary part of such roots since they come in conjugate pairs). This generates a finite partition $\\{\\Omega_\\nu\\}$. On each patch, one $P_j$ satisfies the UCP properties {\\em (i)-(ii)-(iii)}.\n\\end{proof}\n\n\nIn dimension $n=2$, ellipticity of $(P_1,P_2)$ is equivalent to UCP of $(P_1,P_2)$ at each $(x,N)$. In dimension $n\\geq3$, collective ellipticity and collective UCP are different notions. As soon as there is one $j$ such that $2(\\hat F_j\\cdot N)^2\\geq1$, the UCP is satisfied at $(x,\\xi)$ while ellipticity does not necessarily holds.\nHowever, we have the result:\n\\begin{lemma}\n\\label{lem:ElltoUCP}\n In dimension $n=2$, collective UCP and collective ellipticity of operators of the form $P_j$ above is equivalent. In dimension $n=3$, collective ellipticity of $\\{P_j\\}$ such that $q_j(N)\\not=0$ implies collective UCP at $(x,N)$.\n\\end{lemma}\n\\begin{proof}\n The case $n=2$ is handled as in the proof of Lemma \\ref{lem:ucphyper}. Consider now the case $n=3$. Non-UCP at $(x,N)$ implies from \\eqref{eq:Deltaprime} and the fact that $N$ is non characteristic for $P_j$ that\n \\begin{displaymath}\n   (\\hat F_j\\cdot N)^2 + (\\hat F_j\\cdot \\xi')^2 = \\frac12.\n\\end{displaymath}\nIf we decompose $\\hat F_j=\\hat F_j\\cdot N N + \\hat F_j\\cdot \\xi' \\xi' + \\hat F_j\"$ with $\\hat F_j\"$ orthogonal to ${\\rm span}(N,\\xi')$, then we find that \n\\begin{displaymath}\n  |\\hat F_j\"|^2 = \\frac 12.\n\\end{displaymath} \nIn dimension $n=3$, we find that $\\hat F_j\"=\\pm|\\hat F_j| N\\times \\xi'$. This implies that $N\\times\\xi'$ belongs to the null cone of $q_j$, i.e., $q_j(x,N\\times\\xi')=0$. This holds for every $j$, which implies that $N\\times \\xi'=0$, a contradiction.\n\\end{proof}\n\n\\begin{definition}\\label{def:UCP}\nWe say that the family of operators $P_j(x,D)$ or of quadratic forms $q_j(x,\\xi)$ collectively satisfy a global UCP on $\\bar X$ if  for every $(x,N)\\in \\bar X\\times \\Sm^{n-1}$, we can find a family $\\tilde q_j(x,\\xi)=\\mQ_{jk}q_k(x,\\xi)$ with $\\mQ$ an invertible matrix such that $\\{\\tilde q_j(x,\\xi)\\}$ collectively satisfies a UCP at $(x,N)$ in the sense of Lemma \\ref{lem:ucphyper}.\n\\end{definition}\n\nThe above definition involves a linear change of quadratic forms for each $(x,N)$ that allows us to obtain a family of modified forms $\\tilde q_j$ with similar properties to those of $q_j$ and such that $\\tilde q_j(N)\\not=0$; see Remark \\ref{rem:LC}. This will prove useful in the analysis of reconstruction from  power density functionals. We will also need the following Lemma in dimension $n=3$.\n\\begin{lemma}\n\\label{lem:3dUCP}\nLet $F_1$ and $F_2$ be three-dimensional vector fields on $\\bar X$ such that for each $x\\in\\bar X$, ${\\rm rank}(F_1,F_2)=2$. Let $F_3=F_1+F_2$. We denote by $\\{P_j\\}$ and $\\{q_j\\}$ the corresponding operators and quadratic forms defined in \\eqref{eq:qjPj} for $1\\leq j\\leq 3$. Then $\\{q_j\\}$ satisfies a global UCP on $\\bar X$.\n\\end{lemma}\n\\begin{proof}\n  Let us fix $(x,N)\\in \\bar X\\times \\Sm^2$ and choose a basis of $\\Rm^3$ such that  $\\hat F_1(x)=e_1$ and ${\\rm span}(F_1,F_2)=(e_1,e_2)$. The three quadratic forms $q_j$ allow us to obtain by linear combination any $q_c$ corresponding to $F_c=c e_1+se_2$ with $s=\\sqrt{1-c^2}$. It is then not difficult to find three values of $c$ such that the corresponding $\\tilde q_j$ for $1\\leq j\\leq 3$ form an elliptic family and such that $\\tilde q_j(N)\\not=0$. From Lemma \\ref{lem:ElltoUCP}, we obtain that $\\{\\tilde q_j\\}$ satisfies a UCP at $(x,N)$. The hypotheses of global UCP in Definition \\ref{def:UCP} are met.\n\\end{proof}\n\n\n\\subsubsection{Elliptic system in triangular form.} We now present a modification of the linear system for $(\\delta\\gamma,\\{\\delta u_j\\})$ for which a global UCP result as described in the above definition can be obtained.\n\nWe recast \\eqref{eq:pdlinsyst2} as\n\\begin{equation}\n\\label{eq:triangsyst}\n\\left(\\begin{matrix} P_j & \\tilde P_j \\\\ 0 & \\Delta \\end{matrix} \\right) \\left(\\begin{matrix} \\delta\\gamma \\\\ \\delta u_j \\end{matrix} \\right) = l.o.t.\\left(\\begin{matrix} \\delta\\gamma \\\\ \\delta u_j \\end{matrix} \\right)  + S_j,\n\\end{equation}\nwhere $l.o.t.$ means a system of differential operators of order at most $1$. Let $P$ and $\\tilde P$ denote the $J\\times 1$ columns of the second-order, homogeneous, operators $P_j$ and $\\tilde P_j$, respectively. The symbol of $P_j$ is proportional to $q_j(x,\\xi)$.  The operator $\\Delta$ satisfies hypothesis {\\em (iv)} of Theorem \\ref{thm:2by2system} while the operator $\\tilde P$ is of order $2$. If the operators $P$ collectively satisfy the hypotheses of Theorem \\ref{thm:calderonsystem}, then the above system \\eqref{eq:pdlinsyst2} satisfies a UCP, as does the fourth-order operator $\\mA^t\\mA$ as constructed in the preceding section:\n\n\\begin{theorem}\n\\label{thm:uniqpowdens}\nLet $\\mA v= \\mS$ in $X$ be the system for $v=(\\delta\\gamma,\\{\\delta u_j\\})$ described in \\eqref{eq:triangsyst} and augmented with boundary conditions $v=\\partial_\\nu v=0$ on $\\partial X$. Assume that the coefficients in $\\mA$ are sufficiently smooth (see proof of Theorem \\ref{thm:calderonsystem:app}). Let the operators $P_j$ above satisfy a UCP on $\\bar X$ as described in Definition \\ref{def:UCP} and Lemma \\ref{lem:ucphyper}. Then any solution to that system, or to the system $\\mA^t\\mA v=\\mA^t \\mS$ with the same boundary conditions, is unique.\n\\end{theorem}\n\\begin{proof}\n  The proof is a direct application of Theorem \\ref{thm:2by2system} since $\\Delta$ satisfies hypothesis {\\em (iv)}, $\\tilde P$ is of order $2$, and $P$ collectively satisfies a UCP at each $(x,N)$ in $\\bar X\\times \\Sm^{n-1}$. \n\\end{proof}\n\n\nIn dimensions $n=2$ and $n=3$, we have the following sufficient conditions on the internal functionals to guaranty injectivity of $\\mA^t\\mA$:\n\\begin{corollary}\\label{cor:UCP}\n\\label{cor:3d} Assume that $(u_1,u_2)$ are solutions such that $F_1=\\nabla u_1$ and $F_2=\\nabla u_2$ are such that ${\\rm rank}(F_1,F_2)=2$ for each $x\\in\\bar X$.  \n\nIn dimension $n=2$, assume moreover that  $F_1\\cdot F_2\\not=0$ for each $x\\in\\bar X$. Then $\\{P_1,P_2\\}$ satisfies a global UCP property on $\\bar X$ and the results of Theorem \\ref{thm:uniqpowdens} hold.\n\nIn dimension $n=3$, define $F_3=F_1+F_2$. Then $\\{P_1,P_2,P_3\\}$ satisfies a global UCP property on $\\bar X$ and the results of Theorem \\ref{thm:uniqpowdens} hold.\n\\end{corollary}\n\nThe proof is immediate using Lemma \\ref{lem:3dUCP}, remark \\ref{rem:LC} and the preceding Theorem.\n\nWe have thus exhibited a system of equations  $\\mA^t\\mA v=\\mA^t \\mS$ with boundary conditions $v$ and $\\partial_\\nu v$ prescribed on $\\partial X$, which admits a unique solution that verifies the stability estimate \\eqref{eq:pdstab} (generalized as in \\eqref{eq:stabnormal} for non-homogeneous boundary conditions) with $C_2=0$.\n\nAs in the preceding section devoted to the Holmgren uniqueness result, this result comes at the cost of having to impose boundary conditions of the form $v$ and $\\partial_\\nu v$ to both $v=\\delta u_j$ and $v=\\delta\\gamma$. As we saw in the preceding section, the boundary conditions for $\\delta\\gamma$ at $x\\in\\partial X$ may be deduced from those for $\\delta u_j$.\n\nComparing ellipticity and uniqueness criteria for $\\mA$ in \\eqref{eq:triangsyst}, we observe that ellipticity is obtained by imposing conditions $\\delta u_j=g_j$ on $\\dX$ and ensuring that $\\{q_j\\}$ is an elliptic family of quadratic forms. UCP is obtained by imposing the additional boundary conditions $\\partial_\\nu\\delta u_j=\\psi_j$ and by ensuring that $\\{q_j\\}$ satisfy a global UCP on $\\bar X$. In dimension $n=2$, global ellipticity and global UCP for $\\{q_j\\}$ are equivalent. In dimension $n=3$, both are satisfied for the family of three internal functionals described in Corollary \\ref{cor:UCP}.\n\n\n\\section{Local uniqueness for nonlinear inverse problem}\n\\label{sec:nonlinear}\n\nOnce the injectivity of a linear system can be established, standard theories may be applied to obtain local uniqueness results for the nonlinear inverse problems. Let us recast the nonlinear system of PDE \\eqref{eq:syst} as\n\\begin{equation}\n\\label{eq:nonlinearnotmod}\n\\begin{array}{rcl}\n \\tilde\\mF (v) &=& \\tilde \\mH \\quad \\mbox{ in } X \\\\\n \\tilde \\mB v &=& \\tilde g \\quad \\mbox{ on } \\partial X.\n\\end{array}\n\\end{equation}\nHere $v=(\\gamma,\\{u_j\\}_{1\\leq j\\leq J})$ and $\\tilde \\mB$ is an operator that maps $v$ to the trace of $\\{u_j\\}$ on $\\partial X$. We assume here, as is the case for hybrid inverse problems, that the boundary condition operator $\\tilde \\mB$ is linear.\n\nThe linearization of the above system involves the operator $\\tilde\\mF'(v_0)$ for a fixed $v_0$. The analysis of the preceding sections did not allow us to show that the differential operator $\\tilde \\mF'(v_0)$ augmented with the boundary conditions $\\tilde \\mB$ was invertible. Rather, we obtained that for an appropriate linear differential operator $\\mT(v_0)$, then $\\mA:=\\mT(v_0)\\tilde \\mF'(v_0)$ augmented with the augmented boundary conditions $\\mB$ was invertible. More precisely, we obtained that\n\\begin{equation}\n\\label{eq:linearmod}\n\\begin{array}{rcl}\n  \\mA w := \\mT(v_0) \\tilde \\mF'(v_0) w &=& \\mS  \\quad \\mbox{ in } X \\\\\n \\mB w &=&  h\\quad \\mbox{ on } \\partial X,\n\\end{array}\n\\end{equation}\nadmitted a unique solution with a stability estimate given by\n\\begin{equation}\n\\label{eq:stablinear}\n  \\|w\\|_\\mX \\leq C \\|(\\mS,h) \\|_\\mY.\n\\end{equation}\n\nThe above linearized operator is the linearization of the nonlinear operator\n\\begin{equation}\n\\label{eq:nonlinearmF}\n\\mF(v_0+w):=\\mT(v_0)\\tilde \\mF(v_0+w).\n\\end{equation}\n We thus modify the original inverse problem and replace it with an inverse problem that is necessarily satisfied by the exact solution $v=v_0+w$ we wish to reconstruct and is given by\n\\begin{equation}\n\\label{eq:nonlinearmod2}\n\\begin{array}{rcl}\n \\mF (v_0+w) &=& \\mH:=\\mT(v_0) \\tilde\\mH \\quad \\mbox{ in } X \\\\\n \\mB (v_0+w) &=& g \\quad \\mbox{ on } \\partial X.\n\\end{array}\n\\end{equation}\nBorrowing the notation of section \\ref{sec:param}, we observe that the nonlinear operator $(\\mF(\\cdot),\\mB \\cdot)$ maps $\\mX=\\mU(p,l)$ to $\\mY=\\mR(p,l)$ for appropriately defined spaces $\\mU$ and $\\mR$ for $(\\mA,\\mB)=(\\mF'(v_0),\\mB)$.\n\nLet us pause on the definition of the boundary condition $g$ for $v_0+w$. We cannot expect $\\mB v_0 =g$ with $g$ known so that $\\mB w =0$ on $\\partial X$. The reason is that $v_0$ is typically constructed by guessing $\\gamma_0$ and solving the linear elliptic problems for $\\{u_{j,0}\\}$ with imposed Dirichlet conditions. It is for such a construction of $v_0$ that we were able to show that $A=(\\mA,\\mB)$ above was injective for $\\mA=\\mF'(v_0)$. The boundary condition $g_0:=\\mB v_0$ thus partially depends on solving the above problem and is not in general given by the measurements $g$ (unless $v_0$ is the solution $v$). We thus recast the (modified) nonlinear hybrid inverse problem as\n\\begin{equation}\n\\label{eq:nonlinearmod}\n\\begin{array}{rcl}\n \\mF (v_0+w) &=& \\mH:=\\mT(v_0) \\tilde\\mH \\quad \\mbox{ in } X \\\\\n \\mB w &=& g-g_0 \\quad \\mbox{ on } \\partial X.\n\\end{array}\n\\end{equation}\nThe objective of the following section is to provide an iterative algorithm to reconstruct $w$ provided that $v_0$ is sufficiently close to the solution we wish to obtain in the sense that $g-g_0$ and $\\mH-\\mH_0$ are sufficiently small with $\\mH_0:=\\mF(v_0)$. In section \\ref{sec:injectivity}, we obtain an injectivity result stating that if $\\mF(v_0+w)=\\mF(v_0+\\tilde w)$ and $\\mB w=\\mB \\tilde w$, then $w=\\tilde w$.\n\n\\begin{remark}\n\\label{rem:powerdensity}\n\nIn the power density setting, we recast \\eqref{eq:ellj} and \\eqref{eq:pdj} as\n\\begin{displaymath}\n   \\mF_{2j-1}(v_0+w) = 0,\\qquad \\tilde \\mF_{2j}(v_0+w)=H_j,\\qquad 1\\leq j\\leq J,\n\\end{displaymath}\nrespectively. Let then $L_\\gamma$, $F_j$, and $M_j$ constructed from $v_0=(\\gamma,\\{u_j\\})$ as in the derivation of \\eqref{eq:pdlinsyst2}. Finally, let us define\n\\begin{displaymath}\n  \\mF_{2j}(v_0+w) = L_\\gamma \\tilde \\mF_{2j-1}(v_0+w) - M_j \\mF_{2j-1}(v_0+w) = |F_j|^{-2} H_j =: K_j,\\quad 1\\leq j\\leq J.\n\\end{displaymath}\nThe system $\\mF_{2j-1}(v_0+w)=0$ and $\\mF_{2j}(v_0+w)=K_j$ for $1\\leq j\\leq J$ is recast as \n\\begin{displaymath}\n  \\mF(v_0+w) = \\mH,\n\\end{displaymath}\nin the notation of \\eqref{eq:nonlinearmod} and implicitly defines the linear operator $\\mT(v_0)$. We denote by $\\mA=\\mF'(v_0)$ the linearization of $\\mF$ at $v_0$, which agrees with the differential operator defined in \\eqref{eq:pdlinsyst2}.\n\nWhereas the operator $\\tilde \\mB$ maps $v_0$ to the traces of $\\{u_j\\}$ on $\\partial X$, the extended operator $\\mB$ maps $v_0$ to the traces of $v_0$ and $\\nu\\cdot \\nabla v_0$ on $\\partial X$.\n\n\\end{remark}\n\n\\subsection{Iterative fixed point and reconstruction procedure}\n\\label{sec:fixedpoint}\n\nLet us define\n\\begin{equation}\n\\label{eq:taylormF}\n\\mF(v_0+w) = \\mF(v_0) + \\mF'(v_0) w + \\mG(w;v_0),\n\\end{equation}\nwhere $\\mG(w;v_0)$ is quadratic in the first variable in the sense that \n\\begin{equation}\n\\label{eq:quadG}\n  \\|(\\mG(w;v_0),0)\\|_\\mY \\leq C \\|w\\|_\\mX^2.\n\\end{equation}\nThe latter estimate comes from the fact that $\\mF(v)$ is polynomial in $v$ and the partial derivatives $D$ and that $\\mX$ is an algebra. We may thus recast the nonlinear system of equations for $w$ as\n\\begin{equation}\n\\label{eq:nonlinearw}\n\\begin{array}{rcl}\n \\mA w &=& \\mH - \\mH_0 - \\mG(w;v_0) \\quad \\mbox{ in } X \\\\\n \\mB w &=& g-g_0 \\quad \\mbox{ on } \\partial X.\n\\end{array}\n\\end{equation}\nSince the linear operator $A=(\\mA,\\mB)$ is invertible, we may recast the above equation as\n\\begin{equation}\n\\label{eq:integralw}\n w = \\mI(w) :=A^{-1}(\\mH - \\mH_0, g-g_0) - A^{-1}(\\mG(w;v_0),0) .\n\\end{equation}\nFrom the polynomial structure of $\\mF$and the boundedness of $A^{-1}$ from $\\mY$ to $\\mX$, we deduce in addition to \\eqref{eq:quadG} that\n\\begin{equation}\n\\label{eq:constants}\n\\begin{array}{rcl}\n  \\|A^{-1}(\\mG(w;v_0),0)\\|_{\\mX} &\\leq & C_1 \\|w\\|_\\mX^2\\\\\n  \\|\\mI(w)-\\mI(\\tilde w)\\|_\\mX &\\leq& C_2 (\\|w\\|_{\\mX} + \\|\\tilde w\\|_{\\mX}) \\|w-\\tilde w\\|_{\\mX}.\n\\end{array}\n\\end{equation}\nAs a consequence, for $\\|w\\|_\\mX\\leq R$ and $\\|A^{-1}(\\mH-\\mH_0,g-g_0)\\|_\\mX\\leq \\eta$ sufficiently small so that $\\eta+C_1R^2\\leq R$ and $2C_2 R<1$, we find that $\\mI$ is a contraction from the ball $B=\\{w,\\, \\|w\\|_{\\mX}\\leq R\\}$ onto itself.\n\nThis shows that for $(\\mH-\\mH_0,g-g_0)$ sufficiently small, then the solution to \\eqref{eq:nonlinearw} is unique and is obtained by the converging algorithm $w^{k+1}=\\mI(w^k)$ initialized with $w^0=0$.\n\n\\subsection{Injectivity results}\n\\label{sec:injectivity}\n\nThe fixed point algorithm of the preceding section provides us with an explicit local reconstruction procedure for the nonlinear hybrid inverse problem. A similar methodology allows us to obtain local uniqueness results that are more general but not constructive. Using the same notation as in the preceding section, let us assume that \n\\begin{equation}\n\\label{eq:uniqF}\n\\mF(v_0+w) = \\mF(v_0+\\tilde w) = \\mH\\quad\\mbox{ in } X,\\qquad \\mB w=\\mB \\tilde w\\mbox{ on }\\partial X,\n\\end{equation}\nfor a fixed $v_0\\in\\mX$. In other words, the measurements associated with $v_0+w$ and $v_0+\\tilde w$ are identical. Injectivity of the nonlinear problem $\\mF$ locally in the vicinity of $v_0$ then means proving that $w=\\tilde w$.\n\nSince $\\mF$ is a polynomial in $w$, then there is a polynomial function in $w$ and $\\tilde w$ such that \n\\begin{equation}\n\\label{eq:diffF}\n\\mF(v_0+\\tilde w) - \\mF(v_0+w) = \\mJ(w,\\tilde w) \\cdot (\\tilde w -w).\n\\end{equation}\n\nWhen $\\tilde w-w$ is sufficiently small, then $\\mJ(w,\\tilde w)$ satisfies the same properties as $\\mF'(v_0)$, the Fr\\'echet derivative of $\\mF$. For instance, the ellipticity and unique continuation properties of $\\mF'(v_0)$ still hold for $\\mJ(w,\\tilde w)$ for $w$ and $\\tilde w$ small in $\\mX$. As a consequence, we obtain that \\eqref{eq:uniqF} implies that $w=\\tilde w$ and hence that $\\mF$ is injective. More generally, $\\mJ(w,\\tilde w)$ may still admit a left-inverse for $w$ and $\\tilde w$ not necessarily close to each-other, in which case we also deduce that $w=\\tilde w$ for $w$ and $\\tilde w$ not necessarily small.\n\n\nLet us assume more generally that we have two measurements\n\\begin{displaymath}\n   \\mF(v_0+w) = \\mH,\\quad \\mB w=g-g_0 ;\\qquad \\mF(v_0+\\tilde w) = \\tilde\\mH,\\quad \\mB\\tilde w=\\tilde g-g_0.\n\\end{displaymath}\nAssume that $\\mF'(v_0)$ is elliptic and injective. Then, $\\mJ(w,\\tilde w)$ is elliptic and hence admits a left-inverse (since it is injective), at least for $w$ and $\\tilde w$ sufficiently close to $0$. As a consequence,  we obtain as above the stability estimate\n\\begin{equation}\n\\label{eq:stabnonlinear}\n  \\|w-\\tilde w\\|_{\\mX} \\leq C \\|(\\mH-\\tilde\\mH,g-\\tilde g)\\|_{\\mY}.\n\\end{equation}\n\n\nNote that the nonlinear problem \\eqref{eq:uniqF} is invertible generically. Indeed, for $v_0$ analytic, $\\mA^t\\mA$ augmented with Dirichlet boundary conditions $\\mD$ is invertible. As a consequence, the above result shows that \\eqref{eq:uniqF} may be inverted for $(v_0+w)$ in an open set including $v_0$. Since analytic coefficients $v_0$  in $X_0$ restricted to $X$ are dense in the set of sufficiently smooth coefficients on $X$, we obtain that the inverse problem may be inverted {\\em generically}; see \\cite{SU-JFA-09}.\nWhen $v_0$ is not analytic, then we need to find a unique continuation principle based on Carleman estimates to obtain an estimate of the form \\eqref{eq:stabnonlinear} for the fully nonlinear hybrid inverse problem.\n\n\\cout{\nOLD OLD \n\\subsection{Iterative fixed point and reconstruction procedure}\n\\label{sec:fixedpoint}\n\nLet us consider a setting where $(\\mA,\\mB)$ is injective or $(\\mA^t\\mA,\\mD)$ is invertible so that the linearized problem \\eqref{eq:linear} is solved by either $v=(I-T)^{-1}RA(\\mS,0)$ or by $v=(\\mA^t\\mA)^{-1}(\\mA^t\\mS,0)$. We now present standard results concerning the nonlinear hybrid inverse problem \\eqref{eq:syst}.\n\nLet $v_0=(\\gamma,\\{u_j\\})\\in\\mX=\\mU(p,l)$ the point about which \\eqref{eq:syst} was linearized to give \\eqref{eq:linsyst}. We recast both equations in \\eqref{eq:syst} as \n\\begin{equation}\n\\label{eq:inX}\n\\mF(v_0+v) = \\mH,\n\\end{equation}\nwhere $\\mH\\in\\mY$ collects the available data $\\{H_j\\}$ and $\\mY$ is defined as the appropriate subset of $\\mR(p,l)$ defined in section \\ref{sec:param}. We also define $\\mH_0:=\\mF(v_0)$. \n\nIn the procedure of construction of a left-parametrix to $A$, we impose in addition to \\eqref{eq:inX} the boundary condition $\\mB(v_0+v)=\\mB(v_0)$ so that $\\mB(v)=0$. Note that in the setting of power-density measurements, we do not have injectivity of such an operator $A$. Let us assume nonetheless that $1$ is not an eigenvalue of the operator $T$ so that $(I-T)^{-1}R$ is a bounded left-inverse for $A$. \n\nIn the inversion of $(\\mA^t\\mA,\\mD)$, we impose in addition to \\eqref{eq:inX} the boundary condition $\\mD(v_0+v)=\\mD(v_0)$ so that $\\mD(v)=0$. We have obtained in the preceding section several unique continuation results that apply to the differential operator $\\mA$ given in \\eqref{eq:elim2nd} or \\eqref{eq:pdlinsyst2}.\n\nWe recast \\eqref{eq:inX} as \n\\begin{displaymath}\n  \\mF'(v_0) v = \\mH -\\mF(v_0) - \\big( \\mF(v_0+v)-\\mF(v_0)-\\mF'(v_0) v\\big).\n\\end{displaymath}\nWe are in a setting where $\\mF'(v_0)$ augmented with appropriate vanishing boundary conditions admits a left-parametrix, which we denote by $(\\mF')^{-1}(v_0)$, and which takes the expression $(I-T)^{-1}R$ or $(\\mA^t\\mA)^{-1}\\mA^t$ with boundary conditions $\\mB v=0$ and $\\mD v=0$, respectively.\n\nWe then have that\n\\begin{equation}\n\\label{eq:contraction}\nv = \\mG(v) := (\\mF')^{-1}(v_0) ( \\mH-\\mH_0) - (\\mF')^{-1}(v_0)   \\big( \\mF(v_0+v)-\\mF(v_0)-\\mF'(v_0) v\\big).\n\\end{equation}\n\n\nSince $\\mF(w)$ is polynomial in $w$ with smooth coefficients and we have the stability estimate \\eqref{eq:stabell} with $C_2=0$,  we obtain that \n\\begin{displaymath}\\begin{array}{rcl}\n   \\|\\mG(v)-\\mG(w)\\|_{\\mX} &\\leq& C_1 (\\|v\\|_{\\mX}+\\|w\\|_{\\mX}) \\|v-w\\|_{\\mX} \\\\\n     \\| (\\mF')^{-1}(v_0) \\big( \\mF(v_0+v)-\\mF(v_0)-\\mF'(v_0) v\\big) \\|_{\\mX} &\\leq & C_3 \\|v\\|_{\\mX}^2,\n     \\end{array}\n\\end{displaymath}\nfor $\\|v\\|_{\\mX}$ sufficiently small since $\\mY$ forms an algebra. \n\nFor $\\|v\\|_{\\mX}<R$ sufficiently small and $\\| (\\mF')^{-1}(v_0) ( \\mH-\\mH_0)\\|_{\\mX}<\\eta$ sufficiently small so that $\\eta+C_3R^2\\leq R$ and $2C_1R<1$, we find that $\\mG$ is a contraction from $B=\\{v,\\, \\|v\\|_{\\mX}\\leq R\\}$ to itself. This proves that for $\\mH$ sufficiently close to $\\mH_0$, then the solution to \\eqref{eq:inX} is unique and may be obtained by the converging algorithm $v^{k+1}=\\mG(v^k)$.\n\nProvided that $(\\mathcal F'(v))^{-1}$ is shown to exist and to be uniformly bounded as an operator from $\\mY$ to $\\mX$ for $v$ in the vicinity of $v_0$, then a Newton scheme with faster convergence properties may be introduced:\n\\begin{equation}\n\\label{eq:Newton}\n v^{k+1}-v^k = (\\mathcal F'(v_0+v^k))^{-1} \\big(\\mH-\\mF(v_0+v^k)\\big).\n\\end{equation}\nThe analysis of the scheme and its very fast (quadratic) convergence rate are then classical.\n\n\\subsection{Application to power density measurements.}\n\nUnfortunately, the framework we just described does not directly apply to the setting of power density measurements. The reason, as described in section \\ref{sec:pduniq}, is that the operator $\\mA$ introduced in \\eqref{eq:pdlinsyst} is elliptic in the DN sense but $\\mA^t\\mA$ is not elliptic. The uniqueness results presented in the preceding section thus do not apply directly to \\eqref{eq:pdlinsyst}. We concentrate here on \\eqref{eq:pdlinsyst2} for which we were able to prove a unique continuation principle and for which the Holmgren theory of injectivity applies.\n\nWe first realize that \\eqref{eq:pdlinsyst2} is no longer the linearization of the \\eqref{eq:ellj}-\\eqref{eq:pdj}. We need to construct a nonlinear problem for $v$ whose linearization at $v_0=(\\gamma,\\{u_j\\})$ would be given by \\eqref{eq:pdlinsyst2}. This is done as follows. We recast \\eqref{eq:ellj} and \\eqref{eq:pdj} as\n\\begin{displaymath}\n   \\mF_{2j-1}(v_0+v) = 0,\\qquad \\tilde \\mF_{2j}(v_0+v)=H_j,\\qquad 1\\leq j\\leq J,\n\\end{displaymath}\nrespectively. Let then $L_\\gamma$, $F_j$, and $M_j$ constructed from $v_0=(\\gamma,\\{u_j\\})$ as in the definition of \\eqref{eq:pdlinsyst2}. Finally, let us define\n\\begin{displaymath}\n  \\mF_{2j}(v_0+v) = L_\\gamma \\tilde \\mF_{2j-1}(v_0+v) - M_j \\mF_{2j-1}(v_0+v) = |F_j|^{-2} H_j =: K_j,\\quad 1\\leq j\\leq J.\n\\end{displaymath}\nThe system $\\mF_{2j-1}(v_0+v)=0$ and $\\mF_{2j}(v_0+v)=K_j$ for $1\\leq j\\leq J$ is recast as \n\\begin{displaymath}\n  \\mF(v_0+v) = \\mH,\n\\end{displaymath}\nto fit the notation in \\eqref{eq:inX}. We then denote by $\\mA=\\mF'(v_0)$ the linearization of $\\mF$ at $v_0$, which then agrees with the differential operator defined in \\eqref{eq:pdlinsyst2}.\n\n\nIt is for this problem that the preceding theory (almost) applies. The main practical difficulty resides in the imposition of redundant boundary conditions in the equation \n\\begin{displaymath}\n  \\mF(v_0) = \\mH_0, \\quad \\mbox{ with } v_0 \\mbox{ and } \\partial_\\nu v_0 \\mbox{ prescribed on } \\partial X\n\\end{displaymath}\nIt is indeed unlikely that an initial guess $v_0$, obtained for instance from an initial guess for $\\gamma$ and then solving elliptic equations for $u_j$, may satisfy predetermined Neumann conditions. To remedy this problem, we have to leave open the possibility that $\\partial_\\nu v_0$ may not be the measured $\\partial_\\nu (v+v_0)$; in other words $\\partial_\\nu v=g$ may not vanish. \n\n\\medskip\n\n{\\bf OLD OLD}\n\n\\medskip\n\nNote that $v_0$ and $v_0+v$ and their first derivatives need to agree on $\\partial X$ to uniquely solve \\eqref{eq:pdlinsyst2}. Since $\\mF$ is a second-order differential operator, it may be difficult to find an initial guess $v_0$ satisfying the above hypotheses. We therefore replace the above equation for $v_0$ by\n\\begin{displaymath}\n  \\mA^t\\mF(v_0) = \\mA^t \\mH_0\\quad \\mbox{ with } v_0 \\mbox{ and } \\partial_\\nu v_0 \\mbox{ prescribed on } \\partial X.\n\\end{displaymath}\nThis is a fourth-order problem with appropriate Dirichlet conditions. \n\n\nConsider the scheme introduced in \\eqref{eq:pdlinsyst2} or \\eqref{eq:elim2nd}, which we recast as $\\mA v=\\mS$. Then $\\mF'(v_0)$ is an operator from $\\mX=W^l_p(X)\\times W^{l+1}_p(X;\\Rm^J)$ to $\\mY=W^l_p(X;\\Rm^{2J})$, which to $\\mS$ associates $v$ solution of\n\\begin{equation}\n\\label{eq:vquad}\n \\mA^t\\mA v = \\mA^t \\mS\\quad\\mbox{ in }\\, X,\\qquad v=\\partial_\\nu v =0 \\quad\\mbox{ on } \\, \\partial X.\n\\end{equation}\nWe have presented sufficient conditions in the preceding section for the above problem to be uniquely solvable and hence for $(\\mF'(v_0))^{-1}$ to be bounded from $\\mY$ to $\\mX$. \n\n\nWe then find that $\\mF'(v)$ is invertible from $\\mY$ to $\\mX$ for $\\|v-v_0\\|_{\\mX}$ sufficiently small so that the nonlinear iteration schemes in \\eqref{eq:contraction} or \\eqref{eq:Newton} converge. \nThis is because in the setting of power density measurements, elliptic regularity for $v_0\\in \\mX$ shows that $\\delta v\\in\\mX$ as well provided that $\\delta H\\in \\mY$. More precisely, for $\\gamma\\in W^l_{p}(X)$, then $u_j\\in W^{l+1}_p(X;\\Rm^J)$ by elliptic regularity of \\eqref{eq:ellj} \\cite{S-JSM-73} and then $(\\delta\\gamma,\\{\\delta u_j\\})\\in\\mX$ by elliptic regularity of either \\eqref{eq:pdlinsyst2} or \\eqref{eq:elim2nd} since power density measurements for $\\gamma\\in W^l_p(X)$ and $u\\in W^{l+1}_p(X)$ of the form $\\gamma|\\nabla u|^2$ are indeed in $W^l_p(X)$. \n\n\nWhen one of the unique continuation results applies (which may require that $l$ above be quite large so that coefficients are sufficiently smooth; see the proof of Theorem \\ref{thm:calderonsystem:app} in the appendix), then the nonlinear hybrid inverse problem may be solved locally using \\eqref{eq:contraction} or \\eqref{eq:Newton}. Indeed, for \\eqref{eq:Newton}, $v^{k+1}-v^k$ belongs to $\\mX$ as soon as $(\\mF'(v_0))^{-1}$ is bounded from $\\mY$ to $\\mX$ and $\\mH-\\mH_0$ is sufficiently small in $\\mY$. This provides a local uniqueness result and an explicit reconstruction procedure for the power density measurement hybrid inverse problem.\n\nWe refer the reader to \\cite{BNSS-JIIP-13} for the implementation of a similar scheme that displays very fast convergence rates but for which we are unable to prove a unique continuation principle.\n\\subsection{Injectivity results}\n\\label{sec:injectivity}\n\nThe fixed point algorithm of the preceding section provides us with an explicit local reconstruction procedure for the nonlinear hybrid inverse problem. A similar methodology allows us to obtain local uniqueness results that are more general but not constructive. Using the same notation as in the preceding section, let us assume that \n\\begin{equation}\n\\label{eq:uniqF}\n\\mF(v_0+v) = \\mF(v_0) = \\mH,\n\\end{equation}\nin other words the measurements associated with $v_0$ and $v_0+v$ are identical. Injectivity of the nonlinear problem $\\mF$ locally in the vicinity of $v_0$ then means proving that $v=0$.\n\nSince $\\mF$ is a polynomial in $v$ (or more generally a differentiable function in $v$), then there is a polynomial function in $v$ and $\\tilde v$ such that \n\\begin{equation}\n\\label{eq:diffF}\n\\mF(v_0+\\tilde v) - \\mF(v_0+v) = \\mG(v,\\tilde v) \\cdot (\\tilde v -v).\n\\end{equation}\n\nWhen $\\tilde v-v$ is sufficiently small, then $\\mG(v,\\tilde v)$ satisfies the same properties as $\\mF'(v)$, the Fr\\'echet derivative of $\\mF$. For instance, the ellipticity and unique continuation properties of $\\mF'(v)$ still hold for $\\mG(v,\\tilde v)$ for $v-\\tilde v$ small in the appropriate topology. As a consequence, we obtain that \\eqref{eq:uniqF} implies that $v=0$ and hence that $\\mF$ is injective. However, $\\mG(v,\\tilde v)$ may still admit a left-inverse for $v$ and $\\tilde v$ not necessarily close to each-other, in which case we also deduce that $v=\\tilde v$ for $v-\\tilde v$ not necessarily small.\n\n\nLet us assume more generally that we have two measurements\n\\begin{displaymath}\n   \\mF(v_0+v) = \\mH,\\qquad \\mF(v_0+\\tilde v) = \\tilde\\mH.\n\\end{displaymath}\nAssume that we have been able to prove that $\\mF'(v_0)$ is elliptic and injective. Then, $\\mG(v,\\tilde v)$ is elliptic and hence admits a left-inverse (since it is injective), at least for $v$ and $\\tilde v$ sufficiently close to $v_0$. As a consequence, if we assume that the boundary conditions that need to be imposed for $v$ and $v_0$ agree on $\\partial X$, then we obtain the stability estimate\n\\begin{equation}\n\\label{eq:stabnonlinear}\n  \\|v-\\tilde v\\|_{\\mX} \\leq C \\|\\mH-\\tilde\\mH\\|_{\\mY}.\n\\end{equation}\nErrors in the boundary conditions $\\mB(v)$ or $\\mD(v)$ may be accounted for in a similar manner. Using the notation in section \\ref{sec:param}, this is\n\\begin{equation}\n\\label{eq:stabnonlin2}\n\\dsum_{j=1}^{J+M} \\|v_j-\\tilde v_j\\|_{W_p^{l+t_j}(X)} \\leq C  \\dsum_{i=1}^{2J} \\|\\mH_i-\\mH_{0,i}\\|_{W_p^{l-s_i}(X)}.\n\\end{equation}\n\n\nNote that the nonlinear problem \\eqref{eq:uniqF} is invertible generically. Indeed, for $v_0$ analytic, the inversion of $\\mA^t\\mA$ augmented with Dirichlet boundary conditions $\\mD$ is guaranteed. As a consequence, the above result shows that \\eqref{eq:uniqF} may be inverted for $(v+v_0)$ in an open set including $v_0$. Since analytic coefficients $v_0$  in $X_0$ restricted to $X$ are dense in the set of sufficiently smooth coefficients on $X$, we obtain that the inverse problem may be inverted {\\em generically}; see \\cite{SU-JFA-09}.\n\nWhen $v_0$ is not analytic, then we need to find a unique continuation principle based on Carleman estimates; see the paragraph on applications to power density measurements in section \\ref{sec:fixedpoint}, which extends to the setting of injectivity results and not-necessarily local stability estimates of the form \\eqref{eq:stabnonlinear}.\n\n}\n\n\\section*{Acknowledgment} \nI would like to thank the organizers of the Irvine conference for their invitation to a meeting that was a befitting tribute to the exceptional contributions to analysis and inverse problems of Gunther Uhlmann. I am glad to acknowledge many insightful discussions with Shari Moskow on the content of section 4, which borrows some ideas of our joint work in \\cite{BM-LINUMOT-13}. I am also indebted to  Thomas Widlak for noticing and proposing solutions to inconsistencies in the presentation of the Lopatinskii conditions in section 2.6.2.\nThis paper was partially funded by NSF grant DMS-1108608 and AFOSR Grant NSSEFF- FA9550-10-1-0194.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nWe know many things about the structure of our Galaxy, and we have an approximate idea\nof the functional shape of its components: thin and thick disc, bulge, long bar, halo\nor spheroid, spiral arms, ring. In the past decades, different large-area surveys in visible\nor near-infrared wavelengths have allowed us to know our Galaxy much better.\nHowever there are some parts of the Galaxy that are not well known; one of them is the\nouter part of the disc, with Galactocentric distances beyond $R=15$ kpc.\nWe know this component is warped and flared, but the\ndetails of its shape are still a topic to develop further.\n\nWe are interested in analyse this outer part of stellar disc in our Galaxy in this paper,\nusing available visible data of the Sloan Digital Sky Survey (SDSS) in low Galactic latitude regions.\nSeveral authors (Bilir et al. 2006, 2008; Juri\\'c et al. 2008; Jia et al. 2014) \nhave previously used the SDSS to measure the parameters of the disc, but they have explored the high Galactic latitudes, so they could not access the outer disc. Other authors have analyzed the disc \nusing the Two Micron All Sky Survey (2MASS) either at low latitudes \n(L\\'opez-Corredoira et al. 2002, 2004) or at high latitudes using red-clump giants (Cabrera-Lavers et al. 2005, 2007; Chang et al. 2011) or within the whole sky using a given luminosity function (Polido et al. 2013), but, even using near-plane regions, the depth of 2MASS is not enough to analyze the\nouter disc; it is dominated by stars with smaller distances than the outer disc we\nwould like to analyze.\n\nOur purpose here is to use a survey like the SDSS (\\S \\ref{.data}), deeper than 2MASS, \nin near-plane regions, although avoiding the plane itself because of the high extinction. From this we separate a type of dwarf stars as\nstandard candles (\\S \\ref{.method}).\nWe are particularly interested in modelling the flare of the outer disc. There\nis some evidence of its existence in the different components of the Galaxy, but\nthe stellar flare at $R\\gtrsim 15$ kpc is still poorly known.\nThe flare of HI was investigated for instance by Nakanishi \\& Sofue (2003), Levine et al. (2006), or Kalberla et al. (2007). It was modelled by Kalberla et al. (2007) in terms of a dark matter ring, by Saha et al. (2009) in terms of a lopsided halo,\nor by L\\'opez-Corredoira \\& Betancort-Rijo (2009) in terms of accretion of intergalactic matter onto the disc.\nAnother example of modelling is given by Narayan \\& Jog (2002), who\nused a three-component (HI, H$_2$ and stars) disc gravitationally coupled to calculate the scale height of each component and their flares, and they derived a good prediction for the gas flare;\nthe result of mild stellar flaring up to $R=12$ kpc was obtained in the model by Narayan \\& Jog (2002), and it agreed well with the  observational data on flaring provided by Kent et al. (1991). \nThe stellar flare was also observed by\nAlard (2000), L\\'opez-Corredoira et al. (2002), Yusifov (2004), Momany et al. (2006), or Reyl\\'e et al. (2009) for the outer disc, but limited to $R\\lesssim 20$ kpc and with large uncertainties over 15 kpc or for the differences between the flare of the thin and thick disc.\n\nA flare in the thick disc is \nposited (Hammersley \\& L\\'opez-Corredoira 2011; Mateu et al. 2011; L\\'opez-Corredoira et al. 2012), \nand this may constitute an explanation for the Monoceros ring instead of the tidal stream\nhypothesis. This is an additional reason to pursue this research: to know whether we are able\nto fit our star counts without needing some new extragalactic component. The fact that\na flared thin+thick disc alone fits our data (\\S \\ref{.fit}, \\S \\ref{.explor}) gives a positive answer,\nwhich is discussed and compared with other works of the literature in \\S \\ref{.compar}, \\S \\ref{.discu}, together with the considerations derived from the morphological information obtained in this paper.\n\n\\section{Data} \n\\label{.data}\n\nThe SDSS (Sloan Digital Sky Survey)-DR8 release (Aihara et al. 2011) contains imaging of \n14\\,555 deg$^2$ of the sky in five filters (u,g,r,i,z), with a completeness higher than 95\\% \nfor $m_g\\le 22.2$. Within this survey, the subsample SEGUE (Sloan Extension for Galactic Understanding \nand Exploration) includes many Galactic plane regions.\nFor this paper, in which we are interested to explore the outer disc, we used the regions with \n$|\\ell |> 50^\\circ $ (to avoid the regions of the inner Galaxy), \n$|b|\\le 23^\\circ $ (dominated by disc stars), taking all the point-like SDSS sources with\nthe flags and constraints for the $g$ and $r$ filters: ((flags\\_r\\,\\& \\,0x10000000)\\, != 0),\n((flags\\_r\\, \\& \\,0x8100000c00a4)\\, = 0), (((flags\\_r\\, \\& \\,0x400000000000)\\, = 0)\\, or\\, \n(psfmagerr\\_r <= 0.2)),\n(((flags\\_r\\, \\& \\,0x100000000000)\\, = 0)\\, or\\, (flags\\_r\\, \\& \\,0x1000)\\, = 0), plus the same thing for \nflags\\_g;\nthis avoids the stars that are too close to the borders, are too small to determine\nthe radial profile, have saturated pixels or too many interpolated pixels\nto derive a correct flux, stars for which the deblend algorithm finds two or more\ncandidates in cases with a magnitude error in $r$ larger than 0.2, stars with poor detection, or cases with cosmic rays.\nAll the magnitudes were corrected for extinction by the SDSS team using the \nGalactic extinction model of Schlegel et al. (1998). \nIn total, we cover an area of 1\\,745 deg$^2$.\nAs we explain in the next section, we reduced this area by avoiding the regions very close to the plane with high extinction to reduce the errors due to the correction of extinction.\n\n\\section{Method}\n\\label{.method}\n\nThe simplest method of determining the stellar density along a \nline of sight in the disc is by isolating a group of stars with the same colour and absolute magnitude \n$M$ within a colour magnitude diagram. This allows the luminosity function to\nbe replaced by a constant in the stellar statistics equation and the differential star counts for each line of sight, $A(m)$, can be immediately converted into density $\\rho (r)$:\n\n\\begin{equation}\n\\label{diffsc}\nA(m)\\equiv \\frac{dN(m)}{dm}=\\frac{ln\\ 10}{5}\\omega \\rho\n[r(m)]r(m)^3  \n,\\end{equation}\\[\nr(m)=10^{[m-M+5]\/5}\n,\\]\nwhere $\\omega $ is the area of the solid angle in radians and $r$ is the\ndistance in parsecs.\n\nIn the near-infrared, red-clump giants have been successfully used as  standard candles, particularly for the innermost 15 kpc from the Galactic centre\n(L\\'opez-Corredoira et al. 2002). However, red-clump stars in the outer disc at the distance of interest would appear at $m_k\\ge 14$, where the local dwarfs with the same colour would completely dominate the counts \n(L\\'opez-Corredoira et al. 2002). Therefore, we have to use something different here.\n\nFor our extinction-corrected areas, it is possible to use star counts in visible. An examination of the HR diagram shows that when the extinction is low, the late F and early G dwarfs can be isolated using colour with only minimal contamination from other sources with the same colour, but different\nabsolute magnitudes. For this work we selected the \nsources between F8V and G5V. \nThere will be sufficient stars detected in the outer Galaxy to give meaningful statistics, \nwhich would not be the case if a smaller range in absolute magnitudes were used. Earlier sources were\nnot included as these sources would belong to a younger population with a \nfar lower scale height; the absolute magnitude also changes\nfar more rapidly with colour. Later sources would have significant giant contamination, \nand again the absolute magnitude changes more rapidly with colour. \n\n   \\begin{figure}\n\\vspace{1cm}\n   \\centering\n   \\includegraphics[width=8cm]{CMdiagram.eps}\n   \\caption{Extinction-corrected colour-magnitude diagram for the region $\\ell=183^\\circ $, $b=21^\\circ$. The region between the dashed lines contains F8V-G5V dwarfs.}\n   \\label{Fig:CMdiagram}\n   \\end{figure}\n\nThe F8V-G5V dwarfs have a range of $g-r$ of 0.36 to 0.49 \nand a range of absolute magnitudes $M_g$=4.2 to 5.3 (Bilir et al. 2009),\nwhich makes the sources approximately $m_g\\lesssim 21.5$ \nat the distance of interest up to 22 kpc. See the selection example in Fig. \\ref{Fig:CMdiagram}. We adopted a constant absolute magnitude for all of them\n$M_g$=4.8. For a range of absolute magnitudes, when the counts are converted into \ndensity vs. distance, there is some smoothing which is not included in a model that assumes a single absolute magnitude. This smoothing has little effect,  \nand the above approximation remains valid: see L\\'opez-Corredoira et al. 2002, \\S 3.3.1\nfor a calculation of the difference between a narrow Gaussian distribution and a Dirac delta: it leads to an error in the scale length of the order of 2\\% assuming an r.m.s. in the Gaussian distribution of 0.3 mag.; although the application of L\\'opez-Corredoira et al. is for red-clump giants, \nit is valid for any kind of population.\nWe did not take into account the possibility that some of these stars might indeed be binary systems.\nSee Siegel et al. (2002) or Bilir et al. (2009, \\S 3.4) for a calculation of \nthe effects of a high ratio of\nbinary stars to derive the parameters of the disc: they may produce \n$\\Delta M_g\\sim 0.2$ mag.\nThere are some radial and vertical gradients of metallicity for\nthin and thick disc (Rong et al. 2001; Ak et al. 2007; Andreuzzi et al. 2011); \nhere, like in Juri\\'c et al. (2008),\nwe did not take into account the variation of the absolute magnitude due to the variation of metallicity: \nsee Siegel et al. (2002) or Juri\\'c et al. (2008, \\S 2.2.1) for a discussion on it. \nA rough estimate of the maximum difference for the most extreme cases of \nlower metallicity for the highest values of $R$ and $z$ ($[Fe\/H]\\sim 1$ dex\nlower than in the solar neighbourhood) is $\\Delta M_R\\approx 0.4$ [Siegel et al. 2002, using $R-I=0.38$, which is the corresponding transformation from SDSS to Johnson filters (Jordi et al. 2006) of the\ncolor of our population with average $(r-i)=0.13$ (Bilir et al. 2009)]. The variation of $M_g$ is\nsimilar (Juri\\'c et al. 2008, Fig. 3). Therefore, \nthe fact that we did not take this metallicity gradient into account \nmay produce an overestimate of\nthe distance of the stars at the highest $R$ or $z$ of $\\sim 20$\\% . Its effects are explored in \\S \n\\ref{.explor}.\n \nIn the 1\\,745 deg$^2$ of our SDSS data are 6\\,506\\,360 stars with $g-r$ (corrected for extinction) between 0.36 and 0.49. The source densities in these regions are expected to be very low, therefore we require regions of more than 0.5 square degrees of sky to be covered to provide sufficient counts to give reasonable statistics. \nWe divided them into multiple space bins with $\\Delta \\ell \\times \\Delta b=2^\\circ \\times 2^\\circ $, $\\Delta m_g=0.2$ for $15.25<m_g<21.85$, and,\nsince we know their distances and coordinates (hence, we know their position in 3D space), given the differential star counts $A(m)$, we can derive through Eq. (\\ref{diffsc}) the density $\\rho (R,\\phi ,z)$ \n(in cylindrical Galactocentric coordinates). This average stellar density is plotted in Fig.\n\\ref{Fig:dens}, excluding the bins with $|b|<8^\\circ $.\n\n\\begin{figure}\n\\vspace{1cm}\n\\centering\n\\includegraphics[width=8.5cm]{bins_phi.eps}\n\\caption{Average stellar density of F8V-G5V stars from available SDSS data as a function of the \ncylindrical coordinates $R$ (kpc), $\\phi $ (deg.), $z$ (kpc)\nfor the range $0<R<30$ kpc, $|z|<15$ kpc, constrained within Galactic latitudes\n$8^\\circ \\le |b|\\le 22^\\circ $. The circle stands for $R=8$ kpc and $z=0$ (like the\nSun position) for any value of $\\phi $ (for the Sun, it would be $\\phi =0 $).\nThe axes in the bottom left panel apply to all panels in the figure.}\n\\label{Fig:dens}\n\\end{figure}\n\nWe excluded these in-plane regions (bins with $|b|<8^\\circ $), which reduced \nour area to 1\\,396 deg$^2$, to reduce the errors\nin the extinction correction: with extinctions of $\\langle A(g)\\rangle \\lesssim 1.5$ mag \nin filter g at most for $|b|\\ge 8^\\circ $ (Schlegel et al. 1998), $E(g-r)=0.27A(g)$ \nand a relative uncertainty of the reddening of \n$\\sim 10$\\% (Schlegel et al. 1998; M\\\"ortsell 2013), we derive relative\nerrors of $\\Delta (g-r) \\lesssim 0.04$ for each region. With a $\\frac{dM_g}{d(g-r)}\\approx\n\\frac{5.4-4.2}{0.49-0.36}=9.2$, we compute uncertainties in the distance determination\nof $\\lesssim 18$\\%. Although this error is still large, when combining all the bins,\nthe errors compensate for each other and are mostly cancelled on except for some systematic\nones that may lead to some variation of the scales. We discuss this point in \\S \\ref{.discu}. At present, we do not carry out a deconvolve the line of sight distribution with the spread of distances because of the r.m.s. of the reddening, which would provide a better determination of the morphology, because one would need a very accurate expression of the r.m.s. of the extinction, which is not available (see also discussion in \\S \\ref{.discu}). Because the regions are all well off the plane we assumed that \nthis extinction is local.\n\n\n\\section{Fitting free parameters of a disc model}\n\\label{.fit}\n\n\\subsection{Disc model}\n\nAfter deriving the stellar density $\\rho (R,\\phi ,z)$, we can\nfit a disc model to obtain its best free parameters. We\nused the following model, which contains flared axysymmetric thin+thick discs:\n             \n\\begin{equation}\n\\label{rho}\n\\rho _{\\rm disc}(R,z)=\\rho _{\\rm thin}(R,z)+\\rho _{\\rm thick}(R,z)\n,\\end{equation}\n\\[\n\\rho_{\\rm thin} (R,z)= \\rho_\\odot \\frac{h_{\\rm z, thin}(R_\\odot)}{h_{\\rm z, thin}(R)}\\exp\\left( \\frac{R_\\odot}{h_{\\rm r, thin}} + \\frac{h_{\\rm r, hole}}{R_\\odot} \\right)\n\\]\\[\\ \\ \\ \\ \\times \n\\exp\\left( -\\frac{R}{h_{\\rm r, thin}}-\\frac{h_{\\rm r, hole}}{R} \\right) \\exp\\left( -\\frac{|z|}{h_{\\rm z, thin}(R)} \\right)\n,\\]\\[\n\\rho_{\\rm thick} (R,z)= f_{\\rm thick}\\,\\rho_\\odot \\frac{h_{\\rm z, thick}}{h_{\\rm z, thick}(R)}\\exp \\left( \\frac{R_\\odot}{h_{\\rm r, thick}} + \\frac{h_{\\rm r, hole}}{R_\\odot} \\right)\n\\]\\[\\ \\ \\ \\ \\times \\exp \\left( -\\frac{R}{h_{\\rm r, thick}}-\\frac{h_{\\rm r, hole}}{R} \\right) \\exp\n\\left( -\\frac{|z|}{h_{\\rm z, thick}(R)} \\right)\n,\\]\nwhere the respective scale heights are\n\\begin{equation}\nh_{\\rm z, thin}(R) = h_{\\rm z, thin}(R_\\odot)\\left( 1 + \\sum_{i=1}^2 k_{\\rm i, thin}(R-R_\\odot )^i \\right)\n,\\end{equation}\\[\nh_{\\rm z, thick}(R) = \\left\\{ \\begin{array}{lcl}\n      h_{\\rm z, thick}(R_{\\rm ft})&  R<R_{\\rm ft}\\\\\n      h_{\\rm z, thick}(R_{\\rm ft})\\left( 1 + \\sum_{i=1}^2 k_{\\rm i, thick}(R-R_{\\rm ft})^i\\right) & R \\geq R_{\\rm ft}\\\\\n   \\end{array}\n   \\right.\n.\\]\nThis expresses an exponentially flared disc (L\\'opez-Corredoira et al. 2002) with a hole or deficit of\nstars in the inner in-plane region (L\\'opez-Corredoira et al. 2004). Since we did not examine the inner\ndisc, we kept $h_{\\rm r, hole}$ constant: $h_{\\rm r, hole}=3.74$ kpc (L\\'opez-Corredoira et al. 2004).\nWe also kept constant the ratio of thick to thin disc stars in the solar neighbourhood,\n$f_{\\rm thick}=0.09$; and the rest of the parameters were left free in our fit: \nthe scale length of the thin disc $h_{\\rm r, thin}$, the scale height of the thin disc at solar Galactocentric\ndistance $h_{\\rm z, thin}(R_\\odot)$, the scale length of the thick disc $h_{\\rm r, thick}$,\nthe scale height of the thin disc at solar Galactocentric\ndistance $h_{\\rm z, thick}(R_\\odot)$, the two parameters defining the flare of the thin disc\n$k_{\\rm 1, thin}$ and $k_{\\rm 2, thin}$, the two parameters defining the flare of the thick disc $k_{\\rm 1, thick}$, $k_{\\rm 2, thick}$, and the scale at which the thick disc flare starts, $R_{\\rm ft}$.\n\nBovy et al. (2012) suggested that there is no thick disc that sensibly can be \ncharacterized as a distinct component\nbecause mono-abundance sub-populations, defined in the [alpha\/Fe]-[Fe\/H] space, are well described by single-exponential spatial-density profiles in both the radial and the vertical direction; therefore, any star of a given abundance would be associated with a sub-population of a given scale height and there would be a continuous and monotonic distribution of disc thicknesses. This result is contradicted by other authors however (e.g., Haywood et al. 2013), who showed a bimodal distribution in the [alpha\/Fe]-[Fe\/H] space, or that thin and thick disc can be distinguished because of the rotation and vertical dispersion, age, or other indicators of metallicity. Here we assumed that there are morphologically two discs and do not engage in the discussion about their populations.\n\nWe also included a halo, but did not try to fit any of its parameters; we just took\nthe fixed density distribution provided by Bilir et al. (2008) using SDSS data. \nIts contribution to our counts is lower than the disc contribution (20\\% for the highest $R$ and $z$ of\nour range).\n\\begin{equation}\n\\rho_{\\rm halo} (R,z)= 1.4\\times 10^{-3}\\,\\rho_\\odot \\frac{\\exp\\left[10.093\\left( 1 - \\left(\\frac{R_{\\rm sp}}{R_\\odot} \\right)^{1\/4} \\right) \\right]}{(R_{\\rm sp}\/R_\\odot )^{7\/8}} \n,\\end{equation}\\[\nR_{\\rm sp} = \\sqrt{R^2+2.52z^2}\n.\\]\n\nOur model does not take the stellar warp into account.\nThe warp moves the position of the Galactic plane: the plane can be \nshifted away from the expected position of $b=0$. It can be clearly\nseen in the counts within a few kpc of the Sun (L\\'opez-Corredoira et al. 2002, \nMomany et al. 2006, Reyl\\'e et al. 2009) and its effect is stronger towards \nthe outer edge of the disc.\nThe models of the warp are normally simple, based on tilted rings, and in general \nrepresent the star count well. Nonetheless, its average effect is not important for\n$|b|\\ge 8^\\circ $ (see Fig. 15\/top of L\\'opez-Corredoira et al. 2002 with ratio of positive\/negative counts\nclose to unity): some regions are overdense in the positive latitudes with respect to the negative latitude for the northern warp region (the southern warp is not visible with SDSS), but on average for our fit both positive and negative latitude data cancel each other out and only a \nminor second-order effect may remain in the fit, which we neglected. \nIn Fig. \\ref{Fig:dens}, no significant signs of the northern warp are detected\n(the southern warp is not visible in our SDSS data). A more detailed \ndiscussion on the influence of the warp is given in the\nlast paragraph of the next subsection and in \\S \\ref{.explor}.\n\n\\subsection{Best fit}\n\n\\begin{figure*}\n\\vspace{1cm}\n\\centering\n\\includegraphics[width=18cm]{fitres.eps}\n\\caption{Best fit (solid line) for the data of $\\rho (R,z)$ extracted from the star counts of\nF8-G5V stars in the SDSS. The dashed line stands for the best fit of the disc within $R<15$ kpc\nwithout any flare, and extrapolated for $R\\ge 15$ kpc.\nError bars stand only for Poissonian errors in the star counts and\ndo not include other possible factors such as errors in the extinction.\nNote that in the lowest $|z|$ bins there are no \npoints at high $R$ because of our constraint of \n$|b|>8^\\circ $ (see Fig. \\ref{Fig:dens}).}\n\\label{Fig:fitres}\n\\end{figure*}\n\n\\begin{figure}\n\\vspace{1cm}\n\\centering\n\\includegraphics[width=9cm]{err_theta_zpos.eps}\\\\\n\\vspace{1cm}\n\\includegraphics[width=9cm]{err_theta_zneg.eps}\n\\caption{Residuals of $\\rho $ with respect to the best fit for the data extracted from the star counts of\nF8-G5V stars in the SDSS. $R>7.5$ kpc, $|z|<3.5$ kpc. Top: $z>0$. Botton: $z<0$.}\n\\label{Fig:err_theta}\n\\end{figure}\n\n\\begin{table*}\n\\caption{Disc parameters of the best fit (see text). Note that there are nine independent parameters\nin the fit; the last six parameters depend on the previous ones. The constraint \n$|\\phi |\\le 30^\\circ $ explores the region where the warp amplitude is very low. The metallicity gradient\nis modelled following Eq. (\\ref{gradmet}).}\n\\begin{center}\n\\begin{tabular}{cccc}\n\\textrm{parameter} & \\textrm{All data, no grad. metal.} & $|\\phi |\\le 30^\\circ $, no grad. metal. & \\textrm{All data, with grad. metal.}  \\\\\n\\hline\n$h_{\\rm r, thin}$ (kpc) & $2.0^{+0.3}_{-0.4}$  & 2.1  & 2.0  \\\\\n$h_{\\rm z, thin}(R_\\odot)$ (kpc) & $0.24^{+0.12}_{-0.01}$  & 0.28  & 0.21  \\\\\n$h_{\\rm r, thick}$ (kpc) & $2.5^{+1.2}_{-0.3}$  & 2.5  & 2.7  \\\\\n$h_{\\rm z, thick}(R_\\odot)$ (kpc) & $0.71^{+0.22}_{-0.02}$  & 0.60  & 0.55 \\\\ \\hline\n$k_{\\rm 1, thin}$ (kpc$^{-1}$) & -0.037  & 0.090 & -0.190 \\\\\n$k_{\\rm 2, thin}$ (kpc$^{-2}$) & 0.052  & 0.043 & 0.070 \\\\\n$k_{\\rm 1, thick}$ (kpc$^{-1}$) & 0.021  & 1.448 & 0.000 \\\\\n$k_{\\rm 2, thick}$ (kpc$^{-2}$) & 0.006  & -0.055 & 0.010 \\\\\n$R_{\\rm ft}$ (kpc) & $6.9^{+10.1}_{-1.9}$ & 15.8 & 6.9 \\\\ \\hline\n$h_{\\rm z, thin}(15\\,{\\rm kpc})\/h_{\\rm z, thin}(R_\\odot)$   & 3.3$^{+1.8}_{-1.7}$    & 3.7   & 3.1 \\\\\n$h_{\\rm z, thin}(20\\,{\\rm kpc})\/h_{\\rm z, thin}(R_\\odot)$   & 8.1$^{+5.4}_{-5.4}$    & 8.3   & 8.8 \\\\\n$h_{\\rm z, thin}(25\\,{\\rm kpc})\/h_{\\rm z, thin}(R_\\odot)$   & 15.5$^{+10.8}_{-11.3}$  & 14.9  & 17.8 \\\\\n$h_{\\rm z, thick}(15\\,{\\rm kpc})\/h_{\\rm z, thick}(R_\\odot)$ & 1.5$^{+4.8}_{-0.4}$   & 1.0  & 1.6 \\\\\n$h_{\\rm z, thick}(20\\,{\\rm kpc})\/h_{\\rm z, thin}(R_\\odot)$  & 2.3$^{+12.9}_{-0.8}$  & 6.1   & 2.7 \\\\\n$h_{\\rm z, thick}(25\\,{\\rm kpc})\/h_{\\rm z, thick}(R_\\odot)$ & 3.4$^{+25.4}_{-1.7}$   & 9.6  & 4.2 \\\\ \\hline\n\\label{Tab:bestfit}\n\\end{tabular}\n\\end{center}\n\\end{table*}\n\nWe used the selected regions for the fit, excluding bins with a density lower than\n$3\\times 10^{-9}$ star pc$^{-3}$. For the fit of the scale lengths and scale \nheights ($h_{\\rm r, thin}$,\n$h_{\\rm z, thin}(R_\\odot)$, $h_{\\rm r, thick}$, $h_{\\rm z, thick}(R_{\\rm ft})$), we used \nthe regions with $|z|\\le 3$ kpc, $R<15$ kpc, i.e., we neglect the flare; after we obtained these four parameters we fitted the rest of them for the flare in the regions with $1.5<|z({\\rm kpc})|\\le 3.5$ kpc, $7.5<R({\\rm kpc})<30$. Within these ranges, the Galaxy is dominated by the disc rather than the halo star counts.\n\nThe parameters that produce the best weighted fit are given in Table \\ref{Tab:bestfit}.\nFigs. \\ref{Fig:fitres} and \\ref{Fig:err_theta} show how this model fits the data.\nThe error bars (1$\\sigma $) of these parameters were derived using the method of Avni (1976) for four free\nparameters (the first fit of the scales) and\nfive free parameters (the fit of the flare): $\\Delta \\chi^2=4.72$ and 5.89 respectively; \nwhere we normalized $\\chi ^2$ such that \n$\\chi _{\\rm min}^2=N$ (number of data points) to take into account that the Poissonian\nerror is only a small part of the total error.\n\nIt can be observed how the flare becomes strong for high $R$ and $z$. In Fig. \\ref{Fig:flare}, we plot the functional shape of $h_{\\rm z, thin}(R)$ and $h_{\\rm z, thick}(R)$ in this best-fit model.\nThe error bars for the flare amplitude are high, but they exclude the non-flared solution.\nNote that the errors for the thin disc include the free variation of the thick disc to compensate for the\nrespective variations within those errors and vice versa. If we fixed one of the discs, the errors bars of\nthe other disc would be much lower than the case with its variation. When we combine both discs, we also expect\nto reduce the error of the average disc (the errors of the average are also derived through the\n$\\chi ^2$ analysis), from which we conclude that we need a flare to fit our data, although distinguishing between thin and\nthick disc component is only possible with low significance. \nIt is clear from Fig. \\ref{Fig:fitres} that an extrapolation of the\ndisc fitted at $R<15$ kpc without flare does not work at larger $R$: the shape of the density presents\na more complex form than the dashed line in the log-linear plot of Fig. \\ref{Fig:fitres}.\n\n\n\\begin{figure}\n\\vspace{1cm}\n\\centering\n\\includegraphics[width=8cm]{flare.eps} \n\\caption{Scale height of the thin and thick discs according to our best fit.\nThe average is defined as the $-\\left(\\frac{d\\,ln\\,\\rho _{\\rm disc}(R,z=0)}{d|z|}\\right)^{-1}$\nand takes into account the increasing ratio of thick disc stars outwards.}\n\\label{Fig:flare}\n\\end{figure}\n\n\\begin{figure}\n\\vspace{1cm}\n\\centering\n\\includegraphics[width=8cm]{flare2.eps} \n\\caption{Scale height of the average defined as in Fig. \\protect{\\ref{Fig:flare}} for the best fits \nof Table \\protect{\\ref{Tab:bestfit}}.}\n\\label{Fig:flare2}\n\\end{figure}\n\nWe can see how the variation of the density with $z$ for\nhigh $R$ becomes quite weak: green extends to almost all regions with high $R$ in Fig. \\ref{Fig:dens}, and the few blue areas (very low densities) represent the few fluctuations due possibly\nto some errors in the extinction or warp presence\n(because the lowest latitudes are more affected by these\nfluctuations). This is precisely what we obtain in our fit, \nwhich indicates the existence of a conspicuous\nflare: a very significant increase of the scale height of the disc both for the thin and thick discs.\nIn Fig. \\ref{Fig:fitres}, we also observe the effect of the flare: at $R\\approx 15$ kpc and high $z$ the\ndensity becomes almost constant with little decrease with $R$ \nbecause of a combination of the exponential decrease with radius and the increase of the density at high $z$ due to the increase of the scaleheight.\n\nFig. \\ref{Fig:err_theta} shows some residuals in the observed stellar \ndensity with respect to the best fit. If the northern warp effect were significant, it would produce an excess density at $60^\\circ \\lesssim \\phi \\lesssim 100 ^\\circ $ at $z>0$ and a deficit of stars in the same azimuths at $z<0$, and we do not observe that.\nIn general, some deviations from null residuals may be due to different effects such as metallicity gradients, a different\ncontamination of the halo than expected (at high $|z|$), some irregularities in the distribution of the extinction,\nsome degree of lopsidedness, or a variation of $h_z$ with $\\phi $ (L\\'opez-Corredoira \\& Betancort-Rijo 2009).\n\n\\section{Exploring the effects of the warp and metallicity gradients}\n\\label{.explor}\n\nWe carried out two numerical experiments to determine the effects of \nthe warp and the metallicity gradients. Since \nthere are large uncertainties for both, we do not expect to derive direct\nconclusions from the results of the following best fits, but they should serve as an estimate of the typical variation in our parameters under these changes. \n\n\\begin{description}\n\n\\item[Warp:] instead of the whole data set, we used only those with $|\\phi |\\le 30^\\circ $, where the amplitude\nof the warp is lowest. We fitted $\\rho (R,z)$ in the same way as before and the parameters obtained are those given in Table \\ref{Tab:bestfit} and Fig. \\ref{Fig:flare2}. \nThey are compatible with the previous values, confirming our guess that the warp \ndoes not strongly change our results.\n\n\\item[Metallicity gradient:] we attributed to each star a Galactocentric distance $R'$, $\\phi '$ and vertical position $z'$ corresponding to their coordinates and a distance $r'(m)$ given by\n\\begin{equation}\n\\label{gradmet}\nr'(m)=10^{[m-M'+5]\/5}\n,\\end{equation}\\[\nM'=4.8-0.4\\Delta[Fe\/H]\n,\\]\\[\n\\Delta[Fe\/H](R'[{\\rm kpc}],z'[{\\rm kpc}])=\n\\]\\[\n\\left\\{ \\begin{array}{lcl}\n      -0.078(R'-R_\\odot)-0.220|z'| &  , R'\\le f(z') \\\\\n      -1 & , R'>f(z') \\\\\n   \\end{array}\n   \\right.\n,\\]\\[\nf(z')=R_\\odot +12.82-2.82|z'|\n.\\]\n\nAs said in \\S \\ref{.method}, this stems from the variation of the \nabsolute magnitude of F8V-G5V stars \nestimated by Siegel et al. (2002) for a variation of matallicity with respect to the Sun \n[$\\Delta M_R\\approx 0.4$ for $\\Delta[Fe\/H]=-1$ using $R-I=0.38$, which is the corresponding transformation from SDSS to Johnson filters (Jordi et al. 2006) of the\ncolor of our population with average $(r-i)=0.13$ (Bilir et al. 2009); $\\Delta M_g\\approx \\Delta M_R$ \n(Juri\\'c et al. 2008, Fig. 3)], and considering a variation of metallicity from the combination of radial and vertical gradients of metallicity for the disc given by Rong et al. (2001) and Ak et al. (2007); assuming the\nsame gradient for thin and thick discs and that \nit remains constant for the farthest disc for a metallicity lower than the solar one.\nThis is probably an overestimation of $|\\Delta[Fe\/H]|$, which should be lower than unity at least for the\nthin disc (Andreuzzi et al. 2011), but it serves as a limit of the strongest \neffect of this gradient.\nGiven that $\\Delta[Fe\/H](R',z')$ depends on the position and the position depends on this variation of metallicity, we carried out the calculation with an iterative process.\n\nThen, we fitted $\\rho (R',z')$ in the same way as before and the parameters obtained are\nthose given in Table \\ref{Tab:bestfit} and Fig. \\ref{Fig:flare2}. The results for the flare parameters\nare totally compatible within the error bars with those obtained without taking into account any gradient\nof metallicity.\nAs said in \\S \\ref{.method}, a lower metallicity in the farthest parts of the disc may overestimate\nthe distance of the stars (unfortunately, we do not know by how much since there are no \naccurate measurements of the metallicity of stars at those Galactocentric distances; we estimate an error of no more\nthan 20\\% [\\S \\ref{.method}]), but our numerical experiment shows that \nthe variation of the scale lengths and scale heights is slight and\nthe need of the flare is beyond these uncertainties with the metallicity.\n\n\\end{description}\n\n\n\\section{Comparison with other works}\n\\label{.compar}\n\nThe scale lengths and scale heights can be compared with those in other publications, although mainly for low $R$. Table \\ref{Tab:compar} gives some of the values of the literature.\nJia et al. (2014, Table 1) reported many other values.\nIn general, the relative trends between thin and thick disc\nare similar in all these papers and our results, but there\nis some variation in the absolute scales that might arise because of different observed populations, different techniques of distance estimation, or different regions of application, apart, of course, from\npossible systematic errors. Jia et al. (2014) have shown how the parameters, especially the \nscale height of the thin disc, depend on the absolute magnitude of the main-sequence stars used, indicating\nthat different populations have different velocity dispersions. The numbers of\nJuri\\'c et al. (2008) are somewhat higher than ours, possibly because they represent a range of smaller $R$, or possibly because\nof their method of distance determination. Bilir et al. (2006) showed that  \nthe scale length depends on Galactic longitude. Here, we derived its average value but did\nnot examine any dependence on Galactic coordinates. \n\n\\begin{table*}\n\\caption{Some values of the scale lengths and scale heights from the literature (units in kpc), \nderived either with SDSS (visible) or 2MASS (near infrared red).}\n\\begin{tabular}{ccccccc}\nReference & source & spatial range & $h_{\\rm r, thin}$ & $h_{\\rm z, thin}(R_\\odot)$ & $h_{\\rm r, thick}$ & $h_{\\rm z, thick}(R_\\odot)$ \\\\\n\\hline\nL\\'opez-Corredoira et al. (2002) & 2MASS & low Gal. latitudes & 3.3 & 0.28 & -- & -- \\\\ \nCabrera-Lavers et al. (2005) & 2MASS & high Gal. latitudes & -- & 0.27 & -- & 1.1  \\\\\nBilir et al. (2006) & SDSS & interm. Gal. latitudes & 1.9 & 0.22 & -- & --  \\\\\nCabrera-Lavers et al. (2007) & 2MASS & interm. Gal. latitudes & -- & 0.19 & -- & 0.96  \\\\\nBilir et al. (2008) & SDSS & high Gal. latitutes & -- & 0.19 & -- & 0.63  \\\\\nJuri\\'c et al. (2008) & SDSS & $r<1.5$ kpc & 2.6 & 0.30 & 3.6 & 0.90  \\\\\nChang et al. (2011) & 2MASS & interm.-high Gal. latitudes & 3.7 & 0.36 & 5.0 & 1.0  \\\\\nPolido et al. (2013) & 2MASS & whole sky & 2.1 & 0.21 & 3.0 & 0.64 \\\\\nJia et al. (2014) & SDSS+SCUSS & interm. Gal. latitude & -- & 0.20 & -- & 0.60 \\\\ \n\\label{Tab:compar}\n\\end{tabular}\n\\end{table*}\n\nThe flare was previously observed by Alard (2000), \nL\\'opez-Corredoira et al. (2002), Yusifov (2004), Momany et al. (2006), or Reyl\\'e et al. (2009).\nPolido et al. (2013) also introduced a flare in their model, but did no fit their parameters.\nThe values of the scale height at high $R$ from our fit (see Fig. \\ref{Fig:flare}) are high: with\na thin disc scaleheight around 0.8 kpc at R=15 kpc, lower than the\nextrapolation of L\\'opez-Corredoira et al.\n(2002) or that of Yusifov (2004), and\nhigher than the values of Alard (2000), Momany et al. (2006) and Reyl\\'e et al. (2009). Between 20-25 kpc we derive a\nthin disc scale height of 2-4 kpc, which is also higher than the values by\nMomany et al. (2006) and Reyl\\'e et al. (2009)  \n(for the rest of the authors, there are no values at such \nhigh values of $R$).\nFor the thick disc, there are few or no studies to compare our studies with: \nthere is a hint with unconclusive results  \nby Cabrera-Lavers et al. (2007), which was constrained within $R<10$ kpc and\nobtained the opposite sign in the increase of scale height for the solar neighbourhood;\nit is interesting that the values of the flare in the thick disc \nthat we obtained are those that are needed to explain the Monoceros ring in terms of \nGalactic structure (Hammersley \\& L\\'opez-Corredoira 2011).\n\nNote that in our results at $R\\gtrsim 17$ kpc \nthe thin disc becomes ``thicker'' than the so-called thick disc.\nThis should motivate us to change the nomenclature: maybe instead of thin disc + thick disc\nwe should speak of disc 1 and disc 2. In any case, our model is just an exercise of fitting\nstellar densities and, within the error bars we are unable to see which disc has a larger scale height\nat large Galactocentric radius. We do not aim here to distinguish among different populations.\nWe see from our results that we need a flare to interpret the global density, but, with\nthe present analysis, we are not able to distinguish  the populations\nthat are flared at high $R$. We expect that at high $R$ there should be no significant difference between both discs\nthicknesses and we have only one mixed component. An average disc as plotted in \nFig. \\ref{Fig:flare} represents this average old population of type F8V-G5V stars.\nIn this average disc, at high $R$ \nthe thick disc (or better: disc 2) has a higher ratio of stars than at $R_\\odot $:\nwhile at $R_\\odot $ it is 9\\% of the stars of the thin disc (or better: disc 1), at $R=25$ kpc it is 50\\% of the stars of the thin disc, because the scale length of the thick disc is larger than that of the thin disc, and consequently the fall-off of the density is slower.\n \nThe theoretical explanation of the observed features is beyond the scope of this paper.\nThe origin of the thick disc and its flare need to be\npredicted by a model that aims to understand these observations. One of the hypotheses for the formation of thick discs is through minor mergers, which predicts a scale length of the thick disc larger than the scale length of the thin disc, a flare of increasing scale height in the thick disc, and a constant scale height for the stellar excess added by the merger (Qu et al.\n2011). If we assumed that the mixture of the thin disc of the primary galaxy \nplus the stellar excess due to the accreted minor galaxy produces what we observe as the thick disc, our results would be fitted by those predictions. Nonetheless, our analysis is very rough, and without a necessary study of the populations we are unable to confirm this scenario.\n\n\\section{Discussion and conclusions}\n\\label{.discu}\n\nOur method of deriving the 3D stellar distribution is quite straightforward, although it may contain some errors due mainly to inappropriate extinction estimate (some systematic error in the scales may be produced, but we do not expect it to exceed 18\\%; see\n\\S \\ref{.method}), metallicity gradients, or the effect of the warp. \nEven taking into account these factors, \nwe have not observed features that suggested that the derived morphology\nmight be very different: the scales might change slightly, \nbut the presence of the flare is unavoidable.\n\nOur results show that the stellar density distribution of the outer disc (up to $R=30$ kpc) is well fitted by a component of thin+thick disc with flares (increasing scale height outwards). From our diagrams, it is clear that there is no a cut-off of the stellar component at $R=14-15$ kpc\nas stated by Ruphy et al. (1996) or Minniti et al. (2011); we only examined off-plane regions \n($|b|\\ge 8^\\circ $) so we cannot judge what occurs in the in-plane regions, but from our results and by interpolating the results in the $z$-direction one can clearly conclude the reason why Ruphy et al. or Minniti et al. appreciated a significant drop-off of stars at $R=14-15$ kpc: the flare becomes important at those galactocentric distances, and consequently, the stars are distributed in a much wider range of heights, producing this apparent depletion of in-plane stars. Indeed, our Galactic disc does not present a cut-off there but the stars are spread in off-plane regions, even at $z$ of several kpc up to \na Galactocentric distance of 15 scale lengths. Assuming that our fit is correct, for a constant luminosity function along the disc, the flux of the Milky Way seen observed face-on would follow a dependence\n(from Eq. (\\ref{rho}), neglecting the hole of the inner disc, which is totally insignificant for $R>R_\\odot $)\n\\begin{equation}\nF(R)\\propto \\int _{-\\infty}^\\infty dz\\,\\rho_{\\rm disc}(R,z)\n\\end{equation}\\[\\,\\,\\,\\,\\,\\,\n\\approx \\rho _\\odot \\left[\\exp\\left(-\\frac{R}{h_{\\rm r, thin}}\\right)+f_{\\rm thick}\\exp\\left(-\\frac{R}{h_{\\rm r, thick}}\\right)\\right]\n.\\]\nIt is clear that in $F(R)$ there is no radial truncation in the explored range, and if this $F(R)$ did not represent our Galaxy, we would not derive as good a fit as we did. This does not mean that radial truncations are not possible in spiral galaxies: there are other galaxies in which it is observed (van der Kruit \\& Searle 1981; Pohlen et al. 2000), but the Milky Way is not one of them.\n\nThe smoothness of the observed stellar distribution also suggests that there is a continuous structure\nand not a combination of a Galactic disc plus some other substructure or extragalactic component. The\nhypothesis of interpreting the Monoceros ring in terms of a tidal stream of a putative accreted dwarf galaxy\n(Sollima et al. 2011; Conn et al. 2012; Meisner et al. 2012; Li et al. 2012)\nis not only unnecessary (as stated by Momany et al. 2006; Hammersley \\& L\\'opez-Corredoira 2011; L\\'opez-Corredoira et al. 2012), but appears to be quite inappropriate:\nwe see in Fig. \\ref{Fig:dens} no structure overimposed on the Galactic disc.\nInstead, the observed flare explains the overdensity in the Monoceros ring\nobserved in the SDSS fields (Hammersley \\& L\\'opez-Corredoira 2011).\n\nGiven these results, it would be interesting for future works if the dynamicists explained the existence of the observed flares in the Galactic disc, and further observational research on the spectroscopical\nfeatures of the disc stars at very high $R$ and high $z$ is necessary \nto know more about the origin and evolution of this component.\n\n\\begin{acknowledgements}\nThanks are given to A. Cabrera-Lavers, S. Zaggia and the anonymous referee for useful comments that helped us to improve this paper. MLC was supported by the grant AYA2012-33211 of the Spanish Ministry of Economy and Competitiveness (MINECO). Thanks are given to Astrid Peter (language editor of A\\&A) for proof-reading of the text.\n\nFunding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, and the U.S. Department of Energy Office of Science. The SDSS-III web site is http:\/\/www.sdss3.org\/.\nSDSS-III is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS-III Collaboration including the University of Arizona, the Brazilian Participation Group, Brookhaven National Laboratory, Carnegie Mellon University, University of Florida, the French Participation Group, the German Participation Group, Harvard University, the Instituto de Astrofisica de Canarias, the Michigan State\/Notre Dame\/JINA Participation Group, Johns Hopkins University, Lawrence Berkeley National Laboratory, Max Planck Institute for Astrophysics, Max Planck Institute for Extraterrestrial Physics, New Mexico State University, New York University, Ohio State University, Pennsylvania State University, University of Portsmouth, Princeton University, the Spanish Participation Group, University of Tokyo, University of Utah, Vanderbilt University, University of Virginia, University of Washington, and Yale University. \n\\end{acknowledgements}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe study of the details of the fate of a false vacuum plays a key role in\nunderstanding the properties of a variety of systems. It extends from the\nunderstanding the characteristics of various phase transitions in condensed\nmatter and particle physics all the way to quantifying the human angst that\nour universe itself may actually be in a state of a false vacuum. The gross\nfeatures of the decay mechanism, the formation of bubbles and the quantum\nunder the barrier tunneling involve both the classical and quantum aspects\nof the problem. The description of tunneling in the non-relativistic quantum mechanics\nfor a particle moving in many dimensions was considered in \\cite{Banks}.  In quantum field\ntheory the problem of a false vacuum instability was first addressed in \\cite%\n{Kobzarev}, assuming that the thickness of the wall is small compared to the\nsize of the bubble. Coleman \\cite{Coleman} has generalized the results of \n\\cite{Kobzarev} and suggested how to encapsulate and calculate the\nnon-perturbative quantum part of the process by identifying Euclidean\nclassical configurations which give the leading contribution to the\nprobability of the false vacuum decay. In this paper we study cases where\nthe method suggested by Coleman, as is, needs to be reexamined and modified.\nIn particular, we investigate how the theory must be modified by taking into\naccount the inevitable quantum fluctuations of the scalar field.\n\nWe start by a short review of the Coleman theory for the scalar field\npotential $V\\left( \\varphi \\right) ,$ which has a shape shown in Fig. \\ref%\n{Figure1} (for details see, for example, \\cite{Weinberg}, \\cite{Mukhanov1}%\n)). The false vacuum configuration $\\ \\varphi \\left( \\mathbf{x}\\right)\n=\\varphi _{0}$ is unstable and decays via bubble nucleation. The field\ninside the emerged expanding bubble either tends to its value in the true\nminimum or $\\varphi $ $\\rightarrow $ $-\\infty $ for the unbounded potential.\n\n\\vspace{1.0cm}\n\n\\begin{figure}[hbt]\n\\begin{center}\n\\includegraphics[height=60mm]{Fig_MRS_I_1.eps}\n\\end{center}\n\\par\n\\vspace*{-1.11cm} \\hspace*{7.8cm} - $V_{min}$%\n\\par\n\\vspace*{-3.2cm} \\hspace*{5.3cm} $\\varphi_{min}$%\n\\par\n\\vspace*{-3.8cm} \\hspace*{7.85cm}$V$%\n\\par\n\\vspace*{2.8cm} \\hspace*{9.35cm}$\\varphi_{0}$%\n\\par\n\\vspace*{-2.0cm} \\hspace*{8.05cm}$V_{bar}$%\n\\par\n\\vspace*{0.8cm} \\hspace*{13.0cm}$\\varphi$%\n\\par\n\\vspace*{3cm}\n\\caption{}\n\\label{Figure1}\n\\end{figure}\n\n\\vspace{0.5cm}\n\nThe scalar field $\\varphi \\left( \\mathbf{x},t\\right) $ is a system with\ninfinitely many degrees of freedom with spatial coordinates $\\mathbf{x=}%\n\\left( x^{1},x^{2},x^{3}\\right) $ enumerating them, that is, $\\varphi \\left( \n\\mathbf{x},t\\right) =\\varphi _{\\mathbf{x}}\\left( t\\right) $. The scalar\nfield action can be written as \n\\begin{equation}\nS=\\int (\\mathcal{K-V)}\\,dt\\,,  \\label{1}\n\\end{equation}%\nwhere \n\\begin{equation}\n\\mathcal{K=}\\frac{1}{2}\\int \\left( \\frac{\\partial \\varphi }{\\partial t}%\n\\right) ^{2}\\,d^{3}x\\,,  \\label{2}\n\\end{equation}%\nand \n\\begin{equation}\n\\mathcal{V=}\\int \\left( \\frac{1}{2}\\left( \\frac{\\partial \\varphi }{\\partial\nx^{i}}\\right) ^{2}+V\\left( \\varphi \\right) \\right) \\,d^{3}x  \\label{3}\n\\end{equation}%\nare, correspondingly, the kinetic and interaction energies of the field\nconfiguration $\\varphi \\left( \\mathbf{x}\\right) $ and its first time\nderivatives$.$ Note that the functional $\\mathcal{V}\\left( \\varphi \\left( \n\\mathbf{x}\\right) \\right) $ (\\textit{but not} $V\\left( \\varphi \\right) )$\nplays the role of the potential energy when we are considering the\nsub-barrier tunneling in quantum field theory. For the scalar field\npotential $V\\left( \\varphi \\right) ,$ shown in Fig. \\ref{Figure1}, the field\nconfiguration $\\varphi \\left( \\mathbf{x}\\right) =\\varphi _{0}$ has zero\npotential energy, $\\mathcal{V}\\left( \\varphi _{0}\\right) =0,$ while the\nother homogeneous static solution $\\varphi \\left( \\mathbf{x}\\right) =\\varphi\n_{\\min }$ has a lower energy $\\mathcal{V}\\left( \\varphi _{\\min }\\right)\n=V_{\\min }\\times $\\textit{volume }$<$ $0.$ Therefore the local minimum at $%\n\\varphi _{0}$ is unstable (false vacuum) and decays via sub-barrier\ntunneling as a result of which the bubbles of the true vacuum are formed. To\ncalculate the decay rate Coleman has assumed that in the semiclassical\napproximation the dominant contribution to the tunneling rate comes from the\ninstanton - Euclidean solution of the scalar field equation, which matches\nthe metastable vacuum $\\varphi \\left( \\mathbf{x}\\right) =\\varphi _{0}$ to\nsome classically allowed configuration $\\varphi \\left( \\mathbf{x}\\right) $\nwith $\\mathcal{V}\\left( \\varphi \\left( \\mathbf{x}\\right) \\right) =0,$\ndescribing the emerging bubble in Minkowski space. On symmetry grounds it\nwas shown \\cite{Coleman1} that the minimal action has a $O\\left( 4\\right) $%\n-invariant solution of the scalar field equation \n\\begin{equation}\n\\frac{\\partial ^{2}\\varphi }{\\partial \\tau ^{2}}+\\Delta \\varphi -\\frac{dV}{%\nd\\varphi }=0\\,,  \\label{4}\n\\end{equation}%\nfor which $\\varphi $ depends only on $\\varrho =\\sqrt{\\tau ^{2}+\\mathbf{x}^{2}%\n}$ and where $\\tau =it$ is the Euclidean time. This reduces to the ordinary\ndifferential equation \n\\begin{equation}\n\\ddot{\\varphi}(\\varrho )+\\frac{3}{\\varrho }\\,\\dot{\\varphi}(\\varrho )-\\frac{dV%\n}{d\\varphi }\\,=0\\,,  \\label{5}\n\\end{equation}%\nwhere dot denotes the derivative with respect to $\\varrho .$ If we assume\nthat the field was initially in the false vacuum state one of the boundary\nconditions for $\\left( \\ref{5}\\right) $ is \n\\begin{equation}\n\\varphi \\left( \\varrho \\rightarrow \\infty \\right) =\\varphi _{0}\\,.  \\label{6}\n\\end{equation}%\nAs a second condition Coleman has suggested to use \n\\begin{equation}\n\\dot{\\varphi}(\\varrho =0)=0  \\label{7}\n\\end{equation}%\nto avoid the singularity in the center of the bubble. Equation $\\left( \\ref%\n{5}\\right) $ with \\textit{these boundary conditions} has an unambiguous\nsolution $\\varphi \\left( \\varrho \\right) $ called the Coleman instanton. The\naction for this instanton is given by%\n\\begin{equation}\nS_{I}\\,=\\,2\\pi ^{2}\\,\\int_{0}^{+\\infty }d\\varrho \\,\\varrho ^{3}\\,\\left( \n\\frac{1}{2}\\,\\dot{\\varphi}^{2}\\,+\\,V(\\varphi )\\right)  \\label{8}\n\\end{equation}%\nand the false vacuum decay rate \\textit{per unit time per unit volume} can\nbe estimated as%\n\\begin{equation}\n\\Gamma \\simeq \\varrho _{0}^{-4}\\exp \\left( -S_{I}\\right) \\,,  \\label{9a}\n\\end{equation}%\nwhere $\\varrho _{0}$ is the size of the bubble. The pre-exponential factor\nis based on dimensional grounds. Calculating the potential energy $%\n\\left( \\ref{3}\\right) $ at the moment of Euclidean time $\\tau ,$ we infer\nthat $\\mathcal{V}\\left( \\tau \\right) >0$ for $0<\\tau <\\infty .$ Since the\ntotal energy is normalized to zero this means that in this range of $\\tau $\nthe instanton describes the sub-barrier tunneling in the Euclidean time. As $%\n\\tau \\rightarrow \\infty ,$ $\\mathcal{V}\\rightarrow 0,$ corresponding to the\nfalse vacuum state. The potential energy also vanishes at $\\tau =0$ and\nhence at this instant\\ the bubble emerges from under the barrier in\nMinkowski space. To prove that $\\mathcal{V}\\left( \\tau =0\\right) =0$ we\nfirst note that for $\\varphi \\left( \\mathbf{x},\\tau \\right) =\\varphi \\left(\n\\varrho \\right)$ the expression $\\left( \\ref{3}\\right)$, calculated at $\\tau\n=0,$ reduces to \n\\begin{equation}\n\\mathcal{V}\\left( \\tau =0\\right) =4\\,\\pi \\int_{0}^{+\\infty }d\\varrho \\,\\varrho\n^{2}\\,\\left( \\frac{1}{2}\\,\\dot{\\varphi}^{2}\\,+\\,V(\\varphi )\\right) \\,.\n\\label{10b}\n\\end{equation}%\nNext we find that the first integral of $\\left( \\ref{5}\\right) $ can be\nwritten as%\n\\begin{equation}\n\\frac{1}{2}\\dot{\\varphi}^{2}-V=\\int_{\\rho }^{\\infty }\\frac{3}{\\tilde{\\varrho}%\n}\\left( \\frac{d\\varphi }{d\\tilde{\\varrho}}\\right) ^{2}d\\tilde{\\varrho}\\,,\n\\label{11a}\n\\end{equation}%\nwhere the boundary condition $\\left( \\ref{6}\\right) $ has been taken into\naccount. Finally integrating this equation one gets\\footnote{%\nThe second equality in (\\ref{12b}) is the result of integration by parts\ntaking into account (\\ref{6}) and (\\ref{7}) as well as requiring that $%\n\\varphi(\\varrho)$ decays faster than $\\varrho^{-\\frac{1}{2}}$ at $%\n\\varrho\\rightarrow \\infty$.} \n\\begin{equation}\n\\int_{0}^{\\infty }\\left( \\frac{1}{2}\\,\\dot{\\varphi}^{2}-V\\right) \\varrho\n^{2}d\\varrho =\\int_{0}^{\\infty }d\\left( \\varrho ^{3}\\right) \\left(\n\\int_{\\rho }^{\\infty }\\frac{1}{\\tilde{\\varrho}}\\left( \\frac{d\\varphi }{d%\n\\tilde{\\varrho}}\\right) ^{2}d\\tilde{\\varrho}\\right) =\\int_{0}^{\\infty }\\dot{%\n\\varphi}^{2}\\varrho ^{2}d\\varrho  \\label{12b}\n\\end{equation}%\nand hence%\n\\begin{equation}\n\\int_{0}^{\\infty }\\left( \\frac{1}{2}\\,\\dot{\\varphi}^{2}+V\\right) \\varrho\n^{2}d\\varrho =0  \\label{13a}\n\\end{equation}%\nimplying that $\\mathcal{V}\\left( \\tau =0\\right) $ $\\left( \\ref{10b}\\right) $\nreally vanishes.\n\nThere is a class of potentials for which the results obtained using the\nColeman boundary conditions can either lead to a questionable outcome or\ncannot be applied at all, namely,\n\n\\textit{a}) for the very steep unbounded potentials (see Fig. \\ref{Figure3})\nColeman's instantons lead to nearly instantaneous instability of the false\nvacuum, the so called zero size instanton problem,\n\n\\textit{b}) for some unbounded potentials, when false vacuum must be\nunstable, the Coleman instanton does not exist.\n\nWe will show that both problems have common origin and can be resolved by\ntaking into account inevitable quantum fluctuations which induce the\nultraviolet cutoff scale. In that situation the remedy suggested by Coleman\nto avoid a singularity at the origin needs to be and is replaced by an\nultraviolet cutoff scale induced by those very quantum fluctuations. In\nturn, this allows us to abandon the very restrictive boundary condition $%\n\\left( \\ref{7}\\right) $ and obtain a whole class of new instantons which\nwould be singular in the absence of this quantum ultraviolet cutoff.\n\nIn this paper we will consider an unbounded linear potential and clarify\nthe role of quantum fluctuations in resolving small instanton problem. The\ncase of a quartic unbounded potential for which the Coleman instanton does not\nexist is analyzed in the companion paper $\\cite{MRS2}.$\n\n\\section{The model}\n\nLet us consider the classically exactly solvable potential \n\\begin{equation}\nV\\left( \\varphi \\right) =\\left\\{ \n\\begin{array}{cc}\n\\lambda _{-}\\,\\varphi _{0}^{3}\\,\\varphi +\\frac{\\lambda _{+}}{4}\\,\\varphi\n_{0}^{4} & \\text{for }\\varphi <0\\,, \\\\ \n\\frac{\\lambda _{+}}{4}\\,\\left( \\varphi -\\varphi _{0}\\right) ^{4} & \\text{for \n}\\varphi >0\\,.%\n\\end{array}%\n\\right.  \\label{14a}\n\\end{equation}%\nIt has a local minimum, corresponding to the false vacuum at $\\varphi _{0} $%\n. To avoid the problem with one loop quantum corrections, we will assume\nthat the dimensionless coupling constant $\\lambda _{+}$ is always much\nsmaller than unity. At $\\varphi =0$ the $\\varphi ^{4}-$potential is matched\nto a linear unbounded potential for which the dimensionless constant $%\n\\lambda _{-}$ can be taken to be large.\n\nThe results obtained in this paper can also be applied to estimate the\nprobability of tunneling for some potentials with a second true minimum at\nsome negative $\\varphi _{\\min },$ which for $\\varphi <0$ can well be\napproximated by the linear potential $\\left( \\ref{14a}\\right) $ nearly up to \n$\\varphi _{\\min }$ (see dotted line in Fig. \\ref{Figure2}).\n\n\\vspace*{0.9cm}\n\n\\begin{figure}[hbt]\n\\begin{center}\n\\includegraphics[height=60mm]{Fig_MRS_I_2.eps}\n\\end{center}\n\\par\n\\vspace*{-4.4cm} \\hspace*{11.33cm}$V_{min}$%\n\\par\n\\vspace*{1.18cm} \\hspace*{11.2cm} $-\\frac{4V_{bar}}{\\beta }\\left( 1-\\frac{%\n\\beta }{4}-{\\nu}^{2\/3}\\right)$%\n\\par\n\\vspace*{-3.4cm} \\hspace*{10.58cm}$V_{bar}$%\n\\par\n\\vspace*{-0.243cm} \\hspace*{11.56cm}$\\varphi_{0}$%\n\\par\n\\vspace*{-1.8cm} \\hspace*{3.3cm}$(a)$%\n\\par\n\\vspace*{-0.48cm} \\hspace*{7.13cm}$(b)$%\n\\par\n\\vspace*{-1.0cm} \\hspace*{11.1cm}$V$%\n\\par\n\\vspace*{1.6cm} \\hspace*{12.9cm}$\\varphi$%\n\\par\n\\vspace*{4cm}\n\\caption{}\n\\label{Figure2}\n\\end{figure}\n\n\\section{The Coleman Instanton}\n\nFirst we find the explicit exact solution for the Coleman instanton and show\nhow the problem mentioned in the introduction arises in this simple\nparticular case. For positive $\\varphi $ equation $\\left( \\ref{5}\\right) $\ntakes the following form%\n\\begin{equation}\n\\ddot{\\varphi}+\\frac{3}{\\varrho }\\dot{\\varphi}-\\lambda _{+}\\left( \\varphi\n-\\varphi _{0}\\right) ^{3}=0  \\label{15}\n\\end{equation}%\nand its solution with the boundary condition $\\left( \\ref{6}\\right) $ is \n\\begin{equation}\n\\varphi \\left( \\varrho \\right) =\\varphi _{0}\\frac{\\varrho ^{2}-\\varrho\n_{0}^{2}}{\\varrho ^{2}-\\varrho _{0}^{2}\/(1+\\delta )}\\,,  \\label{16}\n\\end{equation}%\nwhere%\n\\begin{equation}\n\\delta =\\frac{4\\left( 1+\\sqrt{1+\\lambda _{+}\\,\\varphi _{0}^{2}\\,\\varrho\n_{0}^{2}\/2}\\right) }{\\lambda _{+}\\,\\varphi _{0}^{2}\\,\\varrho _{0}^{2}}\n\\label{17}\n\\end{equation}%\nand the constant of integration $\\varrho _{0}$ (the size of the bubble) yet\nremains to be fixed with the help of the second boundary condition $\\left( %\n\\ref{7}\\right) .$ The solution $\\left( \\ref{16}\\right) $ is valid only for $%\n\\varrho _{0}<\\varrho <\\infty .$ At $\\varrho =\\varrho _{0}$ the field $%\n\\varphi $ vanishes and then becomes negative. For $\\varphi <0$ the potential\nis linear and equation $\\left( \\ref{5}\\right) $ simplifies to%\n\\begin{equation}\n\\ddot{\\varphi}+\\frac{3}{\\varrho }\\,\\dot{\\varphi}-\\lambda _{-}\\,\\varphi\n_{0}^{3}=0\\,.  \\label{18}\n\\end{equation}%\nThe solution of this equation, which vanishes at $\\varrho _{0}$ and\nsatisfies $\\left( \\ref{7}\\right) ,$ is \n\\begin{equation}\n\\varphi \\left( \\varrho \\right) =\\frac{\\lambda _{-}\\,\\varphi _{0}^{3}}{8}%\n\\,\\left( \\varrho ^{2}-\\varrho _{0}^{2}\\right) \\,.  \\label{19}\n\\end{equation}%\nThe derivative of the field $\\varphi $ at $\\varrho =\\varrho _{0}$ must be\ncontinuous. This allows us to express the size of the bubble in terms of the\nparameters $\\lambda _{+},\\lambda _{-}$ and $\\varphi _{0}.$ Equating the\nderivatives of solutions $\\left( \\ref{16}\\right) $ and $\\left( \\ref{19}%\n\\right) $ at $\\varrho _{0}$ we obtain the following equation \n\\begin{equation}\n1+\\frac{1}{\\delta }=\\frac{\\lambda _{-}\\,\\varphi _{0}^{2}}{8}\\,\\varrho\n_{0}^{2}\\,,  \\label{20}\n\\end{equation}%\nwith $\\delta $ given in $\\left( \\ref{17}\\right) .$ Solving this equation for \n$\\varrho _{0}^{2}$ one gets \n\\begin{equation}\n\\varrho _{0}^{2}=\\frac{8}{\\varphi _{0}^{2}\\,\\lambda _{-}\\,\\left( 1-\\beta\n\\right) }\\,,  \\label{21}\n\\end{equation}%\nwhere we have introduced the parameter \n\\begin{equation}\n\\beta =\\frac{\\lambda _{+}}{\\lambda _{+}+\\lambda _{-}}  \\label{21a}\n\\end{equation}%\ninstead of $\\lambda _{+}.$\n\nAs one can see from $\\left( \\ref{16}\\right) $ for $\\delta \\ll 1$ the bubble\nhas a thin wall. Substituting $\\varrho _{0}^{2}$ from $\\left( \\ref{21}%\n\\right) $ into equation $\\left( \\ref{20}\\right) $ we find \n\\begin{equation}\n\\delta =\\frac{1-\\beta }{\\beta }  \\label{22a}\n\\end{equation}%\nand therefore the thin-wall approximation is valid only if $1-\\beta \\ll 1,$\nthat is, for rather flat potential $\\left( \\lambda _{-}\\ll \\lambda\n_{+}\\right) $ at negative $\\varphi $. As it follows from $\\left( \\ref{19}%\n\\right) $ the value of the scalar field in the center of the bubble is equal\nto%\n\\begin{equation}\n\\varphi \\left( \\varrho =0\\right) =-\\frac{\\varphi _{0}}{1-\\beta }\\,,\n\\label{23}\n\\end{equation}%\nwhich corresponds to \n\\begin{equation}\nV\\left( \\varphi \\left( \\varrho =0\\right) \\right) =-\\frac{\\lambda\n_{-}\\,\\varphi_{0}^{4}}{1-\\beta } \\,\\left( 1-\\frac{\\beta }{4}\\right) =-\\frac{%\n4\\,V_{bar}}{\\beta}\\, \\left( 1-\\frac{\\beta }{4}\\right) ,  \\label{24}\n\\end{equation}%\nwhere%\n\\begin{equation}\nV_{bar}=\\lambda _{+}\\,\\varphi _{0}^{4}\/4  \\label{24a}\n\\end{equation}%\nis the height of the barrier. The instanton action can be easily calculated\nand is equal to%\n\\begin{equation}\nS_{I}=\\frac{8\\,\\pi ^{2}}{3\\,\\lambda _{-}\\left( 1-\\beta \\right) ^{3}}\\,\\left(\n2-\\beta \\right)\\, (2-2\\beta +\\beta ^{2}) \\,.  \\label{25}\n\\end{equation}\nWe note that if one would decide to consider $\\lambda_-\\rightarrow \\infty$, $%\n\\beta\\rightarrow 0$ the value of the scalar field in the center of the\nbubble is equal to $\\varphi(\\varrho=0)\\rightarrow -\\varphi_0$ and $V\\left(\n\\varphi \\left( \\varrho =0\\right) \\right)\\rightarrow -\\infty$.\n\n\\textit{Instantaneous vacuum decay via small bubbles.} For a very steep\nunbounded potential shown in Fig. \\ref{Figure3} and obtained by taking the\nlimit $\\lambda _{-}\\rightarrow \\infty ,$ the bubble size given in $\\left( %\n\\ref{21}\\right) $ shrinks to zero as $\\varrho _{0}\\propto \\lambda\n_{-}^{-1\/2} $ and the action $\\left( \\ref{25}\\right) $ vanishes$.$ The false\nvacuum decay rate $\\left( \\ref{9a}\\right) $ becomes infinite irrespective\nhow high is the potential barrier $V_{bar}$. This instantaneous false vacuum\ndecay happens via infinitely small bubbles, which according to $\\left( \\ref%\n{24}\\right) $ emerge with infinitely large negative potential in the center.\nSuch a conclusion does not sound physically acceptable. Moreover, even for\nfinite but large enough $\\lambda _{-}$ the problem of small instantons still\nremains because they lead to unexpectedly efficient decay with the rate\npractically independent on the shape of the potential at positive $\\varphi $%\n. The situation starts to look even more strange if one assumes that the\npotential in Fig. \\ref{Figure3} gets a second rather sharp minimum (see\ndotted line). Then, irrespective of the depth of this second minimum, it\nlooks like the Coleman instanton ceases to exist and, thus, nearly\nirrelevant modification of the potential abruptly changes the decay rate\nfrom infinity to zero.\n\n\\begin{figure}[tbh]\n\\begin{center}\n\\includegraphics[height=60mm]{Fig_MRS_I_3.eps}\n\\end{center}\n\\par\n\\vspace*{-4.08cm} \\hspace*{5.55cm}$V_{min}$%\n\\par\n\\vspace*{1.6cm} \\hspace*{5.32cm} -$-\\frac{128\\,V_{bar}}{\\sigma^2\\,\\lambda_+}$%\n\\par\n\\vspace*{-4.1cm} \\hspace*{5.58cm}$V_{bar}$%\n\\par\n\\vspace*{0.5cm} \\hspace*{7.6cm}$\\varphi_{0}$%\n\\par\n\\vspace*{-2.5cm} \\hspace*{2.88cm}$(a)$%\n\\par\n\\vspace*{-0.46cm} \\hspace*{3.78cm}$(b)$%\n\\par\n\\vspace*{-0.9cm} \\hspace*{5.4cm}$V$%\n\\par\n\\vspace*{1.7cm} \\hspace*{13.0cm}$\\varphi$%\n\\par\n\\vspace*{4.0cm}\n\\caption{{}}\n\\label{Figure3}\n\\end{figure}\n\\textbf{\\ }\n\nBelow we will show how these problems are resolved if we take into account\nquantum fluctuations.\n\n\\section{Quantum Fluctuations}\n\nThe instanton is a \\textit{classical} solution and it is reliable only if\nboth the field and its derivative exceed the level of the minimal quantum\nfluctuations. The bubble emerges in Minkowski space at $\\tau =0$. At this\ninstant the distance to the center of the bubble is equal to $\\varrho $ and\nthe \\textquotedblleft typical\\textquotedblright\\ amplitude of the quantum\nfluctuations in corresponding scales is about (see for example, \\cite%\n{Mukhanov2}): \n\\begin{equation}\n\\left\\vert \\delta \\varphi _{q}\\right\\vert \\simeq \\frac{\\sigma}{\\varrho }\\,,\n\\label{27}\n\\end{equation}%\nand, respectively, its time derivative is of order%\n\\begin{equation}\n\\left\\vert \\delta \\dot{\\varphi}_{q}\\right\\vert \\simeq \\frac{\\sigma }{\\varrho\n^{2}}\\,,  \\label{28}\n\\end{equation}%\nwhere $\\sigma $ is the numerical coefficient of order unity (we set $\\hbar=1$%\n). The quantum fluctuations grow very fast as $\\varrho \\rightarrow 0$ and,\nhence, Coleman's boundary condition $\\dot{\\varphi}_{\\varrho =0}=0,$\nformulated in the deep ultraviolet limit, must be re-analyzed. The potential \n$V\\left( \\varphi \\right) $ can be arbitrarily shifted along the $\\varphi$%\n--axis. Therefore, to estimate the magnitude of quantum fluctuations it is\nmore appropriate to consider either the typical change of the classical\nfield $\\Delta \\varphi \\simeq \\dot{\\varphi}\\times \\varrho $ in scales $%\n\\varrho $ or it's time derivative. In both cases we obtain (up to a\nnumerical coefficient) the same result and to be concrete we will use $%\n\\left( \\ref{28}\\right) $ to determine when quantum fluctuations become\nrelevant. Equating the derivative of the classical solution $\\left( \\ref{19}%\n\\right) $ to the amplitude of quantum fluctuations $\\left( \\ref{28}\\right) $\nwe find the following ultraviolet cutoff scale%\n\\begin{equation}\n\\varrho _{uv}=\\left( \\frac{4\\,\\sigma }{\\lambda _{-}\\,\\varphi _{0}^{3}}%\n\\right) ^{1\/3}=\\left( \\frac{\\sigma ^{2}}{32}\\right) ^{1\/6}\\lambda\n_{-}^{1\/6}\\,\\left( 1-\\beta \\right) ^{1\/2}\\varrho _{0}\\,,  \\label{29}\n\\end{equation}%\nwhere $\\beta $ is defined in $\\left( \\ref{21a}\\right) $ and $\\varrho _{0}$\nis given in $\\left( \\ref{21}\\right) .$ The classical solution with the\nColeman boundary condition $\\left( \\ref{7}\\right) $ is obviously valid only\nfor $\\varrho >\\varrho _{uv}$ and it is completely spoiled by quantum\nfluctuations on scales smaller than $\\varrho _{uv}$. Let us notice that for\nthe very steep potentials with $\\lambda _{-}\\gg 1$ the ultraviolet cutoff\nscale exceeds the size of the bubble and therefore the instantons which lead\nto the large decay rate cannot be trusted anymore.\n\nThe ultraviolet cutoff $\\varrho _{uv}$ resulted from the fact that the\nquantum fluctuations are characterized by the field derivative of order $%\n1\/\\varrho ^{2}$ for all values of $\\varrho $ while the contribution of the\nclassical solution at small $\\varrho $ decreases as $\\varrho $. Thus, for\nsmall enough $\\varrho $ the quantum fluctuations dominate. For the large\nvalues of $\\varrho $ the quantum fluctuations continue to exhibit a $%\n1\/\\varrho ^{2}$ decay while the classical solution decreases as $1\/\\varrho\n^{3}$. Thus, the quantum fluctuations take over both in the deep\nultra-violet and the large infrared scales carving a range of values of $%\n\\varrho $ for which the classical solutions can be valid.\n\nNext we estimate the infrared cutoff scale above which (in length) quantum\nfluctuations dominate. To find this scale we have to equate the derivative\nof solution $\\left( \\ref{16}\\right) ,$ valid for $\\varrho >\\varrho _{0},$ to \n$\\left( \\ref{28}\\right) .$ As a result one gets%\n\\begin{equation}\n\\varrho _{ir}\\simeq \\frac{2}{\\sigma }\\frac{\\delta }{1+\\delta }\\,\\varphi\n_{0}\\,\\varrho _{0}^{2}\\simeq \\frac{4\\sqrt{2}}{\\sigma }\\lambda\n_{-}^{-1\/2}\\left( 1-\\beta \\right) ^{1\/2}\\varrho _{0}\\,.  \\label{30}\n\\end{equation}%\nTo simplify the result we have assumed that $\\varrho _{ir}\\gg \\varrho _{0}.$\nAs one can check a posteriori this assumption is really valid when both $%\n\\lambda _{+}$ and $\\lambda _{-}$ are much smaller than unity.\n\nFor $\\lambda _{-}>1$ we have $\\varrho _{ir}<\\varrho _{0}$ and moreover $%\n\\varrho _{uv}>\\varrho _{ir}$. Thus, the range \n\\begin{equation}\n\\varrho _{uv}<\\varrho <\\varrho _{ir}  \\label{31}\n\\end{equation}%\nwhere the classical instanton solution is supposed to be valid completely\ndisappears. This tells us that for $\\lambda _{-}>1$ the classical instanton\nwith the Coleman boundary condition can not be applied.\n\nConcluding this section we would like to stress that both the \\textit{%\nultraviolet and infrared cutoff scales} \\textit{are entirely determined by\nthe parameters characterizing the corresponding classical solution.}\n\n\\section{New Instantons}\n\n\\label{section5}\n\nThe existence of the cutoff scales allows us to obtain a new class of $%\nO\\left( 4\\right) $ instantons, which otherwise would be singular and having\nan infinite action would not contribute to the decay rate. Let us first\nconsider the new solutions emerging thanks to the existence of the\nultraviolet cutoff scale. Later on we will estimate how the infrared cutoff\ncorrections influence these new solutions. Thus, we will assume that for $%\n\\varrho >\\varrho _{0}$ solution $\\left( \\ref{16}\\right) -\\left( \\ref{17}%\n\\right) $ is still valid, but abandon the boundary condition $\\dot{\\varphi}%\n_{\\varrho =0}=0.$ Then the most general solution of equation $\\left( \\ref{18}%\n\\right) $, which vanishes at $\\varrho _{0}=0$ and valid for $\\varphi <0$ is%\n\\begin{equation}\n\\varphi \\left( \\varrho \\right) =\\frac{\\lambda_{-}\\,\\varphi _{0}^{3}}{8}\\left(\n\\varrho ^{2}-\\varrho _{0}^{2}\\right) \\left( 1-\\frac{\\varrho _{1}^{2}}{%\n\\varrho ^{2}}\\right) ,  \\label{32}\n\\end{equation}%\nwhere the constant of integration $\\varrho _{1}<\\varrho _{0}$ parametrizes\nour new instantons. These instantons are singular at $\\varrho \\rightarrow 0.$\nHowever, as we have shown above the solution $\\left( \\ref{32}\\right) $ can\nonly be trusted for $\\varrho >\\varrho _{uv}$. This allows us to avoid a\nsingularity in the classical solution. When $\\varrho _{1}\\neq 0$ there is a\nbounce at \n\\begin{equation}\n\\varrho _{b}=\\sqrt{\\varrho_{0}\\,\\varrho _{1}}\\,,  \\label{33}\n\\end{equation}%\nwhere $\\dot{\\varphi}(\\varrho _{b})=0,$ and after that the field $\\varphi $\nmust go back and vanish at $\\varrho _{1},$ which is smaller than $\\varrho\n_{b}.$ However, before this happens the quantum fluctuations become relevant\nat $\\varrho _{uv}>\\varrho _{b}$ and as we will show the classical bubble\nwith a quantum core emerges in Minkowski space$.$ To calculate $\\varrho\n_{uv} $ we equate the derivative of $\\left( \\ref{32}\\right) $ to $\\left( \\ref%\n{28}\\right) $ and obtain the following equation $\\qquad $\n\\begin{equation}\n1-\\frac{\\varrho _{b}^{4}}{\\varrho _{uv}^{4}} =\\frac{4\\,\n\\sigma}{\\lambda_{-}\\varphi _{0}^{3}}\\,\\frac{1}{\\varrho_{uv}^{3}}\\,,  \\label{34}\n\\end{equation}%\nwhich can be solved exactly for $\\varrho _{uv}$. Because we will not need\nthe explicit solution for $\\varrho _{uv}$ we skip it here, but instead let\nus \\ rewrite equation $\\left( \\ref{34}\\right) $ in more convenient form as%\n\\begin{equation}\n\\frac{\\varrho _{uv}^{2}-\\varrho _{b}^{2}}{\\varrho _{uv}^{2}}=\\nu \\left( \n\\frac{\\bar{\\varrho}}{\\varrho _{uv}}\\right) ^{3},  \\label{35}\n\\end{equation}%\nwhere%\n\\begin{equation}\n\\bar{\\varrho}^{2}\\equiv \\frac{8}{\\varphi_{0}^{2}\\,\\lambda _{-}\\,\\left( 1-\\beta\n\\right) }  \\label{35a}\n\\end{equation}%\nand \n\\begin{equation}\n\\nu \\equiv \\frac{\\sigma }{4\\,\\sqrt{2}\\,\\kappa\\, \\left( \\varrho _{uv}\\right) }%\n\\,\\lambda_{-}^{1\/2}\\,\\left( 1-\\beta \\right) ^{3\/2}.  \\label{36}\n\\end{equation}%\nHere, $\\kappa \\left( \\varrho _{uv}\\right) =1+\\varrho _{b}^{2}\/\\varrho_{uv}^{2} $ and since $\\varrho _{b}^{2}\/\\varrho _{uv}^{2}<1$ it\nimplies that $1<\\kappa \\left( \\varrho _{uv}\\right) <2.$ The scale $\\bar{%\n\\varrho}$ entering in $\\left( \\ref{35}\\right) $ is equal to the bubble size $%\n\\varrho _{0}$ (see (\\ref{21})) only for $\\varrho _{1}=0,$ corresponding to\nthe Coleman instanton. To determine $\\varrho _{0}$ in general case we require\nthat $\\dot{\\varphi}(\\varrho )$ is continuous at $\\varrho _{0}$ and equating\nthe derivatives of $\\left( \\ref{16}\\right) $ and $\\left( \\ref{32}\\right) $\nat this point we obtain the following equation \n\\begin{equation}\n1+\\frac{1}{\\delta }=\\frac{\\lambda _{-}\\,\\varphi_{0}^{2}}{8}\\,(\\varrho_{0}^{2}-\\varrho _{1}^{2})\\,,  \\label{38}\n\\end{equation}%\nwhere $\\delta $ is given in $\\left( \\ref{17}\\right) .$ Solving this equation\nfor $\\varrho _{0}^{2}$ one gets \n\\begin{equation}\n\\varrho _{0}^{2}=\\frac{1}{2}\\left( 1+\\sqrt{1+4\\,\\beta \\,\\frac{\\varrho _{1}^{2}}{%\n\\bar{\\varrho}^{2}}}~\\right) \\bar{\\varrho}^{2}+\\varrho_{1}^{2}\\,.  \\label{39}\n\\end{equation}\n\nThus we have found the new instantons with finite action, which are\nparametrized by $\\varrho _{1}$. The main contribution to the decay rate\ncomes from those instantons which have the minimal action while describing\nthe required transition. Let us stress that $\\varrho _{1}^{2}$ must be\npositive because otherwise the growing mode $\\varphi \\propto 1\/\\varrho ^{2}$\nwould be increasing faster than the quantum fluctuations as $\\varrho\n\\rightarrow 0$ and we would end up at a singularity, where $\\varphi\n\\rightarrow -\\infty .$ For positive $\\varrho _{1}^{2}$ the classical field $%\n\\varphi $ bounces at $\\varrho _{b}$ and evolves towards the positive values.\nHowever, as we have noticed above, before the bounce is reached the\nclassical solution fails at $\\varrho _{uv}>\\varrho _{b}$ and the bubble\nemerges from under the barrier materializing in Minkowski space.\n\nTo simplify the formulae it is convenient to parametrize the instantons by\nthe dimensionless parameter $\\chi \\left( \\varrho _{1}\\right) ,$ related to $%\n\\varrho _{1}^{2}$ as%\n\\begin{equation}\n\\varrho _{1}^{2}=\\chi \\left( 1+\\beta \\,\\chi \\right) \\bar{\\varrho}^{2}\\,,\n\\label{40}\n\\end{equation}%\nwhere $\\bar{\\varrho}^{2}$ is defined in $\\left( \\ref{35a}\\right) $. Then, as\nit follows from $\\left( \\ref{39}\\right) $ \n\\begin{equation}\n\\varrho _{0}^{2}=\\left( 1+\\chi \\right) \\left( 1+\\beta\\, \\chi \\right) \\bar{%\n\\varrho}^{2}  \\label{42}\n\\end{equation}%\nand%\n\\begin{equation}\n\\varrho _{b}^{2}=\\varrho _{0}\\,\\varrho _{1}=\\left( 1+\\beta\\, \\chi\\, \\right) \\sqrt{%\n\\chi \\,(1+\\chi )}\\,\\bar{\\varrho}^{2}\\,.  \\label{43}\n\\end{equation}%\nThe expression $\\left( \\ref{32}\\right) $ for $\\varphi \\left( \\varrho \\right) \n$ can be rewritten in the following more convenient form: \n\\begin{equation}\n\\varphi \\left( \\varrho \\right) =-\\frac{\\varphi _{0}}{1-\\beta }\\left( \\frac{%\n\\left( \\varrho _{0}-\\varrho _{1}\\right) ^{2}}{\\bar{\\varrho}^{2}}-\\frac{%\n\\left( \\varrho ^{2}-\\varrho _{b}^{2}\\right) ^{2}}{\\bar{\\varrho}^{2}\\varrho\n^{2}}\\right)  \\label{44a}\n\\end{equation}%\nand using the formulae above as well as equation $\\left( \\ref{35}\\right) $\nwe obtain%\n\\begin{equation}\n\\varphi _{uv}\\equiv \\varphi \\left( \\varrho _{uv}\\right) =-\\frac{\\varphi_{0}%\n}{1-\\beta }\\left( \\frac{1+\\beta \\,\\chi }{\\left( \\sqrt{1+\\chi }+\\sqrt{\\chi }%\n\\right) ^{2}}-\\nu ^{2}\\left( \\frac{\\bar{\\varrho}}{\\varrho_{uv}}\\right)^{4}\\right) \\,.  \\label{45}\n\\end{equation}%\nThe value of the potential at $\\varphi _{uv}$ is \n\\begin{equation}\nV_{uv}\\left( \\chi \\right) =V\\left( \\varphi _{uv}\\right) =-\\frac{4V_{bar}}{%\n\\beta }\\left( \\frac{1+\\beta\\, \\chi }{\\left( \\sqrt{1+\\chi }+\\sqrt{\\chi }\\right)^{2}}\n-\\frac{\\beta }{4}-\\nu^{2}\\,\\left( \\frac{\\bar{\\varrho}}{\\varrho _{uv}}%\n\\right)^{4}\\right) \\,.  \\label{46}\n\\end{equation}%\nAssuming that $\\lambda _{-}\\ll 1$ we find that when $\\chi $ varies from zero\nto infinity $V_{uv}$ changes within the range \n\\begin{equation}\n-\\frac{4V_{bar}}{\\beta }\\left( 1-\\frac{\\beta }{4}-\\nu ^{2\/3}\\right)\n<V_{uv}\\left( \\chi \\right) <0\\,,  \\label{46a}\n\\end{equation}%\nvanishing at $\\chi \\rightarrow \\infty $. The last term in the brackets is\ndue to the quantum corrections and it is much smaller than unity for $%\n\\lambda _{-}\\ll 1$ (the case of $\\lambda _{-}>1$ will be considered\nseparately).\n\nThus, we conclude that as a result of quantum regularization there appears a\nwhole class of new nonsingular instantons which all contribute to the decay\nof the false vacuum. Depending on $\\chi $, which parametrizes these\ninstantons, the value of the potential in the central part of the bubble\ndominated by quantum fluctuations, after subtracting the energy of zero\npoint fluctuations, is equal to $V_{uv}$ and takes its value within the\ninterval $\\left( \\ref{46a}\\right) .$ If there exists a second true vacuum\nwith $V_{\\min }$ in the range $\\left( \\ref{46a}\\right) $ (see curve (b) in\nFig. \\ref{Figure2}), we expect that the dominant contribution to the false\nvacuum decay is given by the instanton with $\\chi $ determined by equation $%\nV_{uv}\\left( \\chi \\right) \\simeq V_{\\min }.$\n\nTo verify that the new instanton really emerges from under the potential\nbarrier in Minkowski space at $\\tau =0$ we have to check that%\n\\begin{equation}\n\\mathcal{V}\\left( \\tau =0\\right) =4\\,\\pi \\int_{\\varrho _{uv}}^{+\\infty\n}d\\varrho \\,\\varrho ^{2}\\,\\left( \\frac{1}{2}\\,\\dot{\\varphi}%\n^{2}\\,+\\,V(\\varphi )\\right) +\\frac{4\\,\\pi }{3}\\,\\varrho_{uv}^{3}V_{uv},\n\\label{47}\n\\end{equation}%\nvanishes up to a possible quantum correction not exceeding the contribution\nof a single quantum. The second term in $\\left( \\ref{47}\\right) $ accounts\nfor the shifted energy inside the bubble quantum core, which remains after\nnormalizing the energy of quantum fluctuations to zero in the false vacuum\nstate. Substituting solutions $\\left( \\ref{16}\\right) $ and $\\left( \\ref{32}%\n\\right) $ in $\\left( \\ref{47}\\right) $ we obtain%\n\\begin{eqnarray}\n&& 4\\,\\pi \\int_{\\varrho _{uv}}^{+\\infty }d\\varrho \\,\\varrho ^{2}\\,\\left( \\frac{1%\n}{2}\\,\\dot{\\varphi}^{2}\\,+\\,V(\\varphi )\\right)  \\notag \\\\\n&=&\\frac{\\pi\\, \\lambda_{-}^{2}\\,\\varphi _{0}^{6}}{24}\\left[ \\frac{\\left(\n\\varrho _{uv}^{4}-\\varrho _{b}^{4}\\right) ^{2}}{\\varrho _{uv}^{3}}-4\\varrho\n_{uv}\\left( \\left( \\varrho _{0}^{2}-\\varrho _{uv}^{2}\\right) \\left( \\varrho\n_{1}^{2}-\\varrho _{uv}^{2}\\right) +\\frac{2\\lambda _{+}}{\\lambda\n_{-}^{2}\\varphi _{0}^{2}}\\varrho _{uv}^{2}\\right) \\right]  \\notag \\\\\n&=&\\frac{4\\,\\pi }{3}\\varrho_{uv}^{3}\\left( \\frac{1}{2}\\left( \\dot{\\varphi}%\n\\left( \\varrho _{uv}\\right) \\right) ^{2}-V\\left( \\varphi \\left( \\varrho\n_{uv}\\right) \\right) \\right) ,  \\label{48}\n\\end{eqnarray}%\nand hence%\n\\begin{equation}\n\\mathcal{V}\\left( \\tau =0\\right) =\\frac{2\\,\\pi }{3}\\varrho _{uv}^{3}\\left( \n\\dot{\\varphi}\\left( \\varrho _{uv}\\right) \\right) ^{2}=\\frac{2\\,\\pi\\, \\sigma ^{2}%\n}{3\\,\\varrho _{uv}}\\,,  \\label{49}\n\\end{equation}%\nwhich corresponds to one quantum of energy in scales $\\varrho _{uv}.$ Thus,\nthe energy balance is satisfied within an accuracy dictated by the\ntime-energy uncertainty relation and the bubble emerges from under the\nbarrier.\n\nTo determine the decay rate we calculate the action%\n\\begin{equation}\nS_{I}\\,=\\,2\\pi ^{2}\\,\\int_{\\varrho _{uv}}^{+\\infty }d\\varrho \\,\n\\varrho^{3}\\,\\left( \\frac{1}{2}\\,\\dot{\\varphi}^{2}\\,+\\,V(\\varphi )\\right) +%\n\\frac{\\pi ^{2}}{2}\\varrho _{uv}^{4}V_{uv}\\,,  \\label{50}\n\\end{equation}%\nwhere the last term accounts for the contribution of the bubble quantum\ncore. The result is%\n\\begin{eqnarray}\nS_{I}\\, &=&\\frac{\\pi \\,\\varphi_{0}^{2}}{96}\\left[\\lambda_{-}^{2}\\,\\varphi_{0}^{4}\\,\n\\left( \\varrho_{uv}^{6}+\\frac{3\\varrho_{b}^{8}}{\\varrho _{uv}^{2}}%\n-\\varrho_{0}^{6}-\\frac{3\\,\\varrho_{b}^{8}}{\\varrho_{0}^{2}}\\right) +\\frac{%\n32\\,\\left( \\lambda _{-}\\varphi _{0}^{2}\\left( \\varrho _{0}^{2}-\\varrho\n_{1}^{2}\\right) -8\\right) }{\\lambda _{+}\\varphi _{0}^{2}}\\right.  \\notag \\\\\n&&\\left. +\\left( 16\\,\\lambda _{-}\\,\\varphi _{0}^{2}\\left( \\left( 1+\\frac{%\n3\\,\\lambda _{+}}{4\\lambda_{-}}\\right) \\varrho _{0}^{2}\n-\\varrho_{1}^{2}\\right) \\right) \\varrho_{0}^{2}+32\\,\\varrho_{0}^{2}\\right]\\,.\n\\label{51}\n\\end{eqnarray}%\nSubstituting here the expression for $\\varrho _{0},\\varrho _{1}$ and $%\n\\varrho _{b}$ from $\\left( \\ref{40}\\right) -\\left( \\ref{43}\\right) $ and\nusing equation $\\left( \\ref{35}\\right) $ for $\\varrho _{uv}$ this action can\nbe represented in the following form:%\n\\begin{eqnarray}\nS_I &=&\\frac{8\\pi ^{2}}{3\\lambda _{-}\\left( 1-\\beta \\right) ^{3}}\\Bigg[ %\n\\left( 2-\\beta \\right) (2-2\\beta +\\beta ^{2})+\\frac{4\\,\\chi^{3\/2}\\left(\n4+3\\,\\chi \\right) \\,\\left( 1+\\beta \\,\\chi \\right) ^{3}}{ 2\\,\\left( 1+\\chi\n\\right) ^{3\/2}+\\chi^{1\/2}\\left( 3+2\\,\\chi \\right) }  \\notag \\\\\n&-&\\beta\\, \\chi^{2}\\left( 1+2\\,\\left( 1+\\beta \\,\\chi \\right) +3\\left( 1+\\beta\\,\n\\chi \\right)^{2}\\right) \\Bigg] +\\frac{\\pi^{2}\\,\\sigma ^{2}}{6}\\left(\n1+2\\,\\left( \\frac{\\varrho_{b}^{2}}{\\varrho_{uv}^{2}+\\varrho_{b}^{2}}\\right)\n^{2}\\right) \\,.  \\label{52}\n\\end{eqnarray}%\nThe last term here is always of order unity and can be neglected.\n\n\\textbf{Remarks on infrared cutoff}\\textit{. } Considering instantons we\nhave ignored the infrared cutoff. Let us estimate how the results above are\nchanged if we take it into account. For $\\chi \\neq 0$ the expression $\\left( %\n\\ref{30}\\right) $ for $\\varrho _{ir}$ is modified as \n\\begin{equation}\n\\varrho _{ir}\\simeq \\frac{4\\sqrt{2}}{\\sigma }\\lambda _{-}^{-1\/2}\\left(\n1-\\beta \\right) ^{1\/2}\\left( \\frac{1+\\chi }{1+\\beta \\chi }\\right)\n^{1\/2}\\varrho _{0}\\,.  \\label{53}\n\\end{equation}%\nIf both $\\lambda _{+},\\lambda _{-}\\ll 1$ the bubble size $\\varrho _{0}$ is\nalways much smaller than $\\varrho _{ir}$ and the infrared effects do not\ninfluence much the instantons for any $\\chi $. However, if $\\lambda _{-}\\gg\n1,$ it follows from $\\left( \\ref{53}\\right) $ that only if \n\\begin{equation}\n\\chi \\gg \\chi _{\\min }=4\\,\\nu ^{2}\\,,  \\label{54}\n\\end{equation}%\nwhere $\\nu $ is defined in $\\left( \\ref{36}\\right) ,$ we have $\\varrho\n_{ir}\\gg \\varrho _{0},$ otherwise $\\varrho _{ir}$ can becomes even smaller\nthan $\\varrho _{uv}$ and, hence, the classical solutions with $\\chi <\\chi\n_{\\min }$ do not make sense. Thus, for the steep potentials with $\\lambda\n_{-}\\gg 1$ the value of $\\chi $ must always exceed $\\chi _{\\min }\\approx\n\\lambda _{-}.$\n\nCalculating the corrections to the potential energy $\\left( \\ref{49}\\right) $\nand action (\\ref{50}) due to the infrared cutoff, we find that%\n\\begin{equation}\n\\Delta \\mathcal{V}_{ir}=-4\\pi \\int_{\\varrho _{ir}}^{+\\infty }d\\varrho \\,\n\\varrho^{2}\\,\\left( \\frac{1}{2}\\,\\dot{\\varphi}^{2}\\,+\\,V(\\varphi )\\right) =%\n\\frac{2\\,\\pi \\,\\sigma ^{2}}{3}\\left( 1+\\frac{\\sigma^{2}\\,\\lambda_{+}}{60}\\right) \n\\frac{1}{\\varrho _{ir}}  \\label{55}\n\\end{equation}%\nand \n\\begin{equation}\n\\Delta S_{I}=-\\frac{\\pi ^{2}\\,\\sigma^{2}}{32}\\left( 1+\\frac{\\sigma^{2}\\,\\lambda _{+}}{4}\\right) \\,,  \\label{56}\n\\end{equation}%\nrespectively. If we would decide to normalize $V\\left( \\varphi \\left(\n\\varrho _{ir}\\right) \\right) $ to zero we would have to subtract from the\naction \n\\begin{equation}\n\\Delta S=-\\frac{\\pi ^{2}\\,\\sigma^{4}\\,\\lambda_{+}}{128} \\,.  \\label{57}\n\\end{equation}%\nThus, as it follows from $\\left( \\ref{55}\\right) -\\left( \\ref{57}\\right) $\nall infrared corrections are at the level of one quantum with the frequency\ncorresponding to the infrared cutoff scale. Hence, except the important\nbound $\\left( \\ref{54}\\right)$ which has to be respected for $\\lambda\n_{-}\\gg 1$, the quantum infrared corrections can be ignored.\n\n\\section{Implications}\n\nWe will now apply the results obtained above in several limiting cases. We\nconsider separately the instantons with $\\chi \\ll 1$ and $\\chi \\gg 1.$ The\naction $\\left( \\ref{52}\\right) $ simplifies to \n\\begin{equation}\nS_{I}\\left( \\chi \\right) =\\frac{8\\,\\pi ^{2}}{3\\,\\lambda _{-}\\left( 1-\\beta\n\\right) ^{3}}\\left[ \\left( 2-\\beta \\right) (2-2\\,\\beta +\\beta^{2})\n+8\\,\\chi^{3\/2}+O\\left(\\chi ^{2}\\right) \\right]  \\label{57a}\n\\end{equation}%\nfor $\\chi \\ll 1$ and up to corrections of order $\\chi ^{3\/2}$ coincides with\nthe Coleman action, while for $\\chi \\gg 1$ it becomes%\n\\begin{equation}\nS_{I}=\\frac{8\\,\\pi^{2}\\,\\chi }{\\lambda_{-}\\left(1-\\beta \\right) ^{3}}\\,\n\\left[\\frac{1}{3}\\left( 1-\\frac{\\beta}{2}\\right) \\left(\\beta \\,\\chi \\right)\n^{2}+\\left( 1-\\frac{\\beta}{4}\\right)^{2}\\,\\beta\\, \\chi +\\left( 1-\\frac{\\beta }{%\n4}\\right) \\left( 1-\\frac{\\beta }{4}+\\frac{\\beta ^{2}}{8}\\right) +O\\left( \n\\frac{1}{\\chi }\\right) \\right] .~  \\label{57b}\n\\end{equation}%\nLet us recall that in order to ignore the quantum loop corrections $\\lambda\n_{+}$ must always be smaller than unity, while $\\lambda _{-}$ can be\n large and therefore we investigate the potentials with $\\lambda\n_{-}\\ll 1$ and $\\lambda _{-}\\gg 1$ separately.\n\n\\textbf{A)} For $\\lambda _{-}\\ll 1$ the parameter $\\nu $ defined in $\\left( %\n\\ref{36}\\right) $ is much smaller than unity and thus from $\\left( \\ref{46}%\n\\right) $ one obtains \n\\begin{equation}\nV_{uv}\\left( \\chi \\right) =-\\frac{4\\,V_{bar}}{\\beta }\\left( 1-\\frac{\\beta }{4}%\n-2\\,\\sqrt{\\chi }-\\nu^{2\/3}+O\\left( \\chi \\right) \\right)  \\label{58}\n\\end{equation}%\nfor $\\chi \\ll 1.$ These instantons with the action $\\left( \\ref{57a}\\right) $\ngive the main contribution to the decay rate for unbounded potential and for\nthe potentials with the second true minimum of depth $V_{\\min }<V_{uv}\\left(\n\\chi =0\\right) .$ They are different from the Coleman instanton ($\\chi =0)$ only\nby higher order corrections. As it follows from $\\left( \\ref{35}\\right) $\nthe size of quantum core%\n\\begin{equation}\n\\varrho _{uv}\\simeq \\left( \\frac{\\sigma ^{2}}{32}\\right) ^{1\/6}\\,\n\\lambda_{-}^{1\/6}\\,\\left( 1-\\beta \\right)^{1\/2}\\varrho _{0}  \\label{60}\n\\end{equation}%\nis much smaller than the bubble size \n\\begin{equation}\n\\varrho _{0}=\\left( 1+\\frac{1+\\beta}{2}\\,\\chi +O\\left( \\chi ^{2}\\right)\n\\right) \\bar{\\varrho}\\,.  \\label{60a}\n\\end{equation}%\nTo calculate the contribution of instantons with the value of the potential\nin the quantum core%\n\\begin{equation}\nV_{uv}\\left( \\chi _{\\varepsilon }\\right) =\\varepsilon \\,V_{uv}\\,\n\\left(\\chi=0\\right) \\simeq -\\frac{4\\,\\varepsilon \\,V_{bar}}{\\beta }\\left( 1-\\frac{\\beta }{4%\n}\\right) \\,,  \\label{60b}\n\\end{equation}%\nwhere $\\varepsilon \\ll 1,$ to the decay rate we have to consider solutions\nwith $\\chi \\gg 1.$ In this case the expression $\\left( \\ref{46}\\right) $\nsimplifies to%\n\\begin{equation}\nV_{uv}\\left( \\chi \\right) =-\\frac{V_{bar}}{\\beta }\\left( \\left( 1-\\frac{%\n\\beta }{2}\\right) \\frac{1}{\\chi }+O\\left( \\frac{1}{\\chi ^{2}}\\right) \\right)\n,  \\label{63}\n\\end{equation}%\nand as it follows from $\\left( \\ref{60b}\\right) $ \n\\begin{equation}\n\\chi _{\\varepsilon }\\simeq \\frac{1}{2}\\left( \\frac{2-\\beta }{4-\\beta }%\n\\right) \\frac{1}{\\varepsilon }\\gg 1\\,.  \\label{64}\n\\end{equation}%\nThe expressions for $\\varrho _{0}^{2}$ becomes%\n\\begin{equation}\n\\varrho _{0}^{2}\\left(\\chi_{\\varepsilon }\\right) \\simeq \\frac{4}{\\left(\n1-\\beta \\right) }\\left( \\frac{2-\\beta }{4-\\beta }\\right) ^{2}\\frac{1}{%\n\\varepsilon \\lambda _{-}\\varphi _{0}^{2}}\\left( \\left( \\frac{4-\\beta }{%\n2-\\beta }\\right) +\\frac{1}{2}\\,\\frac{\\beta }{\\varepsilon }\\right) ,  \\label{65}\n\\end{equation}%\nand the action $\\left( \\ref{57b}\\right) $ is equal to%\n\\begin{equation}\nS_{\\varepsilon }=\\frac{\\pi ^{2}\\left( 2-\\beta \\right) }{\\varepsilon \\,\\lambda\n_{-}\\left( 1-\\beta \\right) ^{3}}\\left[ \\frac{1}{6}\\left( \\frac{2-\\beta }{%\n4-\\beta }\\right) ^{3}\\frac{\\beta^{2}}{\\varepsilon^{2}}+\\frac{\\left(\n2-\\beta \\right) }{8}\\frac{\\beta }{\\varepsilon }+\\left( 1-\\frac{\\beta }{4}+%\n\\frac{\\beta ^{2}}{8}\\right) +O\\left( \\varepsilon \\right) \\right] \\,.\n\\label{67}\n\\end{equation}%\nThe contribution of these $\\chi _{\\varepsilon }$-instantons to the decay\nrate is given by%\n\\begin{equation}\n\\Gamma \\sim \\varrho _{0}^{-4}\\left( \\chi _{\\varepsilon }\\right) \\exp \\left(\n-S_{\\varepsilon }\\right) \\,.  \\label{67a}\n\\end{equation}%\nFor the unbounded potential the instantons with $\\chi \\ll 1$ give the main\ncontribution to the overall decay rate. However, even in this case the whole\nspectrum of instantons is present when we consider the false vacuum decay.\nThe formula $\\left( \\ref{67a}\\right) $ can also be applied to estimate the\nleading contribution to the decay rate for a broad class of potentials with\ntwo minima in the case when the potential is well approximated by the linear\npotential nearly up to the second true minimum of depth\n\\begin{equation}\nV_{\\min }\\simeq -\\frac{4\\,\\varepsilon \\,V_{bar}}{\\beta }\\left( 1-\\frac{\\beta \n}{4}-\\nu ^{2\/3}\\right)\n\\end{equation}%\n(see, for example, curve (b) in Fig. \\ref{Figure2}).\n\nThe expression $\\left( \\ref{17}\\right) $ for the parameter $\\delta ,$\ncharacterizing the thickness of the bubble wall for $\\chi \\neq 0,$ is\nmodified as \n\\begin{equation}\n\\delta =\\frac{1-\\beta }{\\beta \\left( 1+\\chi \\right) },  \\label{70a}\n\\end{equation}%\nand hence the bubble has a thin wall $\\delta \\ll 1$ for all $\\chi $ if $%\n\\lambda _{-}\\ll \\lambda _{+}$ $\\left( 1-\\beta \\ll 1\\right) .$ In the case $%\n\\lambda _{+}\\ll \\lambda _{-}<1$ only bubbles with $\\chi _{\\varepsilon }\\sim\n\\varepsilon ^{-1}\\gg \\lambda _{-}\/\\lambda _{+}$ have a thin wall.\n\nWhen we have two nearly degenerate minima, that is $\\varepsilon \\rightarrow\n0 $ ($\\varepsilon \\ll \\beta $), the probability of the vacuum decay, for\nexample, for $\\beta \\ll 1$ vanishes as \n\\begin{equation}\n\\Gamma \\sim4\\, \\lambda _{+}^{2}\\left( \\varphi _{0}\\frac{\\varepsilon }{\\beta }%\n\\right) ^{4}\\exp \\left[ -\\frac{\\pi ^{2}}{24\\lambda _{+}}\\left( \\frac{\\beta }{%\n\\varepsilon }\\right) ^{3}\\right]  \\label{70b}\n\\end{equation}%\nin complete agreement with our expectations.\n\n\\textbf{B) }$\\lambda _{-}\\gg 1$ (\\textit{zero size instanton problem}).\nFinally let us consider the most interesting case of a very steep unbounded\npotential (see Fig. \\ref{Figure3} where $\\lambda _{-}\\rightarrow \\infty $).\nSince $\\lambda _{+}$ is always smaller than unity $\\beta \\ll 1$ and the\nparameter $\\nu ^{2}$ in $\\left( \\ref{36}\\right) $ is \n\\begin{equation}\n\\nu ^{2}\\simeq \\frac{\\sigma ^{2}}{128}\\,\\lambda _{-}\\gg 1\\,.  \\label{77}\n\\end{equation}%\nAs one can find by inspecting $\\left( \\ref{45}\\right) $ and $\\left( \\ref{46}%\n\\right) ,$ in this case the value of $\\chi $ must be larger than unity\nbecause otherwise the quantum fluctuations dominate over the classical\ninstanton solution everywhere. Therefore, the Coleman instanton ($\\chi =0$)\nis never trustable$.$ The solution of equation $\\left( \\ref{35}\\right) $ is \n\\begin{equation}\n\\varrho _{uv}=\\varrho _{b}\\left( 1+\\frac{1}{2}\\frac{\\nu }{\\left( \\left(\n1+\\beta \\chi \\right) \\chi \\right) ^{3\/2}}+O\\left( \\frac{\\nu ^{2}}{\\chi ^{3}}%\n\\right) \\right)  \\label{77a}\n\\end{equation}%\nand to the leading order the equations $\\left( \\ref{45}\\right) $ and $\\left( %\n\\ref{46}\\right) $ simplify to%\n\\begin{equation}\n\\varphi_{uv}\\left(\\chi \\right) \\simeq -\\varphi_{0}\\,\\left( \\frac{1+\\beta\\,\n\\chi }{4\\,\\chi }-\\frac{\\nu ^{2}}{\\left( 1+\\beta\\, \\chi \\right)^{2}\\chi ^{2}}%\n\\right)  \\label{78}\n\\end{equation}%\nand \n\\begin{equation}\nV_{uv}\\left( \\chi \\right) \\simeq -\\frac{4\\,V_{bar}}{\\beta}\\left( \\frac{1}{%\n4\\,\\chi }-\\frac{\\nu^{2}}{\\left( 1+\\beta\\, \\chi \\right)^{2}\\,\\chi ^{2}}\\right) \\,.\n\\label{79}\n\\end{equation}%\nIt follows from here that only if \n\\begin{equation}\n\\chi >4\\,\\nu ^{2}  \\label{80}\n\\end{equation}%\nboth $\\varphi _{uv}$ and $V_{uv}$ are negative and hence the tunneling\nbecomes possible. This lower bound on $\\chi $ is in complete agreement with\nthe bound $\\left( \\ref{54}\\right) $ obtained by inspecting the relevance of\nthe infrared cutoff scale. Because of this bound the minimal value of the\npotential in the center of the bubble must be always larger than \n\\begin{equation}\nV_{c}\\simeq -\\frac{32\\,\\varphi_{0}^{4}}{\\sigma ^{2}}\\,.  \\label{81}\n\\end{equation}\nLet us consider the $\\chi_{\\varepsilon }$ instanton for which \n\\begin{equation}\nV_{uv}\\left( \\chi _{\\varepsilon }\\right) =\\varepsilon\\, V_{c}\\,,  \\label{82}\n\\end{equation}%\nwhere $\\varepsilon <1.$ We obtain the corresponding $\\chi _{\\varepsilon }$\nsubstituting expansion $\\left( \\ref{79}\\right) $ into (\\ref{82}), where we\nkeep only the first term,%\n\\begin{equation}\n\\chi _{\\varepsilon }\\simeq \\frac{\\sigma^{2}\\,\\lambda _{-}}{128\\,\\varepsilon }\\,.\n\\label{82aa}\n\\end{equation}%\nThe size of the bubble is given by \n\\begin{equation}\n\\varrho_{0}^{2}\\,\\varphi _{0}^{2}\\simeq \\frac{\\sigma ^{2}}{16\\,\\varepsilon }%\n\\left( 1+\\frac{\\sigma ^{2}\\,\\lambda _{+}}{128\\,\\varepsilon }+O\\left( \\frac{%\n\\varepsilon}{\\lambda_{-}}\\right) \\right) .  \\label{82bb}\n\\end{equation}%\nand this size is always larger than some minimal size, that is, \n\\begin{equation}\n\\varrho _{0}\\geq \\frac{\\sigma }{4\\,\\varphi _{0}},  \\label{82cc}\n\\end{equation}%\nirrespectively of the value of $\\lambda _{-}>1.$ This resolves the problem\nof small size instantons for the steep potentials. To calculate the action\nwe have to substitute $\\left( \\ref{82aa}\\right) $ in $\\left( \\ref{57b}%\n\\right) .$ As a result one gets%\n\\begin{equation}\nS\\left( \\chi_{\\varepsilon }\\right) \\simeq \\frac{\\pi^{2}\\sigma ^{2}}{%\n16\\,\\varepsilon }\\left[ 1+\\frac{\\sigma^{2}\\,\\lambda_{+}}{128\\,\\varepsilon }+%\n\\frac{1}{3}\\left(\\frac{\\sigma^{2}\\,\\lambda_{+}}{128\\,\\varepsilon }\\right)^{2}\n+O\\left( \\frac{\\varepsilon}{\\lambda_{-}}\\right) \\right] \\,.\n\\label{82dd}\n\\end{equation}%\nNotice that both the bubble size or the action do not depend on $\\lambda\n_{-} $ in the leading order and therefore we can take a limit $\\lambda\n_{-}\\rightarrow \\infty $ (see Fig. \\ref{Figure3}), when all $\\lambda _{-}$%\n-dependent corrections proportional to $\\varepsilon \/\\lambda _{-}$ vanish.\nAlthough all instantons contribute to the vacuum decay rate, that major\ncontribution to this rate for the unbounded potential and the potentials\nwith the second very deep minimum $V_{\\min }<V_{c}$ (see curve (a) in Fig. %\n\\ref{Figure3}) give instantons with the minimal possible size and action,\ncorresponding to $\\varepsilon \\simeq 1,$ so that%\n\\begin{equation}\n\\Gamma \\sim \\varphi_{0}^{4}\\,\\exp \\left[ -\\frac{\\pi ^{2}\\sigma ^{2}}{16}%\n\\right] \\,.  \\label{83}\n\\end{equation}%\nOne has to stress that in this case we are working on the border of\nsemiclassical approximation and the emerging bubbles contain only one\nquantum. Therefore equation $\\left( \\ref{83}\\right) $ gives only a very rough\nestimate for the probability of decay. To determine the leading contribution\nto the false vacuum decay rate in case when the second minimum of depth $%\nV_{\\min }$ is above $V_{c}$ (see curve (b) in Fig. \\ref{Figure3}) one has to\nuse the formulae $\\left( \\ref{82bb}\\right) $ and $\\left( \\ref{82dd}\\right) $\nwith $\\varepsilon $ determined by equation $V_{\\min }\\simeq \\varepsilon\nV_{c}.$\n\n\\section{Conclusions}\n\nIn this paper we have analyzed the role of quantum fluctuations for the\nclassical $O\\left( 4\\right) $ solutions which describe the false vacuum\ndecay. It was shown that the quantum fluctuations induce the ultraviolet and\ninfrared cutoff scales, beyond which the classical instanton solutions\ncannot be trusted anymore because the level of quantum fluctuations exceed\nthe contribution of the classical field. The cutoff scales are entirely\ndetermined by the parameters characterizing the corresponding classical\nsolutions. This allows one to abandon the boundary condition imposed by\nColeman on the derivative of the field at $\\rho=0$, which is outside\nthe range of validity of the approximation. The regularized solutions are\nindeed not singular. As a result there emerges the whole spectrum of new\ninstantons, which all contribute to the false vacuum decay. The largest\ncontribution comes from the instantons with the minimal possible action. In\nmany cases these instantons (up to small corrections) lead to the same\nresult as instanton with the Coleman boundary conditions. However, in several\nimportant cases the Coleman approach fails. In particular, we have shown\nthat for the very steep unbounded from below potential, shown in Fig. \\ref%\n{Figure3}, the instanton with the Coleman boundary conditions would lead to\nnearly instantaneous vacuum instability via formation of infinitely small\nbubbles. To the contrary, as the size of the new instantons always exceeds\nsome minimal scale determined by the parameters of the original potential,\nthe decay probability obtained by these new instantons remains finite even\nfor case where the unbounded potential is very steep. This removes the\ndifficulties that were associated with the application of \nsmall instantons\ufffd. We have also checked that the results in the\nmethod we suggest agrees with the standard method when they overlap.\n\n\\vspace*{1.0cm}\n\n\\textbf{Acknowledgments}\n\n\\bigskip\n\nThe work of V. M. was supported under Germany's Excellence\nStrategy---EXC-2111---Grant No. 39081486.\n\nThe work of E. R. is partially supported by the Israeli Science Foundation\nCenter of Excellence. E. R. would also like to thank NHETC at Rutgers Physics\nDepartment, the IHES and the CCPP at NYU for hospitality.\n\nA. S. would like express a gratitude to the Racah Institute of Physics of\nthe Hebrew University of Jerusalem and Ludwig Maxmillian University of\nMunich for the hospitality during his visits. The work of A. S. was\npartially supported by RFBR grant No. 20-02-00411.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nAdversarial patches \\citep{brown_2017} are one of the most relevant threat models for attacks on autonomous systems such as highly automated cars or robots. In this threat model, an attacker can freely control a small subregion of the input (the ``patch'') but needs to leave the rest of the input unchanged. This threat model is relevant because it corresponds to a physically realizable attack \\citep{lee_2019}: an attacker can print the adversarial patch pattern, place it in the physical world, and it will become part of the input of any system whose field of view overlaps with the physical patch. Moreover, once an attacker has generated a successful patch pattern, this pattern can be easily shared, will be effective against all systems using the same perception component, and an attack can be conducted without requiring access to the individual system. This makes for instance attacking an entire fleet of cars of the same vendor feasible.\n\nWhile several empirical defenses were proposed \\citep{Hayes18,NaseerKP19,selvaraju_grad-cam:_2019,wu2020defending}), these only offer robustness against known attacks but not necessarily against more effective attacks that may be developed in the future \\citep{chiang2020certified}. In contrast, certified defenses for the patch threat model \\citep{chiang2020certified, levine,zhang_clipped_2020, xiang_patchguard_2020} allow guaranteed robustness against all possible attacks for the given threat model. Ideally, a certified defense should combine high certified robustness with efficient inference while maintaining strong performance on clean inputs. Moreover, the training objective should be based on the certification problem to avoid post-hoc calibration of the model for certification.\n\nExisting defenses do not satisfy all of these conditions:  \\citet{chiang2020certified} proposed an approach that extends interval-bound propagation \\citep{Gowal_2019_ICCV} to the patch threat model. In this approach, there is a clear connection between training objective and certification problem. However, certified accuracy is relatively low and clean performance severely affected (below $50\\%$ on CIFAR10). Moreover, inference requires separate forward passes for all possible patch positions and is thus computationally very expensive. Derandomized smoothing \\citep{levine} achieves much higher certified and clean performance on CIFAR10 and even scales to ImageNet. However, inference is computationally expensive since it is based on separately propagating many differently ablated versions of a single input. Moreover, training and certification are disconnected and a separate tuning of parameters of the post-hoc certification procedure on some hold-out data is required, a drawback shared also by Clipped BagNet \\citet{zhang_clipped_2020} and PatchGuard \\citep{xiang_patchguard_2020}.\n\nIn this work, we propose \\textsc{BagCert}, which combines high certified accuracy ($60\\%$ on CIFAR10 for $5\\times 5$ patches) and  clean performance ($86\\%$ on CIFAR10), efficient inference (43 seconds on a single GPU for the $10.000$ CIFAR10 test samples), and end-to-end training for robustness against patches of varying size, aspect ratio, and location. \\textsc{BagCert} is based on the following contributions:\n\\begin{itemize}[noitemsep,topsep=0pt,parsep=0pt,partopsep=0pt]\n\t\\item We propose three different conditions that can be checked for certifying robustness. One of these corresponds to the  condition proposed by \\citet{levine}. However, we show that an alternative condition improves certified accuracy of the same model typically by roughly $3$ percent points while remaining broadly applicable.\n\t\\item We derive a loss function that directly optimizes for certified accuracy against a uniform distribution of patch sizes at arbitrary positions. This loss corresponds to a specific type of the well known class of margin losses.\n\t\\item Similarly to \\citet{levine}, we classify images via a majority voting over a large number of predictions that are based on small local regions of a single input. However, the proposed model achieves this via a single forward-pass on the unmodified input, by utilizing a neural network architecture with very small receptive fields, similar to BagNets \\citep{brendel2018approximating}. This enables efficient inference with surprisingly high clean accuracy and was concurrently proposed by \\citet{zhang_clipped_2020} and \\citet{xiang_patchguard_2020}.\n\\end{itemize}\n\n\\section{Related Work}\n\n\\paragraph{Adversarial Patch Attacks}\n\nVulnerability of image classifiers to adversarial patch attacks was first demonstrated by \\citet{brown_2017}. They show that a specifically crafted physical adversarial patch is able to fool multiple ImageNet models into predicting the wrong class with high confidence. Numerous patch attacks were proposed for object detection \\citep{LiuYLSCL19, lee_2019, ThysRG19, huang2019adversarial} and optical flow estimation \\citep{ranjan2019attacking}. The versatility of these attacks allows them to perform efficiently in the black box setup \\citep{abs-2006-12834} as well as suppressing detected objects in a scene without overlapping any of them \\citep{lee_2019}. Following \\citet{athalye_synthesizing_2017}, adversarial patches can be printed out and placed in the physical world to fool different models independently from the scaling, rotation, brightness and other visual transformations. These factors make adversarial patch attacks a non-negligible threat for the safety-critical perception systems \\citep{ThysRG19}.   \n\n\\paragraph{Heuristic Defenses Against Patch Attacks}\n\nSeveral heuristic defenses against adversarial patches such as digital watermarking \\citep{Hayes18} or local gradient smoothing \\citep{NaseerKP19} have been proposed. However, similarly to the results obtained for the norm-bounded adversarial attacks \\citep{AthalyeC018}, it was demonstrated that these defenses can be easily broken by white-box attacks which account for the pre-processing steps in the optimization procedure \\citep{chiang2020certified}. The role of spatial context in the object detection algorithms which makes them vulnerable to the patch attacks was investigated by \\citet{Saha_2019} and an empirical defense based on Grad-CAM \\citep{selvaraju_grad-cam:_2019} was proposed. Existing augmentation techniques based on adding Gaussian noise patch \\citep{lopes2020improving} or a patch from a different image \\citep{Yun_2019_ICCV} increase robustness against occlusions caused by adversarial patches. \\citet{wu2020defending} propose a defense that uses adversarial training to increase robustness against occlusion attacks.\n\n\\paragraph{Certified Defenses}\n\nEvaluating defense methods using their performance against empirical attacks can lead to the false sense of security since stronger adversaries might be developed in the future that break the defenses \\citep{AthalyeC018,UesatoOKO18}. Therefore, it is important to have guarantees of robustness. Numerous works were proposed in the field of certified robustness ranging from complete verifiers finding the worst-case adversarial examples exactly \\citep{HuangKWW17, tjeng2017verifying} to faster but less accurate incomplete methods that provide an upper bound on the robust error \\citep{GehrMDTCV18, WongK18,WongSMK18,Gowal_2019_ICCV}. \nAnother line of work is based on Randomized Smoothing \\citep{8835364,NIPS2019_9143, pmlr-v97-cohen19c}, which exhibits strong empirical results and scales to ImageNet, however at the cost of increasing inference time by orders of magnitude. Certified defenses crafted for the patch attacks were first proposed by \\citet{chiang2020certified}. They adapt the IBP method \\citep{Gowal_2019_ICCV} to the patch threat model. Although their approach allows to obtain robustness guarantees, it only scales to small patches and causes a significant drop in clean accuracy. \\citet{levine} proposed (de)randomized smoothing: they train a base classifier for classifying images where all but a small local region is ablated. At inference time, many (or even all) possible ablations are classified and a majority vote determines the final classification. If this majority vote is with sufficient margin, the decision is provable robust against patch attacks because a patch will be completely ablated in most of the inputs and can thus only influence a minority of the votes. This method provides significant accuracy improvement when compared to \\citep{chiang2020certified} and allows training and certifying ImageNet models. However, its inference in block-smoothing mode is computationally expensive. A last line of work is based on using models with small receptive fields such as BagNets \\citep{brendel2018approximating}: \\citet{zhang_clipped_2020} apply a clipping function while \\citet{xiang_patchguard_2020} apply a ``detect-and-mask'' filter to the logits of pretrained BagNets before global averaging. Using small receptive fields limits the number of region scores affected by a local patches while clipping and masking ensure that few very large region scores cannot dominate the global average. We note that these approaches do not train models directly for certified robustness but rather achieve it by applying post-hoc procedures that come with additional hyperparameters that require careful tuning.\n\t\n\n\n\\begin{figure}\n\t\\includegraphics[width=\\linewidth]{graphics\/concept}\n\t\\caption{Illustration of \\textsc{BagCert} training for a 1D input and two classes. An input $X$ is processed by region scorer $f_\\theta$, consisting of a 3-layer CNN with kernel sizes 3, 1, and 3. The resulting continuous region scores are passed through a Heaviside step function (replaced by a sigmoid in the backward pass) to obtain binary region scores $s$ for every class. The differences $\\Delta$ between true and non-true class scores are then processed by spatial aggregation $g$, in this case simply summing them via $g=g_{\\sum}$. The resulting value is maximized by passing it into margin loss $L$.}\n\t\\label{Figure:Framework}\n\\end{figure}\n\n\\section{Method}\nWe introduce \\textsc{BagCert}, a framework which consists of novel conditions for certifying robustness, a specific model architecture, and a new end-to-end training procedure. \\textsc{BagCert} allows end-to-end training of classifiers whose robustness against adversarial patch attacks can be certified efficiently. We outline our approach for the task of image classification but note that it can be extended to other tasks with grid-structured inputs. We refer to Figure \\ref{Figure:Framework} for an illustration of the training phase and to Figure \\ref{Figure:Certification} in the supplementary material for an illustration of certification of \\textsc{BagCert}.\n\n\\paragraph{Threat Model} \\label{sec:threat_model}\nWe consider a threat model in which an attacker can conduct an image-dependent patch attack. Let $x \\in [0, 1]^{w_{in}\\times h_{in} \\times c_{in}}$ be an input image of resolution $w_{in}\\times h_{in}$ with $c_{in}$ channels. Let $p$ be a patch and $l$ be a region of an image $x$ having the same size as patch $p$. We denote a set of feasible regions $l$ as $\\mathcal{L}$. For example, for a patch $p \\in [0, 1]^{n \\times c_{in}}$  consisting of $n$ pixels, $\\mathcal{L}$ could be the set of all $w_p \\times h_p$ rectangular regions $l$ of an image $x$ with $w_p\\cdot h_p=n$.  We define an operator $A$ such that $A(x, p, l)$ is the result of placing a patch $p$ onto an image $x$ over a region $l$. We assume that the attacker has white-box knowledge of the model and conducts an input-dependent attack, that is attack region $l$ and inserted patch $p$ can be chosen for every input independently.\n\n\n\\subsection{Certification} \\label{sec:certification}\nWe base our method for certification on assuming a certain structure of the classifier. More specifically, we decompose the classifier into two components:\n\\begin{itemize}\n\t\\item A region scorer $f_\\theta$ that maps from inputs $x$  to region scores $s \\in \\{0, 1\\}^{w_{out}\\times h_{out} \\times c_{out}}$, where $w_{out}\\times h_{out}$ is the output resolution, $c_{out}$ is the number of classes, and $\\theta$ are trainable parameters. Please note that we allow $\\sum_{c} s_{i,j, c} \\neq 1$. \n\t\\item A spatial aggregator $g$ that maps from region scores $s$ to (global) class scores $S \\in [0, 1]^{c_{out}}$. In this work, we restrict $g$ to be monotonically increasing, that is: for class $c$ and two patch score maps $s^{(1)}$ and $s^{(2)}$  with $s^{(1)}_{i, j, c} \\geq s^{(2)}_{i, j, c} \\,\\forall i, j$, we require $g(s^{(1)})_c \\geq g(s^{(2)})_c \\,\\forall c$.\n\\end{itemize}\n\nGenerally, we base certification on upper bounding the effect of an actual attack in the threat model. For this, we only exploit architectural properties of $f$ and $g$ that are valid for any choice of model parameters $\\theta$. More specifically, we only exploit the output dependency map $R$ of $f$, which we define as $R(l) = \\{(i, j) \\mid \\exists x, \\theta, p: f_\\theta(A(x, p, l))_{i, j} \\neq f_\\theta(x)_{i, j}\\}$. Informally, $R(l)$ is the set of all indices of the score map that can be affected by a patch applied at region $l$, for any choice of input $x$, patch $p$, and parameters $\\theta$. That is: the set of all outputs of $f_\\theta$ whose receptive fields overlap with $l$. We discuss options for $f$ and the resulting $R$ in Section \\ref{sec:model}. \n\nFor input $x$ with class label $c_t$ and $s= f_\\theta(x)$, we define the \"worst-case\" score map $s^{wc}(s, l)$ as\n$$s^{wc}_{i, j, c}(s, l) = \n\\begin{cases} s_{i, j, c} &\\text{ if } (i, j) \\notin R(l) \\\\ \n1           &\\text{ if }  (i, j) \\in R(l) \\wedge c \\neq c_t \\\\\n0           &\\text{ if }  (i, j) \\in R(l) \\wedge c = c_t \\\\\n\\end{cases}$$\nMoreover, we define $\\Delta_{i, j, c} = s_{i, j, c_t} - s_{i, j, c}$ and similarly  $\\Delta^{wc}_{i, j, c} = s^{wc}_{i, j, c_t} - s^{wc}_{i, j, c}$. It follows directly that $\\Delta^{wc}_{i, j, c} = \\Delta_{i, j, c} \\,\\forall (i,j) \\notin R(l)$ and $\\Delta^{wc}_{i, j, c} = -1 \\,\\forall (i,j) \\in R(l), c \\neq c_t$.\n\nFor certifying robustness in the threat model for input $x$ with class label $c_t$, we need to show $g(f_\\theta(A(x, p, l)))_{c_t} > g(f_\\theta(A(x, p, l)))_c \\, \\forall c \\neq c_t \\, \\forall l \\in \\mathcal{L} \\, \\forall p$. For this, it suffices to check \n\\begin{condition} \\label{expensive_condition}\n\t$g(s^{wc}(s, l))_{c_t} > g(s^{wc}(s, l))_{c} \\; \\forall c \\neq c_t, \\forall l \\in \\mathcal{L}$\n\\end{condition}\n\\begin{proof}\nConsider arbitrary $l \\in \\mathcal{L}$ and $p$ and let $s^{adv} = f_\\theta(A(x, p, l))$. With $s^{adv}_{i,j,c} \\in \\{0, 1\\}$  we obtain\\footnote{We would like to note that these are ``trivial'' lower and upper bounds for $s^{adv}$ and we see the potential to improve upon these bounds in future work, for instance by relaxing $s \\in [0, 1]^{w_{out}\\times h_{out} \\times c_{out}}$ and applying interval bound propagation \\citep{Gowal_2019_ICCV}. However, the proposed simple bounds have the advantage of not requiring additional forward passes through the model and thus being computationally efficient.}\n $$\n s^{adv}_{i,j, c}\n \\begin{cases} = s^{wc}_{i, j, c}(s, l) = s_{i, j, c}(s, l)         & \\text{ if } (i, j) \\notin R(l) \\\\ \n \t\t       \\leq s^{wc}_{i, j, c}(s, l) = 1  & \\text{ if } (i, j) \\in R(l) \\wedge c \\neq c_t \\\\\n\t\t       \\geq s^{wc}_{i, j, c}(s, l) = 0  & \\text{ if } (i, j) \\in R(l) \\wedge c = c_t \\\\\n \\end{cases}\n $$\nWith $g$ being monotonically increasing, we obtain $g(s^{adv})_{c_t} \\geq g(s^{wc}(l))_{c_t}$ and for all $c \\neq c_t$ $g(s^{wc}(l))_{c} \\geq g(s^{adv})_{c}$. Condition \\ref{expensive_condition} implies $g(s^{adv})_{c_t} > g(s^{adv})_{c} \\, \\forall c \\neq c_t$. \n\\end{proof}\n\nChecking the Condition \\ref{expensive_condition} requires one forward-pass through $f_\\theta$ to obtain $s=f_\\theta(x)$ and $\\vert \\mathcal{L}\\vert$ times the construction $s^{wc}(s, l)$ and the evaluation of $g$. We now consider a special case where this can be implemented very efficiently.\n\n\\subsubsection{Spatial Sum Aggregation}\nFor the case $g = g_{\\sum}(s) = \\sum_{i=1,j=1}^{w_{out}, h_{out}} s_{i, j}$, Condition \\ref{expensive_condition} simplifies to\n\\begin{condition} \\label{expensive_condition_sum}\n\t$\\min\\limits_{c \\neq c_t} \\sum\\limits_{i,j \\notin R(l)} \\Delta_{i, j, c} > \\vert R(l) \\vert \\quad \\forall l \\in \\mathcal{L}$\n\\end{condition}\n\\begin{proof} For all $c \\neq c_t$, we exploit $ \\forall (i, j) \\in R(l): \\Delta^{wc}_{i, j, c} = -1$. With Condition \\ref{expensive_condition_sum}, we obtain\n\t\\begin{align*}\n\tg_{\\sum}(s^{wc}(l))_{c_t} - g_{\\sum}(s^{wc}(l))_{c} \n\t  &= \\sum_{i=1,j=1}^{w_{out}, h_{out}} s^{wc}_{i, j, c_t} -  \\sum_{i=1,j=1}^{w_{out}, h_{out}} s^{wc}_{i, j, c}\n\t  = \\sum_{i=1,j=1}^{w_{out}, h_{out}} \\Delta^{wc}_{i, j, c} \\\\\n \t  &= \\sum_{i,j \\notin R(l)} \\Delta^{wc}_{i, j, c} + \\sum_{i,j \\in R(l)} \\Delta^{wc}_{i, j, c} \\\\\n \t  &= \\sum_{i,j \\notin R(l)} \\Delta_{i, j, c} - \\vert R(l) \\vert > 0. \\qedhere\n   \\end{align*}\n   \\vspace*{-.5cm}\n\\end{proof}\n\nWe note that $\\sum\\limits_{i,j \\notin R(l)} \\Delta_{i, j, c} = \\sum\\limits_{i=1,j=1}^{w_{out}, h_{out}} \\Delta_{i, j, c} - \\sum\\limits_{i,j \\in R(l)} \\Delta_{i, j, c_t} $.  For the special case that all $R(l)$ are rectangular, \n $\\sum_{i,j \\in R(l)} \\Delta_{i, j, c}$ can be computed efficiently for all $l \\in \\mathcal{L}$ simultaneously via integral images\/summed-area tables \\citep{Crow_1984}. For instance, $R(l)$ is rectangular for $l$ being rectangular input patches and the $R$ resulting from an CNN with grid-aligned kernels.\n\nFor the case that the $R(l)$ are not all rectangular and $\\vert \\mathcal{L}\\vert$ becomes large, checking Condition \\ref{expensive_condition_sum} can become prohibitively expensive.\nFor this case, we derive a condition that corresponds to an upper bound on Condition \\ref{expensive_condition_sum} and can be evaluated in constant time with respect to $\\vert \\mathcal{L}\\vert$:\n\\begin{condition} \\label{cheap_condition}\n\t$\\min\\limits_{c \\neq c_t} \\sum\\limits_{i=1,j=1}^{w_{out}, h_{out}} \\Delta_{i, j, c} > 2 R^{max}(\\mathcal{L}) \\text{  with  } R^{max}(\\mathcal{L}) = \\max\\limits_{l \\in \\mathcal{L}}\\vert R(l) \\vert$\n\\end{condition}\n\\begin{proof}\n$\\Delta_{i, j, c} \\leq 1$ implies $\\sum\\limits_{i,j \\in R(l)} \\Delta_{i, j, c} \\leq \\vert R(l) \\vert \\leq  R^{max}(\\mathcal{L})$. For all $c \\neq c_t$, using Condition \\ref{cheap_condition}:  $\\sum\\limits_{i,j \\notin R(l)} \\Delta_{i, j, c} = \\sum\\limits_{i=1,j=1}^{w_{out}, h_{out}} \\Delta_{i, j, c} - \\sum\\limits_{i,j \\in R(l)} \\Delta_{i, j, c_t} >  2 R^{max}(\\mathcal{L}) -  R^{max}(\\mathcal{L}) \\geq  \\vert R(l) \\vert$  \\end{proof}\n\nWe note that Condition \\ref{cheap_condition} corresponds to the condition proposed by \\citet{levine}. It is, however, a strictly weaker condition than Condition \\ref{expensive_condition_sum}. Thus, Condition \\ref{expensive_condition_sum} is preferable if all $R(l)$ are rectangular or $\\vert \\mathcal{L}\\vert$ is of moderate size. We refer to Figure \\ref{Figure:Certification} in the supplementary material for an illustration of Condition \\ref{expensive_condition_sum} and \\ref{cheap_condition}.\n\n\n\\subsection{Model} \\label{sec:model}\nCrucially, the quality of the certification depends on $R^{max}(\\mathcal{L}) = \\max_{l \\in \\mathcal{L}}\\vert R(l) \\vert$: the larger this quantity becomes, the larger the left-hand side of Condition \\ref{expensive_condition_sum} or Condition \\ref{cheap_condition} needs to be to fulfill the condition. We focus on the specific case where $f_\\theta$ is realized by a convolutional neural network (CNN). In that case, $\\vert R(l) \\vert$ is determined fully by $l$ and the receptive field of the CNN. More specifically, we obtain $R(l) = \\{(i, j) \\mid \\exists (\\tilde{i},\\tilde{j}) \\in l : \\vert i - \\tilde{i}\\vert \\leq \\left \\lfloor w_{rf}\/2 \\right \\rfloor \\wedge \\vert j - \\tilde{j}\\vert \\leq \\left \\lfloor h_{rf}\/2 \\right \\rfloor \\}$ for a receptive field size of $w_{rf} \\times h_{rf}$ and ignoring operation strides.\n\nReceptive field sizes of CNNs are determined by the shapes of the convolutional kernels as well as operation strides. We propose using standard CNN architectures such as ResNets but replacing most $3\\times 3$ convolutions by $1 \\times 1$ convolutions, using stride 1 in (nearly) all operations, and removing all dense layers. This results in a network with very small receptive field sizes and thus small $R(l)$. We note that the proposed architecture is similar to BagNets \\citep{brendel2018approximating} and using this type of model was concurrently proposed for certifying robustness against patch attacks by \\citet{zhang_clipped_2020} and \\citet{xiang_patchguard_2020}. BagNets obtain surprisingly high classification accuracy despite small receptive field sizes \\citep{brendel2018approximating}. Importantly, in contrast to BagNets, we do not apply a global average pooling on the final feature layer. This results in a dense output of shape ${w_{out}\\times h_{out} \\times c_{out}}$. The ratios $w_{in} \/ w_{out}$ and $h_{in} \/ h_{out}$ depend on the strides applied in the network and control mostly the computational overhead. We note that the cost for forward\/backward passes in BagNets are in the same order of magnitude as those of a corresponding residual network. Because of the small receptive fields of BagNets, $\\vert R(l) \\vert$ is small if $l$ is a small contiguous region of the input, such as a rectangular patch.\n\t\nWe apply a Heaviside step function $H(x) =\\begin{cases} 0, \\text{ for } x < 0 \\\\ 1, \\text{ for } x \\geq 0 \\end{cases}$ as final layer of $f_\\theta$, which ensures $f_\\theta(X) \\in \\{0, 1\\}^{w_{out}\\times h_{out} \\times c_{out}}$. Similar to clipping \\citet{zhang_clipped_2020} and masking \\citep{xiang_patchguard_2020} this also ensures that a patch cannot flip the global classification by perturbing a local score so strongly that it dominates the globally aggregated score. However, since $H$ is constant nearly everywhere, it does not provide useful gradient information and thereby precludes end-to-end training. We address this by applying a \"straight-through\" type trick \\citep{Bengio_2013} where we replace $H$ in the backward pass by its smooth approximation, the logistic sigmoid function $s(x) = \\frac{1}{1 + e^{-x}}$. That is, we use $H(x)$ in the forward pass but replace the true gradient of $H$ with $H'(x):=s'(x) = s(x)(1 - s(x))$. We explore alternatives to the Heaviside step function in Section \\ref{section:score_functions} in the appendix.\n\nWhile the proposed model computes $f_\\theta(X)$ in a single forward-pass and controls $\\vert R(l) \\vert$ indirectly via the architecture of $f$, we note that alternative models are compatible with \\textsc{BagCert}. For instance, one could compute every element of the output $s_{i, j}$ via a separate forward pass of an arbitrary model on an ablated \\citep{levine} or cropped version of the input similar to Mask-DS-ResNet \\citep{xiang_patchguard_2020}. This also ensures that a specific element of the output depends only on the cropper\/non-ablated part of the input. While these works are more flexible in terms of model architecture, they require a number forward passes proportional to the resolution of the output $s$, which would make inference (and end-to-end) training computationally much more expensive.\n\n\\subsection{End-to-End Training} \\label{sec:training}\nHaving derived conditions that can be used for certifying robustness against patch attacks in Section \\ref{sec:certification} as well as differentiable model for the region scorer $f$ in Section \\ref{sec:model}, we now define a loss function for end-to-end training. We restrict ourselves to the case of a spatial sum aggregation $g_{\\sum}$.\n\nWe recall Condition \\ref{cheap_condition}: $\\min\\limits_{c \\neq c_t} \\sum_{i=1,j=1}^{w_{out}, h_{out}} \\Delta_{i, j, c} >2 R^{max}(\\mathcal{L})$. The corresponding loss for this can be defined as $L_H(\\Delta, c_t, R^{max}) = H(\\min\\limits_{c \\neq c_t} \\sum_{i=1,j=1}^{w_{out}, h_{out}} \\Delta_{i, j, c} \\leq 2 R^{max})$, that is: the loss is 1 if there is a target class $c$ such that $ \\sum_{i=1,j=1}^{w_{out}, h_{out}}  \\Delta_{i, j, c}$ becomes smaller\/equal two times the size of the maximum affected patch score region. However, this requires choosing $\\mathcal{L}$ and the resulting $R^{max}(\\mathcal{L})$ before training, which is undesirable. Instead, we stay agnostic with respect to the specific $\\mathcal{L}$ and simply assume a uniform distribution\\footnote{We note that other choices than the uniform distribution would be an interesting direction for future work, in particular if the defender has prior knowledge about more likely patch sizes and shapes.} for $R^{max}(\\mathcal{L})$,  that is  $R^{max}(\\mathcal{L}) \\sim \\mathcal{U}(0, R)$. Here, $R$ corresponds to the maximum patch size (in region score space) we consider. This results in the loss \n\\begin{align*} L_R(\\Delta, c_t) \n&= \\int\\limits_0^R p(\\tilde{R}) L_H(\\Delta, c_t, \\tilde{R}) d\\tilde{R}\n= \\int\\limits_0^R \\frac{1}{R}H(\\min\\limits_{c \\neq c_t} \\sum\\limits_{i=1,j=1}^{w_{out}, h_{out}} \\Delta_{i, j, c} \\leq 2 \\tilde{R}) d\\tilde{R} \\\\\n&= 1 - \\frac{1}{R} \\int\\limits_0^R H(\\min\\limits_{c \\neq c_t} \\sum\\limits_{i=1,j=1}^{w_{out}, h_{out}} \\Delta_{i, j, c} > 2 \\tilde{R}) d\\tilde{R} \\\\\n&= 1 -\\frac{1}{R} \\min(\\frac{1}{2}\\min\\limits_{c \\neq c_t} \\sum\\limits_{i=1,j=1}^{w_{out}, h_{out}} \\Delta_{i, j, c}, R) = 1 -\\frac{1}{2R} \\min(\\min\\limits_{c \\neq c_t} \\sum\\limits_{i=1,j=1}^{w_{out}, h_{out}} \\Delta_{i, j, c}, 2R).\n\\end{align*}\nIn practice, we minimize $\\tilde{L}_R(\\Delta, c_t) = -\\min(\\min\\limits_{c \\neq c_t} \\sum_{i=1,j=1}^{w_{out}, h_{out}} \\frac{\\Delta_{i, j, c}}{w_{out} \\cdot h_{out}}, M)$ with $M=\\frac{2R}{w_{out} \\cdot h_{out}}$. This loss can be interpreted as a margin loss with margin $M$, where the margin corresponds to twice the maximum patch size in region score space against which we want to become certifiably robust. \n\n\\paragraph{One-hot penalty} While we do not strictly enforce $\\sum_{c} s_{i,j, c} = 1$, we sometimes found it beneficial to add a term in the loss that encourages $S=g_{\\sum}(s)$ being approximately ``one-hot'', that is $L_{oh}(S) = \\max_{c \\neq c_{max}} S_c - S_{c_{max}}$ with $c_{max} = \\arg\\max_c S_c$. Since $S_c \\in [0, 1]$, it holds that $L_{oh}(S) \\in [-1, 0]$ and $L_{oh}(S) = -1$ iff $S_{c_{max}} = 1$ and $S_c = 0 \\;\\forall c \\neq c_{max}$. The term $L_{oh}(S)$ prevents training from prematurely converging to a solution where $s_{i,j, c}$ is approximately constant for all $i, j, c$, which we observed otherwise for tasks with many classes (e.g. ImageNet). The total loss becomes $L_{total} = \\tilde{L}_R(\\Delta, c_t) + \\sigma L_{oh}(S)$, where $\\sigma$ controls the strength of this one-hot penalty.\n\n\n\\section{Experiments}\nWe perform an empirical evaluation of \\textsc{BagCert} on CIFAR10 \\citep{krizhevsky_cifar10_2009} and ImageNet \\citep{ILSVRC15}. We report clean and certified accuracy and compare to Interval Bound Propagation (IBP) \\citep{chiang2020certified}, Derandomized Smoothing (DS) \\citep{levine}, Clipped BagNet (CBN) \\citep{zhang_clipped_2020}, and PatchGuard \\citep{xiang_patchguard_2020}. For DS, we focus on block-smoothing and for PatchGuard, we focus on the masked BagNet (Mask-BN) because column smoothing for DS (and the derived Mask-DS for PatchGuard) perform poorly for non-square patches that are ``short-but-wide'' (see Figure \\ref{figure:patch_aspect_ratios}). We notice that column smoothing and Mask-DS perform better than column smoothing and Mask-BN against square-patches; however, there is no reason an attacker should prefer square over non-square rectangular patches. Details on the \\textsc{BagCert} model architecture and training can be found in Appendix \\ref{section:experimental_details}. Moreover, we focus on certified accuracy, a lower bound on the actual robustness of a model. Results for accuracy against a strong adversarial patch attack, corresponding to an upper bound on actual robustness, are discussed in Section \\ref{section:patch_attacks} in the appendix.\n\n\\begin{figure}[tb]\n\t\\includegraphics{graphics\/scatter_plot}\n\t\\caption{Clean versus certified accuracy on CIFAR10 and ImageNet for \\textsc{BagCert} with different receptive fields and train margins ($M \\in \\{0.25, 0.5, 0.75, 1.0\\}$ for CIFAR10, $M=0.25$ for ImageNet) when certifying via Condition \\ref{cheap_condition} (circles) and Condition \\ref{expensive_condition_sum} (stars), same setting connected by thin line. Smaller $M$ generally corresponds to larger clean accuracy for CIFAR10. Baselines are Derandomized Smoothing (DS) \\citep{levine}, Masked BagNet (Mask-BN) and Masked DS-ResNet (Mask-DS) \\citep{xiang_patchguard_2020}, and Clipped BagNet (CBN) \\citep{zhang_clipped_2020}. Results for these baselines are taken from the respective papers.\n\t}\n\t\\label{figure:clean_vs_certified}\n\\end{figure}\n\nFigure \\ref{figure:clean_vs_certified} shows results for different methods against $5\\times 5$ patches for CIFAR10 corresponding to $2.4\\%$ of the image size and patches of $2\\%$ of the image size for ImageNet. For CIFAR10, when certifying accuracy via Condition \\ref{cheap_condition}, the Pareto frontier of \\textsc{BagCert} follows closely the one reported for DS with block smoothing and $\\theta=0.3$. This is somewhat surprising given that both model and training procedure are very different and only the condition for certifying robustness is identical. We hypothesize that both approaches have reached close to optimal Pareto frontiers when certifying robustness via  Condition \\ref{cheap_condition}. However, as Table \\ref{table:resource_requirements} shows, \\textsc{BagCert} requires (depending on its receptive field size) only between $39.0$ and $48.5$ seconds for certifying all $10.000$  test examples on a single Tesla V100 SXM2 GPU while DS with block smoothing requires $788$ seconds. \\textsc{BagCert} also clearly dominates Mask-BN and CBN, which utilize a similar model architecture, as well as IBP (not shown) which reaches $47.8\\%$ clean and $30.3\\%$ certified accuracy.  Moreover, when applying Condition \\ref{expensive_condition_sum} for certification, certified accuracy is increased by approx. 3 percent points without changes in clean accuracy or any noticeable increase in certification time. In summary, the strongest \\textsc{BagCert} model with receptive field $7\\times 7$ and margin $M=0.5$ can certify all $10.000$ test examples in $43.2$ seconds, reaching clean accuracy of $86\\%$ and certified accuracy of $60\\%$.\n\n\\begin{table}[tb]\n\t\\begin{tabular}{l|rrrrr|rr}\n\t\t\\toprule\n\t\tRF of \\textsc{BagCert}   \t\t\t           &  5   &  7   &  9   &  11  & 13 & DS (BS) & DS (CS) \\\\\n\t\t\\midrule\n\t\tCertification time (seconds)            & 39.0 & 40.6 & 43.2 & 45.9 & 48.5 & 788.0   & 28.0 \\\\\n\t\tNumber of parameters                           & 28M  & 38M  & 47M  & 57M  & 66M  & 11M    &  11M\\\\\n\t\t\\bottomrule\n\t\\end{tabular}\n\t\\caption{Certification time for 10.000 CIFAR10 test examples and number of model parameters.}\n\t\\label{table:resource_requirements}\n\\end{table}\n\nOn ImageNet, \\textsc{BagCert} also dominates all baselines in terms of certified accuracy, reaching $18.9\\%$ via Condition \\ref{cheap_condition} and $22.9\\%$ via Condition \\ref{expensive_condition_sum} for receptive field size $17$ and margin $M=0.25$. Running certification for the entire validation set of $50.000$ images takes roughly $7$ minutes.\n\n\\begin{figure}[tb]\n\t\\includegraphics{graphics\/patch_sizes}\n\t\\caption{Certified accuracy against square patches of different sizes on CIFAR10. Shown is the performance for different receptive fields of \\textsc{BagCert} (left) and train margins (right). Lines correspond to the same model (without retraining), evaluated against patches of different size.}\n\t\\label{figure:patch_sizes}\n\\end{figure}\n\nFigure \\ref{figure:patch_sizes} shows accuracy of \\textsc{BagCert} certified via Condition \\ref{expensive_condition_sum} for square patches of different sizes on CIFAR10. Again, baselines are dominated for both $2\\times 2$ and $5 \\times 5$ patches. Moreover, a single configuration of $\\textsc{BagCert}$ with receptive field size 7 and margin $M=0.5$ performs close to optimal for all patch sizes and can certify non-trivial performance for up to $10 \\times 10$ patch size. This implies that a single model can be used for a broad range of threat models. Figure \\ref{figure:patch_aspect_ratios} shows a similar analysis for non-square patches of a total size of 24 pixels. While \\textsc{BagCert} with the same configuration as above achieves a certified accuracy of $40\\%$ or more for any patch aspect ratio, performance of DS with column smoothing varies greatly with aspect ratio. In particular, ``short-but-wide'' patches of shape $24 \\times 1$ or $12 \\times 2$ reduce certified accuracy of column smoothing close to $0\\%$. Since there is no reason to assume attackers will restrict themselves to square patches, we do not consider DS with column smoothing or Mask-DS \\citep{xiang_patchguard_2020} general patch defenses, despite good performance for square patches and efficient certification according to Table \\ref{table:resource_requirements}.\n\n\\begin{figure}[tb]\n\t\\includegraphics{graphics\/patch_aspect_ratios}\n\t\\caption{Certified accuracy against non-square patches of total size 24 pixels on CIFAR10. Shown is the performance for different receptive fields of \\textsc{BagCert} (left) and train margins (right) compared to Derandomized Smoothing with Column-Smoothing \\citep{levine}. Lines correspond to the same model (without retraining), evaluated against patches of different aspect rations.}\n    \\label{figure:patch_aspect_ratios}\n\\end{figure}\n\n\n\\section{Conclusion and Outlook}\nWe have introduced a novel framework \\textsc{BagCert} that combines efficient certification with end-to-end training for certified robustness. The main contributions are a model architecture based on a CNN with small receptive field, certification conditions that are applicable to a broad range of models, and a margin-loss based objective that is derived from the certification condition. The resulting model achieves high certified robustness against patches with a broad range of sizes, aspect ratios, and locations on CIFAR10 and ImageNet. Promising directions for future work are the exploration of other choices for the spatial aggregation function $g$ (such as ones using the ``detect-and-mask'' mechanism from PatchGuard \\citep{xiang_patchguard_2020}) and corresponding certification conditions and losses that can be used for end-to-end training. Moreover, the development of alternative choices for models with small receptive fields could be promising, such as ones based on learnable receptive fields or based on self-attention. Moreover, applying \\textsc{BagCert} to other modalities than images would be an exciting avenue.\n\n\\nocite{he_resnet_2016}\n\\nocite{ioffe_batch_2015}\n\\nocite{gastaldi_shake_2017}\n\\nocite{loshchilov_sgdr_2017}\n\\nocite{madry2018towards}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}