diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzmgga" "b/data_all_eng_slimpj/shuffled/split2/finalzzmgga" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzmgga" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\nIn the present paper, we study the obstacle problem related to the following nonlocal and nonlinear operator, defined formally as\n\\begin{equation}\\label{operatore}\n\\mathcal{L}u(x)=p.~\\!v. \\int_{{\\mathds R}^n} K(x,y)|u(x)-u(y)|^{p-2}\\big(u(x)-u(y)\\big)\\,{\\rm d}y, \\qquad x\\in {\\mathds R}^n;\n\\end{equation}\nwe take differentiability of order $s\\in (0,1)$ and growth $p>1$. The kernel $K$ is of order $(s,p)$ (see~\\eqref{hp_k}) with merely measurable coefficients. \nThe integral in~\\eqref{operatore} may be singular at the origin and must be interpreted in an appropriate sense. Since we assume that coefficients are merely measurable, the involved equation has to have a suitable weak formulation; see Section~\\ref{sec_preliminaries} below for the precise assumptions on the involved quantities.\n \n\\smallskip\n \nThe obstacle problem involving fractional powers of the Laplacian operator naturally appears in many contexts, such as in the analysis of anomalous diffusion (\\cite{BG90}), in the so called quasi-geostrophic flow problem (\\cite{CV10}), and in pricing of American options regulated by assets evolving in relation to jump processes (\\cite{CT04}). \nIn particular, the last application made the obstacle problem very relevant in recent times in all its forms; the obstacle problem can be indeed stated in several ways.\nRoughly speaking, a solution $u$ to the fractional obstacle problem is a minimal weak supersolution to the equation\n\\begin{equation}\\label{equazione}\n\\mathcal{L}u=0\n\\end{equation} \nabove an obstacle function $h$.\n\n\\smallskip\n\nIn the linear case when $p=2$ and when the kernel~$K$ reduces to the Gagliardo kernel~$K(x,y)=|x-y|^{-n-2s}$ without coefficients, a large treatment of the fractional obstacle problem can be found for instance in the fundamental papers by Caffarelli, Figalli, Salsa, and Silvestre (see, e.~\\!g.,~\\cite{Sil07,CSS08,CF13} and the references therein). See also~\\cite{Foc10,Foc09} for the analysis of families of bilateral obstacle problems involving fractional type energies in aperiodic settings; \nthe paper~\\cite{PP15b} for the fractional obstacle problems with drift; and the recent papers~\\cite{Gua15,MN15} for related estimates and approximations results.\nThis topic, despite its relatively short history, has already evolved into quite an elaborate theory, with connections to numerous branches of Analysis. It is impossible to provide here a complete list of references. We refer the interested reader to the exhaustive recent lecture notes by Salsa (\\cite{Sal12}), for the obstacle problem in the pure fractional Laplacian case, with the natural connection to the thin obstacle problem in low dimensions (for which we refer to~\\cite{ACS08}).\n\n\\smallskip\n\nHowever, in the more general framework considered here, the panorama seems rather incomplete, if not completely deficient in results. Clearly, the main difficulty into the treatment of the operators~$\\mathcal{L}$ in~\\eqref{operatore} lies in their very definition, which combines the typical issues given by its nonlocal feature together with the ones given by its nonlinear growth behavior; also, further efforts are needed due to the presence of merely measurable coefficients in the kernel~$K$. For this, some very important tools recently introduced in the nonlocal theory, as the by-now classic $s$-harmonic extension (\\cite{CS07}), the strong three-term commutators estimates (\\cite{DLR11}), and other successful tricks as e.~\\!g. the pseudo-differential commutator approach in~\\cite{PP14,PP15}, cannot be plainly applied and seem difficult to adapt to the nonlinear framework considered here (mainly due to the non-Hilbertian nature of the involved fractional Sobolev spaces $W^{s,p}$).\n\n\\smallskip\n\nNevertheless, some related regularity results have been very recently achieved in this context, in~\\cite{BL15,DKP14,DKKP15,DKP15,KMS15,KMS15b,IMS15,IS14,Sch15} and many others, where often a fundamental role to understand the nonlocality of the nonlinear operators~$\\mathcal{L}$ has been played by a special quantity,\n\\begin{equation}\\label{coda}\n{\\rm Tail}(u;x_0,r) := \\bigg(r^{sp} \\int_{{\\mathds R}^n \\setminus B_r(x_0)} |u(x)|^{p-1} |x-x_0|^{-n-sp} \\,{\\rm d}x \\bigg)^{\\frac{1}{p-1}};\n\\end{equation}\nthat is, {\\it the nonlocal tail} of a function $u$ in the ball of radius $r>0$ centered in $x_0 \\in {\\mathds R}^n$. This quantity, introduced by two of the authors with A. Di Castro in~\\cite{DKP15}, have been subsequently became a relevant factor in many instances when one requires a fine quantitative control of the long-range interactions, which naturally arise when dealing with nonlocal operators (see Section~\\ref{sec_preliminaries} below).\n\n\\smallskip\n\nFor what concerns the main topic in the present paper, i.~\\!e., the nonlinear fractional obstacle problem with coefficients, we will prove a series of both qualitative and quantitative results. Amongst them, \nwe will formulate the natural variational framework for the obstacle problem, and we will prove both the existence and uniqueness of the solution~$u$ to this variational formulation (Theorem~\\ref{obst prob sol}). We will show that such a solution is a weak supersolution and that it is the smallest supersolution above the obstacle in a suitable sense (Proposition~\\ref{smallest super}).\nWe will also demonstrate that the solution~$u$ inherits the regularity of the obstacle, namely the boundedness~(Theorem~\\ref{thm:obs bnd}), continuity (Theorem~\\ref{thm:obs cont}), and H\\\"older continuity (Theorem~\\ref{thm:obs H cont}). As a consequence, assuming that the obstacle function~$h$ is continuous,~$u$ is a weak solution to~\\eqref{equazione} in the open set $\\{u>h\\}$ (Corollary~\\ref{obst prob free}). These results are in clear accordance with the aforementioned results for the obstacle problems in the pure fractional Laplacian $(-\\Delta)^s$ case. However, our approach here is different and, though we are dealing with a wider class of nonlinear integro-differential operators with coefficients, the proofs are even somehow simpler, since we can make effort of a new nonlocal set-up together with the recent quantitative estimates obtained in~\\cite{DKP14,DKP15,DKKP15}, by also extending to the fractional framework some important tools in the classical Nonlinear Potential Theory.\n\n\\smallskip\n\nFinally, we will deal with the regularity up to the boundary (Theorems~\\ref{thm:H cont bdry}-\\ref{thm:cont bdry}). As well known, in the contrary with respect to the interior case, boundary regularity for nonlocal equations driven by singular, possibly degenerate, operators as in~\\eqref{operatore} seems to be a difficult problem in a general {\\it nonlinear} framework under natural assumptions on the involved quantities (while we refer to the recent paper~\\cite{Ros14} and to the forthcoming survey~\\cite{Ros16} for the case $p=2$). In this respect, a first (and possibly the solely currently present in the literature) result of global H\\\"older regularity has been obtained very recently in the interesting paper~\\cite{IMS15}, where the authors deal with the equation in~\\eqref{equazione}, in the special case when the operator $\\mathcal{L}$ in~\\eqref{operatore} coincides with the nonlinear fractional Laplacian~$(-\\Delta)^{s}_p$, by considering exclusively zero Dirichlet boundary data, and by assuming a strong $C^{1,1}$-regularity up to the boundary for the domain~$\\Omega$. Indeed, their proof is strongly based on the construction of suitable barriers near $\\partial \\Omega$, starting from the fact that, under their restrictive assumptions, the function $x\\mapsto x^s_+$ is an explicit solution in the half-space. Clearly, one cannot expect to plainly extend such a strategy in the general framework considered here, in view of the presence of merely measurable coefficients in~\\eqref{operatore}. Also, we will allow nonzero boundary Dirichlet data to be chosen, and we will assume the domain~$\\Omega$ only to satisfy a natural {\\it measure density condition} (precisely, just on the complement of~$\\Omega$; see Formula~\\eqref{eq:dens cond} on Page~\\pageref{eq:dens cond}); the latter being as expected in accordance with the classical Nonlinear Potential Theory (that is, when $s=1$).\nFor this, we will need a new proof, that will extend up to the boundary part of the results in~\\cite{DKP14,DKP15} together with a careful handling of the tail-type contributions (see Section~\\ref{sec_boundary}). Once again, it is worth stressing that all these results are new even in the pure fractional $p$-Laplacian case when the operator~$\\mathcal{L}$ does coincide with $(-\\Delta)^s_p$, and in the case of the (linear) fractional Laplacian with coefficients.\n\n\\smallskip\n\nAll in all, let us summarize \n the contributions of the present paper: \n\\noindent\n\\\\ - We prove new regularity results in terms of boundedness, continuity, and H\\\"older continuity for the solutions to a very general class of nonlocal obstacle problems, by extending previous results in the literature valid only for the pure linear fractional Laplacian case $(-\\Delta)^s$ without coefficients, also giving new proofs even in that case;\n\\noindent\n\\\\ - We obtain new regularity results up to the boundary for nonlocal operators, and, since we allow the obstacle function~$h$ to be an extended real-valued function, the degenerate case when $h\\equiv -\\infty$ (i.~\\!e., no obstacle is present) does reduce the problem to the standard Dirichlet boundary value problem, so that the results proven here are new even when $\\mathcal{L}$ does coincide with the fractional $p$-Laplacian $(-\\Delta)^s_p$. Also, since we assume that the boundary data merely belong to an appropriate tail space~$L^{p-1}_{sp}({\\mathds R}^n)$, all the (inner and boundary) results here reveal to be an improvement with respect to the previous ones in the literature when the data are usually given in the whole fractional Sobolev space~$W^{s,p}({\\mathds R}^n)$;\n\\noindent\n\\\\ - By solving the fractional obstacle problem together with some of the expected basic results proven here, we provide an important tool for several further investigations and applications.\nIndeed, as well known, the obstacle problem is deeply related to the study of minimal surfaces and the capacity of a set in Nonlinear Potential Theory. Thus, by means of our framework, we possibly give the basis for the development of a {\\it nonlocal} Nonlinear Potential Theory. This can be already seen in some subsequent forthcoming papers, as, e.~\\!g., in ~\\cite{KKL16} where part of the results here are the key for the viscosity approach for nonlocal integro-differential operators, and in~\\cite{DKKP15} where the whole nonlocal obstacle set-up is needed to extend the classical Perron method to a nonlocal nonlinear setting.\n\n\\smallskip\n\nFinally, let us comment about some immediate open problems naturally arising in this framework. Firstly, one can argue about the optimal regularity for the solutions to the nonlocal obstacle problem. We recall that for the classical obstacle problem, when $\\mathcal{L}$ coincides with the Laplacian operator $-\\Delta$, the solutions are known to belong to~$C^{1,1}$. \nThe intuition behind this regularity result goes as follows: in the contact set one has $-\\Delta u = -\\Delta h$, while where $u >h$ one has $-\\Delta u=0$; since the Laplacian jumps from $-\\Delta h$ to $0$ across the free boundary, the second derivatives of $u$ must have a discontinuity, so that $C^{1,1}$ is the maximum regularity class that can be expected. In the contrary, when $\\mathcal{L}\\equiv(-\\Delta)^s$, despite\nthe previous local argument does suggest that the solutions $u$ belong to $C^{2s}$, the optimal regularity is $C^{1,s}$, and this is quite surprising since the regularity exponent is higher than the order of the equation. \nIn the general nonlocal framework, starting from the H\\\"older regularity proven here, we still expect higher regularity results as for the linear case; nevertheless, in view of the interplay between local and nonlocal contributions and without having the possibility to rely on the $s$-harmonic extension, it is not completely evident what the optimal exponent could be as the nonlinear growth does take its part.\\footnote{For preliminary results in this direction, it is worth mentioning the very recent paper~\\cite{CRS16}, where optimal regularity results of the solution to the obstacle problem, and of the free boundary near regular points, have been achieved for linear integro-differential operators as in~\\eqref{operatore} in the case when~$p=2$.}\n\n Secondly, it could be interesting to investigate the regularity in a generic point of the free boundary (known to be analytic in the case of the Laplacian, except on a well defined set of singular points, and smooth in the case of the fractional Laplacian). \n\n Thirdly, a natural goal is to investigate the parabolic version of the nonlocal obstacle problem, as it is inspired in the so-called optimal stopping problem with deadline, by corresponding to the American option pricing problem with expiration at some given time. \nAn extension in the setting presented here could be of relevant interest as it could describe a situation which also takes into account the interactions coming from far together with a natural inhomogeneity. Accordingly with the optimal stopping problem model, a starting point in such an investigation could be the special case when the obstacle~$h$ coincides with the boundary value~$g$.\n\n \n\n\\medskip\nThe paper is organized as follows. In Section~\\ref{sec_preliminaries} below, we fix the notation by also stating some general assumptions on the quantities we will deal with throughout the whole paper.\nIn Section~\\ref{sec_obstacle}, we introduce the nonlinear fractional obstacle problem, and state and prove the existence and uniqueness of the related solutions.\nThe last two sections are devoted to the proofs of all the aforementioned boundedness and continuity results (Section~\\ref{sec_regularity}), and up to the boundary (Section~\\ref{sec_boundary}).\n \n\n\\medskip\n\n\\section{Preliminaries}\\label{sec_preliminaries}\n\nIn this section, we state the general assumptions on the quantities we are dealing with. We keep these assumptions throughout the paper.\n\\smallskip\n\nFirst of all, we recall that the class of integro-differential equations in which we are interested is the following\n\\begin{equation}\\label{problema}\n\\mathcal{L}u(x)=\\int_{{\\mathds R}^n} K(x,y)|u(x)-u(y)|^{p-2}\\big(u(x)-u(y)\\big)\\,{\\rm d}y = 0, \\quad x\\in {\\mathds R}^n.\n\\end{equation}\nThe nonlocal operator~$\\mathcal{L}$ in the display above (being read a priori in the principal value sense) is driven by its {\\it kernel} $K:{\\mathds R}^n\\times {\\mathds R}^n \\to [0,\\infty)$, which is a measurable function satisfying the following property:\n\\begin{equation}\\label{hp_k}\n\\Lambda^{-1} \\leq K(x,y)|x-y|^{n+sp} \\leq \\Lambda \\quad \\text{for a.~\\!e. } x, y \\in {\\mathds R}^n,\n\\end{equation}\nfor some $s\\in (0,1)$, $p>1$, $\\Lambda \\geq1$. We immediately notice that in the special case when $p=2$ and $\\Lambda=1$, we recover the well-known fractional Laplacian operator~$(-\\Delta)^s$.\nAlso, notice that the assumption on $K$ can be weakened,\nand in \\eqref{problema} the dependence of $u(x)-u(y)$, in turn, can be weakened from $t \\mapsto |t|^{p-2}t$ (see, for instance,~\\cite{KMS15}).\nHowever, for the sake of simplicity, we will take \\eqref{problema} and we will\nwork under the assumption in~\\eqref{hp_k}, since the weaker assumptions would bring no relevant differences in all the forthcoming proofs. \n\n\\medskip\n\nWe now recall the definition of {\\it the nonlocal tail \\,{\\rm{Tail}$(f; z, r)$} of a function $f$ in the ball of radius $r>0$ centered in $z\\in {\\mathds R}^n$}. We have\n\\begin{equation} \\label{def_tail} \n{\\rm Tail}(f;z,r) := \\bigg(r^{sp} \\int_{{\\mathds R}^n \\setminus B_r(z)} |f(x)|^{p-1} |x-z|^{-n-sp} \\,{\\rm d}x \\bigg)^{\\frac{1}{p-1}},\n\\end{equation}\nfor any function $f$ initially defined in $L^{p-1}_{\\textrm{loc}}({\\mathds R}^n)$. As mentioned in the introduction, this quantity will play an important role in the rest of the paper. The nonlocal tail has been introduced in~\\cite{DKP15}, and used subsequently in several recent papers (see e.~\\!g.,~\\cite{BL15,DKP14,HRS15,KMS15,KMS15b,IMS15,IS14} and many others\\footnote{\nWhen needed, our definition of Tail can also be given in a more general way by replacing the ball~$B_r$ and the corresponding~$r^{sp}$ term by an open bounded set~$E\\subset{\\mathds R}^n$ and its rescaled measure~$|E|^{sp\/n}$, respectively. This is not the case in the present paper.\n}), where it has been crucial to control in a quantifiable way the long-range interactions which naturally appear when dealing with nonlocal operators of the type considered here in~\\eqref{problema}.\nWhen having to control the positive and negative interactions separately, we denote the positive part and the negative part of a function $u$ by $u_+:=\\max\\{u,0\\}$ and $u_-:=\\max\\{-u,0\\}$, respectively. \nIn the following, when the center point $z$ will be clear from the context, we shall use the shorter notation \\, Tail$(f; r)\\equiv$ Tail$(f; z, r)$. \n Now, in clear accordance with the definition in~\\eqref{def_tail}, for any $p>1$ and any $s\\in (0,1)$, one can consider the corresponding {\\it tail space} $L^{p-1}_{sp}({\\mathds R}^n)$ given by\n\\begin{equation*} \nL^{p-1}_{sp}({\\mathds R}^n) := \\Big\\{ f \\in L_{\\rm loc}^{p-1}({\\mathds R}^n) \\; : \\; {\\rm Tail}(f;z,r)< \\infty \\quad \\forall z \\in {\\mathds R}^n, \\forall r \\in (0,\\infty)\\Big\\}.\n\\end{equation*}\nNotice that \n\\begin{equation*}\nL^{p-1}_{sp}({\\mathds R}^n) = \\Big\\{ f \\in L_{\\rm loc}^{p-1}({\\mathds R}^n) \\; : \\; \\int_{{\\mathds R}^n} |f(x)|^{p-1} (1+|x|)^{-n-sp} \\,{\\rm d}x < \\infty \\Big\\}.\n\\end{equation*}\nAs expected, one can check that $W^{s,p}({\\mathds R}^n) \\subset L^{p-1}_{sp}({\\mathds R}^n)$, where we denoted by $W^{s,p}({\\mathds R}^n)$ the usual fractional Sobolev space of order $(s,p)$, \ndefined by the norm\n\\begin{align}\\label{def_seminorm}\n\\|v\\|_{W^{s,p}({\\mathds R}^n)} & := \\|v\\|_{L^p({\\mathds R}^n)}\n+ [v]_{{W}^{s,p}({\\mathds R}^n)}\n \\nonumber \\\\\n&\\,\\, = \\left(\\int_{{\\mathds R}^n} |v|^p\\,{\\rm d}x \\right)^{\\frac1p} + \\left(\\int_{{\\mathds R}^n}\\int_{{\\mathds R}^n}\\frac{|v(x)-v(y)|^p}{|x-y|^{n+sp}}\\,{\\rm d}x{\\rm d}y\\right)^{\\frac1p}, \n\\end{align}\nwhere $s\\in (0,1)$ and $p\\geq1$. The local fractional Sobolev space $W^{s,p}(\\Omega)$ for $\\Omega \\subset {\\mathds R}^{n}$ is defined similarly.\nBy $W_0^{s,p}(\\Omega)$ we denote the closure of $C_0^\\infty(\\Omega)$ in $W^{s,p}({\\mathds R}^n)$. Conversely, if $v \\in W^{s,p}(\\Omega')$ with $\\Omega \\Subset \\Omega'$ and $v=0$ outside of $\\Omega$ almost everywhere, then $v$ has a representative in $W_0^{s,p}(\\Omega)$ as well (see, for instance, \\cite{DPV12}). \n\\medskip\n\n\nWe now recall the definitions of sub and supersolutions $u$ to the class of \nintegro-differential problems we are interested in.\nA function $u \\in W^{s,p}_{\\rm{loc}}(\\Omega)\\cap L^{p-1}_{sp}({\\mathds R}^n)$ is a {\\it fractional weak $p$-supersolution} of~$\\eqref{problema}$ if\n\\begin{align} \\label{supersolution}\n\\langle \\mathcal{L}u,\\eta\\rangle & \\equiv \\int_{{\\mathds R}^n} \\int_{{\\mathds R}^n}K(x,y)|u(x)-u(y)|^{p-2}\\big(u(x)-u(y)\\big)\\big(\\eta(x)-\\eta(y)\\big)\\,{\\rm d}x{\\rm d}y \\nonumber\\\\*\n& \\ge 0\n\\end{align} \nfor every nonnegative $\\eta \\in C^\\infty_0(\\Omega)$. Here $\\eta \\in C^\\infty_0(\\Omega)$ can be replaced by $\\eta \\in W^{s,p}_0(\\Omega')$ with\nevery $\\Omega' \\Subset \\Omega$. \nIt is worth noticing that the summability assumption of $u$ belonging to the tail space $L^{p-1}_{sp}({\\mathds R}^n)$ is what one expects in the nonlocal framework considered here (see~\\cite{DKKP15}). \n\\\\ A function $u \\in W^{s,p}_{\\rm{loc}}(\\Omega) \\cap L^{p-1}_{sp}({\\mathds R}^n)$ is a {\\it fractional weak \n$p$-subsolution} if $-u$ is a fractional weak $p$-supersolution. Finally, a function $u$ is a {\\it fractional weak $p$-solution} if it is both fractional weak $p$-sub and supersolution. In the following, we simply refer to those $u$ as (weak) supersolutions, subsolutions and solutions. \n\nMoreover, let us remark that we will assume that the kernel~$K$ is symmetric, and once again this is not restrictive, in view of the weak formulation presented above, since one may always define the corresponding symmetric kernel $K_{\\textrm{\\tiny sym}}$ given by\n$$\nK_{\\textrm{\\tiny sym}}(x,y):=\\frac1{2}\\Big(K(x,y)+K(y,x)\\Big).\n$$ \n\n\n\\medskip\nWe conclude this section by presenting some basic estimates which will be useful in the course of the forthcoming proofs. \nAs customary when dealing with nonlinear operators, we will often have to treat in a different way the superquadratic case when $p>2$ and the subquadratic case $12}\nLet $p \\ge 2$ and $a,b,a',b' \\in {\\mathds R}$. Then \n\\begin{align} \\label{p>2aba'b}\n|L(a,b)-L(a',b)| \\le c\\,|a-a'|^{p-1}+c\\,|a-a'||a-b|^{p-2}\n\\end{align}\nand\n\\begin{align} \\label{p>2abab'}\n|L(a,b)-L(a,b')| \\le c\\,|b-b'|^{p-1}+c\\,|b-b'||a-b|^{p-2},\n\\end{align}\nwhere $c$ depends only on $p$.\n\\end{lemma}\n\\begin{proof}\nDenoting by $f(t):=L\\big(ta'+(1-t)a,\\, b\\big)$, we obtain by the chain rule\n\\begin{align*}\n|L(a,b)-L(a',b)| &= \\Big| \\int_0^1 f'(t) \\,{\\rm d}t \\Big| = \\Big| \\int_0^1 (a'-a)\\partial_a L(ta'+(1-t)a,b) \\,{\\rm d}t \\Big| \\\\[1ex]\n&= (p-1)|a-a'| \\int_0^1 |ta'+(1-t)a-b|^{p-2} \\,{\\rm d}t \\\\[1ex]\n&\\le c\\,|a-a'|^{p-1}+c\\,|a-a'||a-b|^{p-2},\n\\end{align*}\nwhere we also used that\n\\[\n\\partial_a L(a,b)=(p-1)|a-b|^{p-2}.\n\\]\nThus, the inequality in~\\eqref{p>2aba'b} does hold. Similarly, one can prove the inequality in~\\eqref{p>2abab'}.\n\\end{proof}\n\nFinally, we would like to make the following observation. In the rest of the paper, we often use the fact that there is a constant $c>0$ depending only on $p$ such that\n\\begin{align} \\label{a-b bounds}\n\\frac1{c} \\le \\frac{\\big(|a|^{p-2}a-|b|^{p-2}b\\big)(a-b)}{(|a|+|b|)^{p-2}(a-b)^{2}} \\le c,\n\\end{align} \nwhen $a,b \\in {\\mathds R}$, $a \\neq b$. In particular,\n\\begin{align} \\label{a-b positive}\n\\big(|a|^{p-2}a-|b|^{p-2}b\\big)(a-b) \\geq 0, \\quad a,b \\in {\\mathds R}.\n\\end{align}\n\n\n\n\\medskip\n\n\n\\section{The obstacle problem}\\label{sec_obstacle}\nAs mentioned in the introduction, by solving the fractional obstacle problem we will provide an important tool in the development of the fractional Nonlinear Potential Theory, and, in order to present such a topological approach, \nwe start by introducing a necessary set of notation. \nLet $ \\Omega \\Subset \\Omega'$ be open bounded subsets of ${\\mathds R}^n$. Let \n$h \\colon {\\mathds R}^n \\to [-\\infty,\\infty)$ be an extended real-valued function, which is considered to be the obstacle,\nand let $g \\in W^{s,p}(\\Omega') \\cap L^{p-1}_{sp}({\\mathds R}^n)$ be the boundary values. We define\n$$\n\\mathcal K_{g,h}(\\Omega,\\Omega') := \\Big\\{u \\in W^{s,p}(\\Omega') \\,:\\, u \\geq h\\, \\text{ a.~\\!e. in } \\Omega, \\, u = g\\, \\text{ a.~\\!e. on } {\\mathds R}^n \\setminus \\Omega \\Big\\}.\n$$\nThe interpretation for the case $h \\equiv -\\infty$ is that \n$$\n\\mathcal K_{g}(\\Omega,\\Omega') \\equiv \\mathcal K_{g,-\\infty}(\\Omega,\\Omega') := \\Big\\{u \\in W^{s,p}(\\Omega') \\,:\\, u = g\\, \\text{ a.~\\!e. on } {\\mathds R}^n \\setminus \\Omega \\Big\\},\n$$ \ni.~\\!e., the class where we are seeking solutions to the Dirichlet boundary value problem. A few observations are in order. First, a natural assumption for any existence theory is that $\\mathcal K_{g,h}(\\Omega,\\Omega')$ is a non-empty set. This is a property of functions $g$ and $h$. Second, we are assuming that $g$ has bounded fractional Sobolev norm in a set~$\\Omega'$ which is strictly containing the set $\\Omega$, and not necessarily in the whole~${\\mathds R}^n$ as previously in the literature.\n\n\\vspace{0.9mm}\n\\subsection{Existence of solutions}\nThe obstacle problem can be reformulated as a standard problem in the theory of variational inequalities on Banach spaces, by seeking the energy minimizers in the set of suitable functions defined above. For this, by taking into account the nonlocality of the involved operators here,\nit is convenient to define a functional $\\mathcal A \\colon \\mathcal K_{g,h}(\\Omega,\\Omega') \\to \\left[W^{s,p}(\\Omega')\\right]'$ given by\n\\begin{equation}\\label{def_a}\n\\mathcal Au(v) := \\mathcal A_1u(v) + \\mathcal A_2 u(v)\n\\end{equation}\nfor every $u \\in \\mathcal K_{g,h}(\\Omega,\\Omega')$ and $v \\in W^{s,p}(\\Omega')$, where\n\\[\n \\mathcal A_1 u(v) := \\int_{\\Omega'}\\int_{\\Omega'} L(u(x),u(y))\\big(v(x)-v(y)\\big)K(x,y)\\,{\\rm d}x{\\rm d}y\n\\]\nand\n\\[\n \\mathcal A_2 u(v) := 2 \\int_{{\\mathds R}^n \\setminus \\Omega'}\\int_{\\Omega} L(u(x),g(y))v(x)K(x,y)\\,{\\rm d}x{\\rm d}y.\n\\]\nThe motivation for the definitions above is as follows. Assuming that $v \\in W_{0}^{s,p}(\\Omega)$, and $u \\in W^{s,p}(\\Omega')$ is such that $u = g$ on ${\\mathds R}^n \\setminus \\Omega'$, we have that \n\\begin{align} \\label{eq:weak sol vs A} \n\\notag \n& \\int_{{\\mathds R}^n}\\int_{{\\mathds R}^n} L(u(x),u(y))\\big(v(x)-v(y)\\big)K(x,y)\\,{\\rm d}x{\\rm d}y\n\\\\* \\notag & \\qquad = \\int_{\\Omega'}\\int_{\\Omega'} L(u(x),u(y))\\big(v(x)-v(y)\\big)K(x,y)\\,{\\rm d}x{\\rm d}y\n\\\\* \\notag & \\qquad \\quad + 2\\int_{{\\mathds R}^n \\setminus \\Omega'} \\int_{\\Omega} L(u(x),g(y)) v(x) K(x,y)\\,{\\rm d}x{\\rm d}y.\n\\\\* & \\qquad \\equiv \\mathcal A_1 u(v) + \\mathcal A_2 u(v).\n\\end{align}\nIn the following we will use the usual brackets, as e.~\\!g.~$\\langle \\mathcal{A}_1(u)-\\mathcal{A}_1(w), v\\rangle$ to denote $\\mathcal{A}_1u(v) -\\mathcal{A}_1w(v)$, and so on.\n\n\\begin{remark} \nThe functional $\\mathcal A u$ really belongs to the dual of $W^{s,p}(\\Omega')$. Indeed, we have\n\\begin{align} \\label{A1uv}\n| \\mathcal A_1 u(v) | &\\le \\int_{\\Omega'}\\int_{\\Omega'} |u(x)-u(y)|^{p-1}|v(x)-v(y)|K(x,y)\\,{\\rm d}x{\\rm d}y \\nonumber\\\\[1ex]\n&\\le c \\left(\\int_{\\Omega'}\\int_{\\Omega'} |u(x)-u(y)|^p \\frac{{\\rm d}x{\\rm d}y}{|x-y|^{n+sp}}\\right)^\\frac{p-1}{p} \\nonumber\\\\\n&\\qquad \\times \\left(\\int_{\\Omega'}\\int_{\\Omega'} |v(x)-v(y)|^p \\frac{{\\rm d}x{\\rm d}y}{|x-y|^{n+sp}}\\right)^\\frac{1}{p} \\nonumber\\\\[1ex]\n&\\le c\\,\\|u\\|_{W^{s,p}(\\Omega')}^{p-1} \\|v\\|_{W^{s,p}(\\Omega')}\n\\end{align}\nby H\\\"older's Inequality. Also,\n\\begin{align} \\label{A2uv}\n| \\mathcal A_2 u(v)| &\\le 2\\int_{{\\mathds R}^n \\setminus \\Omega'}\\int_{\\Omega} |u(x)-g(y)|^{p-1}|v(x)|K(x,y)\\,{\\rm d}x{\\rm d}y \\nonumber\\\\[1ex] \\nonumber\n&\\le c \\int_{{\\mathds R}^n \\setminus \\Omega'}\\int_{\\Omega} |u(x)|^{p-1}|v(x)||x-y|^{-n-sp}\\,{\\rm d}x{\\rm d}y \\nonumber\\\\\n& \\quad + c \\int_{{\\mathds R}^n \\setminus \\Omega'}\\int_{\\Omega} |g(y)|^{p-1}|v(x)||x-y|^{-n-sp}\\,{\\rm d}x{\\rm d}y \\nonumber \\\\[1ex]\n&\\le c\\,r^{-sp} \\left(\\int_{\\Omega} |u(x)|^p \\,{\\rm d}x\\right)^\\frac{p-1}{p} \\left(\\int_{\\Omega} |v(x)|^p \\,{\\rm d}x\\right)^\\frac{1}{p} \\nonumber \\\\\n& \\quad + c\\,\\bigg(\\int_{{\\mathds R}^n \\setminus \\Omega'}|g(y)|^{p-1}|z-y|^{-n-sp}\\,{\\rm d}y\\bigg)\\int_{\\Omega}|v(x)|\\,{\\rm d}x \\nonumber \\\\[1ex]\n&\\le c\\,r^{-sp}\\Big(\\|u\\|_{W^{s,p}(\\Omega')}^{p-1}+{\\rm Tail}(g;z,r)^{p-1} \\Big)\\|v\\|_{W^{s,p}(\\Omega')}\n\\end{align}\nholds, where $z \\in \\Omega$ and $r:=\\dist(\\Omega,\\partial \\Omega')>0$, and $c$ depends on $n,p,s,\\Lambda,\\Omega,\\Omega'$. \n\\end{remark}\n\n\\begin{remark} \\label{remark:A2}\nIn the definition \\eqref{def_a}, we could replace $A_2 u(v)$ by\n\\begin{equation} \\label{A2 alternative}\n2 \\int_{{\\mathds R}^n \\setminus \\Omega'}\\int_{\\Omega''} L(u(x),g(y))v(x)K(x,y)\\,{\\rm d}x{\\rm d}y\n\\end{equation}\nfor $\\Omega''$ satisfying $\\Omega \\subset \\Omega'' \\Subset \\Omega'$.\nAnyway, we need a strictly positive distance between $\\partial \\Omega''$ and $\\partial \\Omega'$\nto deduce $\\mathcal A u \\in [W^{s,p}(\\Omega')]'$, as seen in the calculations for \\eqref{A2uv} above.\n\\end{remark}\n\nNow, we are ready to provide the natural definition of solutions to the obstacle problem in the general nonlocal framework considered here. \n\\begin{definition}\nWe say that $u \\in \\mathcal K_{g,h}(\\Omega,\\Omega')$ is {\\it a solution to the obstacle problem} in $\\mathcal K_{g,h}(\\Omega,\\Omega')$ if\n\\[\n \\mathcal Au(v-u) \\geq 0\n\\]\nwhenever $v \\in \\mathcal K_{g,h}(\\Omega,\\Omega')$.\n\\end{definition}\nBelow, we state and prove the uniqueness of the solution to the obstacle problem and the fact that such a solution is a weak supersolution to~\\eqref{problema}. Also, under natural assumptions on the obstacle $h$, one can prove that the solution to the obstacle problem is (fractional) harmonic away from the contact set, in clear accordance with the classical results when $s=1$.\n We have\n\n\\begin{theorem} \\label{obst prob sol}\nThere exists a unique solution to the obstacle problem in $\\mathcal K_{g,h}(\\Omega,\\Omega')$. Moreover, the solution to the obstacle problem is a weak supersolution to \\eqref{problema} in $\\Omega$.\n\\end{theorem}\n\n\\begin{corollary} \\label{obst prob free}\nLet $u$ be the solution to the obstacle problem in $\\mathcal K_{g,h}(\\Omega,\\Omega')$. If $B_r \\subset \\Omega$ is such that\n\\[\n\\essinf_{B_r} (u - h) >0,\n\\]\nthen $u$ is a weak solution to \\eqref{problema} in $B_r$. In particular, if $u$ is lower semicontinuous and $h$ is upper semicontinuous in $\\Omega$, then $u$ is a weak solution to \\eqref{problema}\nin $\\Omega_+:=\\big\\{x \\in \\Omega : u(x)>h(x)\\big\\}$.\n\\end{corollary}\n\n\\begin{remark}\nWhen solving the obstacle problem in $\\mathcal K_{g,-\\infty}(\\Omega,\\Omega')$,\nwe obtain a unique weak solution to \\eqref{problema} in $\\Omega$ having the boundary values\n$g \\in W^{s,p}(\\Omega') \\cap L^{p-1}_{sp}({\\mathds R}^{n})$ in ${\\mathds R}^{n} \\setminus \\Omega$.\nThis is a generalization of the existence results stated in \\cite{DKP15}, where $g \\in W^{s,p}({\\mathds R}^{n})$, and -- as already mentioned in the introduction -- in general of all the analyses of fractional obstacle problems in the previous literature when $\\Omega'$ does coincide with the whole~${\\mathds R}^n$.\n\\end{remark}\n\n\nBefore going into the related proofs, we need to state and prove some properties of the operator $\\mathcal{A}$ defined in \\eqref{def_a}. We have the following\n\\begin{lemma} \\label{Amcwc}\nThe operator $\\mathcal A$ is monotone, coercive and weakly continuous on the set~$\\mathcal K_{g,h}(\\Omega,\\Omega')$.\n\\end{lemma}\n\\begin{proof}\nWe start with the monotonicity, that is, we show that $\\langle \\mathcal A u - \\mathcal A v, u-v \\rangle \\ge 0$ holds for every $u,v \\in \\mathcal K_{g,h}(\\Omega,\\Omega')$.\nTo this end, let $u,v \\in \\mathcal K_{g,h}(\\Omega,\\Omega')$. We have\n\\begin{align*}\n& \\langle \\mathcal A_1 u - \\mathcal A_1 v, u-v \\rangle \\\\\n&\\qquad = \\int_{\\Omega'}\\int_{\\Omega'} \\big(|u(x)-u(y)|^{p-2}(u(x)-u(y))-|v(x)-v(y)|^{p-2}(v(x)-v(y))\\big) \\\\\n&\\qquad \\qquad\\qquad \\times \\big(u(x)-u(y)-v(x)+v(y)\\big)K(x,y)\\,{\\rm d}x{\\rm d}y\n\\end{align*}\nand this quantity is nonnegative in view of~\\eqref{a-b positive}.\nSimilarly, for $\\mathcal A_2$,\n\\begin{align*}\n&\\langle \\mathcal A_2 u - \\mathcal A_2 v, u-v \\rangle \\\\\n&\\qquad = 2\\int_{{\\mathds R}^n \\setminus \\Omega'}\\int_{\\Omega} \\Big(|u(x)-g(y)|^{p-2}\\big(u(x)-g(y))-|v(x)-g(y)|^{p-2} \\\\\n&\\qquad \\qquad\\qquad \\times (v(x)-g(y)\\big)\\Big)\\big(u(x)-g(y)-v(x)+g(y)\\big)K(x,y)\\,{\\rm d}x{\\rm d}y \\\\\n&\\qquad \\ge 0.\n\\end{align*}\nHence $\\mathcal A$ is monotone.\n\n\\medskip\nNext, we prove the weak continuity. Let $\\{u_j\\}$ be a sequence in $\\mathcal K_{g,h}(\\Omega,\\Omega')$ converging to $u \\in \\mathcal K_{g,h}(\\Omega,\\Omega')$ in $W^{s,p}(\\Omega')$. The weak continuity condition is that $\\langle \\mathcal A u_j - \\mathcal Au,v \\rangle \\to 0$ for all $v \\in W^{s,p}(\\Omega')$. Thus, let $v \\in W^{s,p}(\\Omega')$.\nThen for $\\mathcal A_1$ and $12$, by \nusing \\eqref{a-b bounds},\nwe have, again by H\\\"older's Inequality, that\n\\begin{align*}\n& |\\langle \\mathcal A_1 u_j - \\mathcal A_1 u, v \\rangle| \\\\*[1ex]\n&\\qquad \\le \\int_{\\Omega'}\\int_{\\Omega'} \\big|L(u_j(x),u_j(y))-L(u(x),u(y))\\big||v(x)-v(y)|K(x,y)\\,{\\rm d}x{\\rm d}y \\\\*[1ex]\n&\\qquad \\le c\\int_{\\Omega'}\\int_{\\Omega'} \\big(|u_j(x)-u_j(y)|+|u(x)-u(y)|\\big)^{p-2} \\\\*\n&\\qquad \\qquad\\qquad \\times |u_j(x)-u_j(y)-u(x)+u(y)||v(x)-v(y)|\\frac{{\\rm d}x{\\rm d}y}{|x-y|^{n+sp}} \\\\[1ex]\n&\\qquad \\le c\\int_{\\Omega'}\\int_{\\Omega'} \\bigg(\\frac{|u_j(x)-u_j(y)|^{p-2}}{|x-y|^{s(p-2)}}+\\frac{|u(x)-u(y)|^{p-2}}{|x-y|^{s(p-2)}}\\bigg) \\\\\n&\\qquad \\qquad\\qquad \\times \\frac{|u_j(x)-u_j(y)-u(x)+u(y)|}{|x-y|^{s}}\\,\\frac{|v(x)-v(y)|}{|x-y|^{s}}\\frac{{\\rm d}x{\\rm d}y}{|x-y|^{n}} \\\\[1ex]\n&\\qquad \\le c\\left(\\int_{\\Omega'}\\int_{\\Omega'} \\frac{|u_j(x)-u_j(y)|^{p}}{|x-y|^{n+sp}}\n+ \\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}\\,{\\rm d}x{\\rm d}y\\right)^{\\frac{p-2}{p}} \\\\\n&\\qquad \\qquad \\times \\left(\\int_{\\Omega'}\\int_{\\Omega'} \\frac{|u_j(x)-u(x)-u_j(y)+u(y)|^{p}}{|x-y|^{n+sp}}\\,{\\rm d}x{\\rm d}y\\right)^{\\frac1{p}} \\\\\n&\\qquad \\qquad \\times \\left(\\int_{\\Omega'}\\int_{\\Omega'} \\frac{|v(x)-v(y)|^{p}}{|x-y|^{n+sp}}\\,{\\rm d}x{\\rm d}y\\right)^{\\frac1{p}} \\\\[1ex]\n&\\qquad \\le c\\,\\big(\\|u_j\\|_{W^{s,p}(\\Omega')}+\\|u\\|_{W^{s,p}(\\Omega')}\\big)^{p-2} \\|u_j-u\\|_{W^{s,p}(\\Omega')} \\|v\\|_{W^{s,p}(\\Omega')},\n\\end{align*}\nwhich vanishes as $j \\to \\infty$.\nSimilarly, for $\\mathcal A_2$ when $1

2$, by using~\\eqref{p>2aba'b}--\\eqref{p>2abab'} we get\n\\begin{align*}\n& |\\langle \\mathcal A_2 u_j - \\mathcal A_2 u, v \\rangle| \\\\\n&\\qquad \\le c\\int_{{\\mathds R}^n \\setminus \\Omega'}\\int_{\\Omega} |u_j(x)-u(x)|^{p-1}|v(x)||x-y|^{-n-sp}\\,{\\rm d}x{\\rm d}y \\\\\n&\\qquad \\quad + c\\int_{{\\mathds R}^n \\setminus \\Omega'}\\int_{\\Omega} |u_j(x)-u(x)||u(x)-g(y)|^{p-2}|v(x)||x-y|^{-n-sp}\\,{\\rm d}x{\\rm d}y \\\\[1ex]\n&\\qquad \\le c\\int_{\\Omega} |u_j(x)-u(x)|^{p-1}|v(x)|\\,{\\rm d}x + c\\int_{\\Omega} |u_j(x)-u(x)||u(x)|^{p-2}|v(x)|\\,{\\rm d}x \\\\\n&\\qquad \\quad + c\\,\\bigg(\\int_{{\\mathds R}^n \\setminus \\Omega'}|g(y)|^{p-2}|z-y|^{-n-sp}\\,{\\rm d}y\\bigg) \\int_{\\Omega} |u_j(x)-u(x)||v(x)|\\,{\\rm d}x \\\\[1ex]\n&\\qquad \\le c\\,\\|u_j-u\\|_{W^{s,p}(\\Omega')}^{p-1}\\|v\\|_{W^{s,p}(\\Omega')} \\\\\n&\\qquad \\quad + c\\,\\|u_j-u\\|_{W^{s,p}(\\Omega')}\\|u\\|_{W^{s,p}(\\Omega')}^{p-2}\\|v\\|_{W^{s,p}(\\Omega')} \\\\\n&\\qquad \\quad + c\\,r^{-sp}\\,{\\rm Tail}(g;z,r)^{p-2}\\|u_j-u\\|_{W^{s,p}(\\Omega')}\\|v\\|_{W^{s,p}(\\Omega')},\n\\end{align*}\nwhich again tends to $0$ as $j \\to \\infty$. \nNotice that in the display above the nonlocal integral has been estimated as follows\n\\begin{align*}\n& \\int_{{\\mathds R}^n \\setminus \\Omega'}|g(y)|^{p-2}|z-y|^{-n-sp}\\,{\\rm d}y \\\\\n&\\qquad \\le \\bigg(\\int_{{\\mathds R}^n \\setminus \\Omega'}|g(y)|^{p-1}|z-y|^{-n-sp}\\,{\\rm d}y\\bigg)^\\frac{p-2}{p-1}\n\\bigg(\\int_{{\\mathds R}^n \\setminus \\Omega'}|z-y|^{-n-sp}\\,{\\rm d}y\\bigg)^\\frac{1}{p-1} \\\\[1.5ex]\n&\\qquad \\le c\\,r^{-sp}\\,{\\rm Tail}(g;z,r)^{p-2},\n\\end{align*}\nwhere $z \\in \\Omega$ and $r:=\\dist(\\Omega,\\partial \\Omega')>0$. Thus, $\\langle \\mathcal A u_j,v \\rangle \\to \\langle \\mathcal A u, v \\rangle$ for every $v \\in W^{s,p}(\\Omega')$ as $j \\to \\infty$, i.~\\!e., the weak continuity holds.\n\n\\medskip\nFinally, we prove the coercivity, which means that there exists a function $v \\in \\mathcal K_{g,h}(\\Omega,\\Omega')$ such that\n\\[\n\\frac{\\langle \\mathcal A u - \\mathcal A v, u-v \\rangle}{\\|u-v\\|_{W^{s,p}(\\Omega')}} \\to \\infty \\quad \\text{as} \\quad \\|u\\|_{W^{s,p}(\\Omega')} \\to \\infty.\n\\]\nSince we are assuming that $\\mathcal K_{g,h}(\\Omega,\\Omega')$ is non-empty, there is at least one $v \\in \\mathcal K_{g,h}(\\Omega,\\Omega')$. Let this be fixed. By \\eqref{A1uv} and \\eqref{A2uv} we see that\n\\begin{align} \\label{AuAvv}\n|\\langle \\mathcal A u - \\mathcal A v, v \\rangle| \\le c\\,\\|u\\|_{W^{s,p}(\\Omega')}^{p-1}+c,\n\\end{align}\nwhere the constant $c$ is independent of $u$. We now show that the contribution from $\\langle \\mathcal A u - \\mathcal A v, u \\rangle$ dominates when $\\|u\\|_{W^{s,p}(\\Omega')}$ is large.\nFor $\\mathcal A_1$, we obtain\n\\begin{align}\\label{A1uA1vu}\n&\\langle \\mathcal A_1 u - \\mathcal A_1 v, u \\rangle \\nonumber\\\\\n&\\qquad= \\int_{\\Omega'}\\int_{\\Omega'} \\big(L(u(x),u(y))-L(v(x),v(y))\\big)\\big(u(x)-u(y)\\big)K(x,y)\\,{\\rm d}x{\\rm d}y \\nonumber\\\\[1ex]\n&\\qquad\\ge \\frac1{c} \\int_{\\Omega'}\\int_{\\Omega'} \\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}} \\,{\\rm d}x{\\rm d}y \\nonumber\\\\\n&\\qquad\\quad -c \\int_{\\Omega'}\\int_{\\Omega'} |v(x)-v(y)|^{p-1}|u(x)-u(y)|\\frac{{\\rm d}x{\\rm d}y}{|x-y|^{n+sp}} \\nonumber\\\\[1ex]\n&\\qquad\\ge \n\\frac 1c \\left[ u-g \\right]_{W^{s,p}(\\Omega')}^{p} - c \\left[ g \\right]_{W^{s,p}(\\Omega')}^{p} \n \\nonumber\\\\\n&\\qquad\\quad -c \\int_{\\Omega'}\\int_{\\Omega'} |v(x)-v(y)|^{p-1}|u(x)-u(y)|\\frac{{\\rm d}x{\\rm d}y}{|x-y|^{n+sp}} \\nonumber\\\\[1ex]\n&\\qquad\\ge \\frac1{c} \\|u-g\\|_{W^{s,p}(\\Omega')}^{p}-c\\,\\|g\\|_{W^{s,p}(\\Omega')}^{p} - c\\,\\|v\\|_{W^{s,p}(\\Omega')}^{p-1}\\|u\\|_{W^{s,p}(\\Omega')} \\nonumber \\\\\n&\\qquad\\ge \\frac1{c} \\|u\\|_{W^{s,p}(\\Omega')}^{p}-c\\,\\|g\\|_{W^{s,p}(\\Omega')}^{p} - c\\,\\|v\\|_{W^{s,p}(\\Omega')}^{p-1}\\|u\\|_{W^{s,p}(\\Omega')},\n\\end{align}\nby using in particular H\\\"older's Inequality and the fractional Sobolev embeddings (see for instance~\\cite[Section~6]{DPV12}, and also \\cite[Appendix 6.3]{PSV13} for a simple proof). \nFor~$\\mathcal A_2$, in turn, we obtain\n\\begin{align}\\label{A2uA2vu}\n&\\langle \\mathcal A_2 u - \\mathcal A_2 v, u \\rangle \\nonumber\\\\\n&\\qquad = 2\\int_{{\\mathds R}^n \\setminus \\Omega'}\\int_{\\Omega} \\big(L(u(x),g(y))-L(v(x),g(y))\\big)\\big(u(x)-v(x)\\big){K}(x,y)\\,{\\rm d}x{\\rm d}y \\nonumber \\\\ \n&\\qquad \\quad +\\, 2\\int_{{\\mathds R}^n \\setminus \\Omega'}\\int_{\\Omega} \\big(L(u(x),g(y))-L(v(x),g(y))\\big)v(x){K}(x,y)\\,{\\rm d}x{\\rm d}y\\nonumber \\\\[1ex]\n&\\qquad \\geq -2\\int_{{\\mathds R}^n \\setminus \\Omega'}\\int_{\\Omega} \\big|L(u(x),g(y))-L(v(x),g(y))\\big||v(x)|{K}(x,y)\\,{\\rm d}x{\\rm d}y \\nonumber\\\\[1ex]\n&\\qquad \\geq -c \\int_{{\\mathds R}^n \\setminus \\Omega'}\\int_{\\Omega} \\Big(|u(x)|^{p-1}|v(x)|+|g(y)|^{p-1}|v(x)|+|v(x)|^{p}\\Big)\\frac{{\\rm d}x{\\rm d}y}{|x-y|^{n+sp}} \\nonumber\\\\[1ex]\n&\\qquad \\geq -c\\,\\|u\\|_{L^p(\\Omega')}^{p-1}\\|v\\|_{L^p(\\Omega')} - c\\,r^{-sp}{\\rm Tail}(g;z,r)^{p-1}\\|v\\|_{L^p(\\Omega')} - \\|v\\|_{L^p(\\Omega')}^p \n\\end{align}\nwith $z\\in\\Omega$ and $r:=\\dist(\\Omega,\\partial\\Omega')$, where we also used that the term on the second line is nonnegative by the monotonicity.\nBy combining the estimates \\eqref{AuAvv}, \\eqref{A1uA1vu} with \\eqref{A2uA2vu}, it yields\n\\begin{align*}\n\\langle \\mathcal A u - \\mathcal A v, u-v \\rangle \\ge \\frac1{c} \\|u\\|_{W^{s,p}(\\Omega')}^{p} - c\\,\\|u\\|_{W^{s,p}(\\Omega')}^{p-1}-c\\,\\|u\\|_{W^{s,p}(\\Omega')}-c,\n\\end{align*}\nfor a constant $c$ independent of $u$, and thus\n\\begin{align*}\n\\frac{\\langle \\mathcal A u - \\mathcal A v, u-v \\rangle}{\\|u-v\\|_{W^{s,p}(\\Omega')}}\n\\to \\infty \\quad \\text{as} \\quad \\|u\\|_{W^{s,p}(\\Omega')} \\to \\infty.\n\\end{align*}\nThis finishes the proof.\n\\end{proof}\n\nNow, we are ready to prove the existence of a unique solution to the obstacle problem.\n\n\\begin{proof}[\\bf Proof of Theorem~\\ref{obst prob sol}]\nWe first notice that $\\mathcal K_{g,h}(\\Omega,\\Omega') \\subset W^{s,p}(\\Omega')$ is nonempty, closed and convex. Also, in view of Lemma~\\ref{Amcwc} the operator $\\mathcal A$ is monotone, coercive and weakly continuous on $\\mathcal K_{g,h}(\\Omega,\\Omega')$. This will permit us to apply the standard theory of monotone operators (see, for instance, Corollary III.1.8 in~\\cite{KS80}, or~\\cite{HKM06}) in order to deduce the existence of a function~$u \\in \\mathcal K_{g,h}(\\Omega,\\Omega')$ such that\n\\[\n\\langle \\mathcal Au,v-u \\rangle \\ge 0,\n\\]\nwhenever $v \\in \\mathcal K_{g,h}(\\Omega,\\Omega')$. In order to show the uniqueness, suppose that there are two functions $u_1$ and $u_2$ solving the obstacle problem. As a consequence, \n\\[\n\\langle \\mathcal Au_1,u_2-u_1 \\rangle \\ge 0 \\quad \\text{and} \\quad \\langle \\mathcal Au_2,u_1-u_2 \\rangle \\ge 0,\n\\]\nand then, by summing the preceding inequalities, we obtain\n\\[\n\\langle \\mathcal Au_1-\\mathcal Au_2,u_1-u_2 \\rangle \\le 0.\n\\]\nThis is possible only if $u_1=u_2$ almost everywhere. Thus, the solution $u$ is unique.\n\nNow we show that the function $u$ is a weak supersolution to \\eqref{problema} in $\\Omega$.\nFirst, clearly $u \\in W^{s,p}_{\\rm loc}(\\Omega) \\cap L^{p-1}_{sp}({\\mathds R}^{n})$.\nThen, notice that for any given nonnegative function $\\phi \\in C_0^{\\infty}(\\Omega )$, the function $v:=u+\\phi$ belongs to $\\mathcal K_{g,h}(\\Omega,\\Omega')$. Consequently, we have\nas in~\\eqref{eq:weak sol vs A} that \n\\begin{align*}\n0 \\le \\langle \\mathcal Au, \\phi \\rangle \n&= \\int_{{\\mathds R}^{n}} \\int_{{\\mathds R}^{n}}L(u(x),u(y))\\big(\\phi(x)-\\phi(y)\\big)K(x,y)\\,{\\rm d}x{\\rm d}y.\n\\end{align*}\nThus, $u$ is a weak supersolution in $\\Omega$.\n\\end{proof}\n\n\n\\begin{proof}[\\rm \\bf Proof of Corollary \\ref{obst prob free}]\nBy Theorem \\ref{obst prob sol} $u$ is a weak supersolution \nin $B_r \\subset \\Omega$. To show that $u$ is also a weak subsolution in $B_r$, let $\\eta \\in C_0^{\\infty}(B_r)$ be a nonnegative test function that is not identically $0$. Set \n$\\varepsilon := \\|\\eta\\|_\\infty^{-1} \\essinf_{B_r}(u-h)>0$. Then $v = u - \\varepsilon \\eta \\in \\mathcal K_{g,h}(\\Omega,\\Omega')$ and $\\langle \\mathcal Au,v-u \\rangle \\geq 0$ yields $\\langle \\mathcal Au,\\eta \\rangle \\leq 0$. Therefore, by~\\eqref{eq:weak sol vs A} we obtain that $u$ is a weak subsolution in $B_r$, and thus a weak solution there. \n\\end{proof}\n\nThe solution to the obstacle problem is the smallest supersolution above the obstacle in the following sense.\n\\begin{proposition} \\label{smallest super}\nLet $\\Omega \\Subset \\Omega'' \\subset \\Omega'$. Let $u$ be the solution to the obstacle problem in $\\mathcal K_{g,h}(\\Omega,\\Omega')$ and let $v$ be a weak supersolution in $\\Omega''$ such that $\\min\\{u,v\\} \\in \\mathcal K_{g,h}(\\Omega,\\Omega')$. Then $u \\leq v$ almost everywhere.\n\\end{proposition}\n\\begin{proof}\nSince $u$ is the solution to the obstacle problem and $\\min\\{u,v\\}=u$ in ${\\mathds R}^{n}\\setminus\\Omega$,\n\\begin{align} \\label{uminuv}\n0 &\\leq \\langle \\mathcal Au, \\min\\{u,v\\}-u \\rangle \\\\*\n&= \\int_{{\\mathds R}^{n}}\\int_{{\\mathds R}^{n}} L(u(x),u(y))\\Big(\\min\\{u,v\\}(x)-u(x)-\\min\\{u,v\\}(y)+u(y)\\Big) \\nonumber \\\\*\n& \\qquad \\qquad \\quad \\times K(x,y)\\,{\\rm d}x{\\rm d}y \\nonumber.\n\\end{align}\nSince $v$ is a weak supersolution in $\\Omega''$ and $u-\\min\\{u,v\\} \\in W^{s,p}_0(\\Omega)$ is nonnegative, we have\n\\begin{align} \\label{vminuv}\n0 &\\leq \\int_{{\\mathds R}^{n}}\\int_{{\\mathds R}^{n}} L(v(x),v(y))\\Big(u(x)-\\min\\{u,v\\}(x)-u(y)+\\min\\{u,v\\}(y)\\Big) \\nonumber \\\\\n& \\qquad \\qquad \\quad \\times K(x,y)\\,{\\rm d}x{\\rm d}y.\n\\end{align}\nSumming the preceding inequalities \\eqref{uminuv} and \\eqref{vminuv}, we obtain\n\\begin{align*}\n0 &\\leq \\int_{{\\mathds R}^{n}}\\int_{{\\mathds R}^{n}} \\big(L(v(x),v(y))-L(u(x),u(y))\\big) \\\\*\n&\\qquad\\qquad\\qquad \\times \\big(u(x)-\\min\\{u,v\\}(x)-u(y)+\\min\\{u,v\\}(y)\\big)K(x,y)\\,{\\rm d}x{\\rm d}y \\\\[1ex]\n&= \\int_{\\{u > v\\}} \\int_{\\{u>v\\}}\\big(L(v(x),v(y))-L(u(x),u(y))\\big) \\\\\n&\\qquad\\qquad\\qquad \\times \\big(u(x)-v(x)-u(y)+v(y)\\big)K(x,y)\\,{\\rm d}x{\\rm d}y \\\\\n&\\quad + 2\\int_{\\{u \\leq v\\}} \\int_{\\{u>v\\}}\\big(L(v(x),v(y))-L(u(x),u(y))\\big)\n\\big(u(x)-v(x)\\big)K(x,y)\\,{\\rm d}x{\\rm d}y \\\\*[1ex]\n& \\leq 0\n\\end{align*}\nsince the first term is nonpositive by \\eqref{a-b positive},\nwhereas in the second term, $L(v(x),v(y))v(x)$.\nConsequently, $|\\{u>v\\}|=0$.\n\\end{proof}\n\n\\medskip\n\n\\section{Interior regularity}\\label{sec_regularity}\nIn this section, we state and prove that the regularity of the solution to the obstacle problem inherits the regularity of the obstacle, both in the case of boundedness and (H\\\"older) continuity. This is in clear accordance with the by-now classical results for the obstacle problems in the pure fractional Laplacian $(-\\Delta)^s$ case, as seen e.~\\!g. in~\\cite{ACS08,CSS08,Sil07}, via the Dirichlet-to-Neumann extension.\nHowever, our approach here is different and, though we are dealing with a wider class of nonlinear integro-differential operators with coefficients, the proofs are new and even simpler, since we can make effort of the recent quantitative estimates presented in the previous sections and in~\\cite{DKP14,DKP15}, by taking care of the nonlocal tail quantities.\n\n\\begin{theorem} \\label{thm:obs bnd}\nLet $u$ be the solution to the obstacle problem in $\\mathcal{K}_{g,h}(\\Omega,\\Omega')$. Assume that $B_r(x_0) \\subset \\Omega'$ and set \n\\[\nM :=\\max\\bigg\\{ \\esssup_{B_r(x_0) \\cap \\Omega}h , \\esssup_{B_r(x_0) \\setminus \\Omega} g \\bigg\\}.\n\\]\nHere the interpretation is that $ \\esssup_{A} f = -\\infty$ if $A = \\emptyset$. \nIf $M$ is finite, then $u$ is essentially bounded from above in $B_{r\/2}(x_0)$ and \n\\begin{equation} \\label{eq_obs_bnd}\n\\esssup_{B_{r\/2}(x_0)}(u-m)_+ \\leq \\delta\\, {\\rm Tail}((u-m)_+;{x_0},r\/2)+c\\, \\delta^{-\\gamma} \\left(\\mean{B_r(x_0)} (u-m)_+^t\\,{\\rm d}x\\right)^{\\frac 1t}\n\\end{equation}\nholds for all $m\\geq M$, $t \\in (0,p)$ and $\\delta \\in (0,1]$ with constants $\\gamma \\equiv \\gamma(n,p,s,t)$ and $c \\equiv c(n,p,s,t,\\Lambda)$. \n\\end{theorem}\n\\begin{proof}\nSuppose that $M<\\infty$. Letting $k\\geq 0$, $m\\geq M$, and $\\phi \\in C_0^\\infty(B_r(x_0))$, $0 \\leq \\phi \\leq 1$, we see that $v = u-m - (u-m-k)_+ \\phi^p$ belongs to $\\mathcal{K}_{g-m,h-m}(\\Omega,\\Omega')$ and that $u_m := u-m$ solves the corresponding obstacle problem. Thus we have that \n\\begin{align*}\n&\\int_{{\\mathds R}^n} \\int_{{\\mathds R}^n} L(u_m(x),u_m(y)) \\big( (u_m(x) {-}k)_+ \\phi^p(x) {-} (u_m(y){-}k)_+ \\phi^p(y)\\big)\\nonumber \\\\\n& \\qquad \\quad \\, \\ \\times K(x,y) \\,{\\rm d}x{\\rm d}y\\ \\leq\\ 0.\n\\end{align*}\nAs observed in the proof of~\\cite[Theorem 1.4]{DKP15}, this is enough to prove first a Caccioppoli-type estimate with tail, and subsequently a local boundedness result (see~\\cite[Theorem 1.1]{DKP15}) which yields\n\\begin{equation} \\label{sup obs 1}\n\\esssup_{B_{\\rho\/2}(y)}(u_m)_+ \\leq \\tilde \\delta\\, {\\rm Tail}((u_m)_+;{y},\\rho\/2)+c\\, \\tilde \\delta^{-\\tilde\\gamma} \\left(\\mean{B_\\rho(y)} (u_m)_+^p\\,{\\rm d}x\\right)^{\\frac 1p},\n\\end{equation}\nwhenever $B_\\rho(y) \\subset B_r(x_0)$, \nfor any $\\tilde \\delta \\in (0,1]$, and with positive $\\tilde\\gamma$ depending only on $n,p,s$ and $c$ only on $n,p,s,\\Lambda$. Now, a covering argument, which goes back to the one in the proof of~\\cite[Theorem 1.1]{DKP14}, will allow us to prove that~\\eqref{sup obs 1} actually implies~\\eqref{eq_obs_bnd}. For this, set $\\rho=(\\sigma-\\sigma')r$ with $1\/2\\leq \\sigma'<\\sigma\\leq 1$, and take $y \\in \\sigma'B\\equiv B_{\\sigma' r}(x_0)$. We can estimate the nonlocal contribution in~\\eqref{sup obs 1} as follows\n\\begin{align}\\label{tailm}\n& {\\rm Tail}( (u_m)_+; y, \\rho\/2)^{p-1} \\nonumber \\\\[1ex]\n& \\qquad \\leq \\left(\\frac{\\rho}{2}\\right)^{sp} \\sup_{\\sigma B} (u_m)_+^{p-1} \\int_{\\sigma B\\setminus B_{\\rho\/2}(y)}|x-y|^{-n-sp}\\,{\\rm d}x \n\\nonumber\\\\\n& \\qquad \\quad + \\left(\n\\frac{\\rho}{2}\\right)^{sp}\n \\sup_{x\\in {\\mathds R}^{\\mathds n}\\setminus\\sigma B}\\left(\\frac{|x-x_0|}{|x-y|}\\right)^{n+sp}\n\\int_{{\\mathds R}^n\\setminus \\sigma B} (u_m)_+^{p-1}|x-x_0|^{-n-sp}\\,{\\rm d}x \\nonumber \\\\[1ex]\n& \\qquad \\leq \\ c\\,\\sup_{\\sigma B} (u_m)_+^{p-1}\n+ c\\,(\\sigma-\\sigma')^{-{n}}{\\rm Tail}((u_m)_+; x_0, r\/2)^{p-1}.\n\\end{align}\nFor what concerns the local contribution in~\\eqref{sup obs 1}, we can apply Young's Inequality (with exponents $p\/t$ and $p\/(p-t)$) to get\n\\begin{align*}\n\\tilde \\delta^{-\\tilde\\gamma}\\left( \\mean{B_{\\rho}(y)} (u_m)_+^{p}\\,{\\rm d}x \\right)^{\\frac{1}{p}}\n&\\leq \\tilde \\delta^{-\\tilde\\gamma} \\sup_{B_{\\rho}(y)}(u_m)_+^{\\frac{p-t}{p}} \\left( \\mean{B_{\\rho}(y)} (u_m)_+^{t}\\,{\\rm d}x \\right)^{\\frac{1}{p}} \\\\[1ex]\n&\\leq \\frac{1}{4}\\sup_{\\sigma B} (u_m)_+ \n+ c\\,\\tilde \\delta^{-\\frac{\\tilde\\gamma p}{t}}\\left( \\mean{B_{\\rho}(y)} (u_m)_+^t\\,{\\rm d}x \\right)^{\\frac{1}{t}} \\\\[1ex]\n&\\leq \\frac{1}{4}\\sup_{\\sigma B} (u_m)_+\n+ c\\,\\tilde \\delta^{-\\frac{\\tilde\\gamma p}{t}}\n(\\sigma-\\sigma')^{-\\frac{n}{t}}\n\\left( \\mean{B_r} (u_m)_+^t\\,{\\rm d}x \\right)^{\\frac{1}{t}}.\n\\end{align*}\nThus, by reabsorbing with $\\tilde \\delta\\leq 1\/4c$ we deduce by three last displays that\n\\begin{align*}\n\\sup_{\\sigma' B}(u_m)_+ &\\leq\n\\frac{1}{2}\\sup_{\\sigma B} (u_m)_+\n+ c\\,\\tilde \\delta^{-\\frac{\\tilde\\gamma p}{t}} (\\sigma-\\sigma')^{-\\frac{n}{t}}\\left( \\mean{B_r} (u_m)_+^t\\,{\\rm d}x \\right)^{\\frac{1}{t}} \\\\\n&\\quad +\\, c\\,\\tilde \\delta (\\sigma-\\sigma')^{-\\frac{n}{p-1}}{\\rm Tail}((u_m)_+; x_0, r\/2),\n\\end{align*}\nso that finally a standard iteration argument, see for instance~\\cite[Lemma 3.38]{HKM06}, and choosing $\\tilde \\delta = \\delta\/c$ will give the desired result~\\eqref{eq_obs_bnd}.\n\\end{proof}\n\nThe solution to the obstacle problem inherits the continuity of the obstacle. \n\n\\begin{theorem} \\label{thm:obs H cont}\nSuppose that $h$ is locally H\\\"older continuous in $\\Omega$. Then the solution $u$ to the obstacle problem in $\\mathcal{K}_{g,h}(\\Omega,\\Omega')$ is locally H\\\"older continuous in $\\Omega$ as well.\nMoreover, for every $x_0 \\in \\Omega$ there is $r_0>0$ such that\n\\begin{align} \\label{eq:obs cont} \n\\osc_{B_\\rho(x_0)} u & \\leq c \\left(\\frac{\\rho}{r}\\right)^{\\alpha} \\left[ {\\rm Tail}(u -h(x_0);{x_0},r) + \\bigg(\\mean{B_{r}(x_0)} |u-h(x_0)|^p \\,{\\rm d}x \\bigg)^{\\frac 1p}\\right] \n\\\\ & \\quad \\nonumber + c \\int_{\\rho}^r \\left(\\frac{\\rho}{t}\\right)^{\\alpha} \\omega_h\\left( \\frac{t}{\\sigma} \\right) \\, \\frac{dt}{t}\n\\end{align}\nfor every $r\\in (0,r_0)$ and $\\rho \\in (0,r\/4]$, where $\\omega_h(\\rho) \\equiv \\omega_h(\\rho,x_0) := \\osc_{B_{\\rho}(x_0)} h$,\nand $\\alpha$, $c$ and $\\sigma$ depend only on $n$, $p$, $s$, and $\\Lambda$.\n\\end{theorem}\n\\begin{proof}\nLet us first analyze the contact set, by which we mean that $x_0$ belongs to the contact set if and only if for every $r \\in (0,R)$, $R:= \\dist(x_0,\\partial \\Omega)$, we have\n\\[\n\\inf_{B_{r}(x_0)} (u-h) = 0.\n\\]\nOur first goal is to show that for any such point $x_0$ and for any\n$r \\in (0,R)$ we find $\\sigma \\in (0,1)$ and $c$, both depending only on $n,p,s,\\Lambda$, such that \n\\begin{align} \\label{eq:osc decay 000}\n& \\nonumber \\osc_{B_{\\sigma r}(x_0)} u + {\\rm Tail}(u - h(x_0);{x_0},\\sigma r) \n\\\\ & \\qquad \\leq \\frac12 \\left(\\osc_{B_{r}(x_0)} u + {\\rm Tail}(u - h(x_0);{x_0},r) \\right) + c\\,\\omega_h(r).\n\\end{align}\nTo this end, observe first that $u \\geq d: = h(x_0) - \\omega_h(r)$ almost everywhere in $B_r(x_0)$. Set $u_d := u-d$, which is then a nonnegative weak supersolution in $B_r(x_0)$ by Theorem~\\ref{obst prob sol}. Now apply Theorem~\\ref{thm:obs bnd} and the weak Harnack estimate in \\cite[Theorem 1.2]{DKP14}. We obtain by \\eqref{eq_obs_bnd} (applied with $m= d + 2\\omega_h(r) \\geq \\sup_{B_{2\\rho}(x_0)}h$) that \n\\begin{equation} \\label{eq:sup u_k 0}\n\\sup_{B_\\rho(x_0)} u_d \\leq 2\\omega_h(r) + \\delta\\,{\\rm Tail}((u_d)_+ ;{x_0},\\rho)+c\\, \\delta^{-\\gamma} \\left(\\mean{B_{2\\rho}(x_0)} u_d^t\\,{\\rm d}x \\right)^{\\frac 1t}\n\\end{equation}\nfor $\\rho \\in(0,r]$, $t\\in(0,p)$ and $\\delta\\in(0,1]$, and the weak Harnack gives\n\\[\n\\left(\\mean{B_{2\\rho}(x_0)} u_d^t\\,{\\rm d}x\\right)^{\\frac 1t} \\leq c \\inf_{B_{4\\rho}(x_0)} u_d + c \\left(\\frac{\\rho}{r}\\right)^{\\frac{sp}{p-1}}{\\rm Tail}( (u_d)_-;{x_0},r)\n\\]\nwhenever $\\rho \\in(0,r\/4]$.\nSince $\\inf_{B_{\\rho}(x_0)}u_d \\leq \\omega_h(r)$ due to $\\essinf_{B_{\\rho}(x_0)} (u-h) = 0$, we obtain from the previous display that \n\\[\n\\left(\\mean{B_{2\\rho}(x_0)} u_d^t\\,{\\rm d}x\\right)^{\\frac 1t} \\leq c\\,\\omega_h(r) +c \\left(\\frac{\\rho}{r}\\right)^{\\frac{sp}{p-1}} {\\rm Tail}(u_d;{x_0},r).\n\\]\nThus, recalling that $u_d \\geq 0$ in $B_r(x_0)$, we arrive at\n\\[\n\\osc_{B_\\rho(x_0)} u \\leq c\\,\\delta^{-\\gamma} \\omega_h(r) + c\\,\\delta\\,{\\rm Tail}(u_d;{x_0},\\rho)\n+ c\\,\\delta^{-\\gamma} \\left(\\frac{\\rho}{r}\\right)^{\\frac{sp}{p-1}} {\\rm Tail}(u_d;{x_0},r).\n\\]\nNow we observe that\n\\begin{equation} \\label{eq:tail u_k}\n{\\rm Tail}(u_d;{x_0},\\rho) \\leq c \\sup_{B_{r}(x_0)} |u_d| + c \\left(\\frac{\\rho}{r}\\right)^{\\frac{sp}{p-1}} {\\rm Tail}(u_d;{x_0},r),\n\\end{equation}\nwhere we can estimate \n\\begin{equation} \\label{eq:sup u_k}\n\\sup_{B_r(x_0)} |u_d|=\\sup_{B_r(x_0)} |u-h(x_0)+\\omega_h(r)| \\leq \\osc_{B_{r}(x_0)} u+2 \\omega_h(r).\n\\end{equation}\nNow, for any $\\varepsilon \\in (0,1)$, we can first choose $\\delta$ small and then $\\widetilde \\sigma \\in (0,1)$, correspondingly, so that we have\n\\[\nc\\, \\delta \\leq \\frac{\\varepsilon}{2} \\qquad \\mbox{and} \\qquad c\\,\\delta^{-\\gamma} \\widetilde \\sigma^{\\frac{sp}{p-1}} \\leq \\frac{\\varepsilon}{2},\n\\]\nand therefore, for $\\rho = \\widetilde \\sigma r$,\n\\begin{equation} \\label{eq:osc tildesigmarho}\n\\osc_{B_{\\widetilde \\sigma r}(x_0)} u \\leq \\varepsilon \\left(\\osc_{B_{r}(x_0)} u + {\\rm Tail}(u - h(x_0);{x_0},r) \\right) + c_\\varepsilon\\, \\omega_h(r)\n\\end{equation}\nholds. Using this together with \\eqref{eq:tail u_k}, we further have that for any $\\sigma \\in (0,\\widetilde \\sigma)$\n\\begin{align*} \n{\\rm Tail}(u - h(x_0);{x_0},\\sigma r) & \\leq c \\osc_{B_{\\tilde \\sigma r}(x_0)} u +c \\left(\\frac{\\sigma}{\\widetilde \\sigma}\\right)^{\\frac{sp}{p-1}} {\\rm Tail}(u - h(x_0);{x_0},\\widetilde \\sigma r)\n\\\\* & \\leq c\\, \\varepsilon \\left(\\osc_{B_{r}(x_0)} u + {\\rm Tail}(u - h(x_0);{x_0},r) \\right) + c\\, c_\\varepsilon\\, \\omega_h(r) \\\\*\n&\\quad + c \\left(\\frac{\\sigma}{\\widetilde \\sigma}\\right)^{\\frac{sp}{p-1}} \\left(\\osc_{B_r(x_0)}u + {\\rm Tail}(u - h(x_0);{x_0},r)\\right).\n\\end{align*}\nTherefore, adding \\eqref{eq:osc tildesigmarho} and taking $\\sigma$ and $\\varepsilon$ so small that\n\\[\nc \\left(\\frac{\\sigma}{\\widetilde \\sigma}\\right)^{\\frac{sp}{p-1}} \\leq \\varepsilon \\qquad \\text{and} \\qquad (c+2)\\,\\varepsilon \\leq \\frac12,\n\\]\nyields \\eqref{eq:osc decay 000}.\n\nNext, iterating \\eqref{eq:osc decay 000} we obtain\n\\begin{align} \\label{eq:osc decay 001}\n& \\nonumber \\osc_{B_{\\sigma^k r}(x_0)} u + {\\rm Tail}(u - h(x_0);{x_0},\\sigma^k r) \n\\\\ & \\qquad \\leq 2^{1-k} \\left(\\osc_{B_{\\sigma r}(x_0)} u + {\\rm Tail}(u - h(x_0);{x_0},\\sigma r) \\right) + c \\sum_{j=0}^{k-2} 2^{-j} \\omega_h(\\sigma^{k-j-1} r)\n\\end{align}\nfor any $k \\in \\mathbb N$.\nUsing finally the fact $\\osc_{B_r} u = \\osc_{B_r} u_d \\leq \\sup_{B_r} u_d$ and the supremum estimate \\eqref{eq:sup u_k 0}, \nwe conclude the contact set analysis with\n\\begin{align} \\label{eq:osc decay 002}\n& \\nonumber \\osc_{B_{\\sigma^k r}(x_0)} u + {\\rm Tail}(u - h(x_0);{x_0},\\sigma^k r) \n\\\\ & \\qquad \\leq c\\, 2^{1-k} \\left( {\\rm Tail}(u -h(x_0);{x_0},r) + \\bigg(\\mean{B_{r}(x_0)} |u-h(x_0)|^t\\,{\\rm d}x\\bigg)^{\\frac 1t}\\right) \\\\\n& \\qquad\\quad + c \\sum_{j=0}^{k-1} 2^{-j} \\omega_h(\\sigma^{k-j-1} r). \\nonumber\n\\end{align}\nNotice here that if $h$ is continuous and uniformly bounded in $B_r$, then\n\\begin{equation*} \\label{eq:omega_h sum}\n\\lim_{k\\to \\infty} \\sum_{j=0}^{k-1} 2^{-j} \\omega_h(\\sigma^{k-j-1} r) = 0,\n\\end{equation*}\nimplying that $\\lim_{r \\to 0} \\osc_{B_{r}(x_0)} u = 0$ in this case. \n\nWe then analyze the continuity properties outside of the contact set. In this case we find $r_0 \\in (0,R)$ such that \n$$\n\\inf_{B_{r_0}(x_0)} (u-h) > 0.\n$$\nThen Corollary \\ref{obst prob free} says that $u$ is a weak solution in $B_{r_0}(x_0)$, and consequently we can use the results in~\\cite{DKP15}, by also noticing that in the proofs there it makes no difference to assume\n$u \\in W^{s,p}_{\\rm loc}(\\Omega) \\cap L^{p-1}_{sp}({\\mathds R}^{n})$ instead of $u \\in W^{s,p}({\\mathds R}^{n})$.\nIn particular, \\cite[Theorem 1.2]{DKP15} implies that\n\\begin{equation*} \\label{eq:osc decay 003}\n\\osc_{B_{\\rho}(x_0)} u \\leq c \\left(\\frac{\\rho}{r}\\right)^{\\alpha} \\left({\\rm Tail}(u-h(x_0);{x_0},r) + \\bigg(\\mean{B_{r}(x_0)} | u -h(x_0)|^p\\,{\\rm d}x\\bigg)^{\\frac 1p} \\right).\n\\end{equation*}\nfor every $r \\in (0,r_0)$ and $\\rho \\in (0,r\/2]$.\nThe claim \nfollows from this and \\eqref{eq:osc decay 002} (with $\\alpha \\leq -\\log 2 \/ \\log \\sigma$) after straightforward manipulations. \n\\end{proof}\n\nSlightly modifying the proof above, we easily obtain the following.\n\\begin{theorem} \\label{thm:obs cont}\nSuppose that $h$ is continuous in $\\Omega$. Then the solution to the obstacle problem in $\\mathcal{K}_{g,h}(\\Omega,\\Omega')$ is continuous in $\\Omega$ as well.\n\\end{theorem}\n\\begin{proof}\nThis plainly follows from the previous theorem, since if $\\omega_h(t) \\to 0$ as $t\\to 0$ and $\\omega_h$ is locally uniformly bounded, then it is easy to check that \n$$\n\\int_{\\rho}^r \\left(\\frac{\\rho}{t}\\right)^{\\alpha} \\omega_h\\left( \\frac{t}{\\sigma} \\right) \\, \\frac{dt}{t} \\to 0 \n$$\nas $\\rho \\to 0$ for small enough $r$. \n\\end{proof}\n\n\n\\medskip\n\n\n\n\\section{Boundary regularity}\\label{sec_boundary}\n\nWe continue our investigation by considering the regularity of the solution to the obstacle problem on the boundary of $\\Omega$. \nIn what follows, we assume $x_0 \\in \\partial \\Omega$.\nFirstly, we would need a Caccioppoli-type estimate with tail, whose proof is\na verbatim repetition of the proof of~\\cite[Theorem~1.4]{DKP15} after noticing that $v = u \\mp w_\\pm \\phi^p$, $\\phi \\in C_0^\\infty(B_r(x_0))$, $0\\leq \\phi \\leq 1$, belongs to $\\mathcal{K}_{g,h}(\\Omega,\\Omega')$ for all indicated $k_+$ and $k_-$. For other fractional Caccioppoli-type inequalities, though not taking into account the tail contribution, see~\\cite{Min07,Min11,FP14}. We have the following\n\\begin{lemma} \\label{lemma:cacc bnd}\nSuppose that $u \\in \\mathcal{K}_{g,h}(\\Omega,\\Omega')$ solves the obstacle problem in $\\mathcal{K}_{g,h}(\\Omega,\\Omega')$.\nLet $x_0 \\in \\partial\\Omega$, let $r \\in (0,r_0)$ with $r_0 := \\dist(x_0,\\partial \\Omega')$, and suppose that \n\\[\nk_+ \\geq \\max\\bigg\\{ \\esssup_{B_r(x_0)} g, \\esssup_{B_r(x_0) \\cap \\Omega} h \\bigg\\} \\quad \\text{and} \\quad\nk_- \\leq \\essinf_{B_r(x_0)} g.\n\\]\nThen, for $w_+ := (u - k_+)_+$ and $w_- := (k_--u)_+$, we have \n\\begin{eqnarray}\\label{cacio1}\n\\nonumber && \\int_{B_r(x_0)}\\int_{B_r(x_0)} |w_{\\pm}(x)\\phi(x)-w_{\\pm}(y)\\phi(y)|^p K(x,y) \\,{\\rm d}x{\\rm d}y \\\\[1ex]\n &&\\qquad \\leq c\\int_{B_r(x_0)}\\int_{B_r(x_0)} w_{\\pm}^p(x) |\\phi(x)-\\phi(y)|^p K(x,y) \\,{\\rm d}x{\\rm d}y \\\\\n&&\\qquad \\quad+\\,c \\int_{B_r(x_0)}w_{\\pm}(x)\\phi^p(x)\\,{\\rm d}x \\left(\\sup_{y\\,\\in\\,{\\rm supp}\\,\\phi}\\int_{\\mathds{R}^n\\setminus B_r(x_0)} w_{\\pm}^{p-1}(x)K(x,y)\\,{\\rm d}x \\right), \n\\nonumber\n\\end{eqnarray}\nwhenever $\\phi \\in C_0^\\infty(B_r(x_0))$ and $0\\leq \\phi \\leq 1$.\n\\end{lemma}\n\\begin{remark} \\label{remark:cacc contact}\nIf the maximum $\\max\\{ \\esssup_{B_r(x_0)} g, \\esssup_{B_r(x_0) \\cap \\Omega} h \\}$ is infinite,\nor $\\essinf_{B_r(x_0)} g = -\\infty$, \nthen the interpretation is that there is no test function of the type $w_+$ or $w_-$, respectively. \n\\end{remark}\n\n\nWhen the obstacle and the boundary values are bounded on the boundary, so is the solution to the obstacle problem.\n\n\\begin{theorem} \\label{thm:boundednessx0}\nSuppose that $u \\in \\mathcal{K}_{g,h}(\\Omega,\\Omega')$ solves the obstacle problem in $\\mathcal{K}_{g,h}(\\Omega,\\Omega')$.\nLet $x_0 \\in \\partial\\Omega$ and suppose that \n\\[\n\\max\\bigg\\{ \\esssup_{B_r(x_0)} g, \\esssup_{B_r(x_0) \\cap \\Omega} h \\bigg\\} < \\infty \\quad \\text{and} \\quad \\essinf_{B_r(x_0)} g > -\\infty\n\\]\nfor $r \\in (0,r_0)$ with $r_0 := \\dist(x_0,\\partial \\Omega')$. Then $u$ is essentially bounded close to $x_0$. \n\\end{theorem}\n\\begin{proof}\nChoose \n\\[\nk_+ \\geq \\max\\bigg\\{ \\esssup_{B_r(x_0)} g, \\esssup_{B_r(x_0) \\cap \\Omega} h \\bigg\\} \\quad \\text{and} \\quad\nk_- \\leq \\essinf_{B_r(x_0)} g.\n\\]\nThen, repeating the proof of \\cite[Theorem 1.1]{DKP15} using the estimate \\eqref{cacio1} in Lemma \\ref{lemma:cacc bnd} with $w_+ := (u-k_+)_+$ and $w_- := (k_--u)_+$, we get\n\\begin{equation*} \n\\esssup_{B_{r\/2}(x_0)} w_\\pm \\leq \n\\delta \\, {\\rm Tail}(w_\\pm ; x_0, r\/2)+ c\\,\\delta^{-\\gamma} \\left( \\mean{B_r(x_0)} w_\\pm^p \\,{\\rm d}x \\right)^{1\/p}.\n\\end{equation*}\nfor any $\\delta \\in (0,1]$ with $\\gamma \\equiv \\gamma(n,p,s)$ and $c\\equiv c(n,p,s,\\Lambda)$. Consequently, $u$ is essentially bounded in $B_{r\/2}(x_0)$.\n\\end{proof}\n\nTo prove the H\\\"older continuity of the solution to the obstacle problem on the boundary, we also need the following logarithmic estimate. \n\n\\begin{lemma} \\label{log lemma w}\nLet $B_r \\subset B_{R\/2}$ be concentric balls and let $w \\in W^{s,p}(B_R) \\cap L^{p-1}_{sp}({\\mathds R}^{n})$ satisfy\n\\[\n\\esssup_{B_R} w \\leq M < \\infty \\quad \\text{and} \\quad \\essinf_{B_R} w \\geq \\varepsilon > 0.\n\\]\nSuppose that\n\\begin{equation*}\n\\int_{{\\mathds R}^{n}}\\int_{{\\mathds R}^{n}}L(w(x),w(y))\\left(\\frac{M-w(x)}{w(x)^{p-1}}\\phi^{p}(x)-\\frac{M-w(y)}{w(y)^{p-1}}\\phi^{p}(y)\\right)K(x,y)\\,{\\rm d}x{\\rm d}y \\geq 0,\n\\end{equation*}\nwhere $\\phi \\in C^{\\infty}_0(B_{3r\/2})$ satisfies $0\\leq\\phi\\leq 1$, $\\phi = 1$ in $B_{r}$ and $|\\nabla\\phi|w(y)$. By \\cite[Lemma 1.3]{DKP15} we have\n\\[\n\\phi^{p}(x) \\leq \\phi^{p}(y)+c\\,\\delta\\phi^{p}(y)+c\\,\\delta^{1-p}|\\phi(x)-\\phi(y)|^{p}\n\\]\nwhenever $\\delta \\in (0,1)$. Choosing\n\\[\n\\delta=\\sigma\\frac{w(x)-w(y)}{w(x)} \\in (0,1), \\quad \\sigma \\in (0,1),\n\\]\nin the display above, implies\n\\begin{align*}\n\\Psi(x,y)&:=(w(x)-w(y))^{p-1}\\left(\\frac{M-w(x)}{w(x)^{p-1}}\\phi^{p}(x)-\\frac{M-w(y)}{w(y)^{p-1}}\\phi^{p}(y)\\right) \\\\[1ex]\n&\\leq (w(x)-w(y))^{p-1}\\left(\\frac{M-w(x)}{w(x)^{p-1}}-\\frac{M-w(y)}{w(y)^{p-1}}+c\\,\\delta\\frac{M-w(x)}{w(x)^{p-1}}\\right)\\phi^{p}(y) \\\\\n&\\quad + c\\,\\delta^{1-p}(w(x)-w(y))^{p-1}\\frac{M-w(x)}{w(x)^{p-1}}|\\phi(x)-\\phi(y)|^{p} \\\\[1ex]\n&= \\left(\\frac{M-w(x)}{w(x)^{p-1}}-\\frac{M-w(y)}{w(y)^{p-1}}+c\\,\\sigma\\frac{(w(x)-w(y))(M-w(x))}{w(x)^{p}}\\right) \\\\\n&\\qquad \\times (w(x)-w(y))^{p-1}\\phi^{p}(y) + c\\,\\sigma^{1-p}(M-w(x))|\\phi(x)-\\phi(y)|^{p} \\\\[1ex]\n&=: \\Psi_1(x,y) + \\Psi_2(x,y).\n\\end{align*}\n\nWe estimate $\\Psi_1$ separately in the cases $w(x)>2w(y)$ and $w(x) \\leq 2w(y)$.\nWhen $w(x)>2w(y)$, we obtain\n\\begin{align*}\n\\Psi_1(x,y) &\\leq \\left(\\frac{w(x)-w(y)}{w(y)}\\right)^{p-1}\\left(2^{1-p}(M-w(x))-(M-w(y))+c\\,\\sigma M\\right)\\phi^{p}(y) \\\\[1ex]\n&\\leq \\left(\\frac{w(x)-w(y)}{w(y)}\\right)^{p-1}\\left((2^{-1}-2^{1-p})w(x)-(1-2^{1-p})M+c\\,\\sigma M\\right)\\phi^{p}(y).\n\\end{align*}\nIf $p \\geq 2$, then\n\\begin{align*}\n(2^{-1}-2^{1-p})w(x)-(1-2^{1-p})M \\leq (2^{-1}-2^{1-p})M-(1-2^{1-p})M = -\\frac12 M.\n\\end{align*}\nIf $1>wy}\n\\Psi_1(x,y) &\\leq -\\frac1c M\\left(\\frac{w(x)-w(y)}{w(y)}\\right)^{p-1}\\phi^{p}(y).\n\\end{align}\n\nWhen $w(x) \\leq 2w(y)$, we can estimate\n\\begin{align*}\n\\Psi_1(x,y) &\\leq \\left(\\frac{w(x)\\left((M-w(x))w(y)^{p-1}-(M-w(y))w(x)^{p-1}\\right)}{w(y)^{p-1}(w(x)-w(y))}+c\\,\\sigma M\\right) \\\\*\n&\\qquad \\times \\left(\\frac{w(x)-w(y)}{w(x)}\\right)^{p}\\phi^{p}(y).\n\\end{align*}\nIf $w(x)wy}\n\\Psi_1(x,y) &\\leq -\\frac1c M \\left(\\frac{w(x)-w(y)}{w(x)}\\right)^{p}\\phi^{p}(y)\n\\end{align}\nwhen choosing $\\sigma$ small enough. If $w(x) \\geq M\/2$, in turn, then\n\\begin{align*}\n&w(x)\\left((M-w(x))w(y)^{p-1}-(M-w(y))w(x)^{p-1}\\right) \\\\[1ex]\n&\\qquad\\qquad\\qquad \\leq w(x)\\left((M-w(x))w(y)^{p-1}-(M-w(y))w(y)^{p-1}\\right) \\\\[1ex]\n&\\qquad\\qquad\\qquad \\leq -\\frac12 M(w(x)-w(y))w(y)^{p-1},\n\\end{align*}\nand again we obtain \\eqref{psi1wx>wy} when choosing $\\sigma$ small enough.\n\nLet us then estimate further to get logarithms visible. In the case $w(x)>2w(y)$, it holds\n\\begin{align} \\label{logwx>>wy}\n\\left(\\log\\frac{w(x)}{w(y)}\\right)^{p} \\leq c\\left(\\frac{w(x)-w(y)}{w(y)}\\right)^{p-1}\n\\end{align}\nsince $(\\log t)^{p} \\leq c\\,(t-1)^{p-1}$ when $t>2$. In the case $w(x) \\leq 2w(y)$, in turn, it holds\n\\begin{align} \\label{logwx>wy}\n\\nonumber \\left(\\log\\frac{w(x)}{w(y)}\\right)^{p} &= \\left(\\log\\left(1+\\frac{w(x)-w(y)}{w(y)}\\right)\\right)^{p} \\\\*[1ex]\n&\\leq \\left(\\frac{w(x)-w(y)}{w(y)}\\right)^{p}\n\\ \\leq \\ 2^{p}\\left(\\frac{w(x)-w(y)}{w(x)}\\right)^{p}\n\\end{align}\nsince $\\log(1+t) \\leq t$ when $t \\geq 0$. Thus, combining \\eqref{psi1wx>>wy} with \\eqref{logwx>>wy} and \\eqref{psi1wx>wy} with \\eqref{logwx>wy}, we obtain\n\\begin{align}\n\\Psi_1(x,y) &\\leq -\\frac1c M \\left(\\log\\frac{w(x)}{w(y)}\\right)^{p}\\phi^{p}(y).\n\\end{align}\n\nFor $\\Psi_2$ we easily get\n\\begin{align*}\n\\Psi_2(x,y) &\\leq c\\,M |\\phi(x)-\\phi(y)|^{p} \\leq c\\,M r^{-p}|x-y|^{p}.\n\\end{align*}\nIn the case $w(x) k_+ \\geq \\max\\bigg\\{\\esssup_{B_R} g, \\esssup_{B_R \\cap \\Omega} h\\bigg\\} \\quad \\text{and} \\quad -\\infty < k_- \\leq \\essinf_{B_R} g.\n\\]\nThen the functions\n\\[\nw_\\pm:=\\esssup_{B_R}(u-k_\\pm)_\\pm-(u-k_\\pm)_\\pm+\\varepsilon\n\\]\nsatisfy the following estimate\n\\begin{align} \\label{log lemma claim u}\n&\\int_{B_{r}}\\int_{B_{r}}\\left|\\log\\frac{w_\\pm(x)}{w_\\pm(y)}\\right|^{p}K(x,y)\\,{\\rm d}x{\\rm d}y \\nonumber \\\\[1ex]\n&\\qquad \\leq c\\,r^{n-sp}\\left(1+\\varepsilon^{1-p}\\left(\\frac rR\\right)^{sp}{\\rm Tail}((w_\\pm)_-,x_0,R)^{p-1}\\right)\n\\end{align}\nfor every $\\varepsilon>0$.\n\\end{lemma}\n\\begin{proof}\nLet $\\varepsilon>0$ and denote $H_\\pm := \\esssup_{B_R}(u-k_\\pm)_\\pm+\\varepsilon$. Notice that $H_\\pm$ is finite by Theorem \\ref{thm:boundednessx0}.\nLet $\\phi \\in C^{\\infty}_0(B_{3r\/2})$ be such that $0\\leq\\phi\\leq 1$, $\\phi \\equiv 1$ in $B_{r}$ and $|D\\phi|k_+$, we simply have $-L(u(x),u(y))=L(w_+(x),w_+(y))$, and consequently\n\\begin{align} \\label{-Luxuyk_+\\geq u(y)$, then $w_+(y)=H_+$ and\n\\[\n-(u(x)-u(y))=-(H_+ -w_+(x) +k_+-u(y)) \\leq w_+(x)-w_+(y),\n\\]\nand \\eqref{-Luxuyk_+\\geq u(x)$, we can just exchange the roles of $x$ and $y$ to obtain \\eqref{-Luxuy0$ such that for every $x_0 \\in \\partial\\Omega$\n\\begin{equation} \\label{eq:dens cond}\n\\inf_{00$ and $\\delta_\\Omega>0$,\nand let $B\\equiv B_r(x_0)$ with $x_0 \\in \\partial\\Omega$ and $r \\in (0,r_0)$.\nSuppose that $f \\in W^{s,p}(B)$ and $f=0$ in $B \\setminus \\Omega$. Then\n\\begin{equation}\n\\mean{B}|f|^{p}\\,{\\rm d}x \\leq c\\left(1-(1-\\delta_\\Omega)^{1-1\/p}\\right)^{-p}r^{sp}\\int_B\\mean{B}\\frac{|f(x)-f(y)|^{p}}{|x-y|^{n+sp}}\\,{\\rm d}x{\\rm d}y.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nSince $f=0$ in $B\\setminus\\Omega$,\n\\begin{align*}\n|f_B| &\\leq \\frac{|B\\cap\\Omega|}{|B|}\\mean{B\\cap\\Omega}|f|\\,{\\rm d}x \\leq \\frac{|B\\cap\\Omega|}{|B|}\\left(\\mean{B\\cap\\Omega}|f|^{p}\\,{\\rm d}x\\right)^{1\/p} \\\\[1ex]\n&\\leq\\left(\\frac{|B\\cap\\Omega|}{|B|}\\right)^{1-1\/p}\\left(\\mean{B}|f|^{p}\\,{\\rm d}x\\right)^{1\/p}\n\\ = \\ (1-\\delta_\\Omega)^{1-1\/p}\\left(\\mean{B}|f|^{p}\\,{\\rm d}x\\right)^{1\/p},\n\\end{align*}\nand we can estimate\n\\begin{align*}\n\\left(\\mean{B}|f|^{p}\\,{\\rm d}x\\right)^{1\/p} &\\leq \\left(\\mean{B}|f-f_B|^{p}\\,{\\rm d}x\\right)^{1\/p} + |f_B| \\\\[1ex]\n&\\leq \\left(\\mean{B}|f-f_B|^{p}\\,{\\rm d}x\\right)^{1\/p} + (1-\\delta_\\Omega)^{1-1\/p}\\left(\\mean{B}|f|^{p}\\,{\\rm d}x\\right)^{1\/p}.\n\\end{align*}\nAbsorbing the last term yields\n\\begin{align*}\n\\mean{B}|f|^{p}\\,{\\rm d}x \\leq \\left(1-(1-\\delta_\\Omega)^{1-1\/p}\\right)^{-p}\\mean{B}|f-f_B|^{p}\\,{\\rm d}x,\n\\end{align*}\nand the claim follows from the fractional Poincar\\'e inequality.\n\\end{proof}\n\n\n\\begin{lemma} \\label{lemma:osc reduction}\nAssume that $x_0 = 0 \\in \\partial \\Omega$ and $g(0)=0$, where $\\Omega$ satisfies \\eqref{eq:dens cond} for all $r\\leq R$. Let $\\omega > 0$. There exist $\\tau_0 \\in (0,1)$, $\\sigma \\in (0,1)$ and $\\theta \\in (0,1)$, all depending only on $n$, $p$, $s$ and $\\delta_\\Omega$, such that\nif\n\\begin{equation} \\label{oscB}\n\\osc_{B_R(0)}u + \\sigma{\\rm Tail}(u;0,R) \\leq \\omega \\quad \\text{and} \\quad \\osc_{B_R(0)}g \\leq \\frac\\omega 8\n\\end{equation}\nhold, then the decay estimate \n\\begin{equation} \\label{osctauB}\n\\osc_{B_{\\tau R}(0)}u + \\sigma{\\rm Tail}(u;0,\\tau R) \\leq (1-\\theta)\\omega\n\\end{equation}\nholds as well for every $\\tau \\in (0,\\tau_0]$.\n\\end{lemma}\n\\begin{proof}\nDenote $H=\\theta\/\\sigma$ and $B\\equiv B_R(0)$. We begin by estimating the tail term to obtain\n\\begin{align*}\n\\sigma^{p-1}{\\rm Tail}(u;0,\\tau R)^{p-1} &= \\sigma^{p-1}(\\tau R)^{sp}\\int_{B \\setminus \\tau B}\\frac{|u(x)|^{p-1}}{|x|^{n+sp}}\\,{\\rm d}x \\\\\n& \\quad + \\sigma^{p-1}\\tau^{sp}{\\rm Tail}(u;0,R)^{p-1} \\\\[1ex]\n&\\leq c\\,\\sigma^{p-1}\\omega^{p-1}+\\tau^{sp}\\omega^{p-1}\n\\end{align*}\nby \\eqref{oscB}. Consequently,\n\\begin{equation} \\label{eq:Tail bndr osc 000} \n\\sigma{\\rm Tail}(u;0,\\tau R) \\leq \\tilde c \\left(\\frac\\theta H + \\tau^{sp\/(p-1)}\\right)\\omega\n \\leq \\frac{2\\tilde c\\,\\theta} H \\omega = \\theta\\omega\n\\end{equation}\nwhen restricting $\\tau_0 \\leq \\sigma^{(p-1)\/(sp)}$ and choosing $H = 2\\tilde c \\geq 1$, where $\\tilde c \\equiv \\tilde c(n,p,s)$.\nThus, it suffices to prove that\n\\begin{equation} \\label{osctauB2}\n\\osc_{\\tau B}u \\leq (1-2\\theta)\\omega\n\\end{equation}\nfor all $\\tau \\leq \\tau_0$. To this end, let\n\\[\nk_+ := \\sup_{B} u- \\frac{\\omega}{4}, \\quad k_- := \\inf_{B}u + \\frac{\\omega}{4} , \\quad \\varepsilon:=\\theta\\omega\n\\]\nand\n\\[\nw_\\pm:=\\sup_{B}(u-k_\\pm)_\\pm-(u-k_\\pm)_\\pm+\\varepsilon, \\quad \\tilde w_\\pm:=\\frac{w_\\pm}{\\sup_{B} w_\\pm}.\n\\] \n\nWe may assume $\\sup_{B}u \\geq \\frac38\\omega$ or $\\inf_{B}u \\leq -\\frac38\\omega$ since otherwise\n$\\osc_{\\tau B}u \\leq \\osc_{B}u \\leq \\frac34\\omega$ and there is nothing to prove if we assume that $\\theta \\leq 1\/8$.\nWe consider the case $\\sup_{B}u \\geq \\frac38\\omega$; the case $\\inf_{B}u \\leq -\\frac38\\omega$ is symmetric.\nNotice that we have $\\tilde w_+ = 1$ in $B\\setminus \\Omega$ due to the condition $u=g \\leq \\omega\/8$ in $B\\setminus \\Omega$. First, we estimate, by Lemmas \\ref{lemma:dens cond} and \\ref{log lemma u} with $r\\equiv 2\\tau R$ and \\eqref{oscB} when restricting $\\tau_0 \\leq 1\/4$ and $\\tau_0 \\leq \\sigma^{2(p-1)\/(sp)}$, to obtain \n\\begin{align*}\n\\mean{2\\tau B}\\left|\\log \\tilde w_+\\right|^{p}\\,{\\rm d}x &\\leq c\\,(\\tau R)^{sp}\\int_{2\\tau B}\\mean{2\\tau B}\\left|\\log\\frac{\\tilde w_+(x)}{\\tilde w_+(y)}\\right|^{p}K(x,y)\\,{\\rm d}x{\\rm d}y \\\\[1ex]\n&\\leq c\\,(\\tau R)^{sp}\\int_{2\\tau B}\\mean{2\\tau B}\\left|\\log\\frac{w_+(x)}{w_+(y)}\\right|^{p}K(x,y)\\,{\\rm d}x{\\rm d}y \\\\[1ex]\n&\\leq c\\,\\Big(1+(\\theta\\omega)^{1-p}\\tau^{sp}{\\rm Tail}((w_+)_-;0,R)^{p-1}\\Big) \\\\[1ex]\n&\\leq c\\left(1+(\\theta\\omega)^{1-p}\\sigma^{2(p-1)}{\\rm Tail}(u;0,R)^{p-1}\\right) \\\\[1ex]\n&\\leq c\\left(1+(\\theta\\omega)^{1-p}\\left(\\frac\\theta H\\right)^{p-1}\\omega^{p-1}\\right) \\\\[1ex]\n&\\leq c.\n\\end{align*}\nConsequently, by Chebyshev's Inequality we have\n\\begin{align} \\label{logtildew>M} \\nonumber\n\\frac{\\left|2\\tau B \\cap \\{|\\log \\tilde w_+| \\geq \\left|\\log(20\\,\\theta)\\right| \\}\\right|}{|2\\tau B|} &\\leq |\\log (20\\,\\theta)|^{-p}\\mean{2\\tau B}|\\log\\tilde w_+|^{p}\\,{\\rm d}x \n\\\\*[1ex]\n &\\leq c\\,|\\log (20\\,\\theta)|^{-p}.\n\\end{align}\n\nLet us estimate the left-hand side of \\eqref{logtildew>M}. Since, by definitions, $0 < \\tilde w_+ \\leq 1$ and $\\sup_{B} (u-k_+)_+ = \\omega\/4$, we have that\n\\begin{align*}\n\\left\\{|\\log \\tilde w_+| \\geq|\\log (20\\,\\theta)| \\right\\} &= \\left\\{\\tilde w_+ \\leq 20\\,\\theta\\right\\} \n\\\\*[1ex]\n & = \\Big\\{\\frac{\\omega}{4} + \\varepsilon - (u-k_+)_+ \\leq 20\\,\\theta \\Big(\\frac{\\omega}{4} + \\varepsilon\\Big) \\Big\\}\n\\\\*[1ex]\n & = \\Big\\{\\frac{\\omega}{4} + \\theta\\omega - u + \\sup_B u - \\frac{\\omega}{4} \\leq 5 \\theta \\omega + 20\\,\\theta^2 \\omega \\Big\\}\n\\\\*[1ex]\n & \\supset \\Big\\{u \\geq \\sup_B u - 4 \\theta \\omega \\Big\\}\n\\end{align*}\nprovided that $\\theta < 1\/20$. Consequently, by defining $\\tilde k \\equiv \\tilde k_+ := \\sup_B u - 4 \\theta \\omega$ and using the above two displays, we get\n\\begin{align*}\n\\left( \\mean{2\\tau B} (u - \\tilde k )_+^p \\,{\\rm d}x \\right)^{1\/p} & \\leq 4 \\theta \\omega \\left( \\frac{ |2\\tau B \\cap \\{ u \\geq \\sup_B u - 4 \\theta \\omega \\}| }{|2\\tau B| }\\right)^{1\/p} \\\\[1ex]\n&\\leq 4 \\theta \\omega \\left( \\frac{ |2\\tau B \\cap \\{ |\\log \\tilde w_+| \\geq|\\log (20\\,\\theta)| \\}| }{|2\\tau B| }\\right)^{1\/p} \\\\[1ex]\n&\\leq \\frac{c\\,\\theta\\omega}{|\\log(20\\,\\theta)|}.\n\\end{align*}\n\nSince $\\tilde k \\geq \\sup_B g$, we have by Theorem \\ref{thm:boundednessx0} that\n\\begin{equation*} \n\\sup_{\\tau B} (u - \\tilde k )_+ \\leq \n\\delta \\, {\\rm Tail}((u - \\tilde k )_+ ; 0, \\tau R)+ c\\,\\delta^{-\\gamma} \\left( \\mean{2\\tau B} (u - \\tilde k )_+^p \\,{\\rm d}x \\right)^{1\/p}\n\\end{equation*}\nfor any $\\delta \\in (0,1]$, and hence\n\\begin{equation} \\label{suputauB0}\n\\sup_{\\tau B} u \\leq \\sup_B u - 4 \\theta\\omega + \\delta \\, {\\rm Tail}((u - \\tilde k )_+ ; 0, \\tau R) + \\frac{c\\,\\delta^{-\\gamma }}{|\\log(20\\,\\theta)|} \\theta\\omega.\n\\end{equation}\nTo estimate the tail term, we proceed similarly as in~\\eqref{eq:Tail bndr osc 000} and obtain\n\\begin{align*}\n{\\rm Tail}((u - \\tilde k )_+; 0,\\tau R)^{p-1} &\\leq (\\tau R)^{sp} \\int_{B \\setminus \\tau B} (u(x) - \\tilde k )_+^{p-1} |x|^{-n-sp}\\,{\\rm d}x \\\\\n&\\quad + \\tau^{sp} {\\rm Tail}(u;0,R)^{p-1} \\\\[1ex]\n&\\leq c\\,(\\theta\\omega)^{p-1} \\left( 1 + \\frac{\\tau^{sp}}{\\theta^{p-1}\\sigma^{p-1}} \\right) \n \\\\*[1ex]\n &\\leq c\\,(\\theta\\omega)^{p-1},\n\\end{align*}\nwhere we also used the facts $(u-\\tilde k)_+ \\leq 4\\theta\\omega$ in $B$, ${\\rm Tail}(u;0,R) \\leq \\omega\/\\sigma$ by \\eqref{oscB}, and $\\tau^{sp} \\leq \\tau_0^{sp} \\leq \\theta^{p-1}\\sigma^{p-1}$. Thus, by choosing first $\\delta$ small and then $\\theta$ small accordingly, we deduce from \\eqref{suputauB0} that \n\\[\n\\sup_{\\tau B} u \\leq \\sup_B u - 2 \\theta\\omega,\n\\]\nand \\eqref{osctauB2} follows, as desired. This finishes the proof.\n\\end{proof}\n \nNow, we have finally collected all the machinery to plainly deduce the H\\\"older continuity up the boundary. We have the following\n\\begin{theorem} \\label{thm:H cont bdry}\nSuppose that $u$ solves the obstacle problem in $\\mathcal K_{g,h}(\\Omega,\\Omega')$ and assume $x_0\\in \\partial\\Omega$ and $B_{2R}(x_0) \\subset \\Omega'$.\nIf $g \\in \\mathcal K_{g,h}(\\Omega,\\Omega')$ is H\\\"older continuous in $B_R(x_0)$ and $\\Omega$ satisfies \\eqref{eq:dens cond} for all $r\\leq R$, then $u$ is H\\\"older continuous in $B_R(x_0)$ as well.\n\\end{theorem}\n\\begin{proof}\nWe may assume $x_0=0$ and $g(0)=0$.\nMoreover, we may choose $R_0$ such that $\\osc_{B_0}g \\leq \\osc_{B_0}u$ for $B_0 \\equiv B_{R_0}(0)$ since otherwise we have nothing to prove, and define\n\\begin{equation} \\label{omega0}\n\\omega_0 := 8\\left(\\osc_{B_0}u + {\\rm Tail}(u;0,R_0)\\right).\n\\end{equation}\nBy Lemma \\ref{lemma:osc reduction} there exist $\\tau_0$, $\\sigma$ and $\\theta$ depending only on $n$, $p$, $s$ and $\\delta_\\Omega$ such that if\n\\begin{equation} \\label{oscB0}\n\\osc_{B_r(0)}u + \\sigma{\\rm Tail}(u;0,r) \\leq \\omega \\quad \\text{and} \\quad \\osc_{B_r(0)}g \\leq \\frac\\omega 8\n\\end{equation}\nhold for a ball $B_r(0)$ and for $\\omega>0$, then\n\\begin{equation} \\label{osctauB0}\n\\osc_{B_{\\tau r}(0)}u + \\sigma{\\rm Tail}(u;0,\\tau r) \\leq (1-\\theta)\\omega\n\\end{equation}\nholds for every $\\tau \\in (0,\\tau_0]$. As we can take $\\tau \\leq \\tau_0$ such that\n\\begin{equation} \\label{osctauBg}\n\\osc_{\\tau^{j}B_0}g \\leq (1-\\theta)^{j}\\frac{\\omega_0}{8} \\qquad \\text{for every } j=0,1,\\dots.\n\\end{equation}\nNow, iterating \\eqref{osctauB0} with \\eqref{oscB0} and \\eqref{osctauBg} noticing also that the initial condition is satisfied by \\eqref{omega0}, we obtain\n\\begin{equation*} \\label{osctauBu}\n\\osc_{\\tau^{j}B_0}u \\leq (1-\\theta)^{j}\\omega_0 \\qquad \\text{for every } j=0,1,\\dots.\n\\end{equation*}\nConsequently, $u \\in C^{0,\\alpha}(B_0)$ with the exponent $\\alpha=\\log(1-\\theta)\/\\log \\tau \\in (0,1)$.\n\\end{proof}\n\nSlightly modifying the proof above, we easily obtain the following.\n\\begin{theorem} \\label{thm:cont bdry}\nSuppose that $u$ solves the obstacle problem in $\\mathcal K_{g,h}(\\Omega,\\Omega')$ and assume $x_0\\in \\partial\\Omega$ and $B_{2R}(x_0) \\subset \\Omega'$.\nIf $g \\in \\mathcal K_{g,h}(\\Omega,\\Omega')$ is continuous in $B_R(x_0)$ and $\\Omega$ satisfies \\eqref{eq:dens cond} for all $r\\leq R$, then $u$ is continuous in $B_R(x_0)$ as well.\n\\end{theorem}\n\nFor the sake of completeness, we gather our continuity results into two global theorems. The first one follows by combining Theorems \\ref{thm:obs H cont} and \\ref{thm:H cont bdry} and the second one by combining Theorems \\ref{thm:obs cont} and \\ref{thm:cont bdry}.\n\\begin{theorem} \\label{thm:H cont up to bdry}\nSuppose that $\\Omega$ satisfies \\eqref{eq:dens cond} \nand $g \\in \\mathcal K_{g,h}(\\Omega,\\Omega')$. Let $u$ solve the obstacle problem in $\\mathcal K_{g,h}(\\Omega,\\Omega')$. \nIf $g$ is locally H\\\"older continuous in $\\Omega'$ and $h$ is locally H\\\"older continuous in $\\Omega$, then $u$ is locally H\\\"older continuous in $\\Omega'$.\n\\end{theorem}\n\n\\begin{theorem} \\label{thm:cont up to bdry}\nSuppose that $\\Omega$ satisfies \\eqref{eq:dens cond} \nand $g \\in \\mathcal K_{g,h}(\\Omega,\\Omega')$. Let $u$ solve the obstacle problem in $\\mathcal K_{g,h}(\\Omega,\\Omega')$.\nIf $g$ is continuous in $\\Omega'$ and $h$ is continuous in $\\Omega$, then $u$ is continuous in $\\Omega'$.\n\\end{theorem}\n\n\n\n\\smallskip\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nHelioseismology has probed the interior of the Sun over the last three decades. Combining the information provided by \nseveral hundred pressure-driven modes (p modes), it has been possible to put constraints on our knowledge of the \nstructure and the dynamics of the solar interior \\citep{JCD2002,ThoJCD2003}. Unfortunately, only a small fraction of \nthese modes reaches the solar core. Moreover, due to the increase in sound-speed velocity with depth, these modes \ngive little information on the deeper layers as they spend less time there than in the convection zone. Let us take as an \nexample the internal rotation rate of the Sun: the rotation profile is very well known in the convective zone \n\\citep{ThoTOO1996,1998ApJ...505..390S,HowJCD2000,2000ApJ...541..442A}, while the uncertainties grow in the radiative \nzone and towards the core of the Sun \\citep{1994ApJ...435..874J,ElsHow1995,CouGar2003,ChaSek2004,GarCor2004}. In order \nto put new constraints inside the solar core other kinds of modes are needed: the gravity (g) modes. For example, by \nmeasuring just a few of such modes, information on the core rotation rate can undoubtably be obtained \n\\citep{2008A&A...484..517M}, whereas the deepest layers that could be probed using only p-modes are around \n0.2 $R_{\\odot}$ \\citep{2008SoPh..tmp...43G}.\n \nGravity (g) modes are buoyancy-driven modes that have the advantage of propagating inside the entire radiative \nregion. However, these waves become evanescent in the convective zone and reach the solar surface with tiny \namplitudes preventing us to detect them easily (see for example \\citet{Belkacem08} and references therein). Indeed, \nseveral claims for g-mode detections have been made in the past \\citep{1983Natur.306..651D,1988IAUS..123...79P,1995Natur.376..139T}; \nhowever, modern and better data sets cannot confirm them. \n\nIn 1995 was launched the Solar and Heliospheric Observatory (SoHO), one of whose scientific objectives \nwas the detection and characterization of gravity modes \\citep{DomFle1995}. Recently, using data from the Global \nOscillation and Low Frequency (GOLF) instrument \\citep{GabGre1995}, the signature of the asymptotic properties of \n $\\ell$=1 dipole g modes has been measured with a high confidence level \\citep{2007Sci...316.1591G,2008AN....329..476G}. \nThis signal was also found \\citep{2006ESASP.624E..23G} using photometric data from the Variability of solar \nIRradiance and Gravity Oscillations (VIRGO) experiment \\citep{1995SoPh..162..101F}. Even if certain constraints \ncan be imposed on the structure \\citep{2008SoPh..tmp...55G} and dynamics \\citep{2007Sci...316.1591G} of the \nsolar core thanks to the study of these asymptotic properties, it is extremely important to detect individual \ng modes. After 10 years of observations, the level of noise at 200 $\\mu$Hz has been established at \n$\\sim$ 4.5 mm\/s when individual peaks are looked for and at 1.5 mm\/s when this research is done for multiplets \n\\citep{2006soho...18E..22E}. Indeed some peaks and patterns could be identified as potential gravity modes or\n mixed modes above the noise level \\citep{GabBau2002,STC04,2007ApJ...668..594M} but it has been impossible \n to tag them unambiguously with the correct $\\ell$, $m$ and $n$.\n\nIn this paper we analyse a peak around 220.7 $\\mu$Hz that has been studied several times as part of a g-mode \ncandidate using different instruments on board SoHO (see for example \\citet{Gab99,Fin01,STC04,2007ApJ...668..594M}) \nbut also from the theoretical side (see for example \\citet{CoxGuz2004} and references therein). To do so, we \nstart in section 2 with a brief description of the helioseismic instruments used in this work and we analyse in \ndetail the data of the VIRGO\/SPM instruments (section 3). In section 4, we look for an instrumental origin\n for this peak without success by analysing all the housekeeping parameters of the VIRGO package as well as \n the SoHO pointing. Once it is stablished that this peak seems to have a solar origin we check for its\n presence \n in all the other instruments of the VIRGO package (section 5) and in the velocity instruments GOLF, \n MDI and GONG (section 6). We then finish by discussing its possible nature.\n\n\\section{Instrumentation and data analysis}\n\nThe data of VIRGO (Variability of IRradiance and Gravity Oscillation), GOLF (Gravity Oscillation at Low \nFrequencies) and MDI (Michelson Doppler Imager) on board SoHO (Solar and Heliospheric Observatory) satellite have \nbeen used in this research together with the GONG (Global Oscillation Network Group) ground-based netwok.\n \n\\subsection{SoHO\/VIRGO}\n\nThe VIRGO package was designed to study the characteristics of pressure and internal gravity modes by \nobserving irradiance and radiance variations, to measure the solar total and spectral irradiance and to quantify \ntheir variability (Fr\\\"{o}hlich et al.\\ 1995, 1997). It is composed of three different types of sensors:\n\\begin{itemize}\n\\item Two types of absolute radiometers (one VIRGO\/DIARAD and two VIRGO\/PMO6-V) for the measurements of solar \ntotal irradiance and its variations with high accuracy and precision. The cadences of VIRGO\/DIARAD and VIRGO\/PMO6-V are 180s and 60s respectively.\n\\item Two 3-channel sunphotometers (SPM), one permanently exposed to sun lght and another for backup, set at at 402 nm (blue), 500 nm (green), \nand 862 nm (red), looking at the Sun as a star with a 60 s cadence. \nThe bandwidth of the filters is 5~nm.\n\\item One Luminosity Oscillation Imager (VIRGO\/LOI) for the measurements of the radiance in 12 pixels over the \nsolar disc. The filter is at 500 nm with a bandwidth of 5 nm.The cadence is 60s.\n\\end{itemize}\n\nIn 1998 June, {\\itshape SoHO} was lost\n for several months, but, after a search campaign, was\nfinally found and resumed operations around 1998 October. The VIRGO data after {\\itshape SoHO}'s \n``vacations'' show the same high quality as before the temporary loss of the probe.\n\n\\subsection{SoHO\/GOLF, SoHO\/SOI\/MDI and GONG}\n\n\\begin{itemize}\n\\item\nSoHO\/GOLF is a resonance scattering spectrophotometer (Gabriel et al.\\ 1995, 1997)\n that measures the line-of-sight velocity using the sodium\ndoublet, similar to the IRIS and BiSON ground-based networks. The\nGOLF window was opened in 1996 January and became fully\noperative by the end of that month. Over the following months,\noccasional malfunctions in its rotating polarizing elements were\nnoticed that led to the decision to stop them in a predetermined\nposition; truly non-stop observations began by 1996 mid-April.\nSince then, GOLF has been continuously and satisfactorily operating\nin a mode unforeseen before launch, showing fewer\nlimitations than anticipated. The signal, then, consists of two close\nmonochromatic photometric measurements in a very narrow band (25\nm\\AA) on a single wing of the sodium doublet.\nThis signal has been calibrated into velocity \\citep{UlrGar2000,garcia05} and is indeed similar in nature\nto other known velocity measurements, such as those of IRIS and BiSON \n(Pall\\' e et al.\\ 1999). The sampling of the\nGOLF data used in this paper is 60 s. Before {\\itshape SoHO}'s vacations (1998 June),\nGOLF data were obtained in the blue wing of the sodium line; thus, after the {\\itshape \nSoHO}\nvacations the GOLF team decided change to the red wing of the sodium line (see Garc\\'\\i a et al. 2005, for the latest report on the GOLF instrument).\n\n\\item The Solar Oscillations Investigation (SOI) uses a Michelson Doppler Imager (MDI) type of instrument \n\\citep{1995SoPh..162..129S}. MDI consists of a pair of tunable Michelson interferometers,\nwhich image the Sun onto a 1024 pixel x 1024 pixel\nCCD camera in five wavelengths across the Ni i 676.8 nm line. These resolved data can be processed by\nforming a weighted combination of the pixel signals to yield a\nproxy for a Sun-as-a-star response (see \\cite{Henney99}).\n\n\\item The ground-based Global Oscillation Network Group (GONG;\nHarvey et al. 1996) consists of six sites, with instruments that use\nthe Fourier tachometer approach to observe the Doppler shift, in\nthe Ni i line, with 1024 pixel resolution. Here, a\nSun-as-a-star proxy was formed from a simple integration over\nall pixels with a cadence of 60s.\n\\end{itemize}\n\n\\section{The 220.7 $\\mu$Hz peak seen in VIRGO\/SPM}\nA long time series of 4098 days of VIRGO\/SPM data has been used in this work starting on 1996 April 11. \nAs our purpose is to study the time evolution of signals at low frequency, a total of five independent subseries \nof 800 days have been computed. Data were available to allow a 50 day shift up to June 2007. These 66 \noverlapped series were used only for plots while non-overlapping data were used as input for the statistical tests we have carried on. \n\nThe slow trends in the time series, due to the degradation in the instruments and long-term solar variability,\n have been removed by applying a running mean filter of one day. To check whether this filter could affect the detected \n signal, we have also used a backwards difference filter, in which every measured point is substituted by the \n difference of two consecutive points $\\delta f_{n}=f_{n+1}-f_{n}$. To recover the correct amplitudes in the \n power spectrum, this latter should be divided by the transfer function of the filter $Q(\\nu)$, defined as follows \\citep{GarBal08}: \n\\begin{equation}\nQ(\\nu)=[2sin(\\pi\\nu\\Delta t)]^2\n\\end{equation}\n $\\nu$ being the frequency and $\\Delta t$ the sampling of the data.\n\nBoth filters gave the same results and, in the rest of this paper, we work only with the time series filtered by the 1-day running mean.\n\nWe therefore computed several power density spectra using a Fast Fourier Transform (FFT) algorithm and\n built the time-evolution power diagrams used in this work. Each of them has been computed from the \n time series which have been extended by four equal time intervals of zero signal . This oversampling makes it easier to detect the bins in which the \n power is concentrated \\citep{GabBau2002}. We also verified that a sine wave fit (SWF), computed \n in steps of 0.0001 $\\mu$Hz between 220.5 and 221 $muHz$, yields the same results. Therefore, we have used the \n normal zero-padded FFT as it is much faster than the SWF. \n\nThe time-evolution power diagrams are built as follows: The 66 power spectra of the overlapped time series are \ncomputed and plotted vertically using a colour scale for the power. The vertical axis is the frequency of the \npower spectra with the colour equivalent to the power as indicated on the right-hand side of the diagram and the \nhorizontal axis the number of the time series from 0 to 65, i.e.\\ the time span corresponding to the \ntime series. Looking these time-evolution power diagrams we know at which frequency and for how long \n a signal can have enough power to be observed above the noise level.\n \n In figure~\\ref{spms} the time-evolution of the three VIRGO\/SPM channels are shown for the frequency range\n 220.5--221.0 $\\mu$Hz. The x-axis spans 11 years of the SoHO mission. A clear signal is observed \n in the blue channel (top) at around 220.7 $\\mu$Hz, which is stable in time with power that goes from 6--7 to \n 16--17 $ppm^2\/\\mu$Hz. Around time series 60 this signal seems to change its frequency slightly by around 0.3 \n $\\mu$Hz. In the green and red channels the same continuous signal is visible as in the blue one but with \n the expected decrease in power with wavelength. It is also important to note that in all these VIRGO\/SPM channels \n a second high-amplitude signal is visible at $\\sim$ 220.64 $\\mu$Hz, parallel to the previous one, from time \n series 20 until the last one but with a small gap between time series 44 and 48.\n\n \n \n\\begin{figure}[!htb]\n\\centering\n\\begin{tabular}{c}\n\t\\includegraphics[width=13pc,angle =90]{f1.eps} \\\\\n \t\\includegraphics[width=13pc,angle =90]{f2.eps} \\\\\n\t\\includegraphics[width=13pc,angle =90]{f3.eps} \\\\\n\\end{tabular}\n\\caption{\\label{spms}VIRGO\/SPM time-evolution power diagram of channels blue, green and red (top to bottom \nrespectively) from April 1996 to June 2007. A stable signal at 220.7 $\\mu$Hz is clearly observed. }\n\\end{figure} \n\n\n \n\\subsection{Confidence levels and Monte-Carlo simulations}\n\nIn the previous section we saw that there is a persistent signal around the target frequency of\n 220.7 $\\mu$Hz. Indeed, the VIRGO\/SPM blue channel power density spectrum of the full length time series \n (see Figure~\\ref{full}) shows the presence of a peak at a precise frequency of 220.667 $\\mu$Hz above the \n 90$\\%$ confidence level computed in a 10 $\\mu$Hz window following \\cite{App00}.\n\n\\begin{figure}[!htb]\n\\includegraphics[width=0.36\\textwidth,angle =90]{f4.eps}\n\\caption{\\label{full}VIRGO\/SPM blue channel power spectrum density computed with 4098-day time \nseries starting on 1996 April 11. The horizontal dotted line corresponds to the 90\\% confidence level \nthat a peak above this line would not be due to noise. }\n\\end{figure} \n\nUsing subseries of 800 days and a frequency window of 10 $\\mu$Hz, the power level above which an observed\n peak has a 90\\% probability not due to noise is 8.87$\\sigma$ (e.g. see \\cite{App00}). In the case of \n zero-padded data, the points are no longer independent and are correlated. Therefore, Monte-Carlo \n simulations should be used to derive a correction for the above-mentioned confidence level. In our case, \n for a padding factor of 5 we have added a correction of $\\mathrm{ln}(2.8)=1.03$ derived by \\cite{GabBau2002} \n to the threshold computed using non-zero-padded data. In this conditions the 90\\% confidence level at around \n 9.9$\\sigma$. Using the VIRGO\/SPM blue channel, we found that the peak we are studying has a maximum power in a\n range between 8.74 and 10.4$\\sigma$ considering only five independent realizations of 800 days. In Figure~\\ref{spms} \n the 90\\% limit is obtained at around 14 $ppm^2\/\\mu$Hz (orange colour in Figure~\\ref{spms}). This means that, for example,\n most subseries between the 28th and the 58th have the peak above the 90\\% confidence level, as well as other subseries \n such as those at the very beginning of the time-span. It is important to notice that in this case the $\\sigma$ has \n been averaged over the 66 time series and the value of 14 $ppm^2\/\\mu$Hz is an averaged magnitude. \n\nWe are interested in knowing the probability of having a signal with the same properties to those that we have found in \nthe VIRGO\/SPM blue channel; i.e.\\ a peak that is above the 90$\\%$ level in the full power density spectrum of more \nthan 4098 days, and that is also present in the five independent subseries of 800 days with similar levels to what we have with \nthis instrument (i.e.\\ not necessary all above a 90\\% confidence level in these individual small subseries but around that \nlevel). This latest condition would be much more restrictive because it means that the peak should maintain a certain \ncoherence during the full time-span. A Monte Carlo simulation of 1 million iterations has been done by simulating Gaussian \nnoise time series of 4000 days that have been cut into five intervals of 800 days. To speed up the procedure we have not computed \nthe full spectrum of the 4000 days but only the average of the power density spectrum of the five independent realizations of 800 \ndays (which have an SNR of $\\sim$ 9.2 $\\sigma$ in the VIRGO\/SPM blue channel). Thus, the algorithm looks first for a signal \nin the average spectrum with $\\sim$0.9 times the level found in the VIRGO\/SPM blue channel (8.3$\\sigma$) and, if it is \nfound, it looks for the presence of that signal in the five subseries (again with levels of 0.9 times those of VIRGO).\n Any signal with these properties found in the 10 $\\mu$Hz window will be flagged as a positive identification. The results \n show that, in a window of 10 $\\mu$Hz, the 220.7 $\\mu$Hz signal has a likelihood of 99.8$\\%$ (which is reduced to 91.3\\% if \n we only consider the constraint on the averaged spectrum). We have also checked how the likelihood is degraded when a bigger \n window is considered. Thus, for the 20 and 30 $\\mu$Hz windows we obtain 99.6 and 99.4\\% respectively.\n\nWe can conclude that it is extremely difficult to find a pure noise signal above the 90\\% confidence level after $\\sim$4000 days \nand with a coherence with time as found in the VIRGO\/SPM instruments.\n\n \\section{Possible instrumental origin of the signal inside VIRGO and SoHO}\n \nOnce this interesting signal has been detected in VIRGO\/SPM the main question is to investigate its origin; in other words, determine \nif it is of solar or instrumental origin. In this section, we study all the possible non-solar \norigins of this signal, from the orbital and pointing corrections of the spacecraft to the housekeeping parameters \n(hereafter HK) of the VIRGO package. \n\nPeriodic manoeuvring of the SoHO probe at this frequency (220.7 $\\mu$Hz, i.e. a period around 1.25 hours) due to orbital \nadjustments or pointing corrections could modulate the signal of the instruments on board as a tracking system can produce guided frequencies.\nOn the other hand, a temperature variation at this frequency could also modulate the observed signal. These temperature \nvariations could originate in the sensor itself or in other instrument subsystems.\n\nFor all these parameters we follow the same analysis we as for the VIRGO data; thus, we build the corresponding \ntime-evolution power diagrams and we compare them with the VIRGO\/SPM ones. If the 220.7 $\\mu$Hz signal is produced by the \ntemporal variations of some of these parameters, the time-evolution power diagrams of both VIRGO\/SPM and the parameter must \nbe highly correlated.\n\nFor this purpose we analyse in the following subsections orbital and pointing corrections of the SoHO spacecraft and the different \nHK parameters of the VIRGO package that might modulate the signal. It is important to note that some of the HK data \nare in the scientific telemetry of VIRGO and have a cadence of 60s, while others are in the HK telemetry and have a cadence of 180s. \n\n\n\\subsection {Orbital corrections}\n\nThe radial distance is reduced to 1AU by the usual quadratic law $S_{0}=S \\cdot r^2$, $r$ being the spacecraft-to-Sun distance in \nastronomical units. This correction normalizes the spectral irradiance to the solar constant definition and removes signal modulations \ndue to movements of the Earth, Moon and planets in their orbits.\n\nThe observed radiation $S$ of a moving blackbody source is \n\\begin{equation}\nS=S_{0} \\frac{(1-v)^2}{(1-v^2)} \n\\end{equation}\nwhere $S_{0}$ is the radiation in motionless conditions and\n v is the speed in units of the light speed, c. With SoHO velocity being a few $10^{-6}$ of the speed-of-light one can safely omit terms in \n $v^2$ and thus approximate the reciprocal formula\n\\begin{equation}\nS=S_{0} \\frac{(1-v)^2}{(1-v^2)}\\sim \\frac{S_{0}}{(1-2v)} \\sim S_{0} (1+2*v)\n\\end{equation}\nThis Doppler correction removes a tiny ($10^{-5}$), slow (Halo orbit period is 6 months) modulation of the measured irradiance.\n\nIn this way, the orbital correction applied to the three channels of VIRGO\/SPM is:\n\\begin{equation}\nSPM_{channel}=SPM_{channel} \\cdot radius^2 \\cdot (1+2 \\cdot vel)\n\\end{equation}\nwhere ``radius\" is the spacecraft-to-Sun distance in astronomical units and ``vel\" is the radial velocity in units of the speed-of-light.\n\n\\begin{figure}[!htb]\n\\centerline{%\n\\begin{tabular}{c@{\\hspace{1pc}}c}\n\\includegraphics[width=13pc,angle =90]{f5.eps}\n\\end{tabular}}\n\\caption{\\label{orbit}Time-evolution power diagram of the orbital corrections applied to the VIRGO data. In addition to the order of\n magnitude which is 6 orders of magnitude smaller than VIRGO\/SPM no correlation is found.}\n\\end{figure} \n\nThe orbital parameters (radius and vel) are provided by NASA in a 10 minute cadence and are linearly interpolated to get the \nsame 60s as VIRGO\/SPM. The time series of the orbital correction applied, i.e.\\ $radius^2(1+2\\cdot vel)$, has been analysed in the same \nway as VIRGO\/SPM and the resulting time-evolution power diagram diagram is shown in Figure~\\ref{orbit}. The orbital correction \nsignal is around 6 orders of magnitude smaller than the VIRGO\/SPM one and no correlation has been found.\n\n\n\\subsection{Spacecraft Pointing}\n\n\\begin{figure}[]\n\\centering\n\\begin{tabular}{c}\n\t\\includegraphics[width=12pc,angle =90]{f6.eps} \\\\\n \t\\includegraphics[width=12pc,angle =90]{f7.eps} \\\\\n\t\\includegraphics[width=12pc,angle =90]{f8.eps} \\\\\n\\end{tabular}\n\\caption{\\label{pointing}Power time-evolution diagrams of the pitch, yaw and roll angles (respectively from top to bottom) \nof the SoHO spacecraft. Only data up to September 2002 (time series 32) are available. No correlation with the first 32 time series of VIRGO\/SPM is found.}\n\\end{figure} \n\nThe three critical flight dynamics parameters are rotations in three dimensions around the vehicle's coordinate-system origin, the \ncentre of mass. These angles are pitch, roll and yaw. Pitch is the rotation around the lateral or transverse axis. Therefore, \nmovements of the spacecraft to the north or south of the Sun. Yaw is the rotation about the vertical axis; thus, movements of \nthe spacecraft to the west or east of the Sun and, finally, roll is a rotation around the longitudinal axis i.e.\\ movements of the \nspacecraft from the north or south to the west or east of the Sun.\n\nFor an instrument that looks at the Sun as a star (integrated light) the most plain pointing correction would be divided \nby $cos(\\sqrt(yaw^2 + pitch^2)$, i.e.\\ the cosine of the angle between instrument optical axis and the line-of-sight\ndirection. Nevertheless, this correction was never applied to VIRGO\/SPM because the correction would have been negligible. \nAlso, in December 2001, NASA discontinued the CDHF (Central Data Handling Facility), which was the facility in charge of processing, \nproducing and distributing the SoHO telemetry and the ancillary data products. The production of all these data were continued in\n others ways but the production of attitude data was stopped in September 2002. Indeed, when SoHO is in normal mode, the attitude\n follows nominal attitude well enough for most purposes, and because the roll determination had large errors (because of certain procedural problems).\n\nEven knowing that, it would be very unlikely that the pointing maneuvres could modulate any signal in the SoHO instruments; the three \nangles have been analysed and their time-evolution diagrams computed (see Figure~\\ref{pointing}).\n \nThe available attitude data concerning pitch, yaw and roll angles were obtained from the NASA archive from 1996 April 11 to 2002 September\n 22 and we built the time series of the three angles. These data have a cadence of 10 minutes and have been used with this sampling rate \n because it is good enough for our purposes. The length of these time series enables us to get 32 time series of 800 days (each shifted 50 days \n with respect to the previous one). This is approximately half of the time-evolution power diagrams used in the VIRGO\/SPM. This length is \n sufficient to see if any correlation exists between pointing and SPM signals during the common period (around 6 years).\n\nThe pitch angle has a very constant value of around -3.3 arcmin during the time span, with spikes of 5 arcmin and only a few of them \nwith higher values, between 6 and 13 arcmin. These latter are probably due to spacecraft maneuvres. The associated time-evolution \ndiagram is shown in Figure~\\ref{pointing} ({\\it top}). Some power density has been found at a level of $10^{-5} (arcmin)^2$\/$\\mu$Hz with\n no visible correlation with the VIRGO\/SPM signal.\n\nThe yaw angle is zero during practically the whole time-span considered with some spikes around 1.7 arcmin and only a few between 6.8 to \n12 arcmin. This yields a pure noise time-evolution diagram (see Figure~\\ref{pointing} ({\\it medium})) with a power density of around $10^{-6} \n(arcmin)^2$\/$\\mu$Hz with also no visible correlation with the VIRGO\/SPM signal.\n\nFinally, as we have already said, the roll angle does not affect the data achieved by instruments that observe the Sun as a star (integrated \nlight) but, in any case, it has also been analysed. The roll angle changes following the Earth orbit between 7.16 and -7.16 degrees with some \nlarge spikes that have been removed (the roll angle sometimes has large errors) and the time-evolution diagram is shown in \nFigure~\\ref{pointing} ({\\it bottom}). The power density is around $1 (arcmin)^2$\/$\\mu$Hz and, once again, no correlation with the measurements \nof VIRGO\/SPM has been found.\n\n\\begin{figure*}[!htb]\n\\centering\n\\begin{tabular}{cc}\n\t\\includegraphics[width=12pc,angle =90]{f1a.eps} \n\t\\includegraphics[width=12pc,angle =90]{f9.eps} \\\\\n \t\\includegraphics[width=12pc,angle =90]{f10.eps} \n\t\\includegraphics[width=12pc,angle =90]{f11.eps}\\\\ \n\t\\includegraphics[width=12pc,angle =90]{f12.eps} \n \t\\includegraphics[width=12pc,angle =90]{f13.eps}\\\\ \n\n\n \n\n\n \n\\end{tabular}\n\\caption{\\label{Temps}Time-evolution power diagram of the HK temperatures of the VIRGO\/SPM package. To simplify the \ncomparison we have repeated, on the same scale, the time-evolution diagram of the VIRGO\/SPM Blue channel.}\n\\end{figure*} \n\n\\subsection{VIRGO\/SPM temperatures}\n\n\\begin {itemize}\n\\item{VIRGO\/SPM sensor temperatures}\n\nThe most important VIRGO\/SPM temperature is the temperature sensor. Each of the three VIRGO\/SPM channels (blue, green and red) are \ncorrected by a quantity proportional to each of the temperature sensors (sensor blue, green and red). This correction is applied in \nthe level 1 software, so the data we are handling are already multiplied by this quantity. This correction is:\n \\begin{equation}\nSPM_{channel}=(1+C_{channel}(TS_{channel}-293.15))\n\\end{equation}\n where ``channel'' means blue, green or red; $C_{channel}$ is a constant for each channel and $TS_{channel}$ is the temperature of each of the three sensors.\n\nIn Figure~\\ref{Temps} the time-evolution power diagram of the temperature sensor of the blue channel is shown (top of the right column) . \nThe fluctuation of this temperature is two orders of magnitudes smaller that the VIRGO\/SPM signals at frequencies around 220$\\mu$Hz and no \nclear correlation with VIRGO\/SPM signals is visible. \n\n\n\n\\item{VIRGO\/SPM electronic temperature}\n\nThe temperature of the SPM electronics has been also analysed to see if there exists some modulation that could produce a periodic variation \nin the output voltage of the low-noise electrometer amplifiers (or in the input current). If this exists, a modulation would go to the \nVoltage Frequency Converters (VFC) of the Data Acquisition System (DAS) and could produce a modulation in the output signal. \n\nThe time-evolution power diagram for the VIRGO\/SPM electronic temperature is shown in Figure~\\ref{Temps} (middle of the left column). \nThe power density is of the same order as in the VIRGO\/SPM channels but no correlation with the signal at 220.7 $\\mu$Hz has been found.\n\n\\item{Data Acquisition System (DAS) temperature}\n\nThe Data Acquisition System (DAS) of VIRGO comprises the Onboard Data Handling System (interface for telemetry, telecommands and timing \nsignals), multiplexers, Voltage Frequency Converters (VFC) and counters. If the DAS temperature, the VFC or the counters have a periodic \nbehaviour, the output number of counts could contain that periodicity. The DAS temperature time-evolution diagram is shown in Figure~\\ref{Temps} \n(middle of right column). The power density is an order of magnitude higher than the SPM signal but again there is no correlation with the signal at 220.7 $\\mu$Hz .\n\n\\item{VIRGO\/SPM Heatsink and DC\/DC temperatures}\n \nThe temperature variations of the VIRGO Heatsink and the VIRGO Power Supply (DC\/DC) have been also analysed for security. The Heatsink \ntime-evolution diagram (Figure~\\ref{Temps})(bottom of the left column) is ten times smaller than the VIRGO\/SPM and that corresponding \nto the DC\/DC is ten times larger (Figure~\\ref{Temps} ,bottom of the right column). In both cases no correlation is found with the 220.7 $\\mu$Hz signal.\n\n\n\\begin{figure*}[!hptb]\n\\centerline{%\n\\begin{tabular}{c@{\\hspace{1pc}}c}\n\\includegraphics[width=35pc,angle =0]{f14.eps}\n\\end{tabular}}\n\\caption{\\label{zoom} Zoom of the time-evolution power diagrams of the VIRGO\/SPM Blue and the VIRGO housekeeping analysed in this research \nfor the time series 35 to 55 where the 220.7 $\\mu$Hz has higher amplitudes. This zoom helps to clarify the darker parts of some time-evolution \ndiagrams produced by the colour scales. None of these temperatures can explain the observed signal at ~220.7 $\\mu$Hz .}\n\\end{figure*} \n\nFinally, if a signal is the result of a certain temperature modulation, the temperature variation would be higher just where the power of \nthe signal is higher. In Figure~\\ref{zoom} the SPM\/Blue and VIRGO HK time-evolution power diagrams are plotted together but only between time \nseries 35 to 55, in which the power of the 220.7 $\\mu$Hz signal is stronger in the VIRGO\/SPM data. This zoom helps us to see the darker parts of \nsome HK time-evolution diagrams produced by the colour scales. None of the temperatures analysed in this section can explain the observed signal \nat 220.7 $\\mu$Hz.\n\n\n\\end{itemize}\n\n\\section{The 220.7 $\\mu$Hz signal in the others VIRGO instruments}\n\nAs was mentioned in section 2.1 the VIRGO package comprises the SPM Sunphotometers and also two types of absolute radiometers (VIRGO\/DIARAD \nand VIRGO\/PMO6-V) and one Luminosity Oscillation Imager (VIRGO\/LOI). In this section we study the 220.7 $\\mu$Hz signal in these instruments.\n\n\\begin {itemize}\n\\item{Luminosity Oscillation Imager (VIRGO\/LOI)}\n\nVIRGO\/LOI measures the radiance in 12 pixels over the solar disc. We convert these 12 pixels into one by simply adding all of them. From \nthe raw time series the same analysis as in VIRGO\/SPM has been carried out. Figure~\\ref{loi} ({\\it Top}) shows its time-evolution diagram. \nIt looks similar to the VIRGO\/SPM and with the same visible signal at 220.7 $\\mu$Hz. In the VIRGO\/LOI observations, the signal is weaker \nthan in VIRGO\/SPM but with the same characteristics, for example, at time series 60 the signal slightly changes its frequency. However, \nwith this instrument, the peak seems to be like a doublet instead of only one signal concentrated in a couple of bins.\n\\begin{figure}[]\n\\centering\n\\begin{tabular}{c}\n\t\\includegraphics[width=12pc,angle =90]{f15.eps} \\\\\n \t\\includegraphics[width=12pc,angle =90]{f16.eps} \\\\\n\t\\includegraphics[width=12pc,angle =90]{f17.eps} \\\\\n\\end{tabular}\n\\caption{\\label{loi}Time-evolution power diagram of the Luminosity Oscillation Imager (VIRGO\/LOI), the VIRGO\/DIARAD absolute radiometer \nand the VIRGO\/PMO6-V absolute radiometer, respectively from top to bottom. All these instruments are part of the VIRGO package.}\n\\end{figure} \n\n\\item{Absolute radiometers (VIRGO\/DIARAD and VIRGO\/PMO6-V)}\n\nAbsolute radiometers use a quite different technique from that of VIRGO\/SPM and VIRGO\/LOI, which are silicon detectors measuring the \nspectral irradiance and the radiance respectively. Absolute radiometers are based on the measurements of the heat flux by using an \nelectrically calibrated heat flux transducer to measure the total solar irradiance (solar constant). Once again, from the raw time \nseries we have performed the same analysis. Figure~\\ref{loi} ({\\it medium and bottom}) shows the results for VIRGO\/DIARAD and VIRGO\/PMO6-V. \nTime-evolution power diagrams are similar to the previous ones, although the 220.7 $\\mu$Hz is weaker in both radiometers but the 220.7 $\\mu$Hz signal is still present.\n\n\\end{itemize}\n\n\\section{Analysis using Doppler velocity instrumentation}\n\nUp to now we have not found any instrumental origin for the 220.7 $\\mu$Hz signal observed in all the VIRGO package. We can now study this \nregion in other helioseismic instruments. We will start by analysing the signal of the other two instruments on board SoHO and we finish by\n using the GONG ground-based network.\n\n\\begin{itemize}\n\\item{GOLF} is the other Sun-as-a-star instrument on board SoHO. We have analysed the velocity time series following the same procedure \nemployed in the VIRGO analysis and we have computed the time-evolution power diagram shown in Figure \\ref{Doppler} {\\it (top)}. As \n mentioned in the introduction, the 220.7 $\\mu$Hz signal was first observed by GOLF during the first years of the mission and it was flagged \nas a ``g-mode'' candidate by \\cite{STC04} and, after 4182 days, it is still visible as part of a quadruplet above a 90\\% confidence level \n\\citep{2008AN....329..476G}. Figure~\\ref{Doppler} {\\it (Top)} shows that the evolution with time of the signal, although weaker than in VIRGO, \nis still there. The signal in GOLF has an interval between time series 12 and 19, where it disappears, and it corresponds to the place where \nthe signal in VIRGO\/SPM is the weakest (see figure~\\ref{spms}). Therefore, we can conclude that the 220.7 $\\mu$Hz is also observed in velocity \nmeasurements using GOLF data but with a smaller signal-to-noise ratio (SNR).\n\\begin{figure}[]\n\\centering\n\\begin{tabular}{c}\n\t\\includegraphics[width=12pc,angle =90]{f18.eps} \\\\\n \t\\includegraphics[width=12pc,angle =90]{f19.eps} \\\\\n\t\\includegraphics[width=12pc,angle =90]{f20.eps} \\\\\n\\end{tabular}\n\\caption{\\label{Doppler}Time-evolution power diagram of the GOLF, MDI and GONG instruments respectively from top to bottom.}\n\\end{figure} \n\n\\item{}Disc-averaged MDI velocity signals from the calibrated level-1.4 MDI LOI-proxy Doppler images were obtained using integrated spatially \nweighted masks following \\cite{Henney99}. These time series from 1996 May 25 till 2007 October 28 have been analysed and the time-evolution \ndiagram plotted in Figure~\\ref{Doppler} {\\it (medium)}. There are no fingerprints of the presence of the 220.7 $\\mu$Hz signal in this data set. \nThis could be due to a lower SNR in the MDI LOI-proxy as compared to GOLF. Indeed, \\cite{HenneyUlrich99} showed that for the lowest measurable \np-modes, the GOLF instrument has a higher SNR than this particular MDI mask. To go further, we need to apply our methodology to specifically \ndesigned g-mode masks such as those derived by \\cite{Watcher}.\n\n\\item{}The radial velocity of GONG used in this work started on 1995 May 7 and finished on 2006 March 9. These series are shifted by a year compared to the \nSoHO data but they were the longest we could use. The disc-integrated data provided by the GONG Team are very noisy at low frequency and were not \nsuited for our studies. Therefore, we have decided to use the $\\ell$=2 spherical-harmonic series. These are optimized for acoustic modes of this \ndegree and they have the advantage of having a much stabler behaviour at low frequency. We preferred these series to the $\\ell$=1 mode because the \nclosest theoretical frequency to the target frequency of 220.7 $\\mu$Hz corresponds to an $\\ell$=2 g mode. In Figure~\\ref{Doppler} {\\it(bottom)} \nthe time-evolution diagram of GONG is shown. Although the SNR at this frequencies in the GONG data is very low, it seems to be a trace of the \n220.7 $\\mu$Hz signal in the GONG data, especially between series 36 and 52, which corresponds to a maximum in the GOLF between series 43 and 60 \n(7 subseries shift, i.e.\\ $\\sim$ 350 days). However, looking only at this time-evolution diagram, it is impossible to disentangle the feature at \n220.7 $\\mu$Hz from others visible in the analysed region. Nevertheless, It would be extremely important to be able to confirm the visibility of \nsuch a peak using ground-based data because that would directly mean that the nature of this 220.7 $\\mu$Hz signal has a solar origin.\n\\end{itemize}\n\nTo compare the averaged behavior of the 220.7 $\\mu$Hz signal in the GOLF, GONG and VIRGO\/SPM data sets we computed the collapsograms of the \ntime-evolution power diagram, i.e.\\ to average the 66 power spectra used to produce the time-evolution power diagrams. The resultant graphs \nare plotted in Figure~\\ref{Colapso}. A similar structure appears around the target frequency of 220.7 $\\mu$Hz, this peak being the highest in \nthe three instruments. However, in the case of GONG, it is at noise level. \n\n\\begin{figure}[!hbt]\n\\centerline{%\n\\begin{tabular}{c@{\\hspace{1pc}}c}\n\\includegraphics[width=18pc,angle =0]{f21.eps}\n\\end{tabular}}\n\\caption{\\label{Colapso} Collapsograms of the time-evolution power diagrams of VIRGO\/blue, GOLF and GONG. The 220.7 $\\mu$Hz structure is\n present in the three different instruments although in GONG it is at noise level.}\n\\end{figure} \n\n\n\n\n\\section{Conclusions} \nIn the present paper we have studied a peak that appears around the frequency of 220.7 $\\mu$Hz in the VIRGO\/SPM data. This peak has a more \nthan 90$\\%$ confidence level of not being due to noise in the full spectrum of 4098 days. A detailed study of its nature revealed that this \npeak existed since the very begining of the mission in a continuous way for the last 11 years and only at the very end of the \ntime series considered does it seem to change slightly in frequency. By Monte Carlo simulations we have computed the confidence level of such kinds of behaviour\n and we found that it is really unlikely (more than 99$\\%$) that it is due to a noise with the same statistical characteristics as the convective\n noise. Therefore, we checked all the available housekeeping data from the VIRGO package as well as a detailed analysis of the SoHO \n spacecraft attitude control, looking for an instrumental origin. None of these studies was able to explain the presence of a peak in the \n region studied. Indeed, this study seems to rule out this possibility. The origin should therefore be solar. We then studied Doppler velocity data \n from another instrument on board SoHO, GOLF, and we found that the peak is also present (with lower SNR). Even though analysis of data from the GONG \n ground-based network revealed a very noisy spectrum, the highest peak in a 10 $\n\\mu$Hz region around the 220.7 $\\mu$Hz signal is precisely that peak. However, it is not significant enough for us to claim that we have a positive \ndetection using this instrument.\n\nThe present study has proved the solar origin of the peak at 220.7 $\\mu$Hz. Two solar phenomena could be responsible of such a peak. The first could be \n convection, in particular granulation motions. However, it is very unlikely that a turbulent displacement of plasma on the solar surface \nwith a typical time scale of 10 minutes give a stable frequency during more than 10 years in the power spectrum of the disc-integrated data. On \nthe other hand, gravity modes propagate inside the radiative region of the Sun and are expected to have long lifetimes (at least longer \nthan the period of measurements). Thus, the properties of the peak that we found are similar to those expected for a g mode. Using the principle of Ockham's \nrazor (or the {\\it Lex Parsimoniae} principle) in which the explanation of any phenomenon should make as few assumptions as possible, we \ncan conclude that if this peak is not noise it should be a component of a g mode. Analysing in detail the structure of this possible g-mode \ncomponent, Figure~2 reveals a peak structure containing several bins. Indeed, Figure~1 might also show the presence of a parallel component at \naround 220.64 $\\mu$Hz with high amplitudes in several of the series considered. Thus, a possible explanation of such behaviour might be the presence \nof an inner magnetic field that could slightly split the component of the g-mode multiplet in some peaks. Another possibility might be that the \ng-mode power could be spread into several bins as a consequence of a smaller than expected lifetime or due to a change in the size of the resonant \ncavity (for example due to a displacement of the position of the tachocline during the activity cycle). This latter effect is particularly interesting \nbecause it seems that the 220.7 signal follows a small change in frequency over the entire time-span with the lowest frequency (220.68 $\\mu$Hz) \nreached around time series 35---corresponding to the maximum of the activity cycle--- and then increasing the frequency again towards the two \nperiods with minimum activity (at the beginning and the end of the series). In any case, assuming a faster rotation in the core than in the \nrest of the radiative envelope (as suggested by Garc\\'\\i a et al. 2007), the 220.7 $\\mu$Hz peak could be or a component of the $\\ell$=2, n=-3 g\n mode, or a component of the $\\ell$=3, n=-5 or a bitting between this latter and the $\\ell$=5, n=-8. Whatever the true answer is, there is still \n an important question to be answered: why is this particular peak so excited when there are no other visible g-mode components? More work will be \n necessary before solving the solar g-mode puzzle. \n\n\n\\begin{acknowledgements}\nThe authors want to thank the members of the PHOEBUS group present at the first ISSI (International Space Science Institute) meetings for \ntheir useful comments and discussions. This work has been partially supported by the Spanish grant PNAyA2007-62651 and the CNES\/GOLF grant at the SAp\/CEA-Saclay.\nThe authors also thank all their colleagues (scientists, engineers and technicians) involved with the GOLF, VIRGO and MDI instruments\n aboard SoHO which is a space mission of international cooperation between ESA and NASA. This work utilizes data obtained by the Global \n Oscillation Network Group (GONG) program, managed by the National Solar Observatory, which is operated by AURA, Inc. under a cooperative \n agreement with the National Science Foundation. The data were acquired by instruments operated by the Big Bear Solar Observatory, High Altitude \n Observatory, Learmonth Solar Observatory, Udaipur Solar Observatory, Instituto de Astrof\\'\\i sica de Canarias, and Cerro Tololo Interamerican Observatory. One of the authors (AJ) would like to thank M.Ortiz for invaluable help with the last part of this article. \n \n\\end{acknowledgements}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[scale=0.37]{pl-overview.pdf}\n \\caption{\n An example of how parallel news documents can be used to train a model that is capable of making educated guesses on what the question is asking, and how it may help to derive a better answer.\n \n }\n \\label{fig:overview}\n\\end{figure}\n\nAnswering questions often involves making educated guesses: we do not necessarily have accurate facts but can use common sense to understand what most questions are asking and what kinds of knowledge are needed. For example (see Fig.~\\ref{fig:overview}), we can understand the question ``\\emph{Is Albany, GA more crowded than Albany, NY?}'' involves comparing the size and population of two cities without knowing the specific numbers to compare.\nIt is often desirable to make such a decomposition because a city's population is usually much easier to acquire than a direct answer to the original question. \n\nExisting approaches to end-to-end question-answering (QA) assume that pre-trained language models (LMs) are capable of both robust question understanding of this type and acquiring the relevant facts. Much recent evidence, however, has revealed limitations in the commonsense and compositional reasoning abilities of current transformers \\cite{zhou2019going,liu2021challenges}, in part due to \\emph{reporting biases} (e.g., relating the semantics of ``more crowded'' and ``overpopulation'' can be difficult given that such contexts rarely co-occur in single document on which models are pre-trained) and other \\emph{dataset artifacts} \\cite{gururangan2018annotation}. This is even more evident in recent datasets with complex questions that are designed to require decomposition. For example, GPT-3 \\cite{Brown2020LanguageMA}, a language model with 175 billion parameters, only achieves mid-60s accuracy on StrategyQA \\cite{geva2021did}, a binary QA benchmark with a random baseline at around 50. Moreover, such datasets are often small in size and scope, which makes it difficult to overcome knowledge gaps in LMs through fine-tuning and developing general-purpose decomposition models. \n\n\n\nIn this paper,\\footnote{\\url{http:\/\/cogcomp.org\/page\/publication_view\/992}} we attempt to bridge the gap of reporting biases, which hinders LMs from learning implicit connections between questions and decompositions (e.g., ``crowded'' and ``population''). We do this through intermediate pre-training on distant supervision, following recent attempts to distill common sense into transformers via distant supervision \\cite{zhou2021temporal}.\nSpecifically, we use collections of article pairs with parallel descriptions of similar news events from different angles as our distant supervision. As illustrated in Fig.~\\ref{fig:overview}, large collections of comparable texts (\\S\\ref{sec:distant-supervision-intuitions}) contain a wide variety of commonsense implications needed for decomposition. We extract 2.6 million sentence pairs (\\S\\ref{sec:pl-extraction}) for this purpose, and then train \\mbox{\\textsc{DecompT5}}{} (\\S\\ref{sec:pretrain}), a T5 \\cite{Raffel2020ExploringTL} model that is further-pre-trained on our distant supervision instances. In \\S\\ref{sec:intrinsic-experiments}, we show that \\mbox{\\textsc{DecompT5}}{}, while simple, serves as a more effective model than the base language model on general question understanding through experiments on Overnight \\cite{Wang2015BuildingAS} and TORQUE \\cite{Ning2020TORQUEAR} semantic parsing tasks, achieving 22-32\\% absolute improvements. \n\nSince smaller language models cannot sufficiently memorize facts (e.g., the exact population of Albany), they are often used in conjunction with external knowledge retrieval for more complicated tasks such as QA. To bridge this gap, we design a novel QA pipeline using \\mbox{\\textsc{DecompT5}}{} at its core (\\S\\ref{sec:e2e-qa-system}). The full model and pipeline, called \\mbox{\\textsc{DecompEntail}}{}, first generates explicit question decompositions, then makes factual corrections on the decomposed statements with GPT-3. As a final step, \\mbox{\\textsc{DecompEntail}}{} employs an entailment model that derives the final answer with the generated decomposition as the premise and the question and candidate answer as the hypothesis. \n\nIn \\S\\ref{sec:e2e-experiments}, we show that \\mbox{\\textsc{DecompEntail}}{}, despite its relatively small size, can generate good decomposition chains and outperforms GPT-3 on both StrategyQA and a binary portion of HotpotQA by 4\\% and 8\\%, respectively.\nThis shows that we can improve baseline language models or even much larger reasoners with explicit decomposition, which has the advantage of enhanced interpretability and transferability. On the other hand, \\mbox{\\textsc{DecompT5}}{} only relies on supporting fact annotations instead of explicit reasoning steps, which is more common in datasets and can be better applied for joint learning.\n\n\\noindent \\textbf{Contributions.} In summary, our contributions are three-fold: 1) we collect distant supervision from parallel news to encourage robust semantic understanding for question decomposition, 2) we train a general decomposition model called \\mbox{\\textsc{DecompT5}}{} with our collected distant supervision that significantly improves over the baseline language models on intrinsic evaluations, and 3) we propose a decomposition-based QA pipeline called \\mbox{\\textsc{DecompEntail}}{} that relies on \\mbox{\\textsc{DecompT5}}{} at its core. We show that \\mbox{\\textsc{DecompEntail}}{} has improved performance over several baselines on decomposition-based QA.\n\n\\section{Related Work}\n\nOur work relates to the literature on multi-hop reasoning \\cite{yang2018hotpotqa}, which has recently produced new annotation schemes (e.g., \\emph{QDMR} from \\citet{wolfson2020break} and \\emph{strategy question decomposition} annotations from \\citet{geva2021did}) and datasets for complex reasoning that target explicit model decomposition \\cite{talmor2018web,wolfson2020break,geva2021did,khot2022hey}. We take inspiration from systems that build explicit reasoning paths, such as semantic parsers \\cite{Liang2011LearningDC, Berant2013SemanticPO}, and their modern variations \\cite{Andreas2016NeuralMN,Gupta2020NeuralMN,khot2020text}. \n\\citet{Min2019MultihopRC,Perez2020UnsupervisedQD} aim to build general question decomposition models, however, focusing on simpler tasks than our study. \n\n\nOur work is also related to sentence-pair datasets collected from comparable texts \\cite{Fader2013ParaphraseDrivenLF, Zhang2019PAWSPA, Reimers2019SentenceBERTSE}. Compared to most of these works, our extraction does not use human annotation, and produces clean and diverse signals for question understanding.\n\nPrevious work has also discussed using large-scale further pre-training to improve language models \\cite{Zhou2020TemporalCS, zhou2021temporal, zhao2021effective}. We follow a similar general scheme with novel extraction sources and focus on a general representation for questions, which resembles some idea in existing work \\cite{Khashabi2020UnifiedQACF}.\n\n\n\n\n\n\\section{Distant Supervision for Decomposition}\nIn \\S\\ref{sec:distant-supervision-intuitions}, we describe our intuitions on why question decomposition is important and what is missing from existing pre-trained language models for them to do well. Following that, we describe how we collect distant supervision signals to improve the process of learning to decompose in \\S\\ref{sec:pl-extraction}. In \\S\\ref{sec:pretrain}, we propose \\mbox{\\textsc{DecompT5}}{}, a T5-based model that is further pre-trained on the collected distant supervision using standard seq-to-seq training objectives.\n\n\\subsection{Intuitions}\n\\label{sec:distant-supervision-intuitions}\n\\noindent \\textbf{Educated Guesses in QA.} We, as humans, need to answer questions all the time, but we may not possess all the facts. For example, an ordinary person may not know the exact populations of Albany to answer ``Is Albany, GA more crowded than Albany, NY'', or the density of corgis to answer `Will a corgi float on water''. However, that person may search for ``population'' or ``density'' instead of the original question to find the answer because we know that it is much easier to find the ``population of a city'' than to find an answer to the original question. The human capacity for guessing what the question is asking and how that question can be decomposed to simpler concepts by associating \\textit{crowded} with \\textit{population}, and \\textit{float} with \\textit{density} is crucial for solving day-to-day tasks. However, making such connections can be very challenging for pre-trained language models because of reporting biases. Written texts rarely make such connections explicit in single documents, as most authors expect readers to make many trivial inferences.\n\n\\noindent \\textbf{Parallel News.} In this work, we aim to bridge this decomposition gap in pre-trained language models through incidental supervision \\cite{Roth2017IncidentalSM} from comparable corpora \\cite{Klementiev2006WeaklySN}. We find news articles reporting the same news event but from different authors and angles. Related sentences in such parallel news often complement each other and provide new information. This complementary information is often more sophisticated and diverse than paraphrasing, because it contains implications and causal relations. Fig.~\\ref{fig:overview} shows an example of how a pair of articles describing Tokyo from slightly different angles may help decompose the running example question. One article mentions that Tokyo is crowded, while the other expresses similar points but focuses on area size and population descriptions. Intuitively, a model may benefit from such connections to learn that ``crowded'' is related to ``size'' and ``count''. It is rare, however, for a single document to contain both aspects, causing difficulties for LMs that primarily learn from single documents.\n\n\\subsection{Parallel News Extraction}\n\\label{sec:pl-extraction}\nWe use the RealNews corpus \\cite{Zellers2019DefendingAN} as the source corpus because it contains cleaned, date-marked new articles from diverse domains. We aim to select news article pairs that describe the same main event and find sentence pairs within these document pairs that are likely to contain complementary information to each other.\n\n\\noindent \\textbf{Filter Article Pairs.} We select article pairs within a 2-day window of publication because the same news events are typically covered within a relatively short period. We then employ a pre-trained entailment model from SentenceBert \\cite{Reimers2019SentenceBERTSE} to check the titles of each article pair and retain those pairs whose titles have a cosine similarity greater than $0.8$.\n\n\\noindent \\textbf{Find Sentence Pairs.} We then find sentence pairs across each selected article pairs that are related and complementary to each other. To do this, we run the same sentence similarity model and retain all sentence pairs with a similarity score between $0.6$ and $0.9$. The lower bound is to make sure the sentences are approximately related. Even though $0.6$ is considerably a loose bound for many tasks (e.g., paraphrasing), it is suitable in our case because we have a strong assumption that the articles are closely related because of date and title similarities. This lower bound is sufficient to guarantee that the vast majority of sentence pairs above this threshold contain complementary information to each other. For example, the similarity score between ``The US Military has already started withdrawal from Syria'' and ``The US is only moving non-essential equipment out of Syria, because precipitous withdrawal would shatter US policy in Syria and allow IS to rebuild'' is only $0.6$. However, the second sentence provides non-paraphrasing but complementary information to the first sentence. A model may learn that troops in other countries are linked with foreign policy, which is the type of information that is often implicit in single documents. The upper-bound $0.9$ is to filter out sentence pairs that are too similar or simply paraphrasing each other, as these pairs do not provide much additional information to facilitate question understanding. \n\n\\noindent \\textbf{Filtering with tf-idf.} We employ an additional filtering process based on sentence topics to keep the final dataset's diversity. To do this, we calculate the inverse document frequency (idf) of each word in the vocabulary based on Wikipedia and multiply that with the term frequency (tf) of each word within the sentence pairs. Next, we use the top three words ranked by td-idf scores of each sentence pair as the ``signature'' and randomly keep ten sentence pairs with identical signatures at most. 2.6 million sentence pairs remain after this step. Finally, we format the data as a standard seq-to-seq training task, where the input sentence is one of the sentences in the pair, while the model is trained to generate the other sentence in the pair. The order is randomly decided. \n\n\\noindent \\textbf{Data for Language Modeling Objective.} Beyond the sentence pairs, we also inject some data from Project Gutenberg\\footnote{\\url{https:\/\/www.gutenberg.org\/}} and format it to the language model pre-training format (e.g., the denoising objective for T5 \\cite{Raffel2020ExploringTL}). We sample around 900K sentences for this purpose.\n\n\\subsection{Comparisons with Similar Data Sources}\n\\begin{table}[t]\n\\centering\n\\small\n\\begin{tabular}{lccccc}\n\\toprule\nMetric \/ Data & Ours & P-auto & P-inc. & NLI & QA \\\\\n\\cmidrule(lr){1-1}\\cmidrule(lr){2-2}\\cmidrule(lr){3-3}\\cmidrule(lr){4-4}\\cmidrule(lr){5-5}\\cmidrule(lr){6-6}\nLength $\\uparrow$ & 52 & 42 & 20 & 31 & 40\\\\ \nLength-diff $\\uparrow$ & 9 & 1 & 2 & 10 & 20\\\\\nEmbed-sim $\\downarrow$ & 0.7 & 1.0 & 0.9 & 0.6 & 0.6\\\\\nSem-sim $\\downarrow$ & 0.7 & 1.0 & 0.9 & 0.7 & 0.7\\\\\nCost $\\downarrow$ & low & low & mid & high & high \\\\\n\\bottomrule\n\\end{tabular}\n\\caption{Comparisons between our data and other sources for reasoning tasks. \\textit{P-auto} is paraphrasing data from automatic (distant) collection, \\textit{P-inc.} is paraphrasing data from incidental supervision. \\textit{Sem-sim} is semantic similarity. $\\uparrow$\/$\\downarrow$ marks the direction for each metric to present a more diverse data source.}\n\\label{tab:naive-comparison}\n\\end{table}\nWe compare our data collected in \\S\\ref{sec:pl-extraction} with other sources that may similarly be used, including paraphrasing, textual entailment (NLI), and question-answering (QA). Paraphrasing data can be collected either automatically (e.g., PAWS \\cite{Zhang2019PAWSPA}), or from incidental but human-involved processes (e.g., Quora duplicated questions\\footnote{\\url{https:\/\/quoradata.quora.com\/}}). We use these two datasets to represent each category respectively. In addition, we use the MNLI dataset \\cite{Williams2018ABC} for NLI, and StrategyQA (question+answer\/supporting-facts) for QA. We randomly sample 10k sentence pairs from each source. We compare basic statistics, including sentence pair length and the length difference between the two sentences. We also compare sentence similarity via averaged word embeddings \\cite{Pennington2014GloVeGV} and sentence-level semantic embeddings \\cite{Reimers2019SentenceBERTSE}.\\footnote{We use the ``average\\_word\\_embeddings\\_glove.840B.300d'' and ``all-MiniLM-L6-v2'' models, respectively.}\nTable~\\ref{tab:naive-comparison} shows that our data source provides richer and more diverse information while not requiring any human annotation. This observation aligns with our intuitions in \\S\\ref{sec:distant-supervision-intuitions}.\n\n\\subsection{Pre-training with Distant Supervision}\n\\label{sec:pretrain}\n\nWe use T5-large \\cite{Raffel2020ExploringTL} as our base language model due to its sequence-to-sequence architecture and relatively small parameter size (containing 770m parameters) for easier pre-training. We train the base language model on our distant supervision dataset for one epoch and call the resulting model \\mbox{\\textsc{DecompT5}}{}. We expect, however, that this pre-training technique with our collected dataset is beneficial to most existing pre-trained language models, as it bridges the reporting bias gap in general language modeling objectives.\n\n\\section{Decomposition-based QA Pipeline}\n\\label{sec:e2e-qa-system}\n\n\nOur proposed model \\mbox{\\textsc{DecompT5}}{} has two uses: it can be \\textbf{directly fine-tuned} on tasks that require query understanding and decomposition, as we later show in \\S\\ref{sec:intrinsic-experiments}. It can also be applied in a pipeline fashion to \\textbf{produce meaningful decompositions} that help with more complicated tasks that require external knowledge, such as general question answering. This section focuses on the design challenges and choices for such a QA pipeline. We first explain the intuitions in \\S\\ref{sec:pipeline-intuition}, then describe and propose \\mbox{\\textsc{DecompEntail}}{} in \\S\\ref{sec:pipeline}. We evaluate our proposed QA pipeline in \\S\\ref{sec:e2e-experiments}.\n\n\\subsection{Intuitions and Design Choices}\n\\label{sec:pipeline-intuition}\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[scale=0.37]{stqa-example.pdf}\n \\caption{\n An example StrategyQA \\citep{geva2021did} instance that includes a question annotated with decomposed questions and their corresponding facts.\n }\n \\label{fig:stqa-example}\n\\end{figure}\n\n\\begin{table}[t]\n\\centering\n\\small\n\\begin{tabular}{lcc}\n\\toprule\nAdditional Information & \\#Train & Accuracy \\\\\n\\cmidrule(lr){1-1}\\cmidrule(lr){2-2}\\cmidrule(lr){3-3}\nNone & 2061 & 53.3 \\\\\nAspects & 2061 & 59.7 \\\\\nFacts (Impossible) & n\/a & n\/a \\\\\nAspects, Facts & 2061 & 84.5 \\\\\nAspects, Facts, Indirect & 15296 & \\textbf{90.2} \\\\\n\\bottomrule\n\\end{tabular}\n\\caption{T5-3B accuracy on StrategyQA dev set when different information is provided for both training and evaluation. \\textit{Aspects} refers to the knowledge dimensions (without values) that are involved with each question. \\textit{Facts} refers to the actual knowledge involved, which is not possible to acquire without knowing the corresponding aspects. \\textit{Indirect} contains additional supervision of paraphrasing and entailment. Details are in \\S\\ref{sec:sanity_check}.}\n\\label{tab:sanity_check}\n\\end{table}\n\n\\begin{figure*}[t]\n \\centering\n \\includegraphics[scale=0.5]{pipeline-overview.pdf}\n \\caption{\n An overview of our proposed \\mbox{\\textsc{DecompEntail}}{} pipeline. The final decomposition is an actual output from the pipeline. See more examples in Fig.~\\ref{fig:manual-analysis-examples}. \n \n }\n \\label{fig:pipeline-overview}\n\\end{figure*}\n\nAs we argue in \\S\\ref{sec:distant-supervision-intuitions}, an agent can decompose complex questions into simpler and more controlled forms by linking a question to all relevant \\textit{aspects} of that question (e.g., the relevant sub-queries related to the input question). With such aspects or components, the agent can make easier knowledge retrieval to acquire the specific values of the aspects, which we call relevant \\textit{facts}. As shown in Fig.~\\ref{fig:stqa-example}, StrategyQA provides two kinds of supporting annotations for each question. The decomposed questions do not contain the answers and thus approximate the \\textit{aspects} of each question. The annotated facts answers the sub-questions with accurate values, so they approximate \\textit{aspects+facts}. \n\nIn \\S\\ref{sec:sanity_check}, we conduct an experiment for sanity checking purposes, with results shown in Table~\\ref{tab:sanity_check}. We see that T5 does not improve much when given only the \\textit{aspects} (+6\\%) but gains much more (31\\%-37\\%) when provided with \\textit{aspects+facts} and additional indirect supervision. When given \\textit{aspects+facts}, the model is in effect doing textual entailment. The 90.2\\% accuracy shows that this entailment part of deciding how to use the facts is a much smaller bottleneck than finding the proper aspects and their values. At the same time, relatively small LMs such as T5 do not gain much from only seeing the \\textit{aspects} because of their poor memorization (e.g., even if the model knows that the population of a city is needed, it cannot produce the correct number without external resources). This observation serves as the motivation for building a binary QA pipeline that first generates accurate \\textit{aspects+facts} (\\textbf{decompose}) and then decides the final answer with an entailment model (\\textbf{entail}).\n\nThe \\textbf{decompose} step can be approached in two ways: i) generating the \\textit{aspects} first, then perform information retrieval (IR) and compose a new statement for \\textit{aspects+facts}; ii) generating \\textit{aspects+facts} directly, then perform some factual correction because small LMs cannot memorize well. We choose the second approach for the following three reasons. 1) Our basis \\mbox{\\textsc{DecompT5}}{} is trained on parallel news, which are natural language statements that approximate the \\textit{aspects+facts} together (see Fig.~\\ref{fig:overview}). 2) Generating \\textit{aspects+facts} together allows the model to adhere to its beliefs and generate self-consistent logic chains because decomposition may be inter-dependent (e.g., in Fig.~\\ref{fig:pipeline-overview}, the country that Cyril represents plays an important role in the next generation step). 3) Supporting facts are a much more common type of annotation (e.g., in HotpotQA) than \\textit{aspects}-only annotations, which allows us to explore transfer and joint learning with other existing datasets.\n\n\\subsection{Factual Correction for Generated Facts}\n\\label{sec:factual-correction}\n\nIn order for generating \\textit{aspects+facts} to work, we need to correct any factual errors in the generated facts. This is crucial because relatively small LMs such as T5 cannot generate accurate facts, and wrong information will hinder the performance of the entailment model when deciding the final answer. Standard information retrieval (IR) approaches aim to find a specific piece of text from a knowledge base \\cite{Karpukhin2020DensePR} and tailor the correct information in the retrieved text to specific needs. However, this will not work well in our scenario because doing IR on \\textit{aspects} and incorrect \\textit{facts} will lead to much noise. Moreover, certain commonsense information, such as the weight of a six-year-old, are often missing from standard IR resources such as Wikipedia. \n\nTo this end, we propose to use large-scale language models such as GPT-3 \\cite{Brown2020LanguageMA} directly as a fact-checker, as we have found that GPT-3 does reasonably well on memorizing and retrieving the majority of well-known facts. Furthermore, when given appropriate prompts, GPT-3 simultaneously performs retrieval and new statement synthesis, allowing us to inspect the reasoning capability of our decomposition model directly and more efficiently. Therefore, we design a prompt that starts with \\textit{``Fix the input sentence with correct facts if there are factual errors''} followed by six examples listed in Appendix~\\ref{sec:appendix-prompts}. \n\nWe emphasize that GPT-3 is only used as a fact-checker in our pipeline. It does not add any information on how to find the \\textit{aspects} because it does not see the original question, rather the output of single-step generated facts. As a result, we view our ``reasoning'' component much smaller than GPT-3 as we disentangle these two parts. We discuss this more in \\S\\ref{sec:manual-analysis} and Appendix~\\ref{sec:appendix-examples}.\n\n\\subsection{\\mbox{\\textsc{DecompEntail}}{} QA Pipeline}\n\\label{sec:pipeline}\n\n\\noindent \\textbf{Decompose.} Since \\mbox{\\textsc{DecompT5}}{} hasn't been pre-trained on questions, we fine-tune it on [question, supporting-fact] annotations from relevant datasets to generate \\textit{aspects+facts} for each question. Because supporting facts are usually composed of multiple sentences, we formulate a step-by-step generation. \nFor $n$ training facts, we formulate $n$ training instances from time $1$ to time $n$. At time $t$, a model sees an input sequence that is the question and all supporting facts with indices smaller than $t$ concatenated. The output sequence (learning target) is the supporting fact at index $t$. During evaluation time, the model generates one fact at a time, which then goes through the factual correction process in \\S\\ref{sec:factual-correction}. At time $t$, the model receives an input sequence including the original question and all current generated facts (after correction) before time $t$, and generates the $t^{th}$ supporting fact.\n\nWe design the specific input sequence as \\keywordCode{[Q]Decompositions:[G(current)]}, and output sequences as \\keywordCode{[G(next)]}. \\keywordCode{[G(current)]} is the concatenation of all current generations, which is empty before generating the first fact. \\keywordCode{[G(next)]} is the immediate next fact to be generated. \n\n\\noindent \\textbf{Entail.} With the generated facts from \\textbf{decompose}, we derive binary answers for questions with the \\textit{aspects+facts+indirect} model as seen in Table~\\ref{tab:sanity_check}.\n\n\n\n\\subsection{Inference}\n\\label{sec:voting}\nWe sample the top five generation candidates at each generation step via diverse beam search \\cite{Vijayakumar2016DiverseBS}. We select one randomly based on their $\\mathrm{softmax}$ probabilities. We generate at most three facts (i.e., $t=3$ as specified in \\S\\ref{sec:pipeline}) or early stops for each chain if all candidates at a generation step are very similar to the current generations, determined by the SentenceBert paraphrasing model with 0.95 as the threshold. We run the three-fact generation five times for each question due to randomness in the underlying generation selection process. As a result, we will have five chains of at most three generated facts for each question. We run the entailment model individually on each chain and derive a final answer based on majority voting from each chain. The majority voting is weighted with the confidence score of the entailment model's decisions on each chains.\n\n\\section{Intrinsic Experiments}\n\\label{sec:intrinsic-experiments}\nIn this section, we conduct two intrinsic experiments with \\mbox{\\textsc{DecompT5}}{} that directly evaluate its general decomposition capability through fine-tuning task-specific input\/output sequences. We compare with T5-large as it is the base LM, and such a comparison reveals how much we improve through pre-training with parallel news distant signals. We do not compare our model with GPT-3 because few-shot learning might not be enough for it to learn the complete grammar of different tasks' decomposition. This is an advantage of fine-tuning relatively small but capable models over directly using much bigger ones in few-shot settings. All experiments use a 5e-5 learning rate, and they are repeated and averaged with three seeds.\\footnote{We use $10,20,30$ as the seeds for all experiments.}\n\n\\subsection{Overnight}\n\n\\begin{table}[t]\n\\centering\n\\begin{tabular}{lccc}\n\\toprule\nSystem & Hit@1 & Hit@5 & Hit@10 \\\\\n\\cmidrule(lr){1-1}\\cmidrule(lr){2-2}\\cmidrule(lr){3-3}\\cmidrule(lr){4-4}\nT5-large & 21.8 & 51.6 & 63.1 \\\\\n\\mbox{\\textsc{DecompT5}}{} & 48.6 & 78.9 & 85.4 \\\\\n\\bottomrule\n\\end{tabular}\n\\caption{Hit@K performances on Overnight decomposition generation. Hit@K is the percentage of instances where the top K generations contains at least one exact match. \\mbox{\\textsc{DecompT5}}{} is from this work.}\n\\label{tab:overnight}\n\\end{table}\n\n\\noindent \\textbf{Dataset and Metrics.} We evaluate and compare our model's capability to produce intermediate decomposition on the Overnight dataset \\cite{Wang2015BuildingAS}. It is a semantic parsing dataset that aims to parse natural language queries into a formal parsing that can be programmatically executed to denotations. In between the natural language query and the formal parsing, it annotates an intermediate ``canonical'' form with semi-formal language, which has recently been used for work on text-based semantic parsing with transformers \\cite{shin2021constrained} that we take inspiration from. For example, the annotated intermediate form of ``biggest housing unit that allows dogs'' is ``housing unit that has the largest size and that allows dogs''.\nWe evaluate the performance of mapping natural language queries to such intermediate forms with three domains that contain 3.8K training instances and 972 test cases. Both models are trained with three epochs. We use the same inference for both T5-large and \\mbox{\\textsc{DecompT5}}{}, which generates ten candidates using beam search. Following previous work, the generation is also constrained by possible ``next words'', that is, we assume that we know controlled output space beforehand.\n\n\\noindent \\textbf{Results and Analysis.} Table~\\ref{tab:overnight} details the performance of our \\mbox{\\textsc{DecompT5}}{} compared to its base model, T5-large. Our model doubles the performance on the exact match of the top prediction, which translates to a much higher denotation accuracy because multiple decompositions can be executed to the same denotation. Our model can find the exact match decomposition 78.9\\% of the time with only five candidates to consider, showing much higher potential for end-to-end tasks that may improve through iterative learning. On the other hand, T5-large can barely cover more than half of the queries with top-five candidates and only improves to 63.1\\% with more candidates (top-ten). This shows that \\mbox{\\textsc{DecompT5}}{} is much better at making commonsense connections (e.g., ``biggest'' to ``largest size'') after fine-tuning, thanks to the pre-training process on our parallel news corpus. \n\n\\subsection{TORQUE}\n\n\\begin{table}[t]\n\\centering\n\\begin{tabular}{lc}\n\\toprule\nSystem & Exact Match \\\\\n\\cmidrule(lr){1-1}\\cmidrule(lr){2-2}\nT5-large & 50.3 \\\\\nT5-large-paraphrase & 72.2 \\\\\n\\mbox{\\textsc{DecompT5}}{} & 82.8 \\\\\n\\bottomrule\n\\end{tabular}\n\\caption{Exact match accuracy of different models on custom-annotated TORQUE. T5-large-paraphrase is first fine-tuned on paraphrasing supervision.}\n\\label{tab:torque}\n\\end{table}\n\n\\noindent \\textbf{Dataset.} TORQUE \\cite{Ning2020TORQUEAR} is a temporal question-answering dataset. For example, ``what happened before...'' asks the model to find all events with a start time before that of the given event, and ``what ended before...'' should be answered with events with end times before the start time of the given event. Compared to traditional temporal relation extraction tasks, this format is more challenging to existing temporal reasoning models, as they now have to parse the question and understand what aspects (e.g., start or end times) the question is asking first. To this end, we evaluate if our proposed model can better parse the question into correct temporal phenomena.\n\n\\noindent \\textbf{Annotate Decomposition.} Because TORQUE does not come with an intermediate annotation specifying the temporal properties required for each question, we need to annotate TORQUE questions with a form of intermediate decomposition to evaluate if a model understands the questions correctly. We adopt Overnight grammar for this purpose. For example, ``what started before [X]'' can be written as ``find all events whose start time is smaller than the start time of [X]''. Luckily, TORQUE uses several question templates during its annotation process. As a result, the intermediate decomposition of many questions can be automatically labeled. We create a training set of 15K question-decomposition pairs from 10 templates that are \\textbf{only} about events' start time comparisons. On the other hand, we create an evaluation set of 624 questions from 11 templates, and 9 of them compare events' end times, which a model will not see during training. We do this to evaluate models' capability of ``universal'' decomposition by generalizing to unseen relations in a ``zero-shot'' fashion. For a model to do well, it must have a pre-existing representation of what the question is asking.\n\n\\noindent \\textbf{Results and Analysis.} Table~\\ref{tab:torque} reports the exact match accuracy on our custom TORQUE evaluation. In addition to the T5 baseline, we use the same hyper-parameters as \\mbox{\\textsc{DecompT5}}{} to fine-tune a T5-large on the distant supervision portion from PAWS \\cite{Zhang2019PAWSPA}, containing 320K sentence pairs. We do this to compare the data quality of our distant supervision and that from paraphrasing since TORQUE requires a higher level of question understanding than Overnight. All models are trained for one epoch because the training data is highly repetitive, and generate one sequence via greedy decoding. We see that our model improves more than 30\\% over the baseline model, and 10\\% over the paraphrasing-supervised model. More importantly, this shows that \\mbox{\\textsc{DecompT5}}{} develops a solid implicit representation for query understanding from the pre-training, which allows it to generalize to end-time queries from supervisions that are only about start times. On the other hand, T5-large cannot correctly produce anything about end-time queries as expected.\n\n\\section{Sanity Check Experiments}\n\\label{sec:sanity_check}\nIn this section, we describe the details of the sanity check experiment mentioned and analyzed in \\S\\ref{sec:pipeline-intuition}.\n\n\\subsection{Dataset and Settings}\n\n\\noindent\\textbf{Dataset.}\nWe use StrategyQA, a QA dataset with high requirements for question understanding and knowledge acquisition. It contains questions that can be answered with either ``yes'' or ``no'', and is divided into 2061\/229 train\/dev, and an additional 490 test questions. Each question in the training and development sets is annotated with two types of supporting evidence as shown in Fig.~\\ref{fig:stqa-example}: decomposed questions and annotated facts. We use the decomposed questions as the \\textit{aspects} of a question \n, and the annotated facts as \\textit{aspects+facts}, as they provide specific values for the aspects. \n\n\\noindent \\textbf{Indirect Supervision.} Under the \\textit{aspects+facts} setting, the model is performing general textual entailment (TE) with the given facts as the premise and the question as the hypothesis, which allows us to use indirect supervision inspired by TE. We first augment each training instance in StrategyQA with five additional instances where all supporting facts are replaced with one of their paraphrases obtained with an off-the-shelf paraphrasing model.\\footnote{\\url{https:\/\/huggingface.co\/tuner007\/pegasus_paraphrase}} We then add additional supervision from e-SNLI's development set \\cite{Camburu2018eSNLINL}. We also add supervision from HotpotQA \\cite{Yang2018HotpotQAAD} with its annotated supporting facts.\n\n\\subsection{Training and Results} \nWe formulate a sequence-to-sequence task with input sequences as \\keywordCode{[Q]Decompositions:[D]} and output sequences of either \\keywordCode{yes\/no}. \\keywordCode{[Q]} is the question, and \\keywordCode{[D]} is the additional information such as supporting facts. We fine-tune T5-3B models for three epochs under each supervision setting and evaluate with the same gold information provided during test time. Each experiment is averaged over three random seeds. Table~\\ref{tab:sanity_check} details the performances on StrategyQA's development set. We have analyzed this result in \\S\\ref{sec:pipeline-intuition}.\n\n\\section{Decomposition QA Experiments}\n\\label{sec:e2e-experiments}\n\nWe detail two experiments that evaluates the QA pipeline \\mbox{\\textsc{DecompEntail}}{} proposed in \\S\\ref{sec:pipeline}. \n\\subsection{Datasets} \n\nAs argued in \\S\\ref{sec:pipeline-intuition}, our proposed pipeline benefits from any question-answering dataset that annotates supporting facts. To demonstrate this property, we use StrategyQA and HotpotQA jointly as supervision, and evaluate on both datasets. Because our pipeline setting is mostly designed for binary questions, we select questions that can be answered with either ``yes'' or ``no'' from HotpotQA, which accounts for 5430 questions from the training set. We use 300 binary questions from the development set of HotpotQA as evaluation. Because the supporting fact annotation in StrategyQA is human-written instead of Wikipedia sentences, it is shorter and more precise. To this end, we want the decomposition model to primarily rely on such annotations, and we duplicate each set of supporting facts in StrategyQA five times with shuffled order. These together produce around 35K decomposition instances for training.\n\n\\subsection{Settings and Baselines} \n\nWe compare with T5-large under the same joint supervision setting (denoted as ``S+H''). We also compare with RoBERTa*-IR as described in \\citet{geva2021did} on StrategyQA. It uses BoolQ \\cite{clark2019boolq} as additional supervision, which is denoted as ``S+B''. We also include GPT-3 baselines, one in a regular few-shot setting and another with a few-shot chain-of-thought \\cite{Wei2022ChainOT} supplement (denoted as GPT-3 CoT). Both prompts are available in Appendix~\\ref{sec:appendix-prompts}. \nWe report an aggregated performance (i.e., voting with all seeds as described in \\S\\ref{sec:voting}) on StrategyQA's development set. However, we report a single best seed's\\footnote{We determine the best seed based on the StrategyQA's development set.} performance on the test set as well as HotpotQA because of both StrategyQA leaderboard's limitation and cost considerations of using GPT-3. Experiments are repeated with three random seeds, trained for three epochs with 5e-5 learning rates.\n\n\\begin{table}[t]\n\\centering\n\\small\n\\begin{tabular}{lcccc}\n\\toprule\nSystem & Source & Dev & Test & Hotpot \\\\\n\\cmidrule(lr){1-1}\\cmidrule(lr){2-2}\\cmidrule(lr){3-3}\\cmidrule(lr){4-4}\\cmidrule(lr){5-5}\nT5-Large & S+H & 55.9 & - & 56.0 \\\\\nRoBERTa*-IR & S+B & 65.8 & 64.9 & -\\\\\nGPT-3 & Few & 62.5 & 64.1 & 70.0 \\\\\nGPT-3 CoT & Few & 65.9 & 63.7 & 73.0 \\\\\nOurs & S+H & \\textbf{70.3} & \\textbf{67.4} & 81.0 \\\\\n\\cmidrule(lr){1-5}\nOurs -pretrain & S+H & 67.2 & - & 80.7 \\\\\nOurs -correction & S+H & 62.9 & - & 69.0 \\\\\nOurs -joint & S or H & 65.5 & - & \\textbf{81.3} \\\\\n\\bottomrule\n\\end{tabular}\n\\caption{Accuracy on StrategyQA and HotpotQA. Ours refers to the \\mbox{\\textsc{DecompEntail}}{} pipeline.}\n\\vspace{-0.2cm}\n\\label{tab:strategyqa_test}\n\\end{table}\n\n\\subsection{Results}\nTable~\\ref{tab:strategyqa_test} shows the performances with different baselines on StrategyQA and HotpotQA. On StrategyQA, \\mbox{\\textsc{DecompEntail}}{} outperforms all baseline models by 4\\%, proving that our model benefits the most, and more efficiently, from existing human-annotated resources on complicated questions. On HotpotQA's binary questions, our proposed pipeline outperforms the chain-of-thought variant of GPT-3 by over 8\\%, and the T5 baseline by 25\\%. This shows that explicit decomposition is better than reasoning in a black box, as we achieve better performances with a decomposition model that is over 200 times smaller.\\footnote{There are 770M parameters in T5-large and 175B parameters in GPT-3.}\n\n\\subsection{Ablation Studies}\nWe conduct ablation studies on three variants of the proposed pipeline: without the further pretraining described in \\S\\ref{sec:pretrain} (-pretrain), without the factual correction in \\S\\ref{sec:factual-correction} (-correction), and without the joint learning with both datasets (-joint). Table~\\ref{tab:strategyqa_test} details the performances of ablation models. Similarly, we evaluate the ablation models on StrategyQA's development set with three random seeds and vote with all seeds, but HotpotQA only once due to cost limitations. We see that pretraining with our parallel news corpus accounts for over 3\\% gain on StrategyQA. This aligns with our intuition and intrinsic experiments in \\S\\ref{sec:intrinsic-experiments} because StrategyQA requires advanced question understanding. Factual correction is also significant in our pipeline, which makes a 7\\% difference on StrategyQA and 12\\% on HotpotQA. On the other hand, joint learning contributes to the performances on StrategyQA but not on HotpotQA, which might be because HotpotQA experiments are run with single seeds.\n\n\\subsection{Manual Analysis}\n\\label{sec:manual-analysis}\nWe argue that the core of our improvement is producing proper decompositions instead of the use of GPT-3. We conduct a manual analysis on 20 questions\\footnote{We use the first 20 questions in the dev set that have agreeable annotated facts, without looking at the predictions.} from StrategyQA's dev set and inspect the raw decomposition before factual correction. We find that \\mbox{\\textsc{DecompT5}}{} fails to produce at least one decomposition with all necessary aspects on only two. This suggests that \\mbox{\\textsc{DecompT5}}{} does well in understanding ~90\\% of the questions without GPT-3, even though we need factual correction for the entailment model to produce the correct answer. Moreover, the analysis shows that GPT-3 does not provide anything beyond correcting any factual errors in the statement generated by \\mbox{\\textsc{DecompT5}}{}, as it only sees one decomposition at a time without seeing the actual question. We provide some actual output examples in Fig.~\\ref{fig:manual-analysis-examples} for more insights.\n\n\\section{Conclusion}\nThis work proposes a novel method that extracts distant and incidental signals from parallel news to facilitate general question representation. Such parallel news signals intuitively bridge the reasoning gap in pre-trained language models due to reporting biases. To support this intuition, we train a model named \\mbox{\\textsc{DecompT5}}{} on such distant supervision and show that it improves 20\\%-30\\% on two semantic parsing benchmarks, namely Overnight and TORQUE, that directly evaluate query understanding. With \\mbox{\\textsc{DecompT5}}{} as the basis, we design a well-motivated question-answering pipeline \\mbox{\\textsc{DecompEntail}}{} that follows a decomposition, correction, and entailment scheme. We show that \\mbox{\\textsc{DecompEntail}}{} improves on StrategyQA and HotpotQA by 3.7\\% and 8\\%, respectively.\n\n\\section*{Acknowledgments}\nThis research is based upon work supported in part by the office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), via IARPA Contract No. 2019-19051600006 under the BETTER Program, and by Contract FA8750-19-2-1004 with the US Defense Advanced Research Projects Agency (DARPA). The views expressed are those of the authors and do not reflect the official policy or position of the Department of Defense or the U.S. Government. We also thank the Aristo team at the Allen Institute for AI for valuable support and feedback throughout the entire research process.\n\n\\section{Limitations}\nIn this section, we discuss some of the limitations of our work, and motivate future works.\n\n\\noindent \\textbf{Limited Question Formats.} Our proposed QA pipeline operates on binary \\textit{yes\/no} questions. While binary questions are very general, as most other questions can be re-written into similar forms, such transformations have not been designed or evaluated, which motivates future works.\n\n\\noindent \\textbf{Limited Factual Correction Coverage.} We use GPT-3 as the backbone for our factual correction step. Although it is shown to be effective, it is not as deterministic as Wikipedia-based IR approaches, and we cannot easily interpret why it makes mistakes and understand how to improve.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sec:introduction}\nThe eigenvalues of fourth-order differential operators are central in\ndetermining mechanical properties of rigid bodies. This paper considers\nthe small amplitude out-of-plane vibrations of a thin elastic\nplate~\\cite{RE}. The vibrational frequencies $\\lambda>0$ and modes\n$u(\\mathbf{x})$ satisfy the bi-Laplacian eigenvalue problem\n\\begin{subequations}\\label{eqn:intro}\n \\begin{equation}\n \\label{eqn:introA} \\Delta^2 u = \\lambda u, \\qquad \\mathbf{x} \\in\\Omega; \\qquad \\int_{\\Omega} u^2\\, d\\mathbf{x} = 1,\n \\end{equation}\nwhere $\\Omega \\subset \\mathbb{R}^2$ is a closed planar region\nrepresenting the extent of the plate, $\\mathbf{x} = (x,y)$, and $\\Delta^2 u:=\nu_{xxxx} + 2 u_{xxyy} + u_{yyyy}$. Conditions on the boundary\n$\\partial\\Omega$ are application specific, with a common condition\nbeing that the plate is \\emph{clamped} on its periphery which stipulates that \n \\begin{equation}\\label{eqn:introB} \n u = \\partial_\\mathbf{n} u = 0, \\qquad \\mathbf{x} \\in\\partial\\Omega,\n \\end{equation} \nwhere $\\partial_{\\mathbf{n}}$ is the outward facing normal derivative. A wide\nvariety of engineering systems utilize thin perforated plates in their\nconstruction. Examples include heat exchangers~\\cite{Nilles95,\nVenkatarathnam1996, Krishnakumar2003}, porous elastic materials, and\nacoustic tilings~\\cite{Atalla2007, wang2010, Jaouen2011}. The specific\nplacement of these perforations permits the manipulation of acoustic\nand vibrational properties of the plate while economizing on weight and\nmaterial cost. Homogenization theories have been proposed to replace\nthe natural elastic modulus of the plate with an effective\nmodulus~\\cite{BH,ADK}, however, an averaging approach omits the\npronounced localizing effects that clamping has on vibrational\nmodes~\\cite{FM}.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width = 0.9\\textwidth]{HoleDiagram.pdf}\n\\parbox{0.75\\textwidth}{\\caption{In the limit of vanishing hole radius ${\\displaystyle \\varepsilon}\\to0$, a point constraint $u(\\mathbf{x}_j)=0$ must be enforced at each of the hole centers for $j = 1,\\ldots, M$.\\label{fig:IntroShrink}}}\n\\end{figure}\n\nIn the present work, we consider a finite collection of $M$ defects or punctures on~\\eqref{eqn:introA} with the conditions\n\\begin{equation}\\label{eqn:introC} \nu(\\mathbf{x}_j) = 0, \\qquad j = 1,\\ldots, M.\n\\end{equation}\n\\end{subequations}$\\!$\nThese \\emph{point constraints} arise in singular perturbation studies\nof~\\eqref{eqn:introA} in the presence of $M$ small circular\nperforations of radius ${\\displaystyle \\varepsilon}$ (cf.~Fig.~\\ref{fig:IntroShrink}). As the\nradius ${\\displaystyle \\varepsilon}$ of the perforations shrink to zero, the behavior of the\nlimiting eigenvalue $\\lambda_{{\\displaystyle \\varepsilon}}$ as\n${\\displaystyle \\varepsilon}\\to0$ satisfies~\\cite{KLW,LWK,LHS,CN01}\n\\begin{equation}\\label{HoleBehavior}\n \\lambda_{{\\displaystyle \\varepsilon}} = \\lambda + 4\\pi \\nu \\sum_{j=1}^M \n |\\nabla u(\\mathbf{x}_j)|^2 + \\mathcal{O}(\\nu^2), \\qquad \\nu = -\\frac{1}{\\log{\\displaystyle \\varepsilon}},\n\\end{equation}\nwhere $(\\lambda,u)$ satisfies (\\ref{eqn:introA}-\\ref{eqn:introB}) plus the point\nconstraints~\\eqref{eqn:introC}. In the degenerate case $\\sum_{j=1}^M\n|\\nabla u(\\mathbf{x}_j)|^2 = 0$, equation~\\eqref{HoleBehavior} is not valid\nand a separate limiting form can be derived~\\cite{CN01,KLW}. The fact\nthat the clamping condition on each perforation leaves an imprint as\nthe radius shrinks to zero (Fig.~\\ref{fig:IntroShrink}) implies that no\nmatter how small a perforation is, the vibrational characteristics are\ndistinct from the no hole problem\n\\begin{equation}\\label{HoleFree}\n \\Delta^2 u^{\\star} = \\lambda^{\\star} u^{\\star}, \n \\quad \\mathbf{x} \\in\\Omega; \n \\qquad u^{\\ast} = \\partial_\\mathbf{n} u^{\\star} = 0, \n \\quad \\mathbf{x} \\in\\partial\\Omega; \n \\qquad \\int_{\\Omega} {u^{\\star}}^2 d\\mathbf{x} = 1.\n\\end{equation}\n\n\nThe discontinuous limiting behavior of~\\eqref{HoleBehavior} is\nqualitatively different from the spectral problem for the Laplacian in\nthe presence of small perturbing holes~\\cite{F,KTW,O,WHK,WK}. A\nconsequence of the point constraints~\\eqref{eqn:introC} is that the\neigenfunctions $u(\\mathbf{x})$ are not necessarily smooth but satisfy local\nconditions\n\\begin{equation}\\label{behaviorLocal}\n u(\\mathbf{x}) = \\alpha_j |\\mathbf{x} - \\mathbf{x}_j|^2 \\log|\\mathbf{x} - \\mathbf{x}_j| + \n \\mathcal{O}(1), \\qquad \\mathbf{x}\\to\\mathbf{x}_j; \\qquad j = 1,\\ldots,M,\n\\end{equation}\nwhere the constants $\\{\\alpha_j\\}_{j=1}^M$ reflect the strength of each\npuncture and depend on the domain $\\Omega$ and the clamping locations\n$\\{\\mathbf{x}_j\\}_{j=1}^M$. The difference between the punctured eigenvalues\n$\\lambda$ of~\\eqref{eqn:intro} and the puncture free eigenvalues\n$\\lambda^{\\star}$ of~\\eqref{HoleFree} satisfies (cf.~\\cite{LHS}) \n\\begin{align}\n \\label{EigDifference}\n (\\lambda - \\lambda^{\\star}) \\braket{u,u^{\\star}} = \n -8\\pi\\sum_{j=1}^M \\alpha_j u^{\\star}(\\mathbf{x}_j), \\qquad \n \\braket{u,u^{\\star}} = \\int_{\\Omega} u(\\mathbf{x}) u^{\\star}(\\mathbf{x})\\, d\\mathbf{x}.\n\\end{align}\nThe presence of clamped locations also has a profound localizing effect\non the eigenfunctions. In a rectangular domain with a single clamped\npoint located along the long axis, the effect of clamping\non~\\eqref{eqn:intro} has been observed (cf.~\\cite{FM}) to partition\n$\\Omega$ into two distinct domains on the left and right of the clamping\nlocation, as shown in Fig~\\ref{fig:introClamp}. One aim of this work is\nto numerically investigate the global effects that point constraints\nhave on the eigenfunctions of~\\eqref{eqn:intro} in a variety of\ndifferent planar geometries.\n \n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width = 0.45\\textwidth]{introClamped.png}\n\\parbox{0.75\\textwidth}{ \\caption{The localization of two eigenfunctions\nby a single clamped point, located at the black point, in a rectangular\ndomain. In each case, the eigenfunction is essentially zero on one side\nof the clamping point. See~\\cite{FM} and Sec.~\\ref{sec:rectangle} for\nmore details. \\label{fig:introClamp} }}\n\\end{figure}\n \nFourth-order eigenvalue problems~(Equations~\\eqref{eqn:intro} and\n\\eqref{HoleFree}) exhibit other qualitatively different properties\ncompared to the well-understood Laplacian counterpart. For example, the\nfundamental eigenfunction of~\\eqref{eqn:intro}, ie.~the mode associated\nwith the lowest eigenvalue, is not necessarily single signed~\\cite{CD1,\nCD2, S, Coffman82, Gazzola2010, Grunau2014}. In contrast, the\nfundamental eigenfunction of the Laplacian is always single signed and\nthe corresponding eigenvalue is simple~\\cite{Evans2010,Gilbarg1998}.\nAn elementary example of this phenomenon is the annular domain\n${\\displaystyle \\varepsilon}762.36$, the fundamental eigenfunction\nhas multiplicity two and one nodal line. Also, in domains with a\ncorner, the first eigenfunction may possess an infinite number of nodal\nlines~\\cite{Coffman82}. Many numerical methods have been developed to\ntreat fourth-order eigenvalue problems in view of these\ncharacteristics~\\cite{brown2000, AA, CD, LC2010,\njia-kro-qua2013,Zhao2002}.\n\nThe main goal of this paper is to introduce a novel high-order boundary\nintegral equation method for the numerical solution of~\\eqref{eqn:intro}\nin the presence of a finite collection of punctures \\eqref{eqn:introC}.\nHigh-order methods for computing eigenvalues of the Laplacian and\nHelmholtz equations in two and three dimensions have been developed with\ndomain decomposition methods~\\cite{bet2007, des-tol1983, dri1997},\nradial basis functions~\\cite{pla-dri2004}, boundary integral\nequations~\\cite{bac2003, ste-ung2009, dur-ned-oss2009}, the method of\nparticular solution~\\cite{bar2009, fox-hen-mol1967, kar2001}, the\nDirichlet to Neumann map~\\cite{bar-has2014}, and chebfun~\\cite{drum}.\nThe method of fundamental solutions has also been used to compute\neigenvalues of the biharmonic equation~\\cite{mar-les2005, AA}. However,\nnone of these works consider the eigenvalue problem with clamped points.\nWe extend the work of one of the previous authors~\\cite{LHS} where a\nfinite difference method coupled with an inexact Newton method is used\nto solve~\\eqref{eqn:intro} in the unit circle with symmetrically chosen\nclamped points. Owing to the accuracy and robustness of the boundary\nintegral equation methods, our new method forms third-order solutions\nof~\\eqref{eqn:intro} in smooth two-dimensional geometries, including\nmultiply-connected geometries (Figure~\\ref{fig:couetteModes}), and with\na large assortment of clamping locations. \n\nUsing our new method, we demonstrate the dependence of $\\lambda$ on the\nnumber $M$ and locations $\\{\\mathbf{x}_1, \\ldots,\\mathbf{x}_M \\}$ of the puncture\nsites for a variety of planar regions $\\Omega \\subset \\mathbb{R}^2$.\nIn particular, we investigate two effects that clamped points have on\nthe vibrational properties of plates with various regular and irregular\ngeometries. Our first observation is that by specific location of\npunctures, the vibrational properties can be dramatically altered---in\nparticular, undesirable frequencies of vibration can be tuned out by\ndeliberate location of clamped points at nodal lines of the unclamped\neigenfunction $u^{\\star}$ of \\eqref{HoleFree}. Our second observation,\nextending previous results in~\\cite{FM} for rectangular domains, is\nthat mode confinement occurs in a variety of two dimensional geometries.\n\nThe outline of the paper is as follows. In Section~\\ref{sec:methods} we\ndescribe the details of a boundary integral method for\nsolving~\\eqref{eqn:intro}. In Section~\\ref{sec:algorithms}, the\nimplementation details are discussed and third-order convergence of the\nmethod is verified for a closed-form solution of~\\eqref{eqn:intro}. In\nSection~\\ref{sec:numerics}, we apply our method to a disk, rectangles,\nan ellipse, a non-symmetric shape, and a multiply-connected region.\nFinally, in Section~\\ref{sec:conclusions} we discuss the results and\nareas of future investigations.\n\n\n\\section{Integral equation formulation of the clamped eigenvalue\nproblem}\n\\label{sec:methods}\nIn this section, we first compute and analyze the fundamental solution\nof the modified biharmonic operator $\\Delta^2 - \\lambda$. We then use\nthe fundamental solution to reformulate equation~\\eqref{eqn:intro} as a\nsystem of second-kind boundary integral equations with compact integral\noperators.\n\n\\subsection{Fundamental solution}\n\\label{sec:fundSoln}\nWe require the fundamental solution $G(\\mathbf{x},\\mathbf{y})$ of the modified biharmonic operator satisfying\n\\begin{align*}\n \\Delta^2 G - \\mu^4 G = \\delta(\\mathbf{x}-\\mathbf{y}), \\qquad \\mathbf{x} \\in \\mathbb{R}^2,\n\\end{align*}\nwhere $\\lambda = \\mu^4$. The factorization $\\Delta^2 - \\mu^4 =\n(\\Delta - \\mu^2)(\\Delta + \\mu^2)$, and the fact the fundamental\nsolution is radially symmetric, imposes that $G(\\mathbf{x},\\mathbf{y})$ is a linear\ncombination of the Bessel functions $J_0(\\mu \\rho)$, $Y_0(\\mu \\rho)$,\n$I_0(\\mu \\rho)$, and $K_0(\\mu \\rho)$, where $\\rho = |\\mathbf{x}-\\mathbf{y}|$. Using\na linear combination of the two singular Bessel functions that decay as\n$r \\rightarrow \\infty$, the fundamental solution centered at $\\mathbf{y}$ is\nof the form\n\\begin{align*}\n G(\\mathbf{x},\\mathbf{y}) = c_{1}Y_{0}(\\mu|\\mathbf{x}-\\mathbf{y}|) + c_{2}K_{0}(\\mu|\\mathbf{x}-\\mathbf{y}|).\n\\end{align*} \nTo find the appropriate constants $c_1, c_2$, we use the identities\n$(\\Delta + \\mu^{2})Y_{0}(\\mu|\\mathbf{x}-\\mathbf{y}|) = -4\\delta(\\mathbf{x}-\\mathbf{y})$ and $(\\Delta\n- \\mu^{2})K_{0}(\\mu|\\mathbf{x} - \\mathbf{y}|) = -2\\pi\\delta(\\mathbf{x}-\\mathbf{y})$, and compute\nthe fundamental solution by solving\n\\begin{align*}\n (\\Delta - \\mu^{2})(\\Delta + \\mu^{2})c_{1}Y_{0}(\\mu|\\mathbf{x}-\\mathbf{y}|) + \n (\\Delta + \\mu^{2})(\\Delta - \\mu^{2})c_{2}K_{0}(\\mu|\\mathbf{x}-\\mathbf{y}|)=\n \\delta(\\mathbf{x}-\\mathbf{y}).\n\\end{align*}\nThis calculation reveals that the fundamental solution of $\\Delta^{2} -\n\\mu^{4}$ centered at $\\mathbf{y}$ is \n\\begin{align}\n \\label{FundamentalGreens}\n G(\\mathbf{x},\\mathbf{y}) = -\\frac{1}{8\\mu^{2}} Y_{0}(\\mu|\\mathbf{x}-\\mathbf{y}|) - \n \\frac{1}{4\\pi\\mu^{2}} K_{0}(\\mu|\\mathbf{x}-\\mathbf{y}|).\n\\end{align}\nWe will be using $G$ in an indirect integral equation formulation, and\nthis will require the behavior of the fundamental solution when $\\mathbf{x}\n\\rightarrow \\mathbf{y}$. Without loss of generality, we take $\\mathbf{y}=\\textbf{0}$\nand expand the fundamental solution for small $|\\mathbf{x}|$. Using small\nargument approximations of the Bessel functions\n(cf.~\\cite{abr-ste1964}), we have\n\\begin{align*}\n G(\\mathbf{x},\\textbf{0}) = \n \\frac{|\\mathbf{x}|^2}{8\\pi}\\log |\\mathbf{x}| \\left(1 +\n \\mathcal{O}(|\\mathbf{x}|^{4})\\right)\n +\\frac{|\\mathbf{x}|^{2}}{8\\pi}\\left(-1 + \\gamma +\n \\log\\left(\\frac{\\mu}{2}\\right) + \\mathcal{O}(|\\mathbf{x}|^{4})\\right),\n \\quad \\mbox{as} \\quad |\\mathbf{x}|\\to0,\n\\end{align*}\nwhere $\\gamma \\approx 0.5772156649$ is Euler's constant. As mentioned\nin the introduction, a key behavior of the solution\nof~\\eqref{eqn:intro} is the local behavior~\\eqref{behaviorLocal} near\neach of the defects. Since the fundamental solution satisfies this\nrequired behavior, the solution of~\\eqref{eqn:intro} can be written as\n\\begin{align}\\label{eqn:LinSep}\n u(\\mathbf{x}) = u_S(\\mathbf{x})+ u_R(\\mathbf{x}), \\qquad u_S(\\mathbf{x}) = 8\\pi\\sum_{j=1}^M \\alpha_j G(\\mathbf{x},\\mathbf{x}_j),\n\\end{align}\nwhere $G(\\mathbf{x},\\mathbf{y})$ is given in~\\eqref{FundamentalGreens}. In Section\n\\ref{sec:Newton}, we describe an inexact Newton method to find the\nstrength of the defects $\\{\\alpha_{j}\\}_{j=1}^M$ and the eigenvalues\n$\\lambda$. The decomposition~\\eqref{eqn:LinSep} of the solution as the\nsum of a singular and regular part allows for the local\nbehavior~\\eqref{behaviorLocal} to be precisely enforced while the\nregular part $u_R$ satisfies the homogeneous fourth-order PDE\n\\begin{subequations}\\label{eqn:regularPDE}\n\\begin{gather}\n \\label{eqn:regularPDE_A}\n \\Delta^{2}u_{R} - \\lambda u_{R} = 0, \\quad \\mathbf{x} \\in \\Omega; \\\\[5pt]\n \\label{eqn:regularPDE_B} u_{R} = -u_{S}, \\qquad \\partial_{n} u_{R} = -\\partial_{n} u_{S}, \n \\quad \\mathbf{x} \\in \\partial \\Omega,\n\\end{gather}\n\\end{subequations}$\\!$\nwhere $u_S$ is specified in \\eqref{eqn:LinSep}. We note that in~\\cite{LHS}, the singular part was chosen to be\n\\begin{align*}\n u_{S}(\\mathbf{x}) = \\sum_{j=1}^{M} \\alpha_{j} |\\mathbf{x} - \\mathbf{x}_{j}\n |^2\\log |\\mathbf{x} - \\mathbf{x}_{j}|.\n\\end{align*}\nWhile this choice has the correct local behavior~\\eqref{behaviorLocal},\nit leads to a forcing term in the PDE for $u_R$ that, for a boundary\nintegral equation method, is prohibitive. However, the boundary\nconditions~\\eqref{eqn:regularPDE_B} in our new formulation depends\nnonlinearly on the unknown eigenvalue $\\lambda$.\n\nOnce the functions $u_S$ and $u_R$ are computed, they can be easily\nevaluated at the locations of the clamped points. This is used to\niteratively solve the non-linear equation (Section~\\ref{sec:Newton})\n\\begin{align}\n\\label{eqn:mismatch}\n F(\\mathbf{z}) = \\left[\n \\begin{array}{c}\n u_{S}(\\mathbf{x}_{1}) + u_{R}(\\mathbf{x}_{1}) \\\\\n \\vdots \\\\\n u_{S}(\\mathbf{x}_{M}) + u_{R}(\\mathbf{x}_{M}) \\\\\n \\alpha_{1}^{2} + \\cdots + \\alpha_{M}^{2} - 1\n \\end{array}\n \\right] = \\left[\n \\begin{array}{c}\n 0 \\\\ \\vdots \\\\ 0 \\\\ 0\n \\end{array}\n \\right],\n\\end{align}\nwhere $\\mathbf{z} = (\\alpha_1,\\ldots,\\alpha_M,\\lambda)$. The particular\nnormalization condition $\\sum_{j=1}^M \\alpha_j^2 = 1$ is chosen purely\nfor ease of implementation. Once a solution is obtained, the\neigenfunction can be normalized according to~\\eqref{eqn:introA} or any\nother condition.\n\n\\subsection{Computing the regular solution $u_R$}\n\\label{sec:layer_pots}\nEquation~\\eqref{eqn:regularPDE} is linear and homogeneous, so it can be\nrecast in terms of a boundary integral equation. In this section, we\ndescribe appropriate layer potentials. Since the PDE is fourth-order, a\nsum of two linearly independent layer potentials must be used. The\nregular part $u_R$ is written as\n\\begin{align}\n u_{R}(\\mathbf{x}) = \\int_{\\partial\\Omega} G_{1}(\\mathbf{x},\\mathbf{y})\\sigma_{1}(\\mathbf{y}) ds_{\\mathbf{y}} +\n \\int_{\\partial\\Omega} G_{2}(\\mathbf{x},\\mathbf{y})\\sigma_{2}(\\mathbf{y}) ds_{\\mathbf{y}},\n \\label{eqn:layerPot}\n\\end{align}\nwhere $G_{1}$ and $G_{2}$ are linear combinations of $G$ and its partial\nderivatives. The choice of $G_{1}$ and $G_{2}$ determines the nature of\nthe boundary integral equation which plays a crucial role on the\nconditioning of the linear system that arises after discretization. In\nparticular, $G_{1}$ and $G_{2}$ should be chosen so that the resulting\nboundary integral equation is of the second-kind with compact integral\noperators. This means that the limiting values of the layer potential\nansatz~\\eqref{eqn:layerPot} must have jumps that are proportional to\n$\\sigma_1$ and $\\sigma_2$ as $\\mathbf{x} \\rightarrow \\partial\\Omega$, and the\nkernels must be integrable.\n\nTo find kernels $G_1$ and $G_2$ with these desired results, we use the\nwork of Farkas~\\cite{far1989} who formulated the desired second-kind\nintegral equations for the fourth-order biharmonic equation. For the\nbiharmonic equation with Dirichlet and Neumann boundary conditions,\nFarkas proposed the kernels\n\\begin{align*}\n G_{1}(\\mathbf{x},\\mathbf{y}) &= G_{\\mathbf{n}\\nn\\mathbf{n}} + 3G_{\\mathbf{n}\\boldsymbol{\\tau}\\ttau}, \\\\\n G_{2}(\\mathbf{x},\\mathbf{y}) &= \\Delta G - 2G_{\\mathbf{n}\\nn},\n\\end{align*}\nwhere the normal vector $\\mathbf{n}$ and tangent vector $\\boldsymbol{\\tau}$ are taken with\nrespect to the source point $\\mathbf{y}$. Since the leading order singularity\nof $G$, $\\frac{1}{8\\pi}|\\mathbf{x}|^{2}\\log|\\mathbf{x}|$, is equal to the fundamental\nsolution of the two-dimensional biharmonic equation, the jumps in the\nlayer potential~\\eqref{eqn:layerPot} agree, to a first approximation,\nwith the jumps found by Farkas. In particular, any additional jumps in\n$G_1$ and $G_2$ will result from the higher-order terms in the expansion\nof $G$. Since the higher-order terms contain singularities of strength\nno less than $|\\mathbf{x}|^{6}\\log|\\mathbf{x}|$, no additional jumps will be present\nas long as $G_1$ and $G_2$ do not involve derivatives of order six or\nhigher. Since the derivatives $G_1$ and $G_2$ are no more than\nthird-order, the jumps of $G_1$ and $G_2$ will agree with those computed\nby Farkas.\n\n\\subsection{Explicit expressions of the kernels}\n\\label{sec:kernels}\nFor $\\mathbf{x},\\mathbf{y} \\in \\partial\\Omega$, we require the four kernels\n\\begin{align*}\n G_{11}(\\mathbf{x},\\mathbf{y}) &= G_{1}(\\mathbf{x},\\mathbf{y}), \\\\\n G_{12}(\\mathbf{x},\\mathbf{y}) &= G_{2}(\\mathbf{x},\\mathbf{y}), \\\\\n G_{21}(\\mathbf{x},\\mathbf{y}) &= \\pderiv{}{\\mathbf{n}_{\\mathbf{x}}}G_{1}(\\mathbf{x},\\mathbf{y}), \\\\\n G_{22}(\\mathbf{x},\\mathbf{y}) &= \\pderiv{}{\\mathbf{n}_{\\mathbf{x}}}G_{2}(\\mathbf{x},\\mathbf{y}).\n\\end{align*}\nSubstituting the fundamental solution~\\eqref{FundamentalGreens} into\nthese expressions, and using the identities\n\\begin{align*}\n \\pderiv{}{\\mathbf{n}}(\\mathbf{r} \\cdot \\nn) = -1, \\qquad\n \\pderiv{}{\\mathbf{n}}\\rho = -2 \\frac{\\mathbf{r} \\cdot \\nn}{\\rho^{2}}, \\qquad\n \\pderiv{}{\\mathbf{n}_\\mathbf{x}}(\\mathbf{r} \\cdot \\nn) = 1, \\qquad\n \\pderiv{}{\\mathbf{n}_\\mathbf{x}}\\rho = -2 \\frac{\\mathbf{r} \\cdot \\nn_{\\mathbf{x}}}{\\rho^{2}},\n\\end{align*}\nwhere $\\mathbf{r} = \\mathbf{x} - \\mathbf{y}$, $\\rho = |\\mathbf{r}|$, and similar identities for\nthe tangential derivatives, the kernels $G_{11}$ and $G_{12}$ are\n\\begin{align*}\n G_{11} = -\\frac{1}{4\\pi\\mu^2} &\\left(\n 3\\mu^{3}K_{1}(\\mu\\rho)\\frac{(\\mathbf{r} \\cdot \\nn)}{\\rho} - \n 2\\mu^{3}K_{1}(\\mu\\rho)\\frac{(\\mathbf{r} \\cdot \\nn)^{3}}{\\rho^3} +\n 6\\mu^{2} K_{0}(\\mu\\rho)\\frac{(\\mathbf{r} \\cdot \\nn)}{\\rho^{2}} \\right. \\\\\n &\\left.\n -8\\mu^{2} K_{0}(\\mu\\rho)\\frac{(\\mathbf{r} \\cdot \\nn)^{3}}{\\rho^{4}} - \n 16\\mu K_{1}(\\mu\\rho)\\frac{(\\mathbf{r} \\cdot \\nn)^{3}}{\\rho^{5}} + \n 12\\mu K_{1}(\\mu\\rho)\\frac{(\\mathbf{r} \\cdot \\nn)}{\\rho^{3}}\n \\right) \\\\\n -\\frac{1}{8\\mu^{2}}&\\left(\n -3\\mu^{3}Y_{1}(\\mu\\rho)\\frac{(\\mathbf{r} \\cdot \\nn)}{\\rho} + \n 2\\mu^{3}Y_{1}(\\mu\\rho)\\frac{(\\mathbf{r} \\cdot \\nn)^{3}}{\\rho^3} -\n 6\\mu^{2} Y_{0}(\\mu\\rho)\\frac{(\\mathbf{r} \\cdot \\nn)}{\\rho^{2}} \\right. \\\\\n &\\left.\n +8\\mu^{2} Y_{0}(\\mu\\rho)\\frac{(\\mathbf{r} \\cdot \\nn)^{3}}{\\rho^{4}} - \n 16\\mu Y_{1}(\\mu\\rho)\\frac{(\\mathbf{r} \\cdot \\nn)^{3}}{\\rho^{5}} + \n 12\\mu Y_{1}(\\mu\\rho)\\frac{(\\mathbf{r} \\cdot \\nn)}{\\rho^{3}}\n \\right), \\\\\n G_{12} = -\\frac{1}{4\\pi}&\\left(1 - 2\\frac{(\\mathbf{r} \\cdot \\nn)^2}{\\rho^{2}}\\right)\n \\left(K_{0}(\\mu\\rho) + \\frac{2}{\\mu\\rho}K_{1}(\\mu\\rho)\\right) \n +\\frac{1}{8}\\left(1 - 2\\frac{(\\mathbf{r} \\cdot \\nn)^{2}}{\\rho^{2}}\\right)\n \\left(Y_{0}(\\mu\\rho) - \\frac{2}{\\mu\\rho}Y_{1}(\\mu\\rho)\\right).\n\\end{align*}\nThe expressions for $G_{21}$ and $G_{22}$ require one additional\nderivative of $G_{11}$ and $G_{12}$. For completeness, these lengthy expressions are given in Appendix~\\ref{app:kernels}.\n\n\\subsection{The boundary integral equation}\n\\label{sec:BIE}\n\nAs discussed in Section~\\ref{sec:fundSoln}, all four kernels $G_{ij}$\nhave the same asymptotic behavior as the fundamental solution of the\nbiharmonic equation. Therefore, the boundary integral equation for\n$\\boldsymbol{\\sigma}$ is identical to the boundary integral equation for the\nbiharmonic equation~\\cite{far1989},\n\\begin{align}\n \\label{eqn:bie}\n D(\\mathbf{x})\\boldsymbol{\\sigma}(\\mathbf{x}) + \\int_{\\partial\\Omega} A(\\mathbf{x},\\mathbf{y})\\boldsymbol{\\sigma}(\\mathbf{y}) ds_{\\mathbf{y}}\n =\\gg(\\mathbf{x}),\n\\end{align}\nwhere\n\\begin{align*}\n D(\\mathbf{x}) = \\left(\n \\begin{array}{cc}\n \\displaystyle\\frac{1}{2} & 0 \\\\ -\\kappa(\\mathbf{x}) & \\displaystyle\\frac{1}{2}\n \\end{array}\n \\right),\n\\end{align*}\n$\\kappa(\\mathbf{x})$ is the curvature of $\\partial\\Omega$ at $\\mathbf{x}$, and \n\\begin{align*}\n \\gg = -\\left(\n \\begin{array}{c}\n u_S \\\\ \\partial_{\\mathbf{n}}u_S\n \\end{array}\n \\right),\n \\quad \n \\boldsymbol{\\sigma} = \\left(\n \\begin{array}{c}\n \\sigma_{1} \\\\ \\sigma_{2}\n \\end{array}\n \\right),\n \\quad\n A = \\left(\n \\begin{array}{cc}\n G_{11} & G_{12} \\\\ \n G_{21} & G_{22}\n \\end{array}\n \\right).\n\\end{align*}\nTo apply quadrature formulae, the limiting values of $G_{ij}$ as $\\mathbf{x}\\to\\mathbf{y}$ are\nrequired. These can be found by applying L'H\\^{o}pital's rule to each\nof the four kernels. For $\\mathbf{x}$, $\\mathbf{y}$ on $\\partial\\Omega$ we have\n\\begin{equation}\n \\label{eqn:limits}\n \\begin{aligned}\n \\lim_{\\mathbf{y} \\rightarrow \\mathbf{x}} G_{11}(\\mathbf{x},\\mathbf{y}) &= 0, \\\\\n \\lim_{\\mathbf{y} \\rightarrow \\mathbf{x}} G_{12}(\\mathbf{x},\\mathbf{y}) &=\n \\frac{1}{4\\pi}\\kappa(\\mathbf{x}), \\\\\n \\lim_{\\mathbf{y} \\rightarrow \\mathbf{x}} G_{21}(\\mathbf{x},\\mathbf{y}) &=\n -\\frac{3}{4\\pi}\\kappa(\\mathbf{x})^2, \\\\\n \\lim_{\\mathbf{y} \\rightarrow \\mathbf{x}} G_{22}(\\mathbf{x},\\mathbf{y}) &=\n \\frac{1}{2\\pi}\\kappa(\\mathbf{x}).\n \\end{aligned}\n\\end{equation}\n\n\n\\section{Numerical Methods}\n\\label{sec:algorithms}\nHere we describe a numerical method for solving the boundary integral\nequation~\\eqref{eqn:bie} (Section~\\ref{sec:bie_dis}), applying an\ninexact Newton method for~\\eqref{eqn:mismatch}\n(Section~\\ref{sec:Newton}), and an algorithm for tracing the first\neigenvalue, $\\lambda$, as clamped points are smoothly moved through the\ngeometry $\\Omega$ (Section~\\ref{sec:initial_guess}). \n\n\\subsection{Discretization of the integral equation}\n\\label{sec:bie_dis}\nWe apply a standard collocation method to solve the second-kind boundary\nintegral equation~\\eqref{eqn:bie}. The boundary, $\\partial\\Omega$, is first\ndiscretized at collocation points $\\mathbf{x}_{i}$,\n$i=1,\\ldots,N$. To satisfy the boundary integral equation at these\ncollocation points, we require\n\\begin{align}\\label{eq:integralMain}\n D(\\mathbf{x}_i)\\boldsymbol{\\sigma}(\\mathbf{x}_i) + \\int_{\\partial\\Omega} A(\\mathbf{x}_i,\\mathbf{y})\\boldsymbol{\\sigma}(\\mathbf{y}) ds_{\\mathbf{y}}\n =\\gg(\\mathbf{x}_i).\n\\end{align}\nThe integral in~\\eqref{eq:integralMain} is approximated with the\ntrapezoid rule where the abscissae are the collocation points which\nyields the dense linear system\n\\begin{align*}\n D(\\mathbf{x}_{i}) \\boldsymbol{\\sigma}_i + \\sum_{j=1}^{N} A(\\mathbf{x}_{i},\\mathbf{x}_{j})\\Delta s_{j}\n \\boldsymbol{\\sigma}_{j} = \\gg_{i},\n\\end{align*}\nwhere $\\boldsymbol{\\sigma}_i = \\boldsymbol{\\sigma}(\\mathbf{x}_{i})$, $\\gg_{i} = \\gg(\\mathbf{x}_{i})$, and\n$\\Delta s_{j}$ is the Jacobian of the curve at point $\\mathbf{x}_{j}$. The\nlimiting values from~\\eqref{eqn:limits} are used for the diagonal terms\n$A(\\mathbf{x}_i,\\mathbf{x}_i)$ of the linear system.\n\nThe convergence order of the method depends on the regularity of the\nkernels $G_{ij}$. The regularity of the kernels can be computed by\ntaking a simple geometry, such as the unit circle, fixing $\\mathbf{x}$, and\ncomputing the limit as $\\mathbf{y} \\rightarrow \\mathbf{x}$ of $G_{ij}(\\mathbf{x},\\mathbf{y})$ and\nits derivatives. These calculations reveal that\n\\begin{align*}\n G_{11} \\in C^3, \\qquad G_{12} \\in C^3, \\qquad G_{21} \\in C^1, \\qquad\n G_{22} \\in C^3.\n\\end{align*}\nThe accuracy of the trapezoid rule for a periodic $C^k$ function is\n$k+2$, so we expect third-order convergence because of the $C^1$\nregularity of $G_{21}$. Higher-order accuracy can be achieved by using\nspecialized quadrature~\\cite{alp1999, kap-rok1997} designed for\nfunctions with weak logarithmic singularities. Once values for the\ndensity function $\\boldsymbol{\\sigma}_j$ are computed, we can compute $u_R(\\mathbf{x})$ for\nany $\\mathbf{x} \\in \\Omega$ with spectral accuracy. In particular, we compute\nthe value at the clamped locations with the trapezoid rule to yield that\n\\begin{align*}\n u_{R}(\\mathbf{x}) &= \\int_{\\partial\\Omega} G_{1}(\\mathbf{x},\\mathbf{y})\\sigma_{1}(\\mathbf{y}) ds_{\\mathbf{y}} +\n \\int_{\\partial\\Omega} G_{2}(\\mathbf{x},\\mathbf{y})\\sigma_{2}(\\mathbf{y}) ds_{\\mathbf{y}}\\\\[5pt]\n &\\approx \\frac{2\\pi}{N} \\sum_{j} \\left( G_{1}(\\mathbf{x},\\mathbf{y}_j) \\sigma_{1_j} +\n G_{2}(\\mathbf{x},\\mathbf{y}_j) \\sigma_{2_j}\n \\right)\\Delta s_j.\n\\end{align*}\nIf a target point $\\mathbf{x}$ is sufficiently close to $\\partial\\Omega$, then the\naccuracy of the trapezoid rule will be diminished due to large derivatives\nin $G_{i}(\\mathbf{x},\\mathbf{y})$. In this case, we simply upsample the geometry and\ndensity functions so that sufficient accuracy can be achieved at the\nclamped locations.\n\n\\subsection{Nonlinear solvers}\\label{sec:Newton}\nTo solve the nonlinear equation~\\eqref{eqn:mismatch} for\n$\\{\\alpha_j\\}_{j=1}^{M}$ and $\\lambda$, we apply one of two strategies.\nFirst, in symmetric cases such as the disk geometry, if the clamped\npoints are equidistributed in the azimuthal direction at a fixed radius,\nthen $\\alpha_1 = \\cdots = \\alpha_M$. Therefore, $\\alpha_j =\nM^{-\\frac{1}{2}}$ for $j = 1,\\ldots,M$, and the only parameter remaining\nis $\\lambda$. For such a case, and any scenario in which symmetry\nconsiderations reduce the unknown to just $\\lambda$, a bisection method\ncan be applied to reliably solve~\\eqref{eqn:mismatch} since convergence\nto the desired root is guaranteed for an appropriately chosen initial\ninterval. This method is generally preferred in cases where all the\n$\\alpha_j$ are equal and the single unknown is the eigenvalue itself.\n\nSecond, when symmetry can not be assumed, we apply an inexact Newton's\nmethod to~\\eqref{eqn:mismatch}. In our calculations, the Jacobian\nmatrix $J$ of $F$ is formed by finite difference approximations which we\nhave found to be accurate and efficient.\n\nWe validate the method with the unit disk geometry. A closed-form\nsolution of~\\eqref{eqn:intro} can be developed in the special case $M=1$\nand $\\mathbf{x}_1 =(0,0)$. In a similar manner to the construction of the\nfundamental solution~\\eqref{FundamentalGreens}, a linear combination of\n$K_0$ and $Y_0$ can be chosen to eliminate the logarithmic singularity\nat the origin. Therefore radially symmetric eigenfunctions\nof~\\eqref{eqn:intro} are a combination of $Y_0(\\mu \\rho),$ $K_0(\\mu \\rho)$, \n$J_0(\\mu \\rho)$ and $I_0(\\mu \\rho)$ with $\\rho = |\\mathbf{x}|$. The eigenfunctions that are finite at the\norigin and satisfy $u(0)=0$ and $u(1) = \\partial_\\rho u(1)=0$ are\n\\begin{equation*}\nu(\\rho) = A \\left[J_0(\\mu_{0,n} \\rho) - I_0(\\mu_{0,n} \\rho) -\n\\left(\\frac{J_0(\\mu_{0,n}) - I_0(\\mu_{0,n})}{\\frac{2}{\\pi} K_0(\\mu_{0,n})\n+ Y_0(\\mu_{0,n})} \\right) \\left(\\frac{2}{\\pi} K_0(\\mu_{0,n} \\rho) +\nY_0(\\mu_{0,n} \\rho)\\right) \\right],\n\\end{equation*}\nwhere $A$ is a normalization constant and the eigenvalues $\\lambda_{0,n}\n= \\mu_{0,n}^4$ satisfy the relationship\n\\begin{equation}\\label{disk_exact}\n\\big(J_0(\\mu_{0,n}) - I_0(\\mu_{0,n})\\big) \\left( \\frac{2}{\\pi}\nK_1(\\mu_{0,n}) + Y_1(\\mu_{0,n}) \\right) = \\big(J_1(\\mu_{0,n}) +\nI_1(\\mu_{0,n})\\big) \\left( \\frac{2}{\\pi} K_0(\\mu_{0,n}) + Y_0(\\mu_{0,n})\n\\right).\n\\end{equation}\nThe smallest positive solution of~\\eqref{disk_exact} gives rise to the\neigenvalue $\\lambda_{\\textrm{true}} \\approx 516.9609.$ This solution\nprovides a benchmark against which the efficacy of our numerical method\ncan be verified. We compute the relative error\n$\\mathcal{E}_{\\textrm{rel}}$ between the numerically determined value of\n$\\lambda_{\\textrm{num}}$ and the exact value $\\lambda_{\\textrm{true}}$.\nIn Fig.~\\ref{fig:ErrorDisk}, the numerical error scales $\\mathcal{O}(N^{-3})$\nas the number of boundary points $N$ increases which agrees with our\nexpected third-order convergence. In this example, the bisection method\nwas used, and the strength of the singularity is $\\alpha = 1$.\n\n\\begin{figure}[htbp]\n\\centering\n\\input{ErrorDisk.tikz}\n\\parbox{0.75\\textwidth}{\\caption{The relative error (black) of our\nnumerical method when using the bisection method to find the first\neigenvalue of \\eqref{eqn:intro} with a single clamped point at the\ncenter of the unit disk. A line of slope $-3$ (red) indicates the\nexpected third-order convergence.\\label{fig:ErrorDisk}}}\n\\end{figure}\n\n\\subsection{Initialization, parameterization of puncture patterns, and\narclength continuation}\n\\label{sec:initial_guess}\n\nThe solution of the nonlinear system~\\eqref{eqn:mismatch} by Newton's\nMethod relies on good initial iterates. In addition, a careful selection\nof the initial guess is necessary to reliably locate the lowest mode of\nthe punctured problem~\\eqref{eqn:intro}. For the unit circle, we start\nwith the clamped points at the center of the circle and initialize\nNewton iterations for~\\eqref{eqn:mismatch} with the known eigenvalue\n$\\lambda \\approx 516.9609$ for a single clamped point at the origin.\nFor other geometries, we start the clamped point near $\\partial\\Omega$. In\nthis scenario, equation~\\eqref{eqn:mismatch} is initialized with a mode\nof the unclamped problem~\\eqref{HoleFree} calculated from a low-accuracy\nfinite element approximation~\\cite{KI78}. Once a solution\nof~\\eqref{eqn:intro} has been generated, the punctures are gradually\nmoved, and~\\eqref{eqn:mismatch} is repeatedly solved until the punctures\noccupy a specified target set.\n\nIn the examples that follow, we compute eigenvalues $\\lambda =\n\\lambda(r)$ of~\\eqref{eqn:intro} for families of puncture patterns\ndescribed by a single parameter $r\\geq0$. For reasons of efficiency\nand to provide robustness to the Newton iterations, we use arc-length\nadaptively to focus resolution at sharp peaks of the curve as compared\nto the surrounding areas. The algorithm to find points on the curve\n$\\lambda = \\lambda(r)$ is initialized with a relatively large step size\n$\\mathrm{d}r$ with the concavity monitored until proximity to an extrema\nis detected. Once an extrema of the curve is detected, $\\mathrm{d}r$ is\nreduced based on the current slope up to a minimum allowable step size. \n\n\n\n\\section{Numerical Examples}\n\\label{sec:numerics}\n\nIn this section we demonstrate the effectiveness of the method on\nregular and irregular domains. To understand the role of clamping in the\neigenvalue problem, and interpret the results obtained with our\nnumerical method, we recall from~\\eqref{EigDifference} that\n\\begin{align*}\n (\\lambda - \\lambda^{\\star}) \\braket{u,u^{\\star}} = \n -8\\pi\\sum_{j=1}^M \\alpha_j u^{\\star}(\\mathbf{x}_j), \\qquad \n \\braket{u,u^{\\star}} = \\int_{\\Omega} u(\\mathbf{x}) u^{\\star}(\\mathbf{x})\\, d\\mathbf{x},\n\\end{align*}\nwhich relates the modes $(\\lambda,u)$ of~\\eqref{eqn:intro} to the\nunclamped modes $(\\lambda^{\\star},u^{\\star})$ of~\\eqref{HoleFree}. In\neach of the examples that follow, we will use a low accuracy finite\nelement method~\\cite{KI78} to obtain the required solutions\nof~\\eqref{HoleFree}. \n\n\n\n\\subsection{Unit circle}\nThe relationship~\\eqref{EigDifference} shows how the distinct\neigenvalues and eigenfunctions of the clamped and unclamped\nproblems,~\\eqref{eqn:intro} and~\\eqref{HoleFree}, respectively, are\nrelated. For each domain it is therefore important to consider the\nsolutions $(\\lambda^{\\star},u^{\\star})$ to understand the effect of\npuncture configurations.\n\nFor the unit disk case, the solutions of problem \\eqref{HoleFree} are\nfound by first factorizing $\\Delta^2 - \\mu^4 = (\\Delta -\\mu^2)(\\Delta +\n\\mu^2)=0$ which indicates that the basis for the space of eigenfunctions\nis\n\\begin{equation*}\n e^{im \\theta} \\{ J_m(\\mu_{m,n} \\rho), Y_m(\\mu_{m,n} \\rho), K_m(\\mu_{m,n} \\rho), I_m(\\mu_{m,n} \\rho) \\}, \\qquad \\mu_{m,n} = \\lambda_{m,n}^{1\/4},\n\\end{equation*}\nwhere $\\rho = |\\mathbf{x}|$. The indices $m = 0, \\pm 1, \\pm2, \\ldots$ indicate\nthe angular wavenumber (and number of angular nodal lines) where as\n$n=0,1,2,\\ldots$ counts the number of radial nodal lines for each\nwavenumber. In the unclamped problem~\\eqref{HoleFree}, the smooth\neigenfunctions satisfying $u^{\\star} = \\partial_{\\rho} u^{\\star} = 0$\non $\\rho=1$ are\n\\begin{align*}\n u^{\\star}_{m,n}(\\rho,\\theta) = e^{im\\theta} \\left[ J_m(\\mu^{\\star}_{m,n}\\rho) - \n \\frac{J_m(\\mu^{\\star}_{m,n})}{I_m(\\mu^{\\star}_{m,n})} \n I_m(\\mu^{\\star}_{m,n}\\rho) \\right],\n\\end{align*}\nwith the eigenvalues $\\mu^{\\star}_{m,n}$ determined by the relationship\n\\begin{align}\\label{NoHolesExact_b}\nJ'_m(\\mu^{\\star}_{m,n})I_m(\\mu^{\\star}_{m,n}) = I'_m(\\mu^{\\star}_{m,n}) J_m(\\mu^{\\star}_{m,n}).\n\\end{align}\nThe first four eigenvalues $\\lambda^{\\star}_{m,n} =\n(\\mu^{\\star}_{m,n})^4$, found from the numerical solution\nof~\\eqref{NoHolesExact_b}, are\n\\begin{equation}\\label{eigsDiskPFree}\n\\lambda^{\\star}_{0,0} = 104.4, \\qquad \\lambda^{\\star}_{1,0} = 452.0, \\qquad \\lambda^{\\star}_{2,0} = 1216.4, \\qquad \\lambda^{\\star}_{0,1} = 1581.7.\n\\end{equation}\nIn Fig.~\\ref{fig:NoHoles}, the first few eigenfunctions are plotted with the nodal lines along which $u^{\\star}=0$ highlighted.\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width = 0.7\\textwidth]{EigsDisc.pdf}\n\\parbox{0.75\\textwidth}{\\caption{The contour lines of the first $15$\nmodes of the unclamped problem~\\eqref{HoleFree} on the unit disk. The\nnodal lines ($u^{\\star}=0$) are plotted in black. Eigenfunctions are\nrepeated according to their multiplicity. \\label{fig:NoHoles}}}\n\\end{figure}\nFor punctures away from the origin, we seek solutions\nof~\\eqref{eqn:intro} with parameterized puncture sets to minimize the\nnumber of unknowns over which nonlinear iterations are processed. In the\ndisk case, our first example is a single ring of punctures given\nexplicitly as\n\\begin{align}\n \\label{singleRing}\n \\mathbf{x}_j = r \\left( \\cos\\frac{2\\pi j}{M}, \n \\sin \\frac{2\\pi j}{M} \\right), \\qquad j = 1,\\ldots, M.\n\\end{align}\n\nThere is now a single parameter $r$ over which various configurations\ncan be investigated from $03,000$) set of (mostly) disk giant stars with accurate\nline-of-sight velocities.\n\nAn equally rewarding, but perhaps yet a more challenging approach is\nto gauge the Galactic escape speed $v_{\\rm esc}=\\sqrt{2|\\Phi|}$ by\nanalyzing the tail of the stellar velocity distribution. The results\nare sensitive to the quality of the distance and the proper motion\ndata, in particular, imperfect proper motion measurements are so\ndetrimental that, normally, they are avoided altogether. Instead, a\nvelocity distribution function is chosen, whose exact shape is\ncontrolled by a small number of parameters that get simultaneously\nconstrained in the process of the likelihood maximization. For\nexample, using a relatively small sample (16) of high-velocity stars\nprovided by the earlier releases of the RAVE survey, \\citet{Smith2007}\nmeasure the local escape speed. Conveniently, given the Galactic\nescape speed and assuming the contributions of the bulge and the disk\nto the total potential, the mass and the concentration of the Milky\nWay's halo also can be extracted. The analysis by \\citet{Smith2007}\nseems to prefer the Galaxy with the mass as low as $0.9 \\times 10^{12}\nM_{\\odot}$ and the concentration as high as 24. While the\napplicability of both the circular speed and the escape speed\ntechniques is restricted to the inner Galaxy, the latter has the\nadvantage of probing the Galactic mass out of the disk plane.\n\nMost of the Milky Way's mass lies beyond the extent of the disk, hence\nat large Galacto-centric distances, a different approach is required.\nGiven enough mass tracers (stars or satellites) in a wide range of\nlocations throughout the Galaxy, the total mass profile can be\nobtained by means of Jeans modelling of the tracer kinematics\n\\citep[see e.g.][]{Battaglia2005}. The terms that enter the spherical\nJeans equation are: the tracer density, the tracer velocity dispersion\nand the tracer velocity anisotropy. At large distances, only one of\nthese might be available, namely the line-of-sight velocity\ndispersion. Making the Jeans analysis of the far reaches of the\nGalactic halo possible clearly falls within the realm of Galactic\nArchaeology which can both deliver the most distant tracers as well as\nconstrain the overall tracer density distribution. The stumbling\nblock, however, is the scarcity of tracers with the tangential\ncomponents of the velocity measured. As a consequence, the anistropy\nis normally treated as a nuisance parameter since the most datasets\navailable lack in accuracy and breadth to constrain it. Even with the\narrival of Gaia, the situation will only improve for the nearby\nobjects, leaving the distant ones wanting in more precise proper\nmotions. While assigning anistropy to a tracer population is a\nsolution far from ideal, presently, it is the Jeans modelling together\nwith its variants that provides the most stringent constraints on the\ntotal mass of the Milky Way \\citep[e.g.][]{Xue2008}.\n\nFinally, a new, conceptually different method to probe the matter\ndistribution in the Galaxy is now coming of age. Compared to the three\napproaches discussed above, it does not rely at all on the\ninstantaneous kinematic properties of large samples of tracers, and\nthus, for example, needs no assumption of their velocity\nanisotropy. Stellar {\\it tidal streams} are shown to align closely\nwith the obit of their disrupting (or disrupted) progenitor and\ntherefore give an almost direct way of measuring the underlying\npotential. Recently, the power of the method has been demonstrated\nbeautifully by \\citet{Koposov2010} who, using the 6D data of the GD-1\nstream, measured the Galactic rotation curve locally. This type of\nanalysis can, in principle, be extended to distances beyond the\npredicted Galaxy's scale radius $r_s$. The prime source of degeneracy\nin recovering the Galactic potential using tidal tails, is the length\nof the stream available. However, to date, for several distant streams\nthere exists sufficient data covering tens \\citep[Orphan Stream with\n the maximal distance of $\\sim 50$ kpc][]{Belokurov2007a,\n Newberg2010} or even hundreds of degrees \\citep[Sagittarius Stream\n with the maximal distance of $\\sim 100$ kpc, e.g.][]{ Majewski2004,\n Newberg2003,Belokurov2006b, Yanny2009, Belokurov2013}. Given the\nmagnitude limit of the on-going imaging surveys like SDSS or\nPan-STARRS, for stellar streams to be detected so far out in the halo,\nthe progenitor's luminosity, and therefore mass, ought to be\nsubstantial. This bias implies that the currently known distant\nstreams can not be appropriately modeled using simple orbit\napproximation, the circumstance that now can be mitigated with the\narrival of more sophisticated modeling techniques\n\\citep[e.g.][]{Eyre2011,Sanders2013}\n\n\\subsubsection{Mass assembly history and environment}\n\nThe computational expense of running numerical simulations of Galactic\nhalos at the resolution adequate to capture the properties of the halo\nsub-structure is prohibitively high. Hence, the comparison between DM\nsub-halos and the observed dwarfs has been based on the analysis of\nonly 8 N-body simulations: a sample of 6 Aquarius halos\n\\citep{Springel2008}, complemented by halos of Via Lactea II\n\\citep{Diemand2008} and GHalo \\citep{Stadel2009}. For this reason, the\nhost-to-host variation of the dark and the luminous sub-structure\nremains largely un-studied. As well as improving the resolution and\nthe speed of the simulations, there is an ongoing effort to quantify\nthe complex diversity of structures forming within $\\Lambda$CDM with a\nhandful of key parameters, e.g. host halo mass, shape of the accretion\nhistory and significance of the overdensity of the local\nvolume. These, of course, are inter-related: the mass of the DM halo\nhosting a Milky Way galaxy at redshift $z=0$ is the sum total over its\naccretion history, which in turn is dictated by the whereabouts of the\nhalo within the cosmic Large Scale Structure. While the importance of\nnot knowing such an elementary property of the Galaxy like its mass is\nnow accepted, the impact of the location of the Milky Way within the\nlarger cosmic structure and the details of its accretion history are\njust beginning to be investigated.\n\nToday, there exist two intriguing constraints on the Milky Way's\naccretion history. First is the observation that the Galactic disk\nprobably has to survive intact for some 7-10 Gyr \\citep[e.g. Figure 18\n of][]{Burnett2011}. This, therefore, potentially excludes any\nsignificant mergers between $z \\sim 1$ and now. Second is the new\nobservational and numerical evidence for the late infall of the\nMagellanic Clouds \\citep[e.g.][]{Besla2010}. This signifies the end of\nthe quiescent phase in the Galactic accretion history and can be\nexploited to place useful constraints on the mass assembly of the\nGalaxy \\citep[e.g.][]{Busha2011}. What happened before the quiescent\nphase, why did it begin and why did it end? How common is this\nparticular shape of the {\\it mass assembly history} (MAH) amongst\nother disk galaxies of similar mass? Was the early accretion\ndominated by small satellite infall and was it synchronized? Or\nperhaps, was the bulk of the Galactic matter instead acquired in one\nor two mergers with massive nearby fragments? Unfortunately, these\nquestions remain largely unanswered and therefore, a variety of loose\nends continues to confuse the current picture of the Galaxy formation\nand muddle the modelling of the nearby dwarfs. For example, if many\nsmall satellites are accreted early on, enough should survive and be\ndetectable today. On the contrary, massive mergers usually lead to an\nentirely different outcome: in this case, the dynamical friction is\nstrong enough to slow the dwarf down thus boosting its plunge into the\ninner Galaxy where it is quickly disrupted. These two scenarios can be\nidentical in terms of the epoch of accretion and the total mass\naccreted, yet they can produce dramatically different dwarf satellite\npopulations at $z=0$.\n\nAn attempt to quantify the amplitude of the host-to-host scatter in\nthe properties of artificial Galactic dwarfs using analytic models is\npresented in \\citet{Purcell2012}. The p\\`iece-de-r\\'esistance of the\u2260\u2260\u2260\u2260\nmethod is the Monte-Carlo sampling of an arbitrary large number of\ndifferent accretion histories \\citep[as described\n in][]{Zentner2005b}. Using this technique, it can be demonstrated\nthat the scatter in the possible MAHs is naturally large enough for\nthe Milky Way-like halo to host a satellite population consistent with\nthe observed one in 10\\%-20\\% of cases. These results, within the\nlimitations of the method, shed light onto the statistical\nsignificance of the ``too-big-to-fail'' problem\n\\citep{Boylan-Kolchin2012}: there does not have to be a serious excess\nof massive invisible sub-halos in the Galaxy. Interestingly, together\nwith the recently invoked lower Galaxy mass\n\\citep[e.g.][]{Vera-Ciro2013} and the strong stellar feedback\n\\citep[e.g.][]{Brooks2013}, this is now the third solution for the\npotential problem identified by \\citet{Boylan-Kolchin2012}. It would\nseem that if all three methods are as efficient as described, there\ncould be very few satellites left around the Galaxy! It is, therefore,\nthe most urgent task for the Galactic Archaeology to provide new\nobservational constraints of the Milky Way's accretion history through\nstudies of the spatial and the chemo-dynamical distributions of the\nancient stellar halo populations.\n\nThe Milky Way is not a solitary field spiral: together with its\nneighbor of approximately the same mass, Andromeda and its satellites,\nit makes up the small slightly over-dense region of the Universe known\nas the Local Group of galaxies. The so-called {\\it assembly bias}\nstipulates an excess of probability of finding a massive satellite\nsub-halo around hosts situated in higher density regions as compared\nto those in under-dense environments\n\\citep[e.g.][]{Wechsler2006}. Possibly, this effect could go some way\nto explaining the presence around the Milky Way satellites as massive\nas the Magellanic Clouds. According to \\citet{Busha2011b}, while for\nthe field halo of Milky Way-like mass, the probability to host LMC\/SMC\npair is of order of $5\\%$-11$\\%$, having another host halo of similar\nmass in the vicinity boosts it up to 25$\\%$. This is good news, but\nare these sub-halos on their first (or perhaps second) passage around\nthe simulated Galaxies as the Milky Way observations seem to indicate?\nA unique investigation is described in \\citet{Forero-Romero2011} who\nuse a suite of so-called constrained simulations of the Local Group\n(CLUES, see http:\/\/www.clues-project.org\/) in which the broad-brush\nfeatures of the Milky Way-Andromeda pair are reproduced, to study the\nassembly history of either host halo. They find that i) both galaxies\nhad their last significant accretion event some 10-12 Gyr ago, and\nthat ii) this particular common accretion history is quite rare (from\n1$\\%$ to 3$\\%$) amongst the pairs of host halos in Bolshoi\nsimulation. This conclusion appears to be in contradiction with the\nstudies in which the Clouds are just being accreted.\n\n\\subsection{Tidal origin of the local dwarf galaxies}\n\nIt is inspiriting that there exists at least one alternative, and,\nimportantly, testable scenario of the formation of dwarf satellites in\nand around the Milky Way. \\cite{Lynden-Bell1976} first pointed out the\nproximity of the several of the Galactic dwarfs to the LMC's orbital\nplane as defined by the gaseous stream leading the Cloud. The\nhypothesis then put forward is of a Greater Magellanic Galaxy that had\nbeen torn apart as it interacted with the Milky Way, giving birth to\nthe Large and Small Clouds, as well as to a litter of smaller\ndwarfs. A quarter of a century later, with the measurement of the\nspace velocities of the satellites in hand, the surprising\njuxtaposition of the orbital planes of the LMC, SMC, UMi and Dra is\nconfirmed \\citep[e.g.][]{Palma2002}. This motivates \\citet{Kroupa2005}\nto claim that the observed distribution of the Galactic satellites is\ntoo anisotropic to fit seamlessly within the CDM paradigm. In the\nauthors' opinion, such alignment (dubbed later as the ``disk of\nsatellites'', DoS) is prohibitively rare in computer simulations of\ngalaxy formation in the Universe full of Dark Matter: the accreted\nsub-halos should have had enough time to relax in the Milky Way's\npotential, thus erasing any signs of coherence.\n\nIt is, however, certainly too naive to believe that in $\\Lambda$CDM\nUniverse, the distribution of dwarf satellites around a Milky Way-like\nhost is always isotropic. \\citet{Zentner2005} show that through the\ncombined effect of i) filamentary accretion and ii) the alignment of\nsub-halo orbits with the major axis of the triaxial host halo, the\nprobability of choosing the simulated sub-halo populations from an\nisotropic distribution is as low as $10^{-4}$. The success of these\nsimulations in assembling anisotropic satellite distributions is\ncurious since these particular host galaxies do not posses disks. The\npresence of a baryonic disk should help to get rid of the satellites\norbiting near it, thus making the satellite distribution more\nanisotropic. \\cite{Libeskind2005} use a slightly different numerical\nsetup to generate their host halos as well as their satellite\ngalaxies but come to the same conclusion: a good fraction of the\nbrightest satellites is bound to end up in a plane-like arrangement\nhaving arrived to the host through 1 or 2 primary filaments.\n\nWhile \\cite{Lynden-Bell1976} only briefly mentions a possible scenario\nin which the parent galaxy dissolves to leave several smaller\nfragments behind to be observed today as dwarf satellites,\n\\citet{Kroupa2005} go further to suggest the exact mechanism\nresponsible for their production. They speculate that the creation and\nthe subsequent compression of the gaseous tidal tails is followed by\ntail fragmentation and active star-formation. It is claimed that the\nstellar systems born in this violent process, also known as {\\it tidal\n dwarf galaxies} can survive long enough. If they do, their\nanisotropic distribution on the sky is merely the consequence of the\nproximity of their birthplaces in the tidal tail that is now\nvanished. This dSph formation mechanism advocated by\n\\citet{Kroupa2005} harks back to their earlier dynamical work\n\\citep{Kroupa1997}, in which a quasi-stable solution for a dSph-like\nDM-free stellar system is discovered. With the help of a suite of\nsimple N-body simulations, it is argued that a tidal dwarf galaxy in\nthe last throws of disruption can posses apparent surface brightness\nand velocity dispersion not unlike those observed in dSphs around the\nMilky Way. As \\citet{Kroupa1997} argues such high velocity dispersions\nwould lead to over-estimated masses and therefore to highly inflated\nmass-to-light ratios, while the actual $M\/L$ remains quite\nlow. \\citet{Metz2007} re-run the experiment and show that their\nsimulated tidal dwarf remnants and the Galactic dwarfs can look alike,\nespecially within the region of the structural parameter space\noccupied by the ultra-faint satellites. Even though the fact of the\nexistence of such out-of-equilibrium satellite configurations in\nnumerical simulations is established, as of today, no evidence has\nbeen found that they can persevere for longer than a 1-2 Gyrs\n\\citep[see e.g.][]{Casas2012}.\n\nAs the census of the sub-structure in the halos of the Milky Way and\nthe Andromeda galaxies is being filled in fast, the growing sample of\nsatellites and streams allows for more rigorous tests of possible\nanistropies in their spatial and kinematic distributions. For example,\n\\citet{Pawlowski2012} extend the study of the Galactic ``disk of\nsatellites'' to include the known stellar and gaseous streams. Their\nargument in support of the previously found DoS orientation is that 7\nout the 14 streams they analyse align well with the disk. With this\nobservation in hand, they claim that it is not merely the ``disk of\nsatellites'' that surrounds the Milky Way, but rather a ``vast polar\nstructure'' (VPOS) appears to dominate the Galactic sub-system\ndistribution at all radii. Once again the conclusion is reached that\nthe presence of such structures is in contradiction with the $\\Lambda\nCDM$ theory. Before the probability of encountering this so-called\nVPOS is worked out for the current galaxy formation paradigm, it is\nworth noting that while the number of the streams contributing to it\nseems large (a half of the total considered), their combined mass is\nminuscule. Therefore, these (in particular stellar) streams contribute\nclose to nothing to the significance of the supposed anisotropy in the\nGalactic halo.\n\nCuriously, in the case of the M31, \\citet{Ibata2013} exhibit plausible\nevidence for the planar alignment of nearly half of the dwarf\nsatellites. Moreover, these appear to be co-rotating around Andromeda\nin a semblance of a disk, which contains the line connecting the host\ngalaxy and the Milky Way. This discovery is responsible for another\nattempt to debunk $\\Lambda CDM$ this time by \\citet{Hammer2013} who\ndevelop their earlier idea of a major merger at the M31 location\n\\citep[see e.g.][]{Hammer2007} and suggest that most of the dwarf\ngalaxies, including the Magellanic Clouds have formed as a result of\nthis upheaval. \n\nOverall, it seems that the hypothesis in which dwarf satellites are\nborn in major merger events can give a convincing account of the\nobserved distribution of satellites on the sky. However, currently the\ntheory does not stack up against the entirety of the observational\nevidence, both locally (e.g. the extended star-formation histories and\nthe extremely old stellar populations of the Milky Way dwarfs) as well\nas outside the Galaxy (e.g. low major merger rates for L$_*$ hosts).\n\n\\section{Archaeologist's toolbox}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.93\\linewidth]{tracers.pdf}\n\\caption{Stellar tracer selection in the SDSS database. {\\it Left:}\n Density of stars in the plane of surface gravity $\\log g$ and\n effective temperature $T_{\\rm eff}$ for $\\sim 180,000$ DR8 spectra\n with $15 < g < 17.5$. {\\it Right:} Stars with spectroscopy from the\n left column are plotted on the plane of $u-g$ and $g-r$ color. {\\bf\n Top:} overview of the sample, darker shades of grey indicate\n higher density. {\\bf Middle:} Selecting the tracers. BHB (blue),\n Blue Straggler (violet), MSTO (green) and M-giant (red) stars are\n chosen in the left column based on their temperature and surface\n gravity. Density of selected stars is then over-plotted in $u-g$,\n $g-r$ space using the same color scheme. {\\bf Bottom:} Metallicity\n distribution in the sample. This shows false RGB images (left and\n right) constructed with 3 grey-scale density distributions of stars\n picked based on their $[Fe\/H]$. Red component is for metal-rich stars\n with $-0.75 <[Fe\/H]< 0$, green (intermediate) $-1.5 <[Fe\/H] <\n -0.75$ and blue (metal-poor) $-3 <[Fe\/H] <\n -1.5.$} \\label{fig:tracers}\n\\end{figure}\n\nLow-mass stars (around $\\sim 1 M_{\\odot}$) shine for billions of\nyears, and therefore keep the record of historical events in the Milky\nWay. To be able to read into the Galactic diary, collections of stars\nwith comparable chemistry, age or, at least, similar luminosity class\nmust be identified. The distributions of such {\\it stellar tracers} in\ntwo (positions on the sky), three (place on the sky and along the line\nof sight), four (location in space and in radial velocity) or even\nseven (configuration space and velocity space coordinates together\nwith chemistry) dimensions are then measured to benchmark, with some\nhelp from Galactic Dynamics, the theories of structure formation.\n\nThe Galaxy endlessly churns the pieces of smaller satellites it\nacquires, continuously smoothing the spatial densities of the debris.\nThe rate at which the Galactic blender operates decreases from the\ncentre outwards. Far out in the halo, where the orbital periods reach\ngiga-years, unbound stellar sub-structures can maintain superficial\nspatial coherence for eons. However, closer to the Solar radius, extra\n(dynamical or chemical) information is required to filter out\nparticular debris from the smooth mess. Therefore, the interplay\nbetween the number of useful stellar tracers, the information content\nper star, and the overall volume probed is what determines the\nrelevance of a Galactic halo survey.\n\nIn the not-so-distant future, with the data from the Gaia astrometric\nspace mission and a host of planned large-area spectroscopic surveys,\nit should be possible to paint the unambiguous picture of the events\nthat took place in the Galaxy between redshift $z=20$ and redshift\n$z=0$. At the moment, we will have to make do with what we have\ngot. The observational advances in Galactic Archaeology made in the\nlast few years happened thanks to a handful of wide area imaging\nsurveys, namely 2MASS and SDSS, and massive spectroscopic efforts such\nas Segue and RAVE.\n\nOf the several sky surveys of past decade, the SDSS appears to have\nbeen operating in a sweet spot: it turns out a 54 second exposure is\nlong enough to reach Main Sequence stars at distances of several tens\nof kpc from the Sun, and thus yield an unprecedented 100 million\nobject database; yet short enough to see plenty of the sky in limited\namount of time. The now classic $ugriz$ filter set encodes the stellar\nspectral energy distribution (SED) into a compact form, but preserves\nenough frequency diversity to study in detail a variety of stellar\npopulations. This section therefore mostly concentrates on the\nobserved properties of the Galactic stellar halo as seen by the SDSS\n(and its extensions) outside the Solar radius.\n\n\\subsection{Stellar tracers of the Galactic halo in the SDSS}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.99\\linewidth]{bhb_msto.pdf}\n\\caption{Absolute magnitude of stellar tracers. {\\it Left:} Blue\n Horizontal Branch star candidates in 11 Galactic star clusters. Each\n dot represents one BHB, stars from different clusters are marked\n with different color. Cluster name and the color convention are\n shown in the inset. Once a model for the slight variation of the\n luminosity with color has been applied, the absolute magnitude of a\n BHB star can be estimated with accuracy $\\lesssim 0.1$ mag. {\\it\n Right} Stars with $g-r < 0.4$ in 11 Galactic star clusters. Apart\n from the variation by $\\pm 0.5$ mag around the mean magnitude of the\n turn-off $M_g\\sim 4$ due to age and metallicity differences between\n clusters, stars on the MS with lower luminosity as well as Sub-giant\n stars bright with higher luminosity are picked up by this $g-r$\n cut. This results in the overall asymmetric spread of $\\sim 3$ mag\n in $M_g$.}\n\\label{fig:bhb_msto}\n\\end{figure}\n\nThere are at least three species of stellar tracers available in the\nSDSS photometric data that a Galactic archaeologist can put to\nwork. In order of decreasing population size, increasing luminosity\nand decreasing contamination, these are: Main Sequence Turn Off (MSTO)\nstars, Blue Horizontal Branch (BHB) stars and M giant\nstars. Figure~\\ref{fig:tracers} gives the whereabouts of each of these\nthree in the space of stellar atmosphere parameters and the space of\nbroad-band colors.\n\nThe left column of the Figure shows the logarithm of density of a\nsample of bright ($150^{\\circ}$ at high $b>30^{\\circ}$ there does not exist a\ncounter-part to the Virgo overdensity.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.99\\linewidth]{fos_dr9_galactic.jpg}\n\\includegraphics[width=0.32\\linewidth]{fos_dr9_galactic_components_0.jpg}\n\\includegraphics[width=0.32\\linewidth]{fos_dr9_galactic_components_1.jpg}\n\\includegraphics[width=0.32\\linewidth]{fos_dr9_galactic_components_2.jpg}\n\\caption{Same as Figure~\\ref{fig:fos_gal} but in Galactic\n coordinates. Galactic $l=0^{\\circ}, b=0^{\\circ}$ is at the centre of\n the Figure.}\n\\label{fig:fos_gal}\n\\end{figure}\n\n\\subsubsection{The big 4}\n\\label{sec_big4}\n\nThe Sagittarius Stream, the Galactic Anti-center Stellar Structure,\nthe Virgo and the Hercules-Aquila Clouds are the four largest stellar\nstructures in the halo of the Milky Way. Out of these four, only the\nSgr Stream lies predominantly outside the Galactic disk making it\npossible to estimate its total extent and the overall stellar\nmass. The Stream consists of two tails, the leading and the trailing,\nflowing from the Sgr dwarf galaxy, which currently lies on the\nopposite side of the Galaxy, behind the bulge, several degrees under\nthe disk. The dwarf is falling onto the disk and has just passed its\npoint of the nearest approach at $\\sim$15 kpc from the Galactic\ncenter. The two tails appear bifurcated \\citep[see\n e.g.][]{Belokurov2006a,Koposov2012} and extend each at least as far\nas $\\sim$ 180$^{\\circ}$ away from the progenitor (see\nFigures~\\ref{fig:fos_equ} and \\ref{fig:fos_gal}). The leading tail is\ntraced as far as 50 kpc from the Galactic center, while the\napo-galacticon of the trailing debris is probably as far as 60-100\nkpc. Both the Sgr remnant and the stream host a range of stellar\npopulations with different ages and metallicities. In particular,\nalong the stream, a substantial population gradient is observed\n\\citep[e.g.][]{Chou2007,Yanny2009,Bell2010,Chou2010,Keller2010,Carlin2012},\nwhich, within any sensible model of the dwarf disruption, would mean a\nsimilarly pronounced abundance and age gradient in the\nprogenitor. Using a variety of stellar tracers across the sky,\n\\citet{No2010b} map the Sgr debris and, correcting for the distance\nand the abundance gradients estimate the total stellar luminosity of\nthe progenitor prior to disruption. They find that, before merging\nwith the Galaxy, the dwarf was as bright as $1.5\\times 10^8 M_{\\odot}$\nor just under $M_V \\sim -16$, but today it has lost as much as $70\\%$\nof its stars to the Galactic tides.\n\nThe Virgo Cloud can be seen as green haze directly underneath the Sgr\nStream at around $RA\\sim 12^h$. While early glimpses of this structure\nare reported in several studies, based on the SDSS DR4 imaging data,\n\\citet{Juric2008} provide the first large scale map of the Cloud and\nemphasize its gigantic extent on the sky of least $\\sim\n1000$ deg$^2$. From the inspection of Figure~\\ref{fig:fos_equ}, it is\nobvious that the portion of the Virgo Cloud analyzed by\n\\citet{Juric2008} is only the tip of the structure that appears to\ncontinue to lower Declinations as far as the SDSS\/Segue imaging\nstripes can go. Accordingly, \\citet{Bonaca2012b} take advantage of the\nextra imaging in the SDSS DR8 and claim that the true extent of the\nCloud is somewhere between 2000 deg$^2$ and 3000 deg$^2$. The\ndebris cover an enormous portion of the sky, but given the typical\ndistance and the low surface brightness, the total luminosity of the\nVirgo Cloud is estimated to be modest $< 10^6 M_{\\odot}$\n\\citep{Bonaca2012b}.\n\nThe Galactic Anti-Center Stellar Structure and the Hercules-Aquila\nCloud have most of their stars at low Galactic latitudes: within $|b|\n< 40^{\\circ}$, GASS can be found at roughly $120^{\\circ} < l <\n240^{\\circ}$ and HAC at $20^{\\circ} < l < 70^{\\circ}$ (see\nFigure~\\ref{fig:fos_gal}. In fact, both of these structures appear to\nbe stuck right in the plane of the disk as their candidate member\nstars are detected in both Northern and Southern hemispheres. Given\nsuch an awkward location in the Galaxy, it is still questioned whether\nall, or at least some of the signal attributed to these two can be\nexplained away invoking variants of the known components of Milky\nWay. For example, it is claimed that parts of the GASS can well be\nascribed to the Galactic flare and\/or the warp\n\\citep[e.g.][]{Ibata2003}, and the HAC is really nothing but the\nasymmetric thick disk \\citep[e.g.][]{Larsen2008,Larsen2011}. However,\nthere exists additional observational data within which stellar\nover-densities are clearly seen in the directions of both the GASS and\nthe HAC in tracers unlikely to populate either of the disks. For\nexample, the distant portion of the GASS, the And-Tri stream is traced\nwith M giants at distances of the order of 30 kpc. HAC can be picked\nup with RR Lyrae in the SDSS Stripe 82 dataset\n\\citep[e.g.][]{Watkins2009, Sesar2010a} at $10 < D < 20$ kpc. As most\nof the light in both GASS and HAC is hidden in the Galactic plane,\nonly very approximate estimates of their total stellar masses exist in\nthe literature. \\citet{Belokurov2007b} give a conservative estimate of\n$\\sim 10^7 L_{\\odot}$ for the Hercules-Aquila Cloud. For the closer\nportion of the GASS, \\citet{Yanny2003} get the total stellar mass in\nthe range of $0.2 - 5 \\times 10^8 M_{\\odot}$, with the larger value\nobtained assuming that i) the GASS follows an exponential profile as a\nfunction of z and ii) encompasses the entire Milky Way. Several\nfollow-up studies present the updated measurements of the structure\nand the stellar populations of the pieces of GASS visible in the SDSS\n\\citep[e.g.][]{Dejong2010,Grillmair2011,Li2012} and in the deeper\nimaging \\citep[e.g.][]{Conn2012}. According to the body of work\npublished so far, the components of the GASS most consistent with the\naccretion scenario \\citep[see e.g.][]{Penarrubia2005} have, overall,\nmuch flatter density distribution as a function of Galactic $|b|$ or\n$|z|$. If true, this observation would lead to the substantial\nreduction of the overall luminosity of the GASS. Perhaps, the\nfollowing simple argument can be constructed to provide a\ncomplementary guess as to the total stellar mass in the GASS. Given\nthat the parts of the GASS detected within the SDSS field of view\ntypically have similar or lower surface brightness as compared to the\nSgr Stream, but are on average closer by a factor of 2-5, it is not\nunlikely that the structure, in fact, contains more than $10^8\nM_{\\odot}$.\n\n\n\\subsubsection{Ultra-faint satellites}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.99\\linewidth]{dwarfs_lb_dr8.jpg}\n\\caption{Distribution of the classical dwarf galaxies (blue filled\n circles) and the SDSS ultra-faint satellites (red filled circles),\n including three ultra-faint star clusters, in Galactic\n coordinates. The SDSS DR8 imaging footprint is shown in grey. Dashed\n line marks the tentative orbit of the Sgr dwarf galaxy. Galactic\n $l=0^{\\circ}, b=0^{\\circ}$ is at the centre of the Figure.}\n\\label{fig:dwarfs_lb}\n\\end{figure}\n\nVisible as bright dots of different colors in the maps in\nFigures~\\ref{fig:fos_equ} and~\\ref{fig:fos_gal} are the compact\nstellar over-densities corresponding to the Galactic satellites that\ngive the impression of being still intact. The brightest of these\n``hot pixels'' correspond to the well-known star clusters and\nclassical dwarf galaxies, while the very faint and barely visible\nsmall-scale over-densities mark the locations of the so-called\nultra-faint satellites of the Milky Way. Although several of these,\nincluding Boo I, Boo III, CVn I and UMa II, are seen in this picture\nwith a naked eye, the rest of the population of these objects is too\ninsignificant and can only be unearthed via an automated over-density\nsearch. The first example of such an automated stellar over-density\ndetection procedure is presented in \\citet{Irwin1994} who apply the\nmethod to the data from the photographic plates of the POSS I\/II and\nUKST surveys scanned at the APM facility in Cambridge. A vast area of\n20,000 square degree of the sky is searched but only one new nearby\ndwarf galaxy is detected, namely the Sextans dSph. A variant of the\nprocedure is used, albeit with a little less luck, by\n\\citet{Kleyna1997}, and subsequently by \\citet{Willman2005a,\n Willman2005b} who actually find the two very first examples of\nultra-faint objects in the SDSS data. The ease with which these\nsystems reveal themselves in a stellar halo density map akin to the\n``Field of Streams'' \\citep[see][]{Zucker2006a, Belokurov2006c} helped\nto re-animate the search for new Milky Way satellites and more than a\ndozen of new discoveries have been reported in quick succession\n\\citep{Zucker2006b,Belokurov2007c,Irwin2007,Koposov2007,Walsh2007,\n Belokurov2008,\n Belokurov2009,Grillmair2009,Belokurov2010}. Figure~\\ref{fig:dwarfs_lb}\nmaps the distribution of all presently known SDSS ultra-faint\nsatellites on the Galactic sky.\n\nThe accuracy and the stability of the SDSS photometry makes it\npossible for the over-density detection algorithms to reach\nexceptionally faint levels of surface brightness across gigantic areas\nof the sky. However, even though genuine Galactic satellites can be\nidentified in the SDSS as groups of only few tens of stars, their\nstructural parameters can not be established with adequate accuracy\nusing the same data. Deep follow-up imaging on telescopes like INT,\nCFHT, LBT, Magellan, MMT, Subaru and most recently HST, has played a\nvital role in confirming the nature of the tiny stellar blobs in the\nSDSS, as well as in pinning down their precise sizes, ellipticities\nand their stellar content. The most recent, deep and wide photometric\nstudies of a significant fraction of the new SDSS satellites are\npublished by \\citet{Okamoto2012} and \\citet{Sand2012}. They point out\nthat even at distances $D>100$ kpc from the Galactic centre, the outer\ndensity contours of CVn II, Leo IV and Leo V display extensions and\nperturbations that are probably due to the influence of the Milky Way\ntides. Similarly, there is now little doubt that both UMa II and Her\nare excessively stretched, as their high ellipticities as first\nglimpsed at discovery \\citep{Zucker2006a, Belokurov2007c} are\nconfirmed with deeper data \\citep{Munoz2010, Sand2009}. Note, however\nthat apart from these two obvious outliers there does not seem to be\nany significant difference in the ellipticity distributions of the\nUFDs and the Classical dwarfs contrary to the early claims of\n\\citet{Martin2008}. This is convincingly demonstrated by\n\\citet{Sand2012} with the help of the imaging data at least 2\nmagnitudes deeper than the original SDSS. They, however, detect a\nmore subtle sign of the tidal harassment: the preference of the\ndensity contours of the SDSS satellites to align with the direction to\nthe Galactic centre.\n\nAs far as the current data is concerned, the SDSS dwarfs do not appear\nto form a distinct class of their own, but rather are the extension of\nthe population of the Classical dwarfs to extremely faint absolute\nmagnitudes. However, as more and more meager luminosities are reached,\nit becomes clear how extreme the faintest of the UFDs are. The\nbrightest of the group, CVn I and Leo T show the usual for their\nClassical counter-parts signs of the prolonged star-formation. For\nexample, CVn I hosts both Blue Horizontal Branch and Red Horizontal\nBranch populations, while Leo T shows off a sprinkle of Blue Loop\nstars. However, the rest of the ensemble appears to have narrow CMD\nsequences with no measurable color spread around the conventional\ndiagnostic features, e.g. MSTO and\/or RGB, thus providing zero\nevidence for stellar populations born at different epochs\n\\citep[e.g.][]{Okamoto2012}. The CMDs of the UFDs have revealed no\nsecrets even under the piercing gaze of the HST: all three objects\nstudied by \\citet{Brown2012} appear to be as old as the ancient\nGalactic globular cluster M92. Yet the low\/medium and high-resolution\nfollow-up spectroscopy reveals a rich variety of chemical abundances\nsomewhat unexpected for such a no-frills CMD structure. The first\nlow-resolution studies of \\citet{Simon2007} and \\citet{Kirby2008}\nalready evince the existence of appreciable $[Fe\/H]$ spreads in the\nSDSS dwarfs with the metallicity distribution stretching to extremely\nlow values. Analyzing the medium and high resolution spectra of the\nBoo I system, \\citet{Norris2010} measure the spread in $[Fe\/H]$ of\n$\\sim$1.7 and the $[Fe\/H]$ dispersion of $\\sim$0.4 around the mean\nvalue of -2.55 at $M_V\\sim -6$. It seems that this behavior of\ndecreasing mean metallicity with luminosity while maintaining a\nsignificant enrichment spread is representative of the UFD sample as a\nwhole \\citep[see also][]{Lai2011,Koch2013,Vargas2013}. Crucially,\nthese spectroscopic observations require that, notwithstanding their\nlow stellar luminosities at the present day, these satellites had\nenough total mass in the past to hold on to some of the enriched gas\nafter the first supernovae explosions and subsequently produce more\nstars. Additionally, in the UFDs, the downwards shift of the mean\nmetallicity with decreasing stellar mass reveals that they can not\nsimply be direct analogs of the Classical dwarfs stripped off the bulk\nof their stellar content.\n\nOf the 16 ultra-faint satellites currently known, only 5 systems have\na handful of stars studied with high-resolution spectroscopy. More\nspecifically, one star in Leo IV \\citep{Simon2010}, two stars in Her\n\\citep{Koch2008}, 3 stars in each of UMa II and Com \\citep{Frebel2010}\nand 7 in Boo I \\citep{Gilmore2013} have been measured so far. It is\nperhaps too early to draw far-reaching conclusions from these highly\nincomplete data, nonetheless an interesting picture seems to be\nemerging from the detailed abundance work. Although wanting in\nquantity, these high-resolution high-quality spectroscopic data do\nrobustly confirm the key properties of the UFD chemical enrichment\nhistories hinted at by the analysis of the low-resolution (and at\ntimes, low-S\/N) samples. The SDSS dwarfs are indeed characterized by\nremarkably low levels of the overall iron enhancement as well as the\nheterogeneity of the individual stellar abundances (in each of the 4\nsatellites that have more than 1 star measured). Additionally, the\nvery first high-resolution study of a UFD by \\citet{Koch2008} reported\na depletion of heavy neutron capture elements. RGB stars with low\nabundance levels of barium are also found in Leo IV, Com, UMa II and\nBoo I \\citep{Simon2010, Frebel2010, Gilmore2013}. Moreover, in Boo I,\nseveral extremely metal-poor stars are demonstrated to have increased\nlevels of carbon \\citep[see e.g.][]{Norris2010}. Potentially, there\nare at least two notable implications of these enrichment\npatters. First, carbon-enhanced metal-poor stars are common denizens\nof the Galactic stellar halo, yet if there occur any in the classical\ndSphs, they have so far eluded the detection. The existence of such\nstars in both the UFDs and the MW stellar halo may signify the\ncommonality of the chemical evolution paths of the halo progenitor(s)\nand the ultra-faint satellites. Second, as several authors have\npointed out \\citep[e.g.][]{Koch2008,Simon2010, Frebel2010}, the\nenhancement in $\\alpha$-elements together with the depletion in\nneutron-capture elements at low metallicities can be linked to the\nproducts of the Population III SNe, therefore implying that a good\nfraction of the stellar content in the UFDs could be direct\ndescendants of the first stars \\citep[see also][]{Frebel2012}.\n\nIt is evidently not possible to come up with a sensible theory of the\nUFD formation without an idea of their total masses. Such a\nmeasurement, which necessarily involves accurate kinematics for a\nlarge enough sample of the satellite members, is, however, not\nstraightforward. This is simply due to the fact that, as illustrated\nby \\citet{Koposov2008}, the majority of these objects are discovered\nvery close to the detection boundary, implying that the over-density\nsignal is dominated by the stars close to the SDSS detection limit of\n$r\\sim 22$. At these magnitudes, only half a handful of facilities in\nthe world are capable of obtaining absorption spectra of\nsignal-to-noise sufficient to measure the line-of-sight velocities of\nindividual stars. Even if the kinematic signal is present in the data,\nwinnowing it out from the low-resolution spectra of low-metallicity\nstars is a challenge. An even harder challenge is figuring out the\nuncertainties of the velocity measurements. For most ultra-faints, the\ntypical member velocity uncertainty is of the order of, or larger\nthan, the intrinsic velocity dispersion of the system. Under or\nover-estimating the measurement error by a small fraction can lead to\na substantial systematic velocity dispersion bias, and as a\nconsequence, a wrong aperture mass. Despite the above mentioned\ndifficulties of the task at hand, several teams report the results of\ntheir heroic attempts to gauge the central masses of the UFDs\n\\citep[e.g.][]{Martin2007b, Simon2007, Walker2009, Belokurov2009,\n Simon2011, Koposov2011}\n\nThe structural parameters of the faintest of the SDSS satellites,\ne.g. Willman 1, Segue 1 and 2, Boo II are dangerously similar to those\nof the most diffuse star clusters in the Milky Way and M31. It is not\nconceivable, purely on the basis of their size or luminosity, to come\nup with the most likely scenario of their formation. Therefore, their\nkinematic and chemical properties are the most important clue. Today,\nfor the faintest objects, it is just possible, after many hours spent\non Keck and VLT, to build datasets with radial velocities for a dozen\nor two of the MSTO members and a trickle of the Red\nGiants. Accordingly, the most recent and the most thorough kinematic\nanalysis of Willman 1, Segue 1 and Segue 2 can be found in\n\\citet{Willman2011, Simon2011} and \\citet{Kirby2013}\ncorrespondingly. Moreover, \\citet{Norris2010} independently carries\nout a thorough chemical study of Segue 1 using a different combination\nof the telescope, the instrument and the analysis techniques. For\nthese three best studies objects, the picture does not appear to be as\nclear-cut as for their more luminous peers. For example, the evolution\nof the line-of-sight velocity with radius in Willman 1, where the\ninner-most stars are offset by some 8 km\/s from the outer-most ones is\nunusual, and is, perhaps, a sign of the advanced stage of tidal\ndisruption. There is also an evidence of the spread in [Fe\/H], but\nunfortunately it is based on the measurements of only two Red Giant\nBranch stars.\n\nSegue 1, the best studied of the three, has a substantial velocity\ndispersion at 3.7$^{+1.4}_{-1.1}$ km\/s and an impressive metallicity\nspread. There are however some quirks with regards to both the\nvelocity and the metallicity dispersion measurements, such as the fact\nthat the velocity dispersion calculated using the brightest members\nonly (5 red giants stars) is essentially consistent with zero, or the\nfact that some of the most metal-poor stars also lie several\nhalf-light radii away from Segue 1's center\n\\citep[see][]{Norris2010}. Perhaps more significant is the observation\nby \\citet{Newberg2010} that the Orphan stellar stream runs at the\nidentical distance and velocity only $\\sim$2 degrees away from the\nposition of Segue 1. Given the width of the stream of 1 degree, a\nsignificant contamination of spectroscopic samples at Segue 1's\nlocation is not very likely. Yet, the dynamical association between\nthe two is, however, quite possible: both the progenitor of the Orphan\nStream and Segue 1 itself might have been parts of a bigger system\nwhich is now completely disrupted.\n\nThe evidence of such an accretion event is even more dramatic in the\ncase of Segue 2. Taking into account the observations reported in\n\\citet{Majewski2004,Rocha2004}, Segue 2 is immersed in the debris of\nthe Triangulum-Andromeda stream, which is interpreted as the distant\n(at $\\sim$ 30 kpc compared to $34$ kpc for Segue 2) counter-part of\nthe Monoceros stream and part of the larger Galactic Anti-Center\nStellar Structure. As published by \\citet{Rocha2004}, the velocities\nof M giant members of Tri-And structure are $0 < V_{GSR} < 60$ in the\nrange of longitudes $160^{\\circ} < l < 130^{\\circ}$ at the Galactic\nlatitudes slightly lower than that of Segue 2. This velocity\ndistribution can be modeled as a Gaussian that peaks around $V_{GSR}\n\\sim 30$ km s$^{-1}$ which is a good match to the measurement of the\nsatellites line-of-sight velocity $V_{GSR} \\sim 40$ km s$^{-1}$. The\ncoverage of the area with the spectroscopic M giants is sparse, and\nthe SDSS spectroscopic footprint is seriously incomplete\nhere. However, \\citet{Belokurov2009} present an unambiguous kinematic\nevidence for the stream's existence using the spectra obtained with 1\ndegree wide field Hectochelle instrument on MMT. They claim that the\nstream's stars are more metal-rich on average and their velocity\ndistribution can be described with a broader Gaussian, namely 15 km\/s\nvs $\\sim$3 km\/s for Segue 2. Most recently, \\citet{Kirby2013}\nre-evaluated the spectroscopic properties of Segue 2 albeit with a\ndifferent observational setup and a smaller field of view as compared\nto the original study of \\citet{Belokurov2010}. They claim no\ndetection of the stream signal, which is perhaps not surprising given\nthe targeting strategy and the area of the sky surveyed. Intriguingly,\nthey measure much lower velocity dispersion (essentially consistent\nwith zero), thus markedly reducing the central mass of the satellite.\n\n\n\\subsubsection{Star cluster streams}\n\nThe large undissolved stellar clouds (Virgo, Hercules-Aquila) and\nbroad long streams (Monoceros, Sagittarius) described earlier are the\nprimary contributors to the Galactic halo in terms of the stellar\nmass. In the past decade, an assortment of much narrower, often\nshorter and significantly less luminous streams has been\nidentified. It seems most likely that these would have originated in\nstar clusters. Some of these wispy tidal tails are discernible in\nFigures~\\ref{fig:fos_equ} and~\\ref{fig:fos_gal}, such as the tidal\ntails of the Palomar 5 globular cluster\n\\citep{Odenkirchen2001,Grillmair2006b}. However, in their majority\nthese feathery streams require a more subtle approach and are best\nseen with the help of the Matched Filter technique. Some of the star\ncluster debris have obvious progenitors like the short stubby tails\nvisible around e.g. NGC 5466 \\citep{Belokurov2006b}, NGC 5053\n\\citet{Lauchner2006}, Pal 14 \\citep{Sollima2011}, Pal 1\n\\citep{no2010}. For the others, typically extending many degrees on\nthe sky, no suitable progenitor has been discovered yet, e.g. the GD-1\nstream \\citep{Grillmair2006a}, a group of four streams Styx, Acheron,\nCocytos, Lethe \\citep{Grillmair2009} and the most recently identified\nPisces Stellar Stream \\citep{Bonaca2012, Martin2013}.\n\nIt is interesting to estimate the total number of star clusters that\nhave disrupted so far and whose stars are now part of the Galactic\nhalo. While such a count is valuable as it gives an idea of the\nfraction of the halo that is comprised of the GC debris, it is not\nstraightforward as it requires the knowledge of the Cluster Initial\nMass Function (CIMF) and a model of the cluster evolution in the Milky\nWay tidal field. An example of such calculation is presented in\n\\citet{Poul2013} who approximate the CIMF with a power-law\ndistribution and apply the semi-analytic model of \\citet{Gieles2011}\nfor the star cluster evolution in a logarithmic Galactic\npotential. They find that, of the several models they consider, the\nRoche volume under-filling model with a flat CIMF (power law index 0)\nreproduces the present day properties of the Milky Way's GCs the\nbest. While the authors do not give the exact number of dissolved\nclusters, it is clear that the flat mass function evolution can only\nproduce a moderate number of star cluster streams in the Galactic\nhalo, perhaps orders of magnitude less as compared to the rising power\nlaws (e.g. -2). Alternatively, the number of the GC streams detected\nso far with the SDSS could be translated into a Galaxy total if there\nexisted an estimate of the stream detection efficiency. However, it is\npossible that a significant fraction of the known long and narrow\nstellar streams may have been produced as a result of only a few\naccretion events. For example, given the noticeable alignment of their\norbital planes, it is feasible that the progenitors of the Styx,\nAcheron, Cocytos and Lethe streams arrived to the Galaxy together with\na much bigger satellite. The fact that the GC accretion is most likely\nlinked to the infall of more massive Galactic satellites is another\nreason to believe that the total number of GC streams is relatively\nlow given the evidence for the uneventful Milky Way's mass assembly\nhistory.\n\n\\subsubsection{Orphan and Styx. Streams from ultra-faint satellites?}\n\nThe tidal stream's cross-section on the sky is normally a giveaway of\nthe progenitor's mass. The low-density disrupting star clusters with\nsmall internal velocity dispersion $\\sigma \\lesssim 1$ km s$^{-1}$\ntypically produce tails that are only $\\sim 0.1^{\\circ}$ wide. On the\nother hand, a galaxy as massive as Sgr dwarf with its current $\\sigma\n\\lesssim 20$ km s$^{-1}$ \\citep[see e.g.][]{Ibata2009} gives rise to\nstreams that are at least 10$^{\\circ}$ across (see\nFigure~\\ref{fig:fos_equ} for example). This rule of thumb of course\nassumes comparable distances to the tidal tails and not hugely\ndifferent dynamical ages. Depending on how aspherical the\ngravitational potential of the Galaxy is and how long ago the debris\nwere stripped, even originally narrow tails can puff up with time.\n\nAmongst the panoply of stellar substructure recently discovered in the\nGalactic halo, there are at least two streams that seem to occupy the\nparameter space intermediate between the star clusters and dwarf\ngalaxies. These are the Orphan stream \\citep{Belokurov2006a,\n Belokurov2007b, Grillmair2006c} visible in Figure~\\ref{fig:fos_equ}\nas almost vertical streak of orange color crossing the Sgr debris at\naround $140^{\\circ} <$ RA $< 160^{\\circ}$, and the Styx stream\n\\citep{Grillmair2009}, the faint blue nebulous smear running at almost\nconstant Dec$=30^{\\circ}$ from RA$=250^{\\circ}$ to RA$=220^{\\circ}$\nwhere it starts to drop in Dec towards the Sgr stream. Curiously, both\nOrphan and Styx run in a close vicinity of the several of the Galactic\nultra-faint satellites. The sky projection of the orbit of the Orphan\nstream takes it right through the position of the UMa II dwarf. The\nfeasibility of such association is explored in \\citet{Fellhauer2007}\nwho conclude that UMa II could well be the stream's\nprogenitor. However, as convincingly shown in \\citet{Newberg2010}, the\nearly tentative estimates of the stream's radial velocity were\nincorrect and that the actual orbit of the stream is much more\nconsistent with the 4D location of Segue 1. As regards to the Styx\nstream, when tracing its debris to the lower RA, \\citet{Grillmair2009}\ndiscovers a pronounced stellar clump within the stream's path. Dubbed\nBootes III and subsequently confirmed with spectroscopy\n\\citep{Carlin2009} this is the most diffuse of all ultra-faints found\nso far.\n\n\\subsubsection{Broad and Invisible}\n\nAs the proper motion, spectroscopy and the variability wide-area\nsurveys slowly catch up with the rapidly advancing sky imaging\ncampaigns, it is possible to gauge the presence of stealth stellar\nstructures, so diffuse that they elude detection in stellar halo maps\nakin to those described above. These detections are reminiscent of the\noriginal discovery of the Sgr dwarf \\citep{Ibata1994} that is too\nfaint and spread out to be seen on a photographic plate but produces a\nbooming signal in radial velocities.\n\nTrinagulum-Andromeda is an extended stellar structure located at\nseveral tens of kpc from the Galactic centre \\citep{Rocha2004}. It is\ninitially picked up as a faint excess of 2MASS M-giant stars, and\nlater confirmed with the help of proper motion data and follow-up\nspectroscopy. As judged by the radial velocities of its members, the\nTri-And cloud seems to be connected to the Southern Galactic\ncounterpart of the Monoceros stream, and thus forms the more distant\nwraps of the Galactic Anti-centre Stellar Structure\n\\citep{Newberg2002, Ibata2003, Rocha2003,\n Yanny2003}. \\citet{Majewski2004} and \\citet{Martin2007} report the\ndetection of the Main Sequence stars in the Tri-And cloud, thus\nridding of the last shreds of doubts as to the reality of its\nexistence. Curiously, the recently discovered ultra-faint satellite\nSegue 2 \\citep{Belokurov2009} appears immersed in the debris of what\nvery well might be the Tri-And cloud.\n\nThe recently discovered Cetus Polar Stream \\citep{Newberg2009} has\navoided detection thanks to its low density and the overlap in\nprojection with much brighter Sagittarius trailing stream. However,\ntaking advantage of the SDSS spectroscopy available over a large\nportion of the Southern Galactic sky, \\citet{Newberg2009} present a\nconvincing argument in favor of a distinct stellar sub-structure,\ncolder and more metal-poor than the Sgr debris. \\citet{Koposov2012}\nprovide the first sky map of the Cetus Polar Stream debris, and having\nobtained accurate measurements of the stream's distance and velocity\ngradients they argue that the sense of direction of the orbital motion\nof the CPS is opposite to that of Sgr. In their maps, the structure\nappears to be at least 20$^{\\circ}$ wide and some 40$^{\\circ}$ long,\nyet with only 0.1 mag width along the line of sight.\n\nThe charting of the Galactic halo at distances beyond 50 kpc has been\nsomewhat sluggish due to the obvious lack of suitably bright tracers\ncovering a large enough area of the sky. A small fraction of the SDSS\nfootprint, so-called Stripe 82 has been imaged repeatedly during the\nSupernovae campaign. \\citet{Watkins2009} explores this multi-epoch\ndataset to identify RR Lyra stars. They find a significant\nover-density of RR Lyrae in the constellation of Pisces at\ngalacto-centric distances of $D\\sim 90$~kpc, thus discovering the most\ndistant sub-structure known in the Milky Way halo. \\citet{Sesar2010a}\nconfirm the discovery with a more sophisticated analysis of the same\nSDSS data, while \\citet{Kollmeier2009,Sesar2010b} present the\nspectroscopic confirmation of the structure by obtaining velocities\nfor several RR Lyra members. As of today the true extent of the Pisces\nOver-density is not known, but from the distribution of the RR Lyrae\nit subtends at least $10^{\\circ}$ on the sky making it some 15 kpc\nwide.\n\n\\subsection{Quantifying the amount of sub-structure}\n\nWithin the $\\Lambda$CDM paradigm, the global properties of the\nGalactic stellar halo, namely the total luminosity, the shape, the\nradial profile as well as the amount of sub-structure are simply the\nconsequences of the Milky Way's accretion history and as such all have\na straightforward interpretation. Observationally, however, these\nproperties are awkward to pinpoint. For example, to gauge the\nflattening and the shape of the radial density profile, data across\nlarge portions of the Northern and the Southern Galactic sky are\nrequired. With pencil-beam surveys, the halo flattening or, more\ngenerally any deviation from spherical symmetry (e.g. triaxliaity), is\nimpossible to determine and there is always a good chance of hitting\nunknowingly a stellar stream or a cloud, hence biasing the estimates\nof the density profile. Yet, in photometric studies, a robust global\ndensity model is vital when quantifying the amount of\nsub-structure. As the density distribution in the 6D phase-space,\nwhere the individual accreted fragments are readily identifiable, is\ncollapsed onto the 3 spatial dimensions (or sometimes 2.5 or 2), the\nsignal is diluted as a result of super-position of many\nstructures. Therefore, even a small bias in the background properties\ncan affect dramatically the amplitude of sub-structure. Of course, the\n``background'' itself, in this picture, is nothing else but the\nstellar debris jumbled up more efficiently. Accordingly, the global\nlaw parameterizing the behavior of the background provides crucial\ninformation in which the mass of the satellites contributing to it and\nthe time of their accretion is encoded.\n\n\\subsubsection{Spatial inhomogeneities}\n\nWith plenty of deep multi-band photometry in both Galactic\nhemispheres, the SDSS is an ideal resource to use to infer the global\nproperties of such an immense structure as the Milky Way's stellar\nhalo. A series of fits to the principal Galactic components as traced\nby the MS stars in the SDSS DR5 is presented in\n\\citet{Juric2008}. This sample is dominated by the faint MS dwarfs\nand, therefore, can not trace the volume density in the Milky Way much\nfurther than 20 kpc. Within this radius, the halo appears to be well\ndescribed by a single power law density model with the index $n\\sim\n2.8$. Importantly, this study confirms earlier indications of a\nsubstantial vertical flattening of the stellar halo $q\\sim0.6$. The\nresults of \\citet{Juric2008} are corroborated by the modelling of the\nSDSS DR8 data with increased Southern Galactic hemisphere coverage\npublished by \\citet{Bonaca2012b}. An attempt to delve deeper into the\nstellar halo can be found in \\citet{Bell2008}, where a simple\ncolor-cut (similar to that illustrated in the right panel of\nFigure~\\ref{fig:bhb_msto}) is used to isolate the brightest of the old\nMS stars in the halo. Using these blue, metal-poor turn-off stars,\nwith typical $M_g \\sim 4$, it is possible to discern halo structures\nas far as 30-40 kpc away from the Sun. However, as explained in\nSection~\\ref{sec:abs_mag}, the spread in the intrinsic luminosities of\nthe stars selected is as large as 3 magnitudes. There are two\nimportant consequences of such blurred vision. First, convolving the\nstellar halo distribution with large non-Gaussian errors in tracer\ndistances can have strong destructive effects on the accuracy of the\nvolume density inference. Second, when estimating the amplitude of\nsmall scale deviations from the background, a debris at one particular\ndistance appears in several apparent magnitude bins (and hence\ndistances), thus biasing high the total amount of sub-structure across\nthe range probed. This effect is exacerbated at magnitudes close to\nthe survey limit, as well as for stars with different age and\/or\nmetallicity.\n\nWhile troubled by a number of issues outlined above, the analysis of\n\\citet{Bell2008} is the first of its kind. Taking advantage of the\nimpressive sky coverage and depth of the SDSS imaging, they provide a\nquantitative interpretation of the inhomogeneous stellar halo glimpsed\nby the earlier works. The main conclusions of the study by\n\\citet{Bell2008} are as follows. First, a smooth density model for the\nMSTO tracers within 40 kpc is not appropriate for the Milky Way halo,\nwith most of the model parameters poorly constrained (see their\nFigures 4, 7 and 9). Second, even after excising the major known\ndebris pile-ups such as Sagittarius stream and Virgo overdensity, the\namount of sub-structure $\\sigma\/{\\rm total}$, parameterized in terms\nof the scaled rms deviation $\\sigma$ of the data around the smooth\nmodel, stays just under $40\\%$ from $r\\sim 19$ mag to $r \\sim 22$\nmag. In the presence of these large stellar halo structures, the\n$\\sigma\/{\\rm total}$ statistic grows with apparent magnitude (roughly\nproportional to distance) and reaches $>50\\%$ at $r\\sim\n21.5$. Finally, \\citet{Bell2008} compare the values of $\\sigma\/{\\rm\n total}$ for the Milky Way halo traced by faint metal-poor MSTO stars\nin the SDSS to those obtained for the semi-analytic stellar halo\nsimulations of the Galaxy by \\citet{Bullock2005}. The 11 model halos\nare made entirely of accreted stars, and show a minimal level of\nsub-structure $\\sigma\/{\\rm total} > 20\\%$. Accordingly, the final\nverdict is: the amount of sub-structure in the Galactic halo matches\nthat in the hierarchical galaxy formation models, and, therefore,\nsatellite accretion is the primary mode of the Milky Way's halo\ncreation.\n\n\\citet{Helmi2011} aims to improve the analysis of \\citet{Bell2008} by\ni) coming up with a more robust sub-structure quantification, and ii)\ncomparing the SDSS data to the most recent stellar halo\nsimulations. Similarly to \\citet{Bell2008} they measure the stellar\ndensity scatter in bins of apparent magnitude and the two celestial\ncoordinates. However, instead of calculating the amount of residual\ndeviation between the data and the best-fit smooth parametric model,\n\\citet{Helmi2011} work out the RMS around the mean stellar density in\nthe bin. Predictably, the amount of sub-structure computed in this\nfashion is lower compared to that obtained by \\citet{Bell2008}, albeit\nonly slightly. According to \\citet{Helmi2011}, across the apparent\nmagnitude range of $18.5 < r < 22.5$, the normalized scatter ${\\rm\n rms}(\\rho)\/<\\rho>$ in the SDSS DR7 MSTO star density is at the level\nof 30\\% to 40\\%. These rather serious levels of inhomogeneity found in\nthe SDSS data nonetheless appear low when contrasted with the degree\nof sub-structure in simulated stellar halos. For the comparison with\nthe data, \\citet{Helmi2011} examine the smoothness of the mock stellar\nhalos produced by \\citet{Cooper2010}. These are built into the\nAquarius DM-only halos \\citep{Springel2008} by tagging 1\\% of the\nmost-bound particles in selected sub-halos and following them to\nredshift 0. Compared to the mock ``Milky Ways'' of\n\\citet{Bullock2005}, these have the obvious advantages of being\nfabricated in the Cosmological setting, and with a superior\nresolution. However, there are disadvantages too. First, the Aquarius\nsuite explores only half as many accretion histories, in fact, in the\nend, there are only 4 stellar halos analyzed in \\citet{Helmi2011},\ncompared to 11 in \\citet{Bullock2005}. Second, these Galaxy analogs do\nnot posses disks. A quick glance at the Figure 5 of \\citet{Helmi2011}\nreveals: all stellar halos of \\citet{Cooper2010} are highly\nirregular, with $50\\% < \\frac{{\\rm rms}(\\rho)}{<\\rho>} < 150\\%$. The\nauthors raise concern that some of the data-model discrepancy could be\ndue to the combined effects of the MSTO sample contamination and the\npresence of a smooth, in-situ formed stellar halo\ncomponent. Nonetheless, they conclude that there exists considerable\ntension between the observations of the Galactic stellar halo\nsub-structure and the predictions of the simple but high-resolution\nmodel. Even though the halos of both the real and the mock Galaxy are\nvery inhomogeneous, the simulations easily reach 2-3 times the\nobserved scatter on scales as small as few degrees.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=1.02\\linewidth]{data_model_lb_deason.pdf}\n\\caption{Stellar halo of the Milky Way traced by the BHB stars. {\\it\n Left} Distribution of the SDSS DR8 BHB candidates in the Galactic\n $l$ and $b$. {\\it Right} Best-fit model of the stellar halo density\n distribution shown in the Left panel, from \\citet{Deason2011a}. The\n model halo is flattened with $q\\sim 0.6$ and has a break in the\n radial density profile at $r\\sim 27$ kpc where the power-law index\n changes from -2.3 to -4.6. Figure courtesy of Alis Deason,\n IoA\/UCSC.} \\label{fig:halo_bhb}\n\\end{figure}\n\nThe picture of the utter chaos in the inner parts of the Galactic\nstellar halo is re-visited in \\citet{Deason2011a}. Instead of using\nthe more abundant MSTO stars, they choose to model the halo volume\ndensity with Blue Horizontal Branch stars. While these stars are\nrarer, their higher intrinsic luminosities, lower levels of\ncontamination and accurate absolute magnitude calibration independent\nof age and chemistry all make these a better fit for the task. There\nare, nonetheless, several limitations to the use of BHBs as\ntracers. For example, being some $\\sim 4$ magnitudes brighter and at\nleast two orders of magnitude less frequent as MSTO, these come in\nparticularly low numbers at bright apparent magnitudes due to the size\nof the volume probed. Additionally, while their blue color makes them\nstand out dramatically compared to most other stellar populations at\nhigh Galactic latitudes, there is one troublesome impostor. Blue\nStragglers (see Figure~\\ref{fig:tracers}) have close to identical\nbroad-band colors but are $\\sim 1.5$ mag fainter. Outnumbering the\nBHBs by a factor of 2 on average, these may pose a serious problem by\nscrambling the tracer counts as a function of apparent\nmagnitude. \\citet{Deason2011a} solve both the problem of the limited\ndynamic range and of the contamination by including the BS stars in\nthe model. For all ``blue'' stars in the SDSS DR8, i.e. $-0.25 < g-r <\n-0$, the probability of belonging to the BHB or the BS population is\nassigned based on their $u-g$ and $g-r$ colors. As a result, the\nnumber density of stars in volume elements of the space spanned by\nposition of the sky, color and apparent magnitude can be modeled\nsimply as the sum of the contributions from BHBs and BSs, weighted by\ntheir conditional probabilities.\n\nThe results of the maximum-likelihood analysis presented in\n\\citet{Deason2011a} are summarized for the impatient reader in the\narticle's title ``Squashed, broken but smooth''. In other words: out\nto 40 kpc, the Galactic stellar halo appears to be highly flattened,\nthe density profile follows closely the broken power law and, most\ninterestingly, the overall level of sub-structure detected using the\nBHB tracers is rather low. At small and intermediate distances,\n$\\sigma\/{\\rm total}$ rises from as low $10\\%$ to at most $20\\%$\nirrespective of the spatial scale of density perturbations. At large\ndistance, $\\sigma\/{\\rm total}$ is close to $20\\%$ on most scales, but\nrises to $40\\%$ for the angular sizes of several hundreds of\ndegrees. These numbers are obtained by excluding from the modeling the\nregions of the sky with known large-scale halo overdensities. Even\nwhen these are included, the small-size inhomogeneities are only $10\n\\% < \\sigma\/{\\rm total} < 30\\%$. While, superficially, these estimates\ndiffer significantly from those quoted in \\citet{Bell2008}, there are\nseveral possible solutions to this discrepancy. Both methods have\ntheir weak points. It is quite likely that some of the halo mess\nobserved by \\citet{Bell2008} is simply due to the limitations of the\nMSTO stars as tracers. On the other hand, the average number of BHBs\nin a $1^{\\circ} \\times 1^{\\circ}$ pixel is small, hence limiting the\nareas of the sky tested by \\citet{Deason2011a} to those towards the\ninner Galaxy where mixing is more efficient. On slightly larger\nangular scales (several degrees or so), it is, however, safe to\nconclude that the inner stellar halo is indeed smooth.\n\n\\subsubsection{Phase-space sub-structure. Spaghetti, ECHOS and SKOs}\n\nAs it is much easier to identify the accreted satellite debris in the\nphase-space compared to simple sky density maps or 3D spatial maps,\nseveral attempts have been made to search for the surviving Galactic\nsub-structure in the datasets of wide area spectroscopic surveys. The\nSpaghetti survey \\citep[e.g.][]{Morrison2000} is the first brave\nendeavor to collect substantial numbers of genuine halo tracers in a\nlarge distance range. It is set up to gather photometry and the\nfollow-up spectroscopy in several tens of ``pencil-beam'' fields over\nthe area covering many tens of degrees. The analysis dealing with the\nquantification of the presence of sub-structure in the final set of\n101 giants with spectra covering distances up to 100 kpc is presented\nin \\citet{Starkenburg2009}. They report the detection of 1 group and 6\npairs of clumped stars and conclude that their findings of 10\\% of\nsub-structure in the halo are consistent with the accretion scenarios\nin which early and\/or massive satellite infall leads to the creation\nof broad phase-space features.\n\nThe SEGUE survey that has taken $\\sim$240,000 spectra in $>$200\npointings spread over $\\sim$11,000 square degrees is the ideal source\nof data to carry out a systematic search for un-relaxed\nsub-structure. \\citet{Schlaufman2009} do exactly that, and detect in\n137 lines of sight studied 10 high-confidence ECHOS, elements of cold\nhalo substructure as traced by metal-poor MSTO stars with distances in\nthe range $10 10$ Gyr \\footnote{In this\n definition, the age marks the time of when the stars became unbound,\n which implies slightly earlier epochs for the arrival of the\n progenitor.}) debris have had plenty of time to mix and therefore at\n$z=0$ the radial profile is comfortably fit with a single\npower-law. Old (7-10 Gyr) debris have spread out over a range of\nGalacto-centric distances but around the progenitor's apo-centre, the\ndrop in stellar density remains. Recent ($<6$ Gyr) mergers have not yet\nfilled the entire volume inward of the apo-centre and their radial\ndistribution still peaks at $R>0$.\n\nThe stellar halo (in this model) is just a superposition of the debris\nfrom the individual events across the entire accretion history. The\ncombined stellar profile can have a distinct break (at the average\napo-centre of the most massive accreted satellites) only if the most\nsignificant merger(s) happened at the right time, i.e. 8-10\nGyr. Additionally, it is required to dampen the accretion rate at the\nsubsequent epochs: as the Galaxy grows, the satellites that arrive\nwith increasingly larger apo-centers thus can flatten out the density\nprofile around and beyond the break radius, thus erasing this feature\naltogether. The hypothesis that the density break in the Galactic\nstellar halo reflects the apo-center(s) of the massive satellite(s)\naccreted at early epochs can be tested with 3D kinematics. Radial\nvelocities of stars tend to zero around the apo-center of the orbit,\ntherefore the radial velocity dispersion of the stellar halo should\nhave a dip around the break radius as well as an increase in the\ntangential anisotropy. Moreover, there exists a potentially powerful\ndiagnostic to decipher the properties of this old merger. Namely, if\nthe metallicities of the stellar halo tracers around the break radius\n(i.e. $20 M_V > -4$ can only\n(or mostly) exist as part of bigger dwarf systems, their distribution\nin the Galaxy is different from that of the accreted field dwarf\npopulation. Their radial density profile should be strongly radially\nconcentrated due to the combination of the two effects. First, their\nparent galaxies were massive enough to end up close to the center due\nto the dynamical friction. Second, in the Galaxy most of the large\nsystems (apart from the Sgr dwarf) were accreted as early as 8-10 Gyr,\nwhen the mass and the virial mass of the Milky Way were much\nsmaller. Taking these effects into account, much lower numbers of\nsatellites as faint as Segue I or II are predicted to be discovered by\nthe future deep all sky surveys.\n\n\n\n\\pagebreak\n\n\\section*{Acknowledgments} \nV. Belokurov thanks The Royal Society the support. The work on this\nreview has received funding from the European Research Council under\nthe European Union's Seventh Framework Programme (FP\/2007-2013) \/ ERC\nGrant Agreement n. 308024. The author has enjoyed conversations with\nA. Deason, W. Evans, A. Helmi, M. Irwin, S. Koposov, P. Kroupa,\nJ. Norris, M. Smith, E. Starkenburg and E. Tolstoy.\n\n\\pagebreak\n\n\\vspace*{2cm}\n\n\\noindent\n\n\\bibliographystyle{elsarticle-harv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}