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+{"text":"\\section{Introduction}\n\nWhen a cosmic ray interacts with particles in the Earth atmosphere, it produces\na shower of elementary particles propagating towards the grounds with almost\nthe speed of light. The first suggestion that these air showers can produce radio\nemission was given by Askaryan \\cite{askaryan} \nbased on a charge-excess mechanism. Recently, Falcke\\&Gorham \\cite{falcke-gorham}\nproposed that the mechanism for radio emission of air showers is coherent geosynchrotron\nradiation. Secondary electrons and positrons produced in the particle cascade are deflected\nin the Earth magnetic field and this produces dipole radiation that is relativistically\nbeamed in the forward direction.\nThe shower front emitting the radiation has a thickness which is comparable to a wavelength\nfor radio emission below 100MHz (around few meters). The emission is coherent which amplifies the \nsignal. \n\nRadio emission of cosmic ray air showers has been detected by LOPES (LOFAR Prototype Station) \n\\cite{falcke-nature}, a phased array of dipole antennas co-located with\nthe KASCADE (Karlsruhe Shower Core and Array Detector) experiment which provides \ncoincidence triggers for LOPES and well-calibrated informations about air-shower properties, like\nelectron number $N_e$, reconstructed muon number $N_{\\mu}$, azimuth and zenith angle of the event.\nThe LOPES experiment and data reduction are described in detail by Horneffer et al. \\cite{horneffer}.\n\nHighly inclined showers are expected to be very well detectable in the radio domain\n\\cite{huege-falcke},\\cite{gousset}. \nHowever, we have to mention that neither LOPES nor KASCADE are optimized for large\nzenith angles. For example, KASCADE reconstruction of electron and muon number can be\nnot accurate especially in cases when the shower core of specific event falls out\nof the KASCADE array. \n\nInclined cosmic ray air showers are specific, since they travel trough few times longer\ndistances in the Earth atmosphere compared to vertical showers, and due to this most of the \nelectromagnetic (particle) component of those showers has been \nabsorbed. So, inclined \nshowers that start high in the atmosphere (initiated by protons, iron nuclei or gamma-photons) will have\nlarge electron deficiency on the ground level compared to vertical showers. \nOn the other hand, neutrino induced showers may be generated at any distance from the \nground \\cite{capelle} so they could clearly\nbe distinguished from those whose primary particle initiated the shower\nhigh in the atmosphere by the number of electrons that reach the ground level.\n\n\n\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics*[width=0.3\\textwidth,angle=270,clip]{guipipeline-All-18.eps}\n\\includegraphics*[width=0.3\\textwidth,angle=270,clip]{guipipeline-X-18.eps}\n\\caption{\\label{fig1}Left: Electric field as a function of time for 10 LOPES antennas for an event \ndetected in March 2004 with zenith angle 53.3$^o$ and azimuth \n54.7$^o$. The angle between the shower core and the Earth magnetic field is 69.8$^o$.\n$N_e$=1.5$\\cdot$10$^6$, $N_{\\mu}$=1.5$\\cdot$10$^6$, reconstructed by KASCADE. Right:\nRadio emission as a function of time after beam forming for the same event.\nDark blue line represents X-beam, light blue line the total power and \npink line the Gaussian fit for X-beam.}\n\\end{center}\n\\end{figure}\n\n\n\\section{Discussion}\n\n\n\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics*[width=0.49\\textwidth,clip]{map18.ps}\n\\caption{\\label{fig3}Radio map of the cosmic ray shower with zenith angle 53.3$^o$ detected in March 2004. \nAzimuth (AZEL longitude) and elevation (AZEL latitude) of the event are given on the axes.}\n\\end{center}\n\\end{figure}\n\nWe made a selection of inclined events from data taken during year 2004 (also detected by the KASCADE\narray where the reconstructed shower cores were less than 100m away from the array center).\nWe found 2017 events with zenith angle larger than 50$^o$.\nThen we introduced an additional condition: $N_{\\mu}$$>$10$^5$.\nIn this way we narrowed the selection to 51 event and more than 40\\% of those are detected\nin the radio domain, many with very large field strengths, even though a threshold\non muon number is lower than the one used for bright events in \\cite{falcke-nature}.\nHowever, as we already mentioned, KASCADE is not optimized for large zenith angles, \nso the reconstruction of the electron and muon number failed for half of the detected events.\nThis leaves us with 10 events with strong radio signal and reliable shower properties\nreconstructed by KASCADE.\n\nAs an example, we show here one of those events, detected in March 2004 with zenith angle \n53.3$^o$ and azimuth 54.7$^o$ with roughly the same number of electrons and \nmuons $N_e$,$N_{\\mu}$$\\approx$1.5$\\cdot$10$^6$ \n(reconstructed by KASCADE). The angle between the shower axis and the Earth magnetic field \n(geomagnetic angle) is 69.8$^o$.\n \nIn Figure 1 (left) we show the electric field as a function of time for each antenna. The field is coherent \nat -1.825$\\mu$s which is the arrival time of the shower. The incoherent noise after is radio emission from\nphotomultipliers and in this case is very weak.\nFigure 1 (right) shows the radio emission as a function of time after X(excess)-beam forming.\nThis beam is formed in the following way. First squared signals of all antennas are summed which gives\nthe total power. Then signals of all two antenna combinations\nare multiplied and summed which gives the CC(cross correlation)-beam. Finally, CC-beam is multiplied \nwith the ratio \nbetween CC-beam and total power. In this way a suppression of incoherent noise is achieved.\n\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics*[width=0.35\\textwidth,clip]{r4776e150302eb1.eps}\n\\includegraphics*[width=0.35\\textwidth,clip]{r4776e150302eb2.eps}\n\\caption{\\label{fig3}\nLeft: Energy deposit of the  \ncosmic ray shower with zenith angle 53.5$^o$ detected in March 2004 for e\/$\\gamma$ KASCADE detectors.\nDark blue color shows energy deposit of 100Mev\/m$^2$, red color 1000Mev\/m$^2$. Maximum energy deposit\nfor this event is $\\sim$4500Mev\/m$^2$.\nRight: Particle arrival time for the same event. Dark blue color represents $\\sim$300ns, red $\\sim$800ns.\n}\n\\end{center}\n\\end{figure}\n\nFigure 2 is a radio map of the example event. The air shower is the brightest point in the sky for\nseveral tens of nanoseconds. The resolution of the map is $\\sim$2$^o$ in azimuth and elevation towards the\nzenith. \n\nFigure 3 gives the energy deposit of the chosen cosmic ray shower over the KASCADE array with \ne\/$\\gamma$ detectors\n(left panel) and particle arrival time (right panel). \nWe can see that the shower core falls within the KASCADE array and that\nthe maximum energy is deposited in the north-western part, within the LOPES array. We can \nnotice elliptical shapes of isolines of energy deposit, which is typical for inclined events.\n\n\n\\section{Conclusions}\n\nEven though neither LOPES nor KASCADE are completely \noptimized for the detection of highly inclined events,\nwe find that in a selection of events with zenith angle larger than\n50$^o$ and $N_{\\mu}$$>$10$^5$ (51 event)\naround 40\\% of all events is detected in the radio domain, and some of them \nwith very high field strengths, like the example we have presented in this paper.\nThe most inclined cosmic ray air shower that we detected with LOPES has\na zenith angle of almost 70$^o$.\n\nAfter checking the resonctruction of $N_e$ and $N_{\\mu}$ different correlations \ncan be considered, for example between radio pulse height and muon number, electron number or \ngeomagnetic angle. This will also give more insight into the nature of primary particles that\ninitiate showers and the possibility that some of detected showers might have been triggered\nby neutrinos.  \n\n\n\n\n \n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn this paper, $X$ denotes a smooth complex projective variety of dimension $n$, \n$D$ a (big) divisor on $X$ and $V \\subseteq X$ an irreducible subvariety of dimension $d$.\nThen the restricted volume of $D$ along $V$ is defined to be \n\\begin{equation*}\n\\V{V}(D) = \\limsup_{k \\to \\infty} \\dfrac\n{\\dim \\mathrm{Im} \\Big( H^{0}\\big(X,\\OX(kD) \\big) \\longrightarrow H^{0}\\big(V,\\OV(kD) \\big) \\Big) }{k^d\/d!}.\n\\end{equation*}\nRoughly speaking restricted volumes measure the number of sections of $\\OV(kD)$ \nwhich can be extended to $X$.\nRestricted volumes have many applications in various situations.\nThey are studied in [ELMNP], [BFJ09] and so on.\n\n\nOn the other hand, it is important problem when $D$ admits a Zariski decomposition.\nHere $D$ is said to admit a Zariski decomposition if \nthere exist a nef $\\RR$-divisor $P$ and an effective $\\RR$-divisor $N$ \nsuch that following map is an isomorphism for every integer $k>0$:\n\\begin{equation*}\nH^{0}(X, \\OX(\\lfloor kP \\rfloor)) \\longrightarrow  H^{0}(X, \\OX(kD)).\n\\end{equation*}\nThe map is multipling the section $e_{k}$, \nwhere $e_{k}$ is the standard section of the effective divisor $\\lceil kN \\rceil$.\nHere $\\lfloor G \\rfloor$ (\\textit{resp}. $\\lceil G \\rceil$) denotes round down\n(\\textit{resp}. round up) of an $\\RR$-divisor $G$.\n\nAssume $V$ is not contained in $\\mathbb{B}_{+}(D)$.\nHere $\\mathbb{B}_{+}(D)$ denotes the augmented base locus of $D$.\nIt is the closed analytic set in $X$, which is the defect of $D$ to be ample.\nThen the restricted volume $\\V{V}(\\cdot)$ along $V$ depends only on the first Chern class (the numerical class) of $D$.\nOne ask whether the restricted volume $\\V{\\cdot}{D}$ of $D$ depends only on the numerical class of $V$.\n[BFJ09] shows the question is affirmative when the codimension of $V$ is one.\nIn this paper, we give a necessary and sufficient condition for $D$, that\nthe restricted volume of $D$ depends only on the numerical class of $V$.\nThe condition is related to the existence of a Zariski decomposition of $D$ as follows:\n\n\\begin{theo}\\label{MaB}\nAssume that $D$ is big. \nThe following two conditions are equivalent.\n\n$(1)$ $D$ admits a Zariski decomposition.\n\n$(2)$ $\\V{V}(D) = \\V{V^{'}}(D)$ holds for any pair of subvarieties $V$ and $V'$ of $X$\nsuch that $V\\sim V^{'}$ and $V, V^{'} \\not \\subseteq \\mathbb{B}_{+}(D)$.\n\\end{theo}\nWhen $V$ and $V^{'}$ are numerically equivalent, we write $V \\sim V^{'}$.\nRoughly speaking, the condition (2) claims that the restricted volume $\\V{\\cdot}(D)$\nof $D$ is determined only by the numerical class of $V$.\nTheorem \\ref{MaB} implies \nthat the restricted volumes along some numerically equivalent subvarieties $V$ and $V'$ can be different \nwhen $D$ does not admit a Zariski decomposition.\nSuch example is given by so-called Cutkosky construction.\n\n\nWhen $V$ is the ambient space $X$, the restricted volume of $D$ is the usual volume $\\mathrm{vol}_{X}(D)$ of $D$.\nThe usual volume has been studied by several authors.\nThe general theory is presented in detail and with full references in \\cite{La04}.\nBoucksom expressed the usual volume $\\mathrm{vol}_{X}(D)$ by current integration,\nusing the Calabi-Yau technique to solve Monge-Amp$\\rm{\\grave{e}}$re equations and the singular \nholomorphic Morse inequalities (see \\cite{Bou02}).\nIn other words, Boucksom expressed the volume of $D$ in terms of only the first Chern class $c_{1}(D)$.\n\nRestricted volumes along $V$ depend only on the first Chern class $c_{1}(D)$ \nif $V$ is not contained in the augmented base locus $ {\\mathbb{B}}_{+}(D)$ of $D$.\nThen Boucksom's formula can be extended to restricted volumes as follows:\n\n\\begin{theo}\\label{MaA}\nAssume that $V$ is not contained in the augmented base locus $ {\\mathbb{B}}_{+}(D)$.\nThen the restricted volume of $D$ along $V$ satisfies the following equality:\n\\begin{equation*}\n\\vol (D) = \\sup_{T\\in c_1(D)} \\int_{V_{\\reg}} { (T\\vert_{V_{\\reg}})_{\\ac}^d} ,\n\\end{equation*}\nfor $T$ ranging among a positive $(1,1)$-current with analytic singularities whose singular locus does not contain \n$V$.\n\\end{theo}\nHere $T\\vert_{V_{\\reg}}$ denotes the restriction of $T$ to the regular locus $V_{\\reg}$ of $V$ and $(T\\vert_{V_{\\reg}})_{\\ac}$ denotes the absolutely continuous part (see subsection 2.4).\nMoreover we can express the restricted volumes with non-pluripolar product, which \nis introduced in \\cite{BEGZ}.\nTheorem \\ref{MaA} implies that we can define the restricted volume of \na transcendental class on a compact K\\\"ahler manifold in natural way.\n\n\n\n\\begin{defi} \\label{defi}\nLet $V$ be an irreducible variety of dimension $d$ on a compact K\\\"ahler manifold $M$ \nand $\\A$ the class in $H^{1,1}(M , \\RR)$.\nAssume that $V$ is not contained in the non-K\\\"ahler locus $E_{nK}(\\A)$.\nThen \\textit{the restricted volume} of $\\A$ along $V$ is defined to be\n\\begin{equation*}\n\\mathrm{vol}_{M|V}(\\A) = \\sup_{T\\in \\A} \\int_{V_{\\reg}}{({T|_{V_{\\reg}}})_{ac}^d}, \n\\end{equation*}\nfor $T$ ranging among a positive $(1,1)$-current with analytic singularities and the singular locus of \n$T$ does not contain $V$.\n\\end{defi}\n\n\n\nHere the non-K\\\"ahler locus is analytic counter part of the augmented base locus. \nWhen $\\A$ is the Chern class of some divisor $D$, the non-K\\\"ahler locus $E_{nK}(\\A)$ coincides \nwith the augmented base locus $\\mathbb{B}_{+}(D)$.\nFor this extended definition, the properties of usual restricted volumes hold. \nFor example, the continuity, log concavity, Fujita's approximations and so on (see subsection 4.2).\nMoreover the analogue of Theorem 1.1 holds for this extended definition as follows.\nThe principle of the proof is essentially same as Theorem 1.1.\nThe proof gives another proof of Theorem 1.1 by using analytic methods (see subsection 4.3).\n\\begin{theo}\\label{MaBB}\nLet $\\A$ be a big class in $H^{1,1}(X, \\RR)$.\nThen the following two conditions are equivalent.\n\n$(1)$ $\\A$ admits a Zariski decomposition.\n\n$(2)$ $\\V{V}(\\A) = \\V{V^{'}}(\\A)$ holds for any pair of subvarieties $V$ and $V'$ of $X$\nsuch that $V$, $V^{'}$ defines the same cohomology class and $V, V^{'} \\not \\subseteq E_{nK}(\\A)$.\n\\end{theo}\nHere the Zariski decomposition of a class $\\A$ is defined by divisorial Zariski decompositions \nintroduced in \\cite{Bou04} (see subsection 2.7).\nWhen $\\A$ is the first Chern class of some divisor, the Zariski decomposition of $\\A$ \ncorrespond with that of the divisor.\n\\begin{rem}\nThe class $\\A$ is not always contained in the N\\'eron-Severi group.\nTherefore Theorem \\ref{MaBB} is essentially stronger statement than Theorem \\ref{MaB}. \n\\end{rem}\n\nIn the proof of Theorem 1.1 and 1.4, to consider the restricted base locus of a divisor is important.\nThe restricted base locus $\\mathbb{B}_{-}(D)$ of $D$ is a countable union of closed analytic set,\nwhich is the defect of $D$ to be nef.\nIn section 5, we accurize the simplest version of Kawamata-Viehweg vanishing theorem\nin response to the dimension of the restricted base locus. \nThis proof is essentially to use Kawamata-Viehweg vanishing theorem (see \\cite{Ka82}, \\cite{Vie82}).\nA divisor $D$ is said to be nef in codimension $k$ if the codimension of non-nef locus greater than $k$.\n\n\n\\begin{theo}\\label{KV} \nAssume that $D$ is nef in codimension $k$ and big.\nThen\n\\begin{equation*}\nH^{q}(X, \\OX(\\K + D))=0\\ \\ \\ for\\ any\\ q\\ \\geq \\ n-k .\n\\end{equation*} \n\\end{theo}\n\n\\begin{rem}\nNote $D$ is nef in codimension $n-1$ if and only if $D$ is nef.\nThen Theorem \\ref{KV} implies $H^{q}(X, \\OX(\\K + D))=0\\ \\ \\ (q \\geq 1)$.\nThis is the simplest version of Kawamata-Viehweg vanishing theorem.\n\\end{rem}\n\nThis theorem can be generalized by using the generalized numerical Kodaira dimension $\\nu(D)$ \nof $D$, which is introduced in \\cite{BDPP}.\nTheorem \\ref{KV} implies an asymptotic estimate of higher cohomology as follows:\n\n\\begin{cor}\\label{asy}\nLet $E$ be an arbitrary divisor on $X$.\nAssume that $D$ is nef in codimension $k$.\nThen the following estimate holds for $q \\geq n-k$ as $\\ell \\to \\infty$:\n\\begin{equation*}\n\\dim H^{q}(X, \\OX (E + \\ell D)) =O(\\ell ^{n-q}) .\n\\end{equation*}\n\\end{cor}\nThis estimate is similar to \\cite[Proposition 2.15]{Ku06}.\nCorollary \\ref{asy} and asymptotic Riemann-Roch formula shows that\nthe inequality $\\V{X}(D) \\geq D^{n}$ holds when $D$ nef in codimension $n-2$.\nIn particular, then $D$ is big when the self-intersection number $D^{n}$ is positive.\nIn the case that $X$ is surface, this fact is well-known.\n\n\n\n\\subsection*{Acknowledgment}\nThe author is very grateful to his supervisor Prof.\\ Shigeharu Takayama \nfor valuable comments and various suggestion.\nHe is indebted to Dr.\\ Tomoyuki Hisamoto for stimulating discussions.\n\n\n\\section{Terminology}\n\n\\subsection{Currents}\nLet $M$ be a compact K\\\"ahler manifold in this section.\nThen $H^{p,p}(M,\\CC)$ is identified with the quotient of the space of $d$-closed $(p,p)$-currents modulo the \n$\\ddbar$-exact currents.\nFor our purpose the case of $p=1$ is important.\nA $d$-closed almost positive $(1,1)$-current $T$ is said to have analytic (\\textit{resp}. algebraic) singularities \n(along a subscheme $V(\\I)$ defined by an ideal sheaf $\\I$), \nif its potential function $\\varphi$ can be locally written as \n\\begin{equation*}\n\\varphi =\\frac{c}{2}\\log(|f_1|^2+...+|f_k|^2) + v\n\\end{equation*}\nfor some $c \\in \\RR_{>0}$ (\\textit{resp}. $c \\in \\QQ_{>0}$), \nwhere $f_1,\\dots,f_k$ are local generators of $\\I$ and $v$ is a smooth function.\n(Refer to \\cite{Dem} for details.)\n\n\\subsection{Pull-back of $(1,1)$-current}\nWe can handle $(1,1)$-currents easier than high degree currents.\nFor example we can define the pull-back of a $d$-closed $(1,1)$-current.\nLet $f : Z\\longrightarrow M$ be a holomorphic map between complex manifolds.\nAssume that the image of $Z$ by $f$ is not contained in the polar set of $T$.\nLet $T= \\theta + \\ddbar \\varphi$ be a $d$-closed $(1,1)$-current in the class $\\A \\in  H^{1,1}(M, \\RR)$.\nHere $\\theta$ is a smooth $(1,1)$-form in the class $\\A$ and $\\varphi$ is a $L^{1}$-function on $M$.\nThen the pullback of $T$ is defined to be $f^{*}T:= f^{*}\\theta + \\ddbar{f^{*}\\varphi }$ \nby using the potential $\\varphi $ of $T$. \nIn particular, we can restrict a $d$-closed $(1,1)$-current to any submanifold  \nwhich is not contained in the polar set.\n\n\n\\subsection{Multiplier ideal sheaves and Skoda's lemma.}\nIn this paper, multiplier ideal sheaves is used.\nWe denote by $\\J(\\Vert kD \\Vert)$ the asymptotic multiplier ideal sheaves \nassociated to a divisor $kD$ and \nby $\\I(kT)$ the multiplier ideal sheaves associated to \na $d$-closed current $kT$.\nWe can refer to \\cite{DEL00}, \\cite{Dem} for more details.\nThe Skoda's Lemma implies that a multiplier ideal sheaf is estimated by the Lelong numbers (see \\cite[Lemme 6.6]{Dem}). \n\n\\begin{lemm}{\\rm  [Skoda's Lemma] }\n$\\mathrm{(a)}$ If $\\nu(T,x) <1$, then $e^{-2 \\varphi}$ is integrable in a neighborhood of $x$.\nIn particular $\\I (T)_{M,x} = \\mathcal{O}_{M,x}$.\\\\\n$\\mathrm{(b)}$ If $\\nu(T,x) > n+s$ for some positive integer $s$, then $e^{-2 \\varphi}\\geq C |z-x|^{-2n-2s}$ in \na  neighborhood of $x$. In particular $\\I (T)_{M,x} \\subseteq  \\mathcal{M}_{M,x}^{s+1}$, \nwhere $\\mathcal{M}_{M,x}$ is the maximal ideal of $\\mathcal{O}_{M,x}$.\n\\end{lemm}\n\n\n\n\n\\subsection{Lebesgue decompositions}\nWe can always locally see a positive current $T$ as the differential form with positive measure coefficients.\nA positive measure admits a Lebesgue decomposition into the absolutely continuous part and \nthe singular part with respect to the Lesbegue measure. \nThis decomposition is local.\nHowever we can globally glue the absolutely continuous part and the singular \npart by the uniqueness of a Lebesgue decomposition. \nSo we obtain the docomposition $T = T_{\\ac} + T_{\\sing}$, where $T_{\\ac}$ is the form \nwith ${L^{1}_{loc}}$-function coefficients. \nThen we have $T_{\\ac} \\geq \\gamma $ when $T \\geq \\gamma $ for some smooth form $\\gamma$. \nIn particular the absolutely continuous part $T_{\\ac}$ is positive, when $T$ is a positive current.\nNote that $T_{\\ac}$ is not $d$-closed in general even if $T$ is $d$-closed.\nThen we can define the wedge $T_{\\ac}^{k}$ almost everywhere \nsince $T_{\\ac}$ is a positive form with ${L^{1}_{loc}}$-function coefficients (Refer to \\cite{Bou02} for more details). \n\n\\subsection{Approximations of currents}\nFix a K\\\"ahler form $\\omega$ on $M$. \nLet $T= \\theta + \\ddbar \\varphi$ be a $(1,1)$-current in the class $\\A \\in  H^{1,1}(M, \\RR)$, \nwhere $\\theta$ is a smooth $(1,1)$-form in the class $\\A$.\nWe assume that $T\\geq \\gamma $ holds for a smooth form $\\gamma $.\nThen we can approximate $T$ by smooth forms in the following sense:\n\n\\begin{theo}\\label{ap-dem82} \\textrm { \\cite[TH\\'EOR\\`EME9.1]{Dem82} }\nThere exist a decreasing sequence of smooth functions $\\varphi _{k}$ converging to $\\varphi $ \nsuch that if we set $T_{k} = \\theta + \\ddbar \\varphi _{k} \\in \\A$, we have\n\n\\ \\ $\\mathrm{(a)}$\\ \\ $T_{k} \\longrightarrow  T$ weakly and $T_{k} \\longrightarrow  T_{\\ac} $ almost everywhere .\n\n\\ \\ $\\mathrm{(b)}$\\ \\ $T_{k} \\geq \\gamma - C\\lambda _{k} \\omega $, \nwhere $C$ is a positive constant depending on $(M,\\omega )$ only, \nand $\\lambda _{k}$ is a decreasing sequences of continuous functions  such that \n$\\lambda _{k} \\searrow \\nu(T,x) $ for all $x \\in M$.\n\n\\end{theo}\nRoughly speaking Theorem \\ref{ap-dem82} says that it is possible to smooth a given current $T$ insides \nthe class $\\A$, but only with the loss of positivity controled by the Lelong numbers of $T$. \nBy the proof of Theorem \\ref{ap-dem82} in \\cite{Dem82}, we can add the following property $\\mathrm{(d)}$.\n\n\\textit{\n$\\mathrm{(c)}$\\ \\ When $T$ is smooth on some neighborhood of $x_{0} \\in M$, \n$T_{k} \\longrightarrow T_{\\ac}$ is a pointwise convergence on some neighborhood of $x_{0}$.\n}\\\\\nThe following theorem says that it is possible to approximate a given current $T$ with currents \nwith analytic singularities without the loss of positivity. \n\n\\begin{theo}\\label{ap-bou02} \\textrm{ \\cite{Dem92}},\\textrm{ \\cite[Theorem 2.4]{Bou02}  }\nThere exists a decreasing sequence of functions $\\varphi _{k}$ with analytic singularities \nconverging to $\\varphi $ such that if we set $T_{k} = \\theta + \\ddbar \\varphi _{k} \\in \\A$, we have\n\n\\ \\ $\\mathrm{(a')}$\\ \\ $T_{k} \\longrightarrow  T$ weakly and $T_{k,\\ac} \\longrightarrow  T_{\\ac} $ almost everywhere.\n\n\\ \\ $\\mathrm{(b')}$\\ \\ $T_{k} \\geq \\gamma -\\e_{k}\\omega $, where $\\e_{k}$ is a positive number converging to zero.\n\n\\ \\ $\\mathrm{(c')}$\\ \\ The Lelong number $\\nu(T_{k},x)$ increase to $\\nu(T,x)$ uniformly with respect to $x \\in M$.\n\\end{theo}\n\nIn the proof of \\cite[Theorem 2.4]{Bou02}, \nthe convergence $T_{k,\\ac} \\longrightarrow  T_{\\ac} $ in $\\mathrm{(a')}$ is proved only from the property \n$\\mathrm{(a)}$ in Theorem 2.2.\nTherefore we can add the following property $\\mathrm{(d')}$ from the property $\\mathrm{(c)}$.\n \n \n\\textit{\n\\ $\\mathrm{(d')}$\\ \\ When $T$ is smooth on some neighborhood of $x_{0} \\in M$, \n$T_{k} \\longrightarrow T_{\\ac}$ is pointwise convergence on some neighborhood of $x_{0}$.}\n\\\\\nThis consideration shows the following corollary.\n\\begin{cor}\nAssume that $T$ has analytic singularities whose polar set does not contain a closed analytic set $V$. \nThen $T_{k}$ in Theorem 2.3 satisfies the following:\n\\begin{equation*}\n\\big( T_{k} \\vert_{V_{\\reg}} \\big)_{\\ac} \\longrightarrow {\\big( T \\vert_{V_{\\reg}} \\big) }_{\\ac}\\ \\ \\ \\ \nfor\\ almost\\ point\\ in\\ V_{\\reg}.\n\\end{equation*}\n\\end{cor}\n\n\n\n\\subsection{Restricted volumes and properties.}\nFor simplicity, we denote by $H^{0}(X|V,\\OX (D))$ $$H^{0}(X|V,\\OX (D)):=\n\\mathrm{Im}\\Big(H^{0}(X,\\OX(D)) \\longrightarrow  H^{0}(V,\\OV(D)) \\Big).$$\nThe restricted volume of $D$ along $V$ is defined to be \n\\begin{equation*}\n\\V{V}(D) = \\limsup_{k \\to \\infty} \\dfrac\n{\\dim H^{0}(X|V,\\OX (D)) }{k^d\/d!}.\n\\end{equation*}\nNote that when $V$ is the ambient space $X$, the restricted volume of $D$ is the usual volume.\nWhen $V$ is not contained in $\\mathbb{B}_{+}(D)$, \nmany results for usual volumes are extended to restricted volumes.\nIn this paper, we refer the basic properties of restricted volumes to [ELMNP].\nHere $\\mathbb{B}_{+}(D)$ (\\textit{resp}. $\\mathbb{B}_{-}(D)$) means the augmented base locus \n(\\textit{resp}. the restricted base locus) of $D$\n(see \\cite{ELMNP2}, \\cite{ELMNP3}).\nThe non-K\\\"ahler locus $E_{nK}(\\A)$ (\\textit{resp}. non-nef locus $E_{nn}(\\A)$) \nof a class $\\A$ in $H^{1,1}(X,\\RR)$ is the analytic counter part of the augmented base locus \n(\\textit{resp}. restricted base locus) (see \\cite{Bou04}).\nWhen $\\A $ is the first Chern class $c_{1}(D)$ of some divisor $D$, the the augmented base locus \n(\\textit{resp}.restricted base locus ) coincide with the non-K\\\"ahler locus (\\textit{resp}. the non-nef locus).\nIf $\\A$ is a big class, $E_{nn}(\\A)$ coincide with $E_{+}(T_{\\min}):=\\big\\{x \\in X \\big| \\nu(T_{\\mathrm{min}}, x) >0   \\big\\}$, where \n$T_{\\min}$ is a minimal singular current in $\\A$ (see \\cite[proposition 3.8]{Bou04}).\n\n\\subsection{Divisorial Zariski decomposition.}\n\nIn this subsection, we recall the definition of divisorial Zariski decompositions.\nDivisorial Zariski decomposition for a big divisor coincides with $\\sigma$-decompositions.\nDivisorial Zariski decompositions are studied in \\cite{Bou04} and $\\sigma$-decompositions are studied in \\cite{Nak}.\n\n\nLet $\\A$ be a pseudo-effective class in $H^{1,1}(M,\\RR)$.\nThen an effective $\\RR$-divisor $N$ is defined as follows: \n\\begin{equation*}\n N:=\\sum_{F:\\mathrm{prime\\ div}} \\nu(T_{\\min} , F) F,\n\\end{equation*}\nwhere $T_{\\min}$ is a minimal singular current in $\\A$.\nHere $\\nu(T_{\\min} , F)$ is the Lelong number along $F$, \nwhich is defined by $\\inf_{x \\in F} \\nu(T_{\\min}, x)$.\nThe class $P$ is defined by $P:= \\A -\\{ N \\}$.\nHere $\\{N\\}$ denotes the class of $N$.\nThen the decomposition $\\A = P + \\{ N \\}$ is said to be the divisorial Zariski decomposition of $\\A$.\n$P$ is called the positive part and $\\{ N \\}$ is called the negative part of $\\A$.\nIn general the positive part $P$ is nef in codimension 1.\n(that is, the codimension of the non-nef locus is greater than 1.)\nWe say that $\\A$ admits a Zariski decomposition if the positive part $P$ is nef.\nIf $\\A$ is the class of some divisor, this definition coincides with that of the divisor (see section 1).\nFor example if $M$ is surface, any big class admits a Zariski decompoition (see \\cite[section 4]{Bou04}).\nBy the construction of $N$, positive currents in $\\A$ and positive currents in $P$ are identified  \nby the following correspondence:\n\\begin{equation*}\n\\A\\ni T  \\longmapsto T-[N] \\in P.\n\\end{equation*}\n\n\n\\section{Restricted volumes and Zariski decompositions }\n\n\\subsection{Positive parts and restricted volumes }\nThe main purpose in this section is to prove Theorem \\ref{MaB}.\nIn this section, we assume $D$ is big.\nWe begin to prepare for the proof of Theorem \\ref{MaB}.\nConsider the divisorial Zariski decomposition $D = P+N$ of $D$.\nSince $D$ is big, the divisorial Zariski decomposition coincide with the $\\sigma$-decomposition.\nNote that $\\mathbb{B}_{+}(D) = \\mathbb{B}_{+}(P) $ and $\\mathrm{Supp}(N) \\subseteq \\mathbb{B}_{+}(D)$.\nThe following proposition shows that the restricted volume of $D$ can be computed by the positive part $P$.\n\\begin{prop}\\label{positive}\nLet $W\\subseteq X$ be an irreducible subvariety which is not contained in $\\mathbb{B}_{+}(D)$.\nThen the equality $\\V{W} (D) = \\V{W} (P)$ holds.\n\\end{prop}\n\n\\begin{rem}\nIn general $P$ is an $\\RR$-divisor. \nThen $\\V{W} (P)$ is defined by the limit of the restricted volumes of \n$\\QQ$-divisors which converges to $P$ in N\\'eron-Severi group.\nBy the continuity of restricted volumes (\\cite[Theorem 5.2]{ELMNP}), $\\V{W} (P)$ does not depend on the choice\n of $\\QQ$-divisors which converges to $P$.\n\\end{rem}\n\n\\begin{proof}\nBy the bigness of $D$, we can take an effective $\\QQ$-divisor $D^{'}$ with $D\\sim_{\\QQ}  D^{'}$.\nBy multipling positive integer, we can assume that $D$ is effective divisor.\nSince $W$ is not contained in $\\mathbb{B}_{+}(D)$, we can assume \n$W \\not\\subseteq \\mathrm{Supp}(D) $. \nMoreover $W$ are not contained in $\\mathrm{Supp}(N \\cup P)$, \nsince $ \\mathrm{Supp}(N \\cup P)$ are contained in $\\mathrm{Supp}(D)$.\nFrom \\cite[Theorem 5.5]{Bou04}, there exists a canonical isomorphism \n$H^{0}(X, \\OX(\\lfloor kP \\rfloor)) \\cong H^{0}(X, \\OX(kD))$ by multipling the section $e_{k}$ for a positive intger $k$, \nwhere $e_{k}$ is the standard section of the effective divisor $\\lceil kN \\rceil$. \nWe consider the following commutative diagram:\n\\[\n\\begin{CD} H^0(X,\\OX (\\lfloor kP \\rfloor)) @>\\cdot e_{k}>>H^0(X,\\OX (kD))\\\\\n@V{f}VV  @V{g}VV\\\\\nH^0(W,\\OW (\\lfloor kP \\rfloor)) @> \\cdot e_{k} \\vert_{W}>>H^0(W,\\OW (kD))\n\\end{CD}\n\\]\n\n$f$ and $g$ are the restriction maps.\nThis diagram induces the map $\\mathrm{Im}(f) \\longrightarrow \\mathrm{Im}(g)$.\nThis map is surjective since the vertical map is an isomorphism.\nMoreover the map is injective since ${e_{k}} \\vert_{W}$ is a nonzero section \nfrom $W\\not\\subseteq \\mathrm{Supp}(N)$.\nSo we have $$ \\V{W}(D) = \\limsup _{k \\to \\infty}\\dfrac{h^{0}(X|W,\\OX (\\lfloor kP \\rfloor))}{k^{d}\/d! }.$$\nWhen $P$ is a $\\QQ$-divisor, Proposition \\ref{positive} follows from this equation and \nthe homogeneity of restricted volumes (\\cite[Lemma 2.2]{ELMNP}).\nWhen $P$ is an $\\RR$-divisor, we can reduct to a $\\QQ$-divisor by the continuity of restricted volumes.\n\\big(Note that $\\lfloor kP \\rfloor \/k \\to P$ $(k \\to \\infty)$.\\big)\n\\end{proof}\n\n\\begin{cor}\\label{positive-co}\nLet $W \\subseteq X$ be an irreducible subvariety of dimension $d$ which is not contained in $\\mathbb{B}_{+}(D)$.\nWhen $D$ admits a Zariski decomposition (that is, the positive part $P$ is nef), \nthe equality $\\V{W} (D) = (W\\cdot P^{d})$ holds.\n\\end{cor}\n\n\n\\begin{proof}\nBy Theorem \\ref{positive}, we obtain $\\V{W} (D) = \\V{W}(P)$.\nSince $P$ is nef, there exists ample $\\QQ$-divisors $B_{k}$ such that \n$B_{k} \\longrightarrow  P$ in the N\\'eron-Severi group.\nThe Ampleness of $B_{k}$ give $\\V{W}(B_k) = (V \\cdot B_{k}^{d})$.\nTherefore the continuity of restricted volumes shows \n\\begin{align*}\n\\V{W} (D) &= \\V{W}(P) \\\\\n&= \\lim_{k \\to \\infty} \\V{W} (B_k) \\\\\n&= \\lim_{k \\to \\infty} (W \\cdot B_{k}^{d}) = (W \\cdot P^{d}).\n\\end{align*}\n\\end{proof}\n\n\n\\subsection{Proof of the main Theorem}\n\nIn this subsection we complete the proof of Theorem \\ref{MaB}.\nWe take subvarieties $V$ and $V^{'}$ on $X$ such that $V \\sim V^{'}$ and $V, V^{'} \\not\\subseteq \\mathbb{B}_{+}(D)$. \nAssume that $D$ admits a Zariski decomposition.\nBy Corollary \\ref{positive-co} the restricted volume of $D$ can be computed by the intersection number \nof the positive part.\nThat is, $\\vol(D) =(V\\cdot P^{d}) $ and $\\vo(D) =(V^{'}\\cdot P^{d}) $ holds.\nSo $V \\sim V^{'}$ shows $(V\\cdot D^{d}) = (V^{'}\\cdot D^{d})$.\nHence $\\vol(D) = \\vo (D)$ holds if $D$ admits a Zariski decomposition.\n\n\nNext we prove the converse direction.\nAssume the condition (2).\nMoreover we assume that $P$ is not nef.\nThen the restricted base locus $\\mathbb{B}_{-}(P)$ is not empty.\nFix a very ample divisor $A$ on $X$ and an arbitrary point $x_{0}$ in $\\mathbb{B}_{-}(P)$.\nThen we can take irreducible smooth curves $C$ and $C^{'}$ with the following properties:\\\\\n\\ \\ \\ (1)\\ \\ \\ $C$ and $C^{'}$ are not contained in the augmented base locus $\\mathbb{B}_{+}(D)$. \\\\\n\\ \\ \\ (2)\\ \\ \\ $C$ intersects with the restricted base locus $\\mathbb{B}_{-}(P)$ at $x_{0} \\in X$.\\\\\n\\ \\ \\ (3)\\ \\ \\ $C^{'}$ does not intersect with the restricted base locus $\\mathbb{B}_{-}(P)$.\\\\\n\\ \\ \\ (4)\\ \\ \\ $C$ and $C^{'}$ are the complete intersections of the linear system of $A$.\n\n\nWe can take such curves. \nSince the codimension of the restricted base locus $\\mathbb{B}_{-}(P)$ is greater than one,\na very general complete intersection curve $C^{'}$ satisfies the properties (1), (3) by Bertini's Theorem. \n(We can not take $C^{'}$ which does not intersect $\\mathbb{B}_{-}(D)$ in general.)\n\\cite[Theorem 2.5]{Zha09} assure that a general hyperplane passing through $x_{0}$ \nis irreducible and smooth. \nTherefore we can take a smooth curve $C$ which satisfies the properties (1), (2).\n\n\nNow $C$ and $C^{'}$ are numerically equivalent \nsince $C$ and $C^{'}$ are the complete intersections of the same linear system.\nSo the equality $\\V{C}(P) = \\V{C^{'}}(P) $ holds by the assumption and Theorem \\ref {positive}.\nTherefore it is sufficient to prove the following claim for the contradiction to the assumption that $\\mathbb{B}_{-}(P)$ is not empty.\n\n\n\\begin{lemm}\n$\\mathrm{(A)}$\\ \\ \\ $\\V{C^{'}}(P) = (C^{'} \\cdot P) \\ \\big(=(C\\cdot P) \\big)$. \\\\\n\\ \\ \\ $\\mathrm{(B)}$\\ \\ \\ $ \\V{C}(P) < (C\\cdot P)$.\n\\end{lemm}\n\n\\begin{proof}\nFor simplicity, we assume $P$ is a $\\QQ$-divisor. \nWhen $P$ is an $\\RR$-divisor, the similar argument give the same results by using\ncontinuity of restricted volumes.\nFix a positive integer $a$ such that $aP$ is a $\\ZZ$-divisor.\n\\\\\n(A): Then from \\cite[Theorem 2.13]{ELMNP}, the restricted volume of $P$ can be computed as follows:\n\\begin{equation*}\n\\V{C^{'}}(P) = \\limsup_{\\ell \\to \\infty}\n\\dfrac{h^{0} \\big( C^{'}, \\mathcal{O}_{C^{'}} \\big( \\ell a P \\big) \\otimes \\J \n\\big(\\Vert \\ell  aP \\Vert\\big)\\big\\vert_{C^{'}}  \\big)}{\\ell a},\n\\end{equation*}\nSince $P$ is a big divisor, the restricted base locus of $P$ equals to the set \n$\\big\\{x \\in X \\ \\big| \\ \\nu(T_{\\mathrm{min}}, x) >0 \\big\\}$, \nwhere $T_{\\mathrm{min}}$ is a minimal singular current in $c_{1}(P)$.\nSo the Lelong number of every point $x$ on $C^{'}$ is zero by the property (3).\nThis implies $\\I_{+}(\\ell T_{\\mathrm{min}})$ and $\\I(\\ell T_{\\mathrm{min}})$ are \ntrivial along $C^{'}$ by Skoda's Lemma.\nThis shows $\\J(\\Vert\\ell P \\Vert)$ is trivial along $C^{'}$ by \\cite[Theorem 1.1]{DEL00}.\nHence we obtain \n\\begin{equation*}\n \\V{C^{'}}(P) = \\limsup_{\\ell \\to \\infty} \n\\dfrac{h^{0} \\big( C^{'}, \\mathcal{O}_{C^{'}} ( \\ell aP) \\big)}{\\ell a} .\n\\end{equation*}\n\nSince $C^{'}$ is not contained in $\\mathbb{B}(P)$, $P$ \nis an ample divisor on $C^{'}$.\nBy Riemann-Roch formula on compact Riemann surfaces, \nwe have $\\V{C^{'}} (P+A_{k}) = \\big((P+A_{k}) \\cdot C^{'} \\big)$.\nIn fact, Riemann-Roch formula shows \n\\begin{equation*}\nh^{0} \\big( C^{'}, \\mathcal{O}_{C^{'}} ( \\ell aP ) \\big) =\nh^{1} \\big( C^{'}, \\mathcal{O}_{C^{'}} ( \\ell aP) \\big) + \n\\big( \\ell a P \\cdot C^{'} \\big) -g +1,\n\\end{equation*}\nwhere $g$ is a genus of $C^{'}$.\nBy the ampleness of $P$, the first cohomology vanishes \nfor sufficietly large $\\ell$. \nSo we obtain $\\V{C^{'}} (P) = \\big( P \\cdot C^{'} \\big)$. \nTherefore (A) in Lemma 3.4 holds.\\\\\n\n(B):\\ By the same argument of (A), we get the following equality:\n\n\\begin{equation*}\n\\V{C}(P) = \\limsup_{\\ell \\to \\infty}\n\\dfrac{h^{0} \\big( C, \\mathcal{O}_{C} \\big( \\ell a P \\big) \\otimes \\J \\big(\\Vert \\ell  aP \\Vert\\big)\n\\big\\vert_{C}  \\big)}{\\ell a},\n\\end{equation*}\nNow we estimate the asymptotic multiplier ideal by using Skoda's Lemma. \nFirst of all, we obtain $\\J (\\Vert k P \\Vert) \\subseteq  \\I( k T_{\\mathrm{min}})$ for every positive integer $k$\nfrom the minimal singularity of $T_{\\mathrm{min}}$, \nwhere $T_{\\mathrm{min}}$ is a minimal singular current in $c_{1}(P)$.\nSince $\\nu(T_{\\mathrm{min}},x_{0}) $ is positive by the property (2), \nwe can take a positive rational number $p\/q$ which is smaller than $\\nu(T_{\\mathrm{min}},x_{0})$.\nSo we obtain $p \\ell a < \\nu(q \\ell a T_{\\mathrm{min}}, x_{0})$.\nSkoda's Lemma implies $\\nu(k q T_{\\mathrm{min}})_{x_{0}} \\subseteq {\\mathcal{M}^{{p \\ell a -n+1}}_{x_{0}, X}}$, \nwhere $\\mathcal{M}_{x_{0},X}$ is the maximal ideal in $\\mathcal{O}_{x_{0,X}}$.\nSo we obtain \n\\begin{align*}\n\\V{C}(P) \n&\\leq  \\limsup_{k \\to \\infty} \\dfrac{h^0 \\big( C,\\mathcal{O}_{C} (\\ell a P )\\otimes\n {\\mathcal{M}^{{p \\ell a -n+1}}_{x_{0}, X}} \\vert_{C} \\big) }{\\ell a}  \\\\\n &\\leq  \\limsup_{k \\to \\infty} \\dfrac{h^0 \\big( C,\\mathcal{O}_{C} (\\ell a P )\\otimes\n{\\mathcal{M}^{{p \\ell a -n+1}}_{x_{0}, C}}\\big) }{\\ell a}  \\\\\n& = \\limsup_{k \\to \\infty} \\dfrac{h^0 \\big( C,\\mathcal{O}_{C} (\\lfloor \\ell a P \\rfloor - (p \\ell a -n+1)[x_{0}] )\\big) } {\\ell a} ,\n\\end{align*}\nwhere $[x_{0}]$ is a divisor on $C$ defined by $x_{0}$.\nBy the same argument of (A), we obtain $\\V{C}(P) < (C\\cdot P) -{p}\/{q}$.\n\\end{proof}\nBy the proof of Lemma 3.5 (B), the following corollary is proved.\n\\begin{cor}\nLet $C$ be an irreducible smooth curve. \nAssume that $C$ is not contained in the augmented base locus of $D$ \nThen the following inequality holds.\n\\begin{equation*}\n\\V{C}(D) \\leq (C\\cdot D) - \\sum_{x \\in C \\cap \\mathbb{B}_{-}(D)} \\nu(T_{\\mathrm{min}},x)\n\\end{equation*}\n\\end{cor}\n\n\n\n\n\\section{Expression of restricted volumes by current integration}\n\\subsection{Proof of the expression formula}\nThe main purpose of this subsection is to prove Theorem\\ref{MaA}.\nBefore the proof of Theorem \\ref{MaA}, we must show that the integral \n$\\int_{V_{\\reg}}{({T\\vert_{V_\\reg}})_{\\ac}^d}$ is always finite.\n\n\\begin{prop}\\label{F}\nLet $W$ be an irreducible subvariety of dimension $d$ on a compact K\\\"ahler manifold $Y$ \nand $S$ a $d$-closed positive $(1,1)$-current on $Y$.\nAssume that the polar set of $S$ does not contain $W$.\nThen the integral $\\int_{W_{\\reg}}({S\\vert_{W_{\\reg} } })_{\\ac}^d$ is finite.\n\\end{prop} \n\n\\begin{proof}\nNote the restriction $S\\vert_{W_{\\reg}}$ is $d$-closed positive.\nHence the integral $\\int_{W}({S\\vert_{W}})_{\\ac}^d$ is finite by \\cite[Lemma2.11]{Bou02} when $W$ is non-singular.\nTherefore it is enough to show the integral is finite when $W$ has singularities. \nLet $\\mu :\\widetilde {W} \\subseteq  \\widetilde {X}  \\longrightarrow W \\subseteq X$ be the embedded resolution \nof $W \\subseteq X$.\nThat is, $\\mu:  \\widetilde {X}  \\longrightarrow X$ is a modification and the restriction\n$\\mu :  \\widetilde {W}  \\longrightarrow W$ is the resolution of singularities of $W$.\nThen the following lemma assures Proposition \\ref{F} holds.\nIn fact, $\\int_{\\widetilde{W} } { \\big(  (\\mu^{*} S)\\vert_{\\widetilde{W}} \\big)_{\\ac}^{d} }$ is finite, \nsince $\\widetilde{W}$ is non-singular.\n\\end{proof}\n\n\\begin{lemm}\\label{f}\n\\[ \\ \\ \\ \\ \\int_{W_{\\reg}}{({S\\vert_{W_{\\reg}}})_{\\ac}^d} = \n\\int_{\\widetilde{W} } { \\big(  (\\mu^{*} S)\\vert_{\\widetilde{W}} \\big)_{\\ac}^{d} }\\]\n\\end{lemm}\n\n\\begin{proof}\nThe map $\\widetilde {W}\\xrightarrow{\\ \\ \\mu \\ \\ } W$ is an isomorphism except some subvarieties.\nTherefore $(\\mu ^{*}S) \\vert _{\\widetilde{W}}$ is identified with $S\\vert_{W_{\\reg}}$  by $\\mu$ on Zariski open set.\n$\\big( (\\mu ^{*}S) \\vert _{\\widetilde{W}} \\big)_{\\ac}$ and $\\big( S\\vert_{W_{\\reg}} \\big)_{\\ac}$\n are absolutely continuous with respect to the Lebesgue measure by the definition. \nHence we obtain $\\int_{W_{\\reg}}{({S\\vert_{W_{\\reg}}})_{\\ac}^d} = \n\\int_{\\widetilde{W} } { \\big(  (\\mu^{*} S)\\vert_{\\widetilde{W}} \\big)_{\\ac}^{d} }$ \nsince a Zariski closed set is measure zero with respect to the Lebesgue measure .\n\\end{proof}\n\\textit{Proof of Theorem \\ref{MaA}.)}\n{\\bf{ (Step1)} } First of all we show the inequality $\\geq$ in Theorem \\ref{MaA}\nby using the singular holomorphic Morse inequalities (see [Bon93]) and Proposition \\ref{min}.\nProposition \\ref{min} is proved in the end of this subsection.\nAt first we consider the case when $V$ is non-singular. \nFor a positive $(1,1)$-current $T$ with analytic singularities in the Chern class $c_1(D)$ \nwhose singular locus does not contain $V$, we obtain the following inequality:\n\\begin{align*}\n\\vol (D) &= \\limsup_{k \\to \\infty} \\frac{\\dim\n H^{0}\\bigl(V,\\mathcal{O}_{V} (kD) \\otimes \\mathcal{I} (kT_{\\mathrm{min}})\\vert_{V} \\bigr)}{{ k^d}\/{d!} } \\\\\n& \\geq \\limsup_{k \\to \\infty} \\frac{\\dim \nH^{0}\\bigl(V,\\mathcal{O}_{V} (kD) \\otimes \\mathcal{I} (kT)\\vert_{V} \\bigr)}{{ k^d}\/{d!} } \\\\\n& \\geq \\limsup_{k \\to \\infty} \\frac{\\dim \nH^{0}\\bigl(V,\\mathcal{O}_{V} (kD) \\otimes \\mathcal{I} (kT\\vert_{V})\\bigr)}{{ k^d}\/{d!} }.\n\\end{align*}Here $T_{\\min}$ is a minimal singular current in $c_{1}(D)$.\nThe first equality follows from Proposition \\ref{min} and the second inequality follows from the restriction formula.\nBy using the singular holomorphic Morse inequality we have \n\\begin{align*}\n\\vol (D) &\\geq \\limsup_{k \\to \\infty} \\frac{\\dim \nH^{0}\\bigl(V,\\mathcal{O}_{V} (kD) \\otimes \\mathcal{I} (kT\\vert_{V})\\bigr)}{{ k^d}\/{d!} }\\\\\n& \\geq \\int_{V}{{(T\\vert_{V})}_{\\ac}^d}.\n\\end{align*}Therefore when $V$ is non-singular, Step 1 is proved.\n\nNext we consider the case when $V$ has singularities.\nThen we consider the embedded resolution \n$\\mu: \\widetilde V \\subseteq \\widetilde X \\longrightarrow V\\subseteq X$.\nNote the augmented base locus of the pull back $\\mu ^{*}D$ does not contain $\\widetilde V$ since \nthe restriction $\\mu : \\widetilde V \\longrightarrow V$ is isomorphism at a generic point.\nThen by applying the singular holomorphic Morse inequality and \nrestriction formula to $\\mu ^{*}D$, $\\widetilde V$ and $\\mu ^{*}T$ again, we obtain\n\\begin{equation*} \n\\voll (\\mu ^{*}D) \\geq \\int_{\\widetilde V} { ((\\mu ^{*}T)\\mid _{\\widetilde V})_{\\ac}^{d}}.\n\\end{equation*}\n\nBy Lemma \\ref{F} and \\cite[Lemma 6.7]{ELMNP} shows the following lemma.\nBy this lemma and the above inequalities, Step 1 is proved even if $V$ has singularities.\n\\begin{lemm}\\label{cc}\n\\[ (1)\\ \\ \\ \\ \\vol (D)=\\voll (\\mu ^{*}D) \\]\n\\[ \\ \\ (2)\\ \\ \\ \\ \\int_{V} \\T=\\int_{\\widetilde{V}} \\Tm \\]\n\\end{lemm}\n{\\bf(Step2)}\\ \\ \nWe show the converse inequality $\\leq$ to complete the proof of Theorem \\ref{MaA}.\nBy \\cite[Proposition 2.11]{ELMNP}, for an arbitrary number $\\e >0$, we can find \nthe modification $\\pi_{\\e}:X_{\\e} \\longrightarrow X$\n and the expression ${\\pi_{\\e}}^{*} (D)=A_{\\e}+E_{\\e}$ \nsuch that $({A_{\\e}}^{d}\\cdot V_{\\e}) \\geq \\vol(D) - \\e$.\nHere $A_{\\e}$ is an ample $\\QQ$-divisor and $E_{\\e}$ is an effective $\\QQ$-divisor such that\nthe proper transformation $V_{\\e}$ is not contained in $\\mathrm{Supp}(E_{\\e})$. \nWe express the intersection number $({A_{\\e}}^{d}\\cdot V_{\\e})$ by current integration.\nLet $\\omega _{\\e}$ be a smooth positive $(1,1)$-form on $X_{\\e}$ in the first Chern class $c_{1}(A_{\\e})$.\nSince $\\mathrm{Supp}(E_{\\e})$ does not contain $V_{\\e}$, \nwe can restrict  $[E_{\\e}]$ to $V_{\\e}$.\nSince $[E_{\\e}]$ is a positive current, we obtain\n\\begin{align*}\n({A_{\\e}}^{d} \\cdot {V}_{\\e}) &= \\int_{V_{\\e}}{({\\omega _{\\e}}\\vert_{V_{\\e}})^{d}} \\\\\n&\\leq \\int_{V_{\\e}}{ \\big( { (\\omega _{\\e} + [E_{\\e}])\\vert_{V_{\\e} } } \\big) _{\\ac}^{d}} \\\\\n&= \\int_{V_{\\reg}} { \\big\\{ \\big ({\\pi _{\\e}}_{*}(\\omega _{\\e} + [E_{\\e}])\\big)\\vert_{V_{\\reg} } \\big\\} _{\\ac}^{d} } . \\\\\n\\end{align*}\nThe third equality follows from the property that $\\pi_{\\e}$ is an isomorphism at a general point of $V$ and the same \nargument of Lemma \\ref{f}.\n\nSince $\\pi_{\\e}$ is a modification, the push-forward ${\\pi _{\\e}}_{*}(\\omega _{\\e} + [E_{\\e}])$ is \nthe positive current in the Chern class $c_{1}(D)$.\nHowever the push-forward may not have analytic singularities.\nSo we want to approximate the push-forward by positive currents with analytic singularities. \nWhen we use Theorem \\ref{ap-bou02}, the positivity of a current can be lost.\nNote $(\\omega _{\\e} + [E_{\\e}])$ is a K\\\"ahler current but \nthe push-forward may not be a K\\\"ahler current.\nHence we can not always take the sequence of approximation by positive currents.\nNow we consider the approximations of the push-forward ${\\pi _{\\e}}_{*}(\\omega _{\\e} + [E_{\\e}])$\nafter we change the current for the K\\\"ahler current.\n\n\nFor simplicity we write $T_{\\e} := {\\pi _{\\e}}_{*}(\\omega _{\\e} + [E_{\\e}])$. \nSince $V$ is not contained in the augmented base locus $ {\\mathbb{B}}_{+}(D)$, \nwe can take a K\\\"ahler current $S$ with analytic singularities such that the pole does not contain in $V$. \nFatou's Lemma assures\n\\begin{align*}\n\\vol(D) - \\e & \\leq  ({A_{\\e}}^{d}\\cdot V_{\\e}) \\\\\n&= \\int_{\\Vr} \\liminf _{\\del \\to 0} \\big\\{ (1-\\del)({T_{\\e}}\\vert_{\\Vr}) + \\del (S\\vert_{V_{\\reg}})  \\big\\}_{\\ac}^{d} \\\\\n&\\leq \\liminf _{\\del \\to 0} \\int_{\\Vr} \\big\\{ (1-\\del)({T_{\\e}}\\vert_{\\Vr}) + \\del (S\\vert_{V_{\\reg}})  \\big\\}_{\\ac}^{d} .\n\\end{align*}\nHence there exists a sufficiently small $\\del_{0} >0$ with the following:\n\\begin{equation*}\n\\vol(D) - 2\\e \\leq \\int_{\\Vr} \\{ (1-\\del_{0})({T_{\\e}}\\vert_{\\Vr})_{\\ac} + \\del_{0} (S\\vert_{V})_{\\ac}  \\}^{d}.\n\\end{equation*}\nNote that $(1-\\del_{0})T_{\\e} + \\del_{0} S$ is a K\\\"ahler current in $c_{1}(D)$.\nBy applying the approximation theorem (Theorem 2.3 and Corollary 2.4) to $(1-\\del_{0})T_{\\e} + \\del_{0} S$, \nwe can take positive currents $\\big\\{ U_{k} \\big\\}_{k=1}^{\\infty}$ in $c_1(D)$ with the following properties.\n\\\\\n\\ \\ \\ (1)\\ \\ \\ $U_{k}$ has an analytic singularities for every integer $k$. \\\\\n\\ \\ \\ (2)\\ \\ \\ $({U_{k}}\\vert_{\\Vr})_{\\ac} \\longrightarrow \n\\big\\{ (1-\\del_{0})T_{\\e} \\vert_{\\Vr}+ \\del_{0} S\\vert_{\\Vr} \\big\\}_{\\ac}$\\ \\ \\ for almost point in $V_{\\reg}$  \\\\\n\\ \\ \\ (3)\\ \\ \\ $U_{k}$ is a positive current for a sufficiently large $k$\n\n\nThe property (2) and Fatou's Lemma show \n\\begin{align*}\n\\vol(D) - 2\\e &\\leq  \\int_{\\Vr} \\big\\{ (1-\\del_{0})({T_{\\e}}\\vert_{\\Vr})_{\\ac} + \\del_{0} (S\\vert_{V_{\\reg}})_{\\ac}  \\big\\}^{d} \\\\\n&= \\int_{\\Vr} \\liminf_{k \\to \\infty}({U_{k}}\\vert_{\\Vr})_{\\ac}^{d}\\\\\n&\\leq   \\liminf_{k \\to \\infty} \\int_{\\Vr} ({U_{k}}\\vert_{\\Vr})_{\\ac}^{d}.\\\\\n\\end{align*}\nTherefore we have $$\\vol(D)-3\\e\\leq \\int_{\\Vr} ({U_{k_{0}}}\\vert_{\\Vr})_{\\ac}^{d}$$ for a sufficiently large $k_{0}$.\nSince $\\e$ is an arbitrary positive number and $U_{k_{0}}$ is a positive current with analytic singularities \nin the Chern class $c_1(D)$ whose pole of $U_{k_{0}}$ does not contain $V$.\nHence Step 2 is finished.\nTheorem \\ref{MaA} is proved.\n\\begin{flushright}\n$\\square$\n\\end{flushright}\nIn the end of this subsection, we prove the following proposition.\nIt is the variation of \\cite[Theorem 2.13]{ELMNP}.\n\n\\begin{prop}\\label{min}\nUnder the assumption that $V$ is not contained in $ {\\mathbb{B}}_{+}(D)$, the following equality holds.\n\n\\begin{equation*}\n\\vol (D) = \\limsup_{k \\to \\infty} \\frac{h^{0}\\bigl(V,\\mathcal{O}_{V} (kD) \\otimes \\mathcal{I} \n(kT_{\\mathrm{min}})|_{V} \\bigr)}{{ k^d}\/{d!} } ,\n\\end{equation*}\nwhere $T_{\\mathrm{min}}$ is a minimal singular current in the class $c_1(D)$.\n\\end{prop}\n\n\\begin{proof}\n\\cite[Theorem 2.13]{ELMNP} assures the following equality under the assumption. \n\\begin{equation*}\n\\vol (D) = \\limsup_{k \\to \\infty} \\frac{h^{0}\\bigl(V,\\mathcal{O} \n(kD) \\otimes \\mathcal{J} ( \\Vert kD \\Vert)|_{V} \\bigr)}{{ k^d}\/{d!} } \n\\end{equation*}\nTo prove Proposition \\ref{min}, we compare the multiplier ideal sheaf \n$\\mathcal{I}(kT_{\\mathrm{min}})$ with the asymptotic multiplier ideal sheaf $\\mathcal{J}(\\Vert kD\\Vert)$.\nBy the definition of a minimal singular current, we obtain \n$\\mathcal{I}(kT _{\\mathrm{min}}) \\supseteq \\mathcal{J}(\\Vert kD\\Vert)$ for all positive integer $k$.\nHence we obtain the inequality $ \\leq $ in Proposition \\ref{min}.\nTo prove the converse inequality we show the following Lemma.\n\\begin{lemm}\\label{c}\nThere is an effective divisor $E$ (independent of $k$) which satisfies the following properties:\n\n\n\\ \\ \\ \\ $\\mathrm{(i)}$\\ \\  \\ \\ \\ $\\mathrm{Supp} (E)$ does not contain V.\n\n\\ \\ \\ \\ $\\mathrm{(ii)}$\\ \\ \\ \\ \\ $\\mathcal{I}(kT_{\\mathrm{min}})\\cdot \\mathcal{O}_{X}(-E) \n\\subseteq \\mathcal{J}(\\Vert kD\\Vert )$\n for all $k$ sufficiently large.\n\\end{lemm}\n\n\\begin{proof}\nThis proof is essentially based on the argument in \\cite{DEL00}.\nFix a very ample divisor $A$.\nFor an arbitrary point $x \\in X$, there exists a zero dimensional complete intersection $P_{x}$ \nof the linear system $\\mid A \\mid$ containing $x$. \nThe Ohsawa-Takegoshi-Manivel $L^2$-extension theorem shows \nfor every divisor $F$ with a singular hermitian metric $h$ with the nonnegative curvature $T_{h}$,\nthere exists an ample divisor $B$ ($B$ depends only $A$) and \nthe following restriction map is surjective (see \\cite{OT87}, \\cite{Man93}).\n\n\\begin{equation*}\nH^{0}\\bigl(X,\\mathcal{O}_{X} (F+B) \\otimes \\mathcal{I}(T_{h} )\\bigl) \n\\longrightarrow H^{0}\\bigl(P_{x},\\mathcal{O}_{P_{x}} (F+B)\\otimes \\mathcal{I}(T_{h} \\vert_{P_{x}})\\bigl )\n\\end{equation*}\nMoreover the Ohsawa-Takegoshi-Manivel $L^2$-extension claims that for every section on $P_{x}$, the extension is satisfied an $L^{2}$-estimate with a constant independent of $F$.\nNow since the dimension of $P_{x}$ is zero, the $L^{2}$-estimate does not depend on $P_{x}$.\nThat is, $L^{2}$-estimate of the extension of a section on $P_{x}$ depend only on $B$.\nSince $B$ depends only $A$, the $L^{2}$-estimate of the extension of a section on $P_{x}$ depend only on $A$.\n\n\nSince $D$ is big and $V$ is not contained in the augmented base locus of $D$, \nwe can take $E \\in \\big| k_{0}D - B \\big|$ with the property (i) by choosing a sufficiently large $k_{0}$.\nWe apply the Ohsawa-Takegoshi-Manivel $L^{2}$-extension theorem to \n$F_{k}:=(k-k_0)D + E$ equipped with the singular hermitian metric $h_{\\mathrm{min}}^{\\otimes k-k_{0}}\\otimes h_{E}$.\nHere $h_{\\mathrm{min}}$ is a minimal singular hermitian metric \nand $h_{E}$ is a singular hermitian metric defined by the standard section of the effective divisor $E$.\nThen for a sufficiently large $k$ and a point $x\\in X$, we obtain the global section $s_{x}$ of  \n$F_{k}+B \\equiv _{\\mathrm{lin}} kD$ satisfied the following estimates:\n \n\\begin{equation*}\n \\int_{X}{ \\|  s_{x} {\\|^{2}}_{{h_{\\mathrm{min}}^{\\otimes k-k_{0}}\\otimes h_{E}\\otimes h_{B} } } \\leq C\\ \\ \\ \\ \\ \\ and\\ \\ \\ \\ \\  \n| s_{x}(x) |^{2}}_{h_{\\mathrm{min}}^{\\otimes k-k_{0}}\\otimes h_{E}\\otimes h_{B} }  =1,\n \\end{equation*}\nwhere $C$ is a constant depending only $A$ and $h_{B}$ is a smooth hermitian metric of B with the positive curvature.\nFrom the second equality, we infer the following equality:\n\\begin{equation*}\n|s_{x}(x)|^{2}e^{-2(k-k_{0}) \\varphi _{\\mathrm{min}} -2\\varphi _{E}-2\\varphi _{B} }=1,\n\\end{equation*} \nwhere $\\varphi _{\\mathrm{min}}$, $\\varphi _{E}$, $\\varphi _{B}$ is the weight of the hermitian metric \n$h_{\\mathrm{min}}$,$h_{E}$,$h_{B}$ respectively.\nSince $\\varphi _{B}$ is a smooth function and $X$ is compact, we obtain\n \\begin{equation*}\n \\varphi _{\\mathrm{min}}+\\frac{1}{k-k_0}\\varphi _{E} \\leq \\frac{1}{k-k_{0}} \\log|s_{x}(x)| +C.\n \\end{equation*}\nThe evaluation map $H^{0}\\bigl(X,\\mathcal{O}_{X} (kD))\\longrightarrow \\CC$ is \na bounded operator on the Hilbert space \n$H^{0}\\bigl(X,\\mathcal{O}_{X} (kD))$ with $L^{2}$-norm. \nMoreover the operation norm equals to the Bergman kernel\n ${\\log\\sum_{j}{|f_{j}}|}$, \nwhere $\\{f_{j}\\}$ is the orthonormal basis of $H^{0}\\bigl(X,\\mathcal{O}_{X} (kD))$.\n\nTherefore we get\n\\begin{equation*}\n \\frac{1}{k-k_{0}} \\log|s_{x}(x)|+C \\leq  \\frac{1}{k-k_{0}} \\log\\sum_{j}{|f_{j}}| +C \n\\end{equation*}\nThis inequality implies that the function $\\frac{1}{k-k_{0}} \\log\\sum_{j}{|f_{j}}|$ has \nless singularities than $\\varphi _{\\mathrm{min}}+\\frac{1}{k-k_0}\\varphi _{E} $.\nConsidering the definition of the asymptotic multiplier ideal sheaf,  we get the property (ii). \n\\end{proof}\n\n\nWe complete the proof of Proposition \\ref{min} by using Lemma \\ref{c}.\nFrom the property (i) in the previous Lemma, we can consider the following short exact sequence:\n\\begin{equation*}\n0 \\longrightarrow  \\mathcal{O}_{V}(kD-E)     \\longrightarrow    \\mathcal{O}_{V}(kD)  \\longrightarrow\n     \\mathcal{O}_{V\\cap E}(D)     \\longrightarrow\t0  \n\\end{equation*}\nSince the dimension of the intersection $V\\cap E$ is smaller than $\\dim{V}=d$, \nwe have\n\\begin{equation*}\n\\limsup_{k \\to \\infty} \\dfrac{\\dim \nH^{0}\\big(V \\cap E,\\mathcal{O}_{V \\cap E} (kD) \\big)}{k^{d}\/d!} =0.\n\\end{equation*}\nHence we obtain the following inequality by twisting the restriction to $V$ of the multiplier ideal \nsheaf $\\mathcal{I}(kT_{\\mathrm{min}})$.\n\\begin{equation*}\n\\limsup_{k \\to \\infty} \\frac{\\dim H^{0}\\bigl(V,\\mathcal{O}_{V} (kD) \n\\otimes \\mathcal{I} (kT_{\\mathrm{min}})|_{V} \\bigr)}{{ k^d}\/{d!} } \\leq\n  \\limsup_{k \\to \\infty} \\frac{\\dim H^{0}\\bigl(V,\\mathcal{O}_{V} (kD-E) \n\\otimes \\mathcal{I} (T_{\\mathrm{min}})|_{V}  \\bigr)}{{ k^d}\/{d!} } \n\\end{equation*}\nSince we chose the effective divisor $E$ satisfied the property (ii) of the previous Lemma, \nwe obtain\n \\begin{equation*}\n\\limsup_{k \\to \\infty} \\frac{\\dim H^{0}\\bigl(V,\\mathcal{O}_{V} (kD-E) \\otimes \\mathcal{I}\n (kT_{\\mathrm{min}})|_{V}  \\bigr)}{{ k^d}\/{d!} } \\leq  \n\\limsup_{k \\to \\infty} \\frac{\\dim H^{0}\\bigl(V,\\mathcal{O}_{V} (kD) \\otimes \\mathcal{J}\n(\\Vert kD \\Vert )|_{V} \\bigr)}{{ k^d}\/{d!} }.\n\\end{equation*}\nThese inequalities assure\n\\begin{equation*}\n\\vol (D) \\geq  \\limsup_{k \\to \\infty} \\frac{\\dim H^{0}\\bigl(V,\\mathcal{O}_{V} (kD) \n\\otimes \\mathcal{I} (kT_{\\mathrm{min}})|_{V} \\bigr)}{{ k^d}\/{d!} } .\n\\end{equation*}\nThe proof of Proposition \\ref{min} is finished.\n\\end{proof}\n\n\n\n\n\\subsection{Properties of restricted volumes.}\nTheorem \\ref{MaA} implies that the definition of restricted volumes can be extended to a big class on \na compact K\\\"ahler manifold (see Definition \\ref{defi}).\nLet $M$ be a compact K\\\"ahler manifold, $V\\subseteq M$ a irreducible subvariety of dimension $d$ and \n$\\A$ a big class in $H^{1,1}(M,\\RR)$ in this section. \nIn this subsection, we study the properties of the restricted volumes of a class on \ncompact K\\\"ahler manifold.\n.\n\n\n\\begin{prop}\\label{nef}\nAssume $\\A$ is a nef class and $V$ is not contained in the non K\\\"ahler locus $E_{nK}(\\A)$.\nThen the restricted volume $\\volm(\\A)$ is computed by the intersection number  $(\\A^{d}\\cdot V)$.\nThat is, the equality $\\volm(\\A) = (\\A^{d}\\cdot V)$ holds.\n \\end{prop}\n\n\\begin{proof}\nWhen $V$ is non-singular, this proposition is proved by using the similar argument with \\cite[Theorem 4.1]{Bou02}.\nThe case when $V$ has singularities is reduced to the non-singular case by same \nthe argument in Lemma \\ref{f}.\n\\end{proof}\n\nProposition \\ref{div} is generalization of Lemma \\ref{positive} to a class on a compact K\\\"ahler manifold. \nMoreover the proof gives another proof of Lemma \\ref{positive}. \nLet $\\A = P + \\{N\\}$ be the divisorial Zariski decomposition of $\\A$.\n\n\\begin{prop}\\label{div}\nAssume that $V$ is not contained in the non-K\\\"ahler locus $E_{nK}(\\A)$. \nThen $V$ is not contained in $E_{nK}(P)$ and \nthe equality $\\volm(\\A) = \\volm(P) $ holds.\n\\end{prop}\n\n\n\n\\begin{proof}\nThis proposition is based on the following fact. \nPositive currents in $\\A$ and positive currents in $P$ are identified  \nby the following correspondence.\n\\begin{align*}\n&\\A \\longrightarrow P \\\\\n&T \\longmapsto   T-[N]\n\\end{align*}\nNote $T-[N]$ is a K\\\"ahler current if $T$ is a K\\\"ahler current in $\\A$.\nIn fact if $T \\geq \\delta \\omega $ holds, $T-\\delta \\omega -[E] $ is a positive current\nby considering the Siu decomposition for the positive current $T-\\delta \\omega$.\nHere $E$ is an effective divisor defined by $ E:=\\sum_{D} \\nu(T-\\delta \\omega, D)D$.\nSince $\\omega$ is a smooth form, we get $\\nu(T-\\delta \\omega, D) = \\nu(T, D)$.\nSince $E-N$ is an effective divisor,  $T-[N] \\geq \\delta \\omega$ holds.\n\n\n\nFirst of all we show the following Lemma to prove that \n$V$ is not contained in $E_{nK}(P)$.\n\\begin{lemm}\\label{augment}\nThe equality $E_{nK}(\\A) = E_{nK}(P)$ holds.\n\\end{lemm}\n\\begin{proof}\nNote $\\mathrm{Supp}(N)$ is clearly contained in $E_{nK}(\\A)$ from the definition.\nFor a point $x \\not\\in E_{nK}(\\A) $, we can take a K\\\"ahler current $T$ in $\\A$ with analytic singularities \nsuch that $T$ is smooth at $x$.\nThen $T-[N]$ is a K\\\"ahler current in $P$.\nSince $x$ is not contained in $\\mathrm{Supp}(N)$, the K\\\"ahler current $T-[N]$ is smooth at $x$.\nSo $x \\not\\in E_{nK}(P)$ holds.\n\n\n\nConversely for a point $x \\not\\in E_{nK}(P)$, we take a K\\\"ahler current $S$ such that $S$ is smooth on $x$ .\nWe can assume that $S\\geq \\omega $.\nWe prove that $x$ is not contained in $\\mathrm{Supp}(N)$.\nTo prove this, we consider the surjective map:\n\\begin{equation*}\n\\big\\{ \\mathrm{smooth\\ real\\ \\textit{d}\\textit{-}closed\\ (1,1)\\textit{-}form} \\big\\} \\longrightarrow  H^{1,1} (M,\\mathbb{R}),\n\\ \\ \\theta  \\mapsto \\big\\{ \\theta \\big\\}.\n\\end{equation*}\nWe regard the space of smooth real $d$-closed $(1,1)$-forms as topological space \nwith Fr\\'echet topology. \nFor a smooth $(n-1,n-1)$-form $\\gamma $, the integral \n$\\int_{M}{\\theta _{k}\\wedge \\gamma } $ converges to $ \\int_{M}{\\theta  \\wedge \\gamma} $ \nif $\\theta _{k} \\longrightarrow \\theta $ in Fr\\'echt topology.\nHence the above map $\\theta  \\longmapsto \\big\\{ \\theta \\big\\}$ is continuous by Serre duality .\nFrom open mapping theorem the  map is a open map.\n\n\nSince the map is open, for a positive number $\\e$ we can take a sufficietly small $\\delta >0$ such that \nthe Chern class $\\delta c_{1}(N)$ contains a smooth form $\\eta $ with $-\\e \\omega \\leq  \\eta \\leq \\e\\omega $.\nSo $S+\\eta + (1-\\delta)[N]$ is a positive current in the class $\\A$ and \nthe Lelong number equals to $\\nu \\big((1-\\delta)[N] , x \\big)$ since $S$ is smooth at $x$.\nIf $\\nu([N], x)$ is positive, it is contradiction to minimality of $N$.\nSo $x$ is not contained in $\\mathrm{Supp}(N)$.\nThis implies that the positive current $S+[N]$ lies in $\\A$ and is smooth at $x$.\nHence $x$ is not contained in $E_{nK}(\\A)$.\n\\end{proof}\nBy this lemma we can define the restricted volume of $P$.\nNote that $\\mathrm{Supp} \\big( [E] \\vert_{\\Vr} \\big)$ is contained in $E\\cap V$. \nThis assures Lemma \\ref{N}.\nThe Lemma \\ref{N} implies the exceptional divisor part $[N]$ does\nnot effect the integration on $V$.\n\\begin{lemm}\\label{N}\n$( [E] \\vert_{\\Vr})_{\\mathrm{ac}}=0$\n\\end{lemm}\n\nLemma \\ref{N} shows $\\int_{\\Vr}(T\\vert_{\\Vr})_{\\mathrm{ac}}^{d} =\n \\int_{\\Vr}  (T\\vert_{\\Vr}-[E] \\vert_{\\Vr})_{\\mathrm{ac}}^{d}$.\nTherefore Theorem \\ref{div} is proved by correspondence between positive currents in $\\A$ and in $P$. \n\\end{proof}\nThe following Proposition says that \nFujita's approximation theorem for the restricted volume holds for a class $\\A$.\nThis implies the continuity of a restricted volume.\n\n\\begin{theo}\\label{Fujita}\n\nThe restricted volume of the class $\\A$ along $V$ can be approximated by intersection numbers of semi-positive classes.\nThat is, the following equality holds.\n\\begin{equation*}\n\\volm (D) = \\sup_{\\pi^{*}T =B +[E]} \\big( \\{B \\} ^{d} \\cdot \\widetilde{V} \\big)\n\\end{equation*}\n, where the supremum is taken over all resolution $\\pi : \\widetilde{M}\\longrightarrow M$ of the positive current\n $T \\in \\A$ with analytic singularities such that $\\pi$ is an isomorphism at a generic point of $V$\nand $\\widetilde{V} \\not\\subseteq \\mathrm{Supp}(E)$.\n(Here $\\widetilde{V}$ denotes the proper transformation of $V$.)\n\\end{theo}\n\n\\begin{rem}\nBy subtracting a class of exceptional divisors from semi-positive class $\\{ B \\}$, \nwe can assume that $\\{ B \\}$ ranges among K\\\"ahler classes.\n\\end{rem}\n\n\\begin{proof}\nLet $T$ be a positive current with analytic singularities in the class $\\A$\n whose polar set is not contained $V$.\nThen We take the modification $\\mu$ such that $\\mu^{*}(T) = (B + [E])$.\n(We can assume that the map is an isomorphism for generic point on $V$.)\nBy Lemma \\ref{f} and Lemma \\ref{N} we obtain\n\\begin{align*}\n\\int_{V_{\\reg}}{\\T} &= \\int_{\\widetilde{V}}{ (\\mu^{*}T \\vert_{\\widetilde{V}})_{\\ac}^{d}} \\\\\n&= \\int_{\\widetilde{V}}{\\big( (B + [E]) \\vert_{\\widetilde{V}} \\big)_{\\ac}^{d}} \\\\\n&= \\int_{\\widetilde{V}}{ B _{\\ac}^{d}} = \\big( \\{B \\} ^{d} \\cdot \\widetilde{V} \\big).\n\\end{align*}\nTherefore we get $\\volm (\\A) =\\mathrm{sup} \\big(\\{B\\} ^{d} \\cdot \\widetilde{V} \\big)$ \nfrom the definition of a restricted volume of the class $\\A$ along $V$.\n\\end{proof}\nTo show the continuity of a restricted volume, we consider the domain of a restricted volume for\n a subvariety $V$.\nMoreover we prove the convexity of the domain and concavity of restricted volumes.\n\n\\begin{defi}\nFor a subvariety $V$, \\textit{the domain of a restricted volume} is defined to be \n$\\mathrm{Big}^{V}(M) := \\big\\{ \\B  \\in H^{1,1} (M,\\mathbb{R}) \\ \\big|\\ V \\not\\subseteq  E_{nK}(\\B) \\big\\}$.\n\\end{defi}\n\n\n\\begin{prop}\\label{concave}\n\n$(1)$\\ \\ $\\mathrm{Big}^{V}(M)$ is a open convex domain in $H^{1,1} (M,\\mathbb{R})$ \\\\\n$(2)$\\ \\ For $\\B_{1}, \\B_{2} \\in \\mathrm{Big}^{V}(M)$, the following inequality holds. \n$$ \\volm(\\B_{1}+\\B_{2})^{1\/d} \\geq \\volm(\\B_{1})^{1\/d} + \\volm(\\B_{2})^{1\/d}$$ \n\n\\end{prop}\n\\begin{proof}\n(1):\\ \\ \nThe concavity is directly followed from $E_{nK}(\\B + \\B^{'}) \\subseteq \nE_{nK}(\\B^{'}) \\bigcup E_{nK}(\\B)$ \nand $E_{nK}(\\B) = E_{nK}(k\\B)$ for $k \\geq 0$.\nFor given a class $\\B$, we can prove $E_{nK}(\\B^{'}) \\subseteq E_{nK}(\\B)$ for every class $\\B^{'}$ in a suitable open neighborhood of $\\B$\nby using the argument in Lemma \\ref{augment}.\nThis assures the domain is open.\n\n\n$(2)$\\ \\ Boucksom shows the log concavity for the volume of a transcendental class\n in \\cite{Bou02}.\nHence Proposition \\ref{nef} shows a restricted volume has log concavity\n for nef classes. \nWe can reduct to the case of nef classes by using the Proposition \\ref{Fujita}.\n \n\\end{proof}\n\n\n\\begin{cor}The following map is continuous.\n\\begin{align*}\n \\volm : \\mathrm{Big}^{V}(M) & \\longrightarrow  \\mathbb{R} \\\\\n \\B & \\longmapsto   \\volm(\\B)\n\\end{align*}\n\\end{cor}\n\\begin{proof}\nIt is well-known fact that a concave function on an open convex set $\\mathbb{R}^{N}$ is continuous.\nTherefore Corollary followed from Proposition \\ref{concave}.\n\\end{proof}\n\n\\subsection{Another proof of Theorem \\ref{MaB}}\nIn this subsection we prove Theorem \\ref{MaBB} by using the expression of a restricted volume with current integration. \nThis proof gives another proof of  Theorem \\ref{MaB}.\nLet $\\A \\in H^{1,1}(X,\\RR)$ be a big class on $X$ and $\\A = P + \\{N \\}$ be the divisorial Zariski decomposition. \nNote $E_{nK} (\\A) = E_{nK} (P)$ holds by Lemma \\ref{augment}.\nHence we can consider the restricted volume of $P$ along $V$.\n\n\\textit{Proof of Theorem \\ref{MaBB}.)}\nThe principle of the proof is essentially one of Theorem \\ref{MaB}.\nBy Proposition \\ref{div} we get $\\vol(\\A) =\\vol(P) $ for an irreducible subvariety $V$ \nwith $V\\not\\subseteq E_{nK}(\\A)$. \nMoreover Proposition \\ref {nef} claim that $ \\vol (P) = (V\\cdot P ^{d} )$ holds when $P$ is nef.\nHence the restricted volume are unchangeable for cohomologous \nsubvarieties which is not contained the non-K\\\"ahler locus  \nif $\\A$ admits a Zariski decomposition.\n\n\n \nWe prove the converse direction.\nAssume the non-nef locus $E_{nn}(P)$ is not empty. \nFix a very ample divisor $A$ on $X$. \nWe take smooth curves $C$ and $C^{'}$ with the following properties:\\\\\n\\ \\ \\ (1)\\ \\ \\ $C^{'}$ does not intersect with the non-nef locus $E_{nn}(P)$.\\\\\n\\ \\ \\ (2)\\ \\ \\ $C$ and $C^{'}$ are not contained in the non-K\\\"ahler locus $E_{nK}(\\A)$. \\\\\n\\ \\ \\ (3)\\ \\ \\ $C$ intersects with the non-nef locus $E_{nn}(P)$ at $x_{0} \\in X$.\\\\\n\\ \\ \\ (4)\\ \\ \\ $C$ and $C^{'}$ are the complete intersections of the linear system of $A$.\\\\\nThen we lead the contradictions by proving the following lemma. \n\n\\begin{lemm}\n$(A)$ \\ \\ $\\V{C^{'}}(\\A) = (C^{'}\\cdot P) \\big( = (C \\cdot P) \\big)$ \\\\\n$(B)$ \\ \\ $\\V{C}(\\A) < (C\\cdot P)$ \n\\end{lemm}\nBefore the proof of Lemma 4.15, we prove Lemma 4.16.\nLemma 4.16 assures that the restricted volume is computed by the Lelong number.\n\\begin{lemm}\nLet $S$ be a positive current with analytic singularities on a smooth curve $C$.\nThen the Siu decomposition coincide with the Lebesgue decomposition of $S$.\nThat is, $S_{\\ac} = S- \\displaystyle \\sum_{x \\in C}\\nu(S,x)[x] $ holds.\n\\end{lemm}\n\\begin{proof}\nA current $S$ is locally expressed by $S=\\dfrac{c}{2} \\log \\big( \\sum_{i=1}^N \\big| f_{i} \\big|^{2} \\big) + \\theta $, \nwhere $\\theta $ is a smooth $(1,1)$-form and $f_{i}$ is a holomorphic function.\nWe take the maximal effective divisor $D$ with $\\mathrm{div}(f_{i}) \\geq D \\geq 0$ for all $i=1,2,\\dots,N$ and \nthe holomorphic function $g$ with $\\mathrm{div}(g) = D$.\nWe denote by $g_{i}$ a holomorphic function defined by $g_{i} =f_{i} \/ g$.\nNote $\\frac{c}{2} \\ddbar \\log \\big| g \\big|^{2} =[cD]$ by Poincar\\`e-Lelong formula.\nHence the following decomposition is the Siu decomposition.\n\\begin{equation*}\nS =\\Big\\{ \\frac{c}{2} \\log (\\sum_{i=1}^N \\big| g_{i} \\big|^{2} \\big) + \\theta \\Big\\} + \n\\frac{c}{2} \\log \\big| g \\big|^{2}\n\\end{equation*}\nSince $g_{i}$ does not have common zero point, $\\log (\\sum_{i=1}^N \\big| g_{i} \\big|^{2} )$ is smooth.\nThe absolutely continuous part of $[cD]$ is zero. \nHence the decomposition is the Lebesgue decomposition.\n\\end{proof}\nBy using Lemma 4.16, we prove (B) in Lemma 4.15.\nBy the expressions of a restricted volume with current integration and Theorem \\ref {div} we obtain\n\\begin{equation*}\n \\V{C}(\\A) = \\V{C} (P) = \\sup_{T \\in P} \\int_{C}{( T\\vert_{C} ) _{\\ac}} \n\\end{equation*}\nWe take a positive current $T$ with analytic singularities in the class $P$ \nand whose singular locus does not contain $C$.\n$T \\vert_{C}$ is a positive current with analytic singularities on $C$.\nBy Lemma 4.16 we obtain ${ (T \\vert_{C})_{\\ac} = T \\vert_{C} -\\sum_{x \\in C} \\nu(T \\vert_{C} , x)[x] }$.\nNote that the restriction $T \\vert_{C}$ lies in $P \\vert_{C}$.\nThis implies that \n\\begin{equation*}\n\\int_{C} {T \\vert_{C}} = \\mathrm{deg}_{C} (P \\vert_{C}) = (C\\cdot P).\n\\end{equation*}In fact $T \\vert_{C}$ is expressed $T \\vert_{C} = \\theta \\vert_{C} + \\ddbar \\varphi \\vert_{C} $, \nwhere $\\theta$ is a smooth $(1,1)$-form in $P$ and $\\varphi $ is a $L^{1}$-function on $X$.\nHence we obtain  \n\\begin{equation*}\n\\int_{C} {T \\vert_{C}}  = (C\\cdot P) + \\int_{C} {\\ddbar \\varphi \\vert_{C}}.\n\\end{equation*}\nBy applying the approximation theorem (Theorem 2.2) to $\\varphi \\vert_{C}$, we get smooth functions $\\varphi _{k}$ on $C$ \nsuch that $\\ddbar \\varphi _{k} $ converges to $\\ddbar \\varphi \\vert_{C}$ weakly.\nSo $\\int_{C}{\\ddbar \\varphi _{k}} $ converges to $ \\int_{C} {\\ddbar \\varphi \\vert_{C}}$.\nHere $\\int_{C}{\\ddbar \\varphi _{k}}$ equals to zero by smoothness of $\\varphi _{k}$ and Stokes's theorem.\nHence we have\n\\begin{align*}\n\\V{C}(\\A) &= \\sup_{T \\in c_{1}(P)} \\{ (C \\cdot P) - \\sum_{x \\in C} \\nu(T \\vert_{C} , x)\\}  \\\\\n& = (C \\cdot P) - \\inf_{T \\in P} \\sum_{x \\in C} \\nu(T \\vert_{C} , x).\n\\end{align*}The Lelong number of the restriction of a current is more than that of the current.\nMoreover $\\nu(T_{\\mathrm{min}}, x) \\leq \\nu(T , x) $ holds from the definition of a minimal singular current. \nTherefore we obtain\n\\begin{equation*}\n\\V{C}(\\A)  \\leq (C \\cdot P) -\\sum_{x \\in C} \\nu(T_{\\mathrm{min}} , x). \n\\end{equation*}\nThe curve $C$ intersects with the non-nef locus $E_{nn}(P)$ at $x_{0}$ from the property (3). \nHence $\\nu(T_{\\mathrm{min}} , x_{0})$ is positive.\nThis implies $\\V{C}(\\A)  \\leq (C \\cdot P) - \\nu(T_{\\mathrm{min}} , x_{0}) < (C \\cdot P) $.\nTherefore Lemma (B) holds. \n\nWe prove (A).\nBy the first half argument we get $\\V{C^{'}}(\\A) \\leq (C^{'} \\cdot P) $.\nTo show the converse inequality we take a K\\\"ahler current $S \\in \\A$ with analytic singularities.\nSo We can assume $S \\geq \\omega$, where $\\omega$ is a K\\\"ahler form on $X$. \nBy applying the approximation Theorem (Theorem 2.3) to a minimal singular current $T_{\\min}$ in $P$, \nWe get positive currents $T_{k}$ with analytic singularities with the following properties.\n\\\\\n\\ \\ \\ $\\mathrm{(b')}$\\ \\ \\ $T_{k} \\geq -\\e _{k} \\omega$ and $\\e_{k}$ converges to zero.\\\\\n\\ \\ \\ $\\mathrm{(c')}$\\ \\ \\ The Lelong number $\\nu(T_{k},x)$ increase to\n$\\nu(T_{\\mathrm{min}} , x)$ for every point $x \\in X$.\\\\\nFor every positive number $\\delta $, there is a $k(\\delta )$ such that \n$(1-\\delta ) T_{k(\\delta)} + \\delta S$ is a positive current. \nSince the current has an analytic singularities, \nthe inequality $$\\V{C^{'}} (\\A) \\geq \\int_{C^{'}} \\big( ((1-\\delta ) T_{k(\\delta)} + \\delta S )\\vert_{C^{'}} \\big)_{\\ac}$$ holds\nfor every $\\delta>0$ by the definition of restricted volumes. \nThe Lelong number of $T_{k}$ is zero for every point on $C$ by the property (3), (b). \nThis shows $T_{k}$ is smooth on $C$.\nSo we obtain \n\\begin{equation*}\n\\V{C^{'}} (\\A) \\geq  (1-\\delta)(C^{'} \\cdot P) - \\delta\\int_{C^{'}} \\big(  S \\vert_{C^{'}} \\big)_{\\ac}.\n\\end{equation*} holds for every $\\delta$. \nWhen $\\delta$ converges to zero, we get $\\V{C^{'}} (\\A) \\geq (C^{'} \\cdot P) $. \n\n\n\n\\section{Asymptotic estimates on higher cohomology}\n\n\\subsection{Kawamata-Viehweg type vanishing}\nThe purpose of this section is to prove the asymptotic estimates of higher cohomology of $\\ell D$.\nWe recall the definition. Let $k$ be an integer $k = 0,1,2,\\ldots, n-1$.\n\\begin{defi}\nA divisor $D$ is said to be \\textit{nef in codimension $k$} if the codimension of the restricted base locus $\\mathbb{B}_{-}(D)$\n is greater than $k$.\n\\end{defi}\n\\begin{rem}\n$D$ is nef in codimension $0$ if and only if the Chern class is a pseude-effective class.\n$D$ is nef in codimension $n-1$ if and only if the Chern class is a nef class.\n\\end{rem}\nTheorem \\ref{KV} is the accurization of the simplest version of Kawamata-Viehweg vanishing theorem.\nLemma \\ref{Key} is usefully worked in the proof of Theorem \\ref{KV} and Corollary \\ref{asy}.\nFirst of all we prove Lemma \\ref{Key}.\n\n\n\\begin{lemm}\\label{Key}\nAssume that $D$ is nef in codimension $k$ and $k$ is smaller than $(n-1)$.\nFix a very ample divisor $A$ on $X$.\nThen a very general hyperplane $H\\in |A|$ satisfies the following properties.\\\\\n\\ \\ \\ $\\mathrm{(i)}$\\ \\ \\ \\ $H \\cap X$ is a smooth projective variety of dimension $(n-1)$. \\\\\n\\ \\ \\ $\\mathrm{(ii)}$\\ \\ \\ \\ The restriction $D\\vert_{H}$ is nef in codimension $k$ on $H \\cap X$.\n\\end{lemm}\n\\begin{proof}\nSince $D$ is nef in codimension $k$, for an arbitrary number $\\e>0$, we can take\n a positive current $T_{\\e}$ in the class $c_{1}(D)$ with the following properties \nby the approximation theorem (Theorem 2.3).\\\\\n\\ \\ \\ \\ (1)\\ \\ \\ $T_{\\e}$ has analytic singularities. \\\\\n\\ \\ \\ \\ (2)\\ \\ \\ The codimension of the singular locus of $T_{\\e}$ is greater than $k$. \\\\\n\\ \\ \\ \\ (3)\\ \\ \\ $T_{\\e} \\geq -\\e \\omega $ holds, where $\\omega $ is a K\\\"ahler form on $X$.\n\\\\\nNow we define a countable union of closed analytic sets by  \n\\begin{equation*}\n  Q := \\bigcup _{\\QQ \\ni \\e >0} \\bigcup_{\\QQ\\ni c>0} E_{c}(T_{\\e}) .\n\\end{equation*}\nHere $E_{c}(T_{\\e})$ is a Lelong number upper level set, which is defined by \n$\\big\\{ x \\in X \\big| \\nu(T_{\\e}, x) \\geq c \\big\\}$.\nA theorem of \\cite{Siu74} asserts this is a closed analytic set.\n\\begin{lemm}\nFor a very general hyperplane $H\\in |A|$, $H$ does not contain analytic\n set with the codimension $(k+1)$ in $Q$ and $H\\cap X$ is smooth and irreducible.\n\\end{lemm}\n\\begin{proof}\nLet $\\{ C_{i}\\}_{i \\in I}$ be a family of analytic sets with codimension $(k+1)$ in $Q$.\nFor each $i \\in I$, we define $Q_{i}$ by $Q_{i} := \\big\\{ H \\in |A| \\ \\big|\\ C_{i} \\subseteq  H  \\big\\}$.\nThen $Q_{i}$ is a (proper) analytic set in $|A|$ for each $i$.\nNote $I$ is a countable set.\nEvery hyperplane $H \\in |A|-\\bigcup_{i \\in I}{Q_{i}} $ does not \ncontained an analytic set with the codimension $(k+1)$ in $Q$.\nMoreover Bertini's Theorem shows that $H\\cap X$ is smooth and irreducible for a general member \n$|A|$.\nLemma 5.4 is concluded. \n\\end{proof}\n\n\nFor a very general hyperplane $H\\in |A|$, \n$T_{\\e} \\vert_{H} \\in c_{1}(D \\vert_{H})$ and $T_{\\e} \\vert_{H} \\geq -\\e \\omega\\vert_{H} $ holds.\n$\\omega\\vert_{H}$ is a K\\\"ahler form on a smooth projective variety $X \\cap H$.\nHence following lemma concludes Lemma \\ref{Key}.\n\n\\begin{lemm}\nThe codimension in $X \\cap H$ of the singular locus of $T_{\\e}\\vert_{H}$ greater than $k$.\n\\end{lemm}\n\nLet $C$ be an analytic set with codimension $\\ell =0, 1,2,\\ldots,k$ on $X \\cap H$.\nThen $C$ is the analytic set with codimension $\\ell+1$ on $X$.\nWhen $\\ell+1$ is smaller than $k+1$, \n$\\nu (T_{\\e} , C) = 0$ for every positive number $\\e$ by the choice of $T_{\\e}$.\nWhen $\\ell+1$ equals to $k+1$, \n$\\nu (T_{\\e} , C) = 0$ by the choice of $H$.\nSince $T_{\\e}$ has analytic singularities, \nthe Lelong number is zero if and only if $T_{\\e}$ has smooth at $x \\in X$.\nThis shows that the singular locus of $T_{\\e}\\vert_{H}$ does not contain an analytic set $C$ with \ncodimension $\\ell =0, 1,2,\\ldots,k$ on $X \\cap H$.\nHence Lemma \\ref{Key} holds. \n\\end{proof}\n\n\nWe prove Theorem \\ref{KV} by using Lemma \\ref{Key}.\nThe principle of the proof is to reduct to the ordinary Kawamata-Viehweg vanishing theorem\nby using Lemma \\ref{Key} and the induction on $\\dim {X} =n$.\\\\\n\\textit{Proof of Theorem \\ref{KV}.)}\nWhen $n$ is one, bigness of $D$ implies that $D$ is ample.\nIn this case, Theorem \\ref{KV} follows from Kodaira vanishing theorem.\nHence we can assume that $n$ is greater than one.\nIf $k$ is $(n-1)$, $D$ is nef.\nThen Theorem \\ref{KV} follows from the ordinary Kawamata-Viehweg vanishing theorem.\nTherefore we can assume that $k$ is smaller than $(n-1)$.\nIt is sufficient to show $H^{p}(X, \\OX( -D))=0\\ \\ (0 \\leq p \\leq k)$ by Serre duality.\nWe obtain the following short exact sequence for a hyperplane $H \\in |A|$ with the properties in Lemma \\ref{Key}.\n\\[\n\\begin{CD} \n0 @>>> \\OX(-D-A) @>>> \\OX(-D) @>>> {\\mathcal{O}}_{H}(-D) @>>> 0\\\\\n\\end{CD}\n\\]\nWe consider the long exact sequence induced by the above exact sequence.\nBy choosing $A$ sufficiently positive, we can assume that $A+D$ is ample on $X$.\nThen Serre duality and Kodaira vanishing theorem show that $H^{\\ell}(X,\\OX(-D-A) )=0,\\ \\ (0\\leq \\ell \\leq n-1)$ .\n$D\\vert_{H} $ is nef in codimension $k$ on $X \\cap H$ by the choice of $H$.\nTherefore $H^{p}(H,\\mathcal{O}_{H}(-D) )=0\\ \\ (0\\leq p \\leq k)$ holds by the hypothesis of the induction.\nThese show that $H^{p}(X,\\mathcal{O}_{X} (-D))=0$ $\\ \\ (0\\leq p \\leq k)$.\n\\begin{flushright}\n$\\square$\n\\end{flushright}\n\\textit{Proof of Corollary \\ref{asy}.)}\nCorollary \\ref{asy} is also proved with the induction on $\\dim X =n$.\nWhen $n$ is one, pseudo-effective line bundle is nef.\nWhen the degree of $D$ is positive, Kodaira vanishing theorem shows that  $H^{1}(X, \\OX (M + \\ell D)) =0$\n for a sufficietly large $\\ell$.\n When the degree of $D$ is zero, Riemann-Roch formula shows \n$\\dim H^{1}(X, \\OX (M + \\ell D)) =O(1)$.\n\nWe consider when the dimension of $X$ is greater than one.\nBy choosing a sufficietly positive line bundle $A$, we can assume that $\\OX (M+\\ell D-K_{X} +A)$\n is big for all $\\ell$, since $D$ is pseudo-effective. \nMoreover we can take a member $H$ of $|A|$ such that $D\\vert_{H}$ is nef in codimension $k$ on $X\\cap H$\nby Lemma \\ref{Key}.\nThen we obtain the following short exact sequence.\n\\[\n\\begin{CD} \n0 @>>> \\OX(M+\\ell D) @>>> \\OX(M+\\ell D +A) @>>> {\\mathcal{O}}_{H}(M+\\ell D + A) @>>> 0\\\\\n\\end{CD}\n\\]\nWe consider the long exact sequence induced by the above short exact sequence.\n\nNote that $(E+\\ell D-K_{X} +A)$ is big and nef in codimension $k$ for all $\\ell>0$ .\nSo Theorem \\ref{KV} shows $H^{q} \\big( X,\\OX(E+\\ell D +A) \\big) = 0$ for $q \\geq n-k$.\nTherefore we obtain from the long exact sequence\n\\begin{align*}\n&h^{q} \\big( X,\\OX(E+\\ell D) \\big)= h^{q-1} \\big( H , \\mathcal{O}_{H}(\\OX(E+\\ell D+A) \\big)\\ \\ \\mathrm{for} \\ \\ q \\geq n-k +1  \\\\\n\\mathrm{and}\\ \\ \\   &h^{n-k} \\big( X,\\OX(E+\\ell D \\big) \\leq  h^{n-k-1} \n\\big( H , \\mathcal{O}_{H}(\\OX(E+\\ell D+A) \\big).  \\\\\n\\end{align*}By hypothesis of induction we have \n\\begin{align*}\n&h^{q-1}(H , \\mathcal{O}_{H}(\\OX(E+\\ell D+A))=O(\\ell^{(n-1)-(q-1)}) =O(\\ell^{n-q})\\ \\ \\  \\mathrm{for}\\ \\  q \\geq n-k +1 \\\\\n\\mathrm{and}\\ \\ \\  &h^{n-k-1}(H , \\mathcal{O}_{H}(\\OX(E+\\ell D+A))= O(\\ell^{(n-1)-(n-k-1)}) =O(\\ell^k)\n\\end{align*}\nTherefore Corollary \\ref{asy} holds.\n\n\\begin{flushright}\n$\\square$\n\\end{flushright}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nDialogue state tracking (DST) modules, which aim to extract dialogue states during conversation \\cite{DBLP:journals\/pieee\/YoungGTW13}, is an important component for task-oriented dialog systems to understand users' goals and needs \\cite{DBLP:conf\/eacl\/Rojas-BarahonaG17,lei2018sequicity}. Dialogue states are sets of slots and their corresponding values. Collecting state labels can be costly \\cite{DBLP:conf\/emnlp\/BudzianowskiWTC18}, requiring experts to indicate all \\textit{(slot, value)} information for each turn in dialogues. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.45\\textwidth]{figs\/DST.pdf}\n\\caption{Dialogue state tracking (DST) task. U and A represent user's and system's utterances respectively. DST aims to extract dialogue states pairs (\\textcolor{orange}{slot}, {\\color{OliveGreen}{value}}), for each user's utterance. Values are usually the explicit needs expressed in the utterances.}\n\\label{DST}\n\\end{figure}\n\nTo reduce the dependency on large amounts of training data, some few-shot methods are proposed recently for low-resource DST. Most of them apply domain transfer-based methods \\cite{trade,DBLP:conf\/aaai\/LeeJ19,DBLP:conf\/aaai\/RastogiZSGK20} which rely on the assumption about the similarity among different domains. \nAnother series of approaches has tried to exploit external knowledge. \\citet{DBLP:conf\/asru\/ChenWR13} and \\citet{ddswws} consider slots and frames are similar semantic units and use the FrameNet semantic parsers to automatically induce slots. \\citet{todbert} fine-tune BERT with a task-oriented dialogue dataset and utilize it for the downstream DST task. Recently, a new paradigm, \"\\textit{Pre-train, Prompt and Predict}\" \\cite{prompt-survey}, which aims at utilizing PLM in a more effective way, has aroused the public's attention. It is supposed to be useful in few-shot scenarios as it can ``probe'' the task-related knowledge from PLM efficiently using a prompt.\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=0.9\\textwidth]{figs\/train-arch.pdf}\n\\caption{Overview of the prompt-learning framework for few-shot DST. While training, prompt function $f(v)$ accepts {\\color{OliveGreen}value} candidate $v$ and get the {\\color{RoyalBlue}\\textbf{value-based prompt}}, which is concatenated to dialog history for PLM to generate the corresponding {\\color{orange}slot} $s$. Dashed arrows demonstrate the process of {\\color{RoyalBlue}\\textbf{inverse prompt}} mechanism which gets inverse prompt via the inverse prompt function $I(s)$ for the generated $s$ and aims to generate value $v'$ which is supposed to be close to the original input value $v$.}\n\\label{train-arch}\n\\end{figure*}\nIn 2021, \\citet{sup-dst-prompt} introduce prompt learning for DST task. This work aims to encode slot information into PLM via prompt and improve prediction performance when there are plenty of labeled data. However, the potential of prompt learning for low-resource DST is still under-explored. To further exploit this paradigm, we design a prompt learning framework for low-resource DST, which has value-based prompt module and inverse prompt module.\n\n\n\n\nFirst, we design \\textbf{value-based prompt functions} which use values as inputs to construct prompts. Slots are the target outputs. For few-shot scenarios, the slots that appear in the few labeled dataset may not include all possible needs a user will propose. Defining all possible slots in advance is also difficult because of the rapid application in different domains and users' continuous needs. \\%\\% Thus, we rethink about DST task and consider values as prompts to probe the corresponding slots. We design value-based prompt functions which equip the textual prompt with values. As shown in Figure \\ref{train-arch}, a prompt function is a textual template, e.g., ``\\textit{belief states: value = [v], slot = [s]}''. Given the dialogue history ``\\textit{...Plan a train to London}'', after extracting the value candidate \n\\textit{London}, the prompt becomes ``\\textit{belief states: value = London, slot = [s]}'' where \\textit{[s]} is supposed to be generated as \\textit{destination} by the PLM. Value-based prompting method can alleviate the dependency on pre-defined slot types and generate unseen slots.\n\n\nIn addition, values and slots are all semantic units that describe the users' needs during conversations. Prompting values via slots (value generation) can be seen as a dual-task of prompting slots with values (slot generation). Thus, we design \\textbf{inverse prompt} mechanism as shown in Figure \\ref{train-arch}. While training, after generating slots $s$ via value-based prompts, slots are presented to the inverse prompt function $I$. The inverse prompt process aims to generate the corresponding value $v'$ which is supposed to be close to the original input $v$. Naturally, there exists an internal correlation between these two types of prompt tasks and they can benefit each other, especially under the few shot settings. The inverse prompt mechanism can also help to self-check and restrict the output of the prompt process: if a generated slot can be used to prompt the original value, the value belongs to the slot with a larger probability. Finally, a simple but effective ensemble method is used to leverage the complementary advantages of different prompt functions while testing.\n\nThe main contributions of our work can be summarized as (1) We reformulate DST as a language modeling task for slots generation and design value-based prompt functions to probe DST-related knowledge from PLM for low-resource scenarios. The framework doesn't rely on the known ontology of target slots. (2) We propose an inverse prompt mechanism to further help the PLM understand the essence of DST with few labels and tune the generated slots. (3) Experimental results show that our model can generate unseen slots and significantly outperforms state-of-the-art few-shot approaches.\n\n\\section{Preliminary}\n\\subsection{Prompt Learning}\nPrompt learning, which aims to utilize pre-trained language models more effectively with the help of \\textit{prompt}, is a new NLP paradigm (``\\textit{Pre-train, Prompt and Predict}'') proposed recently. Usually, the original task input $x$ is used to construct a prompt that can reformulate the original task into a language modeling task. Take the emotion classification task as an example, when recognizing the emotion of a social media post, \\textit{\"I missed the bus today\"}, we may continue with a prompt \\textit{\"I felt so \\_\\_\"}, and ask the PLM to fill the blank with an emotion-bearing word. With the appropriate prompts, PLM can be pushed to generate the task-related output directly.\n\n\nGiven the \\textit{prompt function f} which maps original input x to the prompt, the goal is to learn:\n\\begin{equation}\nP\\left(y \\mid  x, f(x)\\right)\n\\end{equation}\nwhere $y$ is the \\textit{answer} to be generated\/filled. In DST, $y$ can be a word in dialogue state $B$. \n\n\n\\subsection{Dialogue State Tracking}\nWe consider each conversation with $T$-turn utterances alternating between the user and system: $c=\\left\\{a_1, u_1, ...a_T, u_T\\right\\}$ where $u_t$ and $a_t$ represent the user's and system's utterance respectively. Given the dialog history $c_t$ (including current user utterance $u_t$ and the former utterances $h_t = \\left\\{a_1, u_1, ...u_{t-1}, a_t\\right\\}$), a DST model aims to extract the dialogue state (belief state) $B_t$ for $u_t$ which comprises multiple tuples of slots $s$ and their associated values $v$ ($B_t = \\left\\{(s_1, v_1),...(s_n, v_n)\\right\\}$). For example, given the dialog history $c_t$ (``\\textit{...Plan a train to London on this Tuesday}''), DST model is supposed to generate belief states $B_t = \\left\\{(destination, London), (day, this\\ Tuesday)\\right\\}$). The goal is to learn probability distribution $P$ \\cite{soloist} for $t$-th turn:\n\\begin{equation}\nP\\left(B_{t} \\mid c_t\\right)\n\\end{equation}\n\nIf $B_t$ is considered as a word sequence \\cite{simple-tod}, DST is essentially a language modeling task. Large-scale pre-trained language models (PLM) show outstanding language modeling and generation ability. Following the existing paradigm (\\textit{Pre-train and Fine-tune}), we need to fine-tune PLM with the task-related dataset. Fine-tuning with few labeled dataset may lead to over-fitting. Thus, an effective way to help PLM understand DST task in their familiar way (language modeling) and utilize the generation ability is important, inspiring the exploration of prompt learning for few-shot DST.\n\n\\section{Prompt Learning for Few-Shot DST}\n\n\n\nIn the following section, we will describe the main components of the designed prompt learning framework for few-shot DST, including \\textbf{value-based prompt} (Section \\ref{prompt function}), \\textbf{inverse prompt} (Section \\ref{inverse prompting}) and \\textbf{prompt ensemble} (Section \\ref{prompt ensemble}).\n\n\\subsection{Value-based Prompt}\\label{prompt function}\nPrevious work \\cite{DBLP:journals\/tacl\/JiangXAN20,DBLP:conf\/eacl\/SchickS21,template-ner} show that the design of prompting function $f$ is a key factor that influences the final performance. \nUsually, the function function contains a \\textit{template} which is a textual string and remains two blanks to be filled (the input and the answer). \n\n\n\nA natural idea is using slots as prompts to generate corresponding values, which is called slot-based prompt \\cite{sup-dst-prompt}. For example, given $c$ (``\\textit{...plan a train to London.}'') and slot (\\textit{destination}), the input of PLM becomes ``\\textit{...Plan a train to London. destination [y]}'' where \\textit{[y]} is supposed to be generated as \\textit{``London}''. This method relies on the known ontology of slot type. For few-shot DST, the slots that appear in the few labeled datasets may not include all possible needs. In addition, defining all possible slots are difficult as the rapid application in different new-rising domains and user's continuous need. In the real-world application, the candidate set of $s$ may be unknown and changeable.\n\nThus, we rethink about DST task and consider values as prompts to generate the corresponding slots as shown in Figure \\ref{train-arch}. Such a method doesn't require any knowledge about the target slot types and thus can generate unseen slots under the generative framework. Four intuitive templates for value-based prompts are shown in Table \\ref{prompts}. Take the first template for an example (\\textit{``belief states: value = [v], slot = [s]''}), given value candidate $v$ = \\textit{``London''}, $f(v)$ = \\textit{``belief states: value = London, slot = [s]''}. The goal is to learn the probability of slots in $t$-th turn given $c_t$ and the value $v$:\n\\begin{equation}\nP\\left(s \\mid  c_t, f(v)\\right)\n\\end{equation}\nThe overall learning objective of this generation processing is maximizing the log-likelihood of slots in the training dataset $D$:\n\\begin{equation}\n\\mathcal L = \\sum_{t}^{|D|} log\\ P\\left(s_t \\mid c_t, f(v_t)\\right)\n\\label{loss_p}\n\\end{equation}\n\n\nAs a turn may contain multiple values and slots, each pair of (slot, value) constructs an instance for training and testing. While training, the values are known with annotations. Real values are utilized to construct prompts and train the slot-generation process. While testing, values are unknown. Referring to \\cite{DBLP:conf\/interspeech\/GoelPH19,latent}, candidate values are extracted from current user's utterance. We extract POS tags \\cite{DBLP:conf\/emnlp\/CuiZ19} and consider adjective and adverb as possible values as well as their previous negator (e.g., ``\\textit{not expensive}''). Named entities (e.g., name of place, time) and co-references are extracted via Stanford CoreNLP toolkit \\cite{corenlp}. Stop words and repeated candidates are filtered. Each value candidate is used to construct the value-based prompt and generate the slot. Actually, value generation can also be modeled as a summarization task and take advantage of few-shot summarization methods, which can be improved in future work but is not the focus of this paper.\n\n\n\n\n\\begin{table}\n\\centering\n\\begin{tabular}{cl}\n\\hline\n\\multicolumn{2}{c}{\\textbf{Prompt Functions}}\\\\\n\\hline\n$f_1$ & belief states: value = [v], slot = \\textbf{[s]} \\\\\n$f_2$ & belief states: [v] = \\textbf{[s]} \\\\\n$f_3$ & [v] is the value of \\textbf{[s]} \\\\\n$f_4$ & What is the slot type of [v] ? \\textbf{[s]} \\\\\n\\hline\n\\end{tabular}\n\\caption{\\label{prompts}\nDifferent prompt functions $f$. [v] is the input of value candidate and [s] is the slot to be generated.\n}\n\\end{table}\n\n\n\n\n\n\n\\subsection{Inverse Prompt}\\label{inverse prompting}\nValues and slots are both core semantic units in utterances that describe users' needs. Thus, prompting values with slots (value generation) can be seen as a dual-task of prompting slots with values (slot generation). Naturally, these two types of prompt tasks are supposed to hold an intrinsic correlation correlation and can benefit each other, especially in the few-shot settings.\n\n\nThus, we design \\textit{inverse prompt} as shown in Figure \\ref{train-arch}. While training, after generating the slot ($s$) via value-based prompts, $s$ is presented to inverse prompt function ($I$). The inverse prompt process aims to answer the corresponding value $v'$ which is supposed be close to the original input one $v$. We take ``\\textit{belief states: [s] = [v]}'' as the template in $I$. The loss function $\\widetilde\\mathcal{L}$ for inverse prompt mechanism is:\n\\begin{equation}\n\\widetilde\\mathcal{L} = \\sum_{t}^{|D|} log\\ P\\left(v_t \\mid c_t, I(s_t)\\right)\n\\label{loss_ip}\n\\end{equation}\nThe final loss function $\\mathcal L^*$ consists of loss functions in prompt learning $\\mathcal L$ and inverse prompt learning $\\widetilde\\mathcal{L}$:\n\\begin{equation}\n\\mathcal L^* = \\mathcal L + w * \\widetilde\\mathcal{L}\n\\label{loss}\n\\end{equation}\nwhere $w$ is a decimal in $(0,1)$ and used to adjust the influence of inverse prompt.\n\nThere are two main differences between our work and existing work using slots as prompts: (1) Inverse prompt is used as an auxiliary task for the original prompt learning. Our goal is to utilize it to help PLM understand the task and tune the output further: if a slot can be used to prompt the original value, it means there is a larger probability that the value belongs to the generated slot. (2) In our work, the slot types are not static. We generate the slots in the whole vocabulary space, making generating unseen slots is possible. \n\n\n\\subsection{Prompt Ensemble}\\label{prompt ensemble}\nIn the previous section, we described methods to generate a set of value-based prompt functions as shown in Table \\ref{prompts}. Each of these prompts may be more or less effective at eliciting knowledge from the PLM, and thus it is necessary to decide how to use these generated prompts at test time. Unfortunately, under few-shot settings, it's hard to get enough training and development set to automatically select or generate the best-performing prompt \\cite{DBLP:journals\/tacl\/JiangXAN20,DBLP:conf\/acl\/GaoFC20,ben2021pada,DBLP:conf\/emnlp\/DavisonFR19,DBLP:journals\/corr\/abs-2103-10385}. We introduce a multi-prompt learning method (\\textit{prompt ensemble}) for few-shot DST task in this section to effectively utilize different prompts.\n\n\nPrompt ensemble methods use multiple unanswered prompts for input at inference time to make predictions \\cite{prompt-survey}. It can leverage the complementary advantages of different prompts and alleviate the cost of choosing one best-performing prompt. There is relatively little work on prompt ensemble for generation tasks. A simple way for ensemble in this case it to train a separate model for each prompt and generate the output based on the vocabulary distribution learned by several models while testing. The probability of slot $s_t$ is calculated via:\n\n\n\\begin{equation}\nP\\left(s_t \\mid c_t \\right)=  \\sum_{k}^{K} \\alpha_k * P\\left(s_t \\mid c_t, f_k(v_t)\\right)\n\\label{eq:ensemble}\n\\end{equation}\nwhere $f_k$ is the $k$-th prompt and $\\alpha_k$ is its weight. $K$ is the number of prompt functions.\n\n\n\\section{Experiments}\n\\subsection{Experimental Setup}\n\n\\paragraph{Datesets}\nWe evaluate our methods on MultiWOZ 2.1 \\cite{multiwoz2.1} dataset. It's a multi-domain task-oriented dialog dataset and contains 8438\/1000\/1000 dialogues for training\/validation\/testing, respectively. Following existing work~\\cite{trade}, we keep five domains (\\textit{Restaurant, Hotel, Attraction, Taxi, Train}) because the other two domains only appear in the training set. Each turn can include multiple slots. \n\n\\paragraph{Evaluation Metrics}\nWe adopt the standard metric in DST \\cite{trade}: joint goal accuracy (JGA). The metric compares the entire predicted belief states to the gold one at each dialog turn. The prediction is considered correct if and only if all the predicted states exactly match the ground truth states. Only when the values and slots are both correct, the prediction is correct. To omit the influence of the effect of value extraction, we also present the corresponding accuracy (JGA*) while the values are correctly identified for our methods.\n\n\\paragraph{Implement Details}\nWe choose SOLOIST \\cite{soloist} as our base model. SOLOIST is initialized with the 12-layer GPT-2 \\cite{gpt2} and further trained on multiple task-oriented dialog corpora (Schema \\cite{sgd} and Taskmaster \\cite{taskmaster} ) for two dialogue-related tasks (belief prediction and response generation). Specifically, the belief prediction task accepts utterance as input and generates the belief states as a word sequence (e.g., ``\\textit{Belief state: destination = London}''). Thus, we suppose that knowledge about DST may be learned via SOLOIST, and what we need to do is to find an effective way to ``probe'' the knowledge and apply it to few-shot scenarios. In addition, the moderate size of SOLOIST (117M parameters) makes fine-tuning for the task-related prompts computationally efficient. $\\alpha$ for each prompt function in Eq.\\ref{eq:ensemble} is set to the same value (1\/4). $w$ in Eq.\\ref{loss} is 0.1. Inverse prompt is used for training. While testing, value-based prompts are given to the models for target slot generation.\n\n\\paragraph{Baselines}\nWe compare our methods with several strong baselines capable of few-shot inference, which achieve SoTA on MultiWOZ 2.1 dataset. (1) \\textbf{TRADE} \\cite{trade} requires the embedding of slots as inputs and uses a soft copy mechanism to either copy the corresponding values from utterance pairs or generate them using RNN. (2) \\textbf{Self-Sup} \\cite{self-sup} adds two self-supervised objectives: preserving latent consistency and modeling conversational behavior for TRADE. (3) \\textbf{TOD-BERT} \\cite{todbert} trained BERT with several task-oriented dialogue relevant tasks: masked language modeling and response generation with large-scale corpora (100k dialogues across over 60 different domains). For DST, it learns a classifier to predict the value over the pre-defined possible value set for each known slot.\n(4) \\textbf{DSI} \\cite{latent} introduces latent variable models for DST which consider dialogue states and slots as latent variables. Although the original paper didn't focus on few-shot scenarios, we compare it here as both of us use the same ways of extracting value candidates as inputs and aim to generate slots. (5) \\textbf{SOLOIST}. It's the base model of our method. To investigate the effect of prompt learning, we fine-tune SOLOIST on the DST task which accepts dialog history and uses values as prompts directly to generate slots. \n\nIt's worthy to mention that except SOLOIST, most baselines need pre-defined slots while inference and some need the entire ontology including both slots and their all possible value sets. Although DSI doesn't need pre-defined slot types, the number of target slots should be given.\n\n\\subsection{Experiment Results}\nTo simulate the few-shot scenarios, we randomly select a limited quantity of labeled training data for training. Similar to the previous few-shot works \\cite{todbert,self-sup}, we conduct experiments on the data ratio of 1\\%, 5\\%, 10\\% and 25\\%. Specifically, to observe the performance in an extreme low-resource scenario, we also compare their performances while the training ratio is only 0.1\\% (8 dialogues and less than 60 users' utterances). \n\n\\begin{table}[t]\n\\small\n\\centering\n\\begin{tabular}{l|c|c|c|c|c}\n\\hline\n& 0.1\\% & 1\\% & 5\\% & 10\\% & 25\\% \\\\\n\\hline\n& \\multicolumn{4}{c}{JGA}\\\\\n\\hline\nTRADE &1.4 & 10.4 & 27.7  & 32.6  & 38.5  \\\\\nSelf-Sup & 15.4  & 19.5  & 30.6  & 34.5 & 40.2  \\\\\nTOD-BERT & 6.3 & 9.9 & 28.6  & 37.0 & 44.3  \\\\\nDSI & 1.0 & 1.1 & 1.3 & 1.3  & 2.6  \\\\\nSOLOIST & 26.8 & 36.4  & 37.1  & 37.9  & 39.4 \\\\\n\\textbf{Ours} & \\textbf{36.1} &  \\textbf{44.3}  & \\textbf{44.7}& \\textbf{44.7}  & \\textbf{45.4} \\\\\n\\hline\n& \\multicolumn{4}{c}{JGA*}\\\\\n\\hline\n\\textbf{Ours} & \\textbf{75.4} &  \\textbf{92.4}  & \\textbf{94.7}& \\textbf{94.7}  & \\textbf{95.1} \\\\\n\\hline\n\\end{tabular}\n\\caption{\\label{fewshot}\nComparison of different models under the different ratios of training data (0.1\\% corresponds to less than 60 users' utterances).\n}\n\\end{table}\n\nFew-shot performances are shown in Table \\ref{fewshot}. We can observe that our model outperforms existing works by a large margin. From 0.1\\% to 10\\%, our model increases JGA by more than 7\\%, indicating the superiority of our model in low-resource scenarios. For 25\\%, the improvement is relatively small (1\\%). It may be owing to that using 1\\% data almost performs the same as using 25\\% data (1.2\\% difference) for our model. Using a few labeled data can help the PLM understand DST task. \n\nWe then analyze the results of value extracting and find it can only achieve an accuracy less than 61\\%. Table \\ref{errors} lists the three main types of wrong extraction: missed, partially extracted and totally wrong. However, our model still outperforms others as JGA in Table \\ref{fewshot} shows. It attributes to the high accuracy of slot generation while turn-level values are correctly extracted. When the ratio of training data is 1\\%, 92.4\\% turn-level states are correctly generated given correct value candidates as JGA* shown in Table \\ref{fewshot}.\nThe high values of JGA* indicate the potential of performance along with the improvement of value extraction in future work. \n\n\\begin{table}[t]\n\\small\n\\centering\n\\begin{tabular}{p{0.95\\columnwidth}}\n\\hline\n\\textbf{Dialogue history:}  ...[user]i need a taxi to pick me up at regency gallery and take me to {\\color{OliveGreen}{Don Pasquale Pizzeria}}.\\\\\n\\textbf{Gold value:}  {\\color{OliveGreen}{Don Pasquale Pizzeria}} \\\\\n\\textbf{Our value:}  (\\textbf{Missed})\\\\\n\\hline\n\\textbf{Dialogue history:}  ...[user] i am looking for {\\color{OliveGreen}city centre north bed and breakfast}\\\\\n\\textbf{Gold value:}  {\\color{OliveGreen}city centre north bed and breakfast}\\\\\n\\textbf{Our value:}  {\\color{OliveGreen} north} (\\textbf{Partially extracted})\\\\\n\\hline\n\\textbf{Dialogue history:} ...[system] any preference on star rating [user] no , that s not important to me .\\\\\n\\textbf{Gold value:}  {\\color{OliveGreen}don't care}\\\\\n\\textbf{Our value:} {\\color{OliveGreen}not important} (\\textbf{Totally wrong})\\\\\n\\hline\n\\end{tabular}\n\\caption{\\label{errors}\nError analyses for extracting {\\color{OliveGreen}value candidates}. Three main types of errors are listed (missed, partially extracted and totally wrong).\n}\n\\end{table}\n\n\n\\subsection{Unseen Slot Generation}\nFor MultiWOZ2.1, we present the slots of each domain in Table \\ref{slots}. We find that some domains share some slots with other domains. For example, all slots of \\textit{Attraction} can be found in \\textit{Hotel}. On the contrary, some domains hold some slots that are not seen in other domain. For \\textit{Hotel}, it has four unseen slots: \\textit{parking}, \\textit{book stay}, \\textit{stars} and \\textit{internet}. \\textit{Restaurant} has two unseen slots (\\textit{food} and \\textit{time}). Here, we consider ``unseen slots'' as both ``unseen'' in the labeled training data and ``unseen'' in the slot names of the source domains.\n\nTo observe the extension and generation ability for unseen slots, we design two zero-shot experiments: leave \\textit{Hotel} or \\textit{Restaurant} as held-out-domain respectively, and train on other four domains. We present slots accuracy which evaluates the slot-level accuracy of correctly generated slots while values are correctly extracted. From the results in Figure \\ref{hotel}, we find that: \n\n(1) For seen slots that have the same names as that of source domains, our model can generate them with high accuracy. For example, \\textit{area} in \\textit{Hotel} domain is a common slot for other two source domains (\\textit{Attraction} and \\textit{Restaurant}), which can be generated with 96.92\\% accuracy. It indicates the good transfer ability across domains.\n\n(2) For some unseen slots (\\textit{book stay} and \\textit{stars} in \\textit{Hotel} of Figure \\ref{hotel}, \\textit{book time} and \\textit{food} in \\textit{Restaurant} of Figure \\ref{restaurant}), our model can generate them with more than 87\\% accuracy. \nFor example, given the dialogue history ``\\textit{...yes, please book it for 1 person and for 5 nights starting Friday.}'' The model successfully generates ``\\textit{book stay}'' for ``\\textit{5}'' even it has never seen the instances of book stay while training. Without known slot types, our model can infer the hidden semantic from the value and contexts, which is supposed to be the slot.\n\n(3) Figure \\ref{hotel} misses two unseen slots (\\textit{parking} and \\textit{internet}). For these two slots, the value extraction module fails to extract its gold value (``\\textit{yes}'') from utterance. The accuracy of value extraction is 0. So, the slot accuracy is also 0. However, we extract the adjective ``\\textit{free}'' from the user utterance as shown in Table \\ref{a-example}. Then PLM model can infer that the word ``\\textit{free}'' is the property of ``\\textit{internet}''. Actually, we think ``\\textit{internet: free}'' can also clearly describe a user's need. The case indicates a possible drawback of the existing annotation system. \n\n\\begin{table}[t]\n\\small\n\\centering\n\\begin{tabular}{p{0.95\\columnwidth}}\n\\hline\nslots\\\\\n\\hline\n$area^{123}$, $arrive\\ by^{45}$, $day^{235}$, $departure^{45}$, $destination^{45}$, \n$food^3$, $internet^2$, $leave^{45}$, $name^{123}$, $people^{235}$,\n$parking^2$, $price^{23}$, $stars^2$, $stay^2$, $time^3$, $type^{12}$\\\\\n\\hline\n\\end{tabular}\n\\caption{\\label{slots}\nAll slots in MultiWOZ2.1. The upper script on slot indicates the domain it belongs to (1: \\textit{Attraction}, 2: \\textit{Hotel}, 3: \\textit{Restaurant}, 4: \\textit{Taxi}, 5: \\textit{Train}).\n}\n\\end{table}\n\n\\begin{figure*}\n\t\\subfigure[\\textit{Hotel}]{\n\t\t\\includegraphics[width=0.46\\textwidth]{figs\/hotel.pdf}\n\t\\label{hotel}\n\t}\n\t\\subfigure[\\textit{Restaurant}]{\n\t\t\\includegraphics[width=0.47\\textwidth]{figs\/restaurant.pdf}\n\t\\label{restaurant}\n\t}\n\t\\caption{Slot accuracy of each slot in \\textit{Hotel} and \\textit{Restaurant} domain under zero-shot settings. X-axis is the slot accuracy and y-axis is the slot. Pink bars mark {\\color{CarnationPink}unseen slots}.}\n\t\\label{slot-acc}\n\\end{figure*}\n\n\\begin{table}[t]\n\\small\n\\centering\n\\begin{tabular}{p{0.95\\columnwidth}}\n\\hline\n\\textbf{Dialogue history:} ... [system] could you please give me any preferences for internet and parking ? [user] I would like it to be a guesthouse that has {\\color{OliveGreen}free} WiFi. \\\\\n\\textbf{Gold states:} {\\color{OliveGreen}yes}, \\textcolor{orange}{internet} \\\\\n\\textbf{Our states:} {\\color{OliveGreen}free}, \\textcolor{orange}{internet}\\\\\n\\hline\n\\end{tabular}\n\\caption{\\label{a-example}\n A test instance that has a wrong value candidate (``{\\color{OliveGreen}free}'') and a correct generated slot (``\\textcolor{orange}{internet}''). However, we think ({\\color{OliveGreen}free}, \\textcolor{orange}{internet}) can also describe the user's need here.\n}\n\\end{table}\n\n\\subsection{Ablation Studies}\nWe further observe the performances of different components including different value-based prompt functions, inverse prompt mechanism and prompt ensemble. We first train separate models with each value-based prompt (``Orig'' for $f_1, ...f_4$), then add inverse prompt mechanism for each of them (``+Inv''). Finally, we apply prompt ensemble (``En'') for the trained models with and without inverse prompt respectively. Experiments with 0.1\\% training data are shown in Table \\ref{ablstudy}.\n\nWe observe that:\n(1) The first four numerals in the first row show the original performance with different prompt functions (no inverse prompt mechanism and prompt ensemble). Among the four prompts, $f_2$ performs best without inverse prompt which may attribute to the similar format of $f_2$ compared with the output sequences in a pre-training task of SOLOIST (Considering dialogue history as inputs and generate dialogue states in the format as ``\\textit{belief states: [s1] = [v1], [s2] = [v2]}''). It's also the reason why inverse prompt doesn't improve its performance much as the prompt (``\\textit{belief states: [v] = [s]}'') and inverse prompt (``\\textit{belief states: [s] = [v]}'') are too similar to learn complementary knowledge.\n(2) Inverse prompt brings improvement for all prompt functions, especially for the ``weakest'' prompt function $f_1$.\n(3) The prompt ensemble enables further improvement for both models with and without inverse prompt. Under few-shot settings, prompt ensemble is a simple but efficient way of utilizing different prompt functions.\n\n\n\\begin{table}[t]\n\\centering\n\\begin{tabular}{c|c|c|c|c||c}\n\\hline\n& $f_1$ & $f_2$ & $f_3$ & $f_4$ & $En$ \\\\\n\\hline\nOrig & 24.3 & 30.9 & 25.7 & 27.7 & 31.9 \\\\\n+ Inv & 32.4 & 31.4 & 29.6 & 33.6 & \\textbf{36.1} \\\\ \n\\hline\n\\end{tabular}\n\\caption{\\label{ablstudy}\nJGA results for our models trained with 0.1\\% data. ``+Inv'' means adding inverse prompt mechanism for the original models (``orig'') given different prompt functions (from $f_1$ to $f_4$). ``En'' means the ensemble of models trained on different prompt functions with and without inverse prompt.\n}\n\\end{table}\n\n\\subsection{Analyses about the Weight of Inverse Prompt}\nWe conduct experiments to observe the influence of weight $w$ in Eq.\\ref{loss}. $w$ is set to {0, 0.1, 0.3, 0.5}. Experiments using 0.1\\% training data and different value-based prompts are shown in Figure \\ref{weight}. We find that the JGA performance always increases with the value of $w$ first and then begins to decrease. It means that the inverse prompt is actually an auxiliary task and can provide useful knowledge when the weight is relatively small. All experiments for the four prompts perform best when the $w$ is 0.1. So we set it to 0.1 in the former experiments. \n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{figs\/weight.pdf}\n\\caption{The influence of weight $w$ for inverse prompt using different prompt functions $f$. Experiments with $w=0.1$ always perform best for all prompt functions.}\n\\label{weight}\n\\end{figure}\n\n\\section{Related Work}\n\n\\subsection{Few-Shot Dialogue State Tracking}\n\nSome few-shot methods used data augmentation to get more labeled data for training. \\citet{DBLP:conf\/acl\/CampagnaFML20} and \\citet{HouCCCL21} propose to synthesize dialogues for a new domain using the small number of domain templates derived from observing a small dataset and the ontology of the domain. These methods depend on the ontology about slots of the target domain.\n\nMost of the existing work focuses on transferring from other resource-rich DST domains. \\citet{DBLP:conf\/aaai\/LeeJ19} and \\citet{DBLP:conf\/aaai\/RastogiZSGK20} utilize the slot description for transferring reusable concepts across domains. \\citet{todbert} learn similarity functions between slots and values, and transfer them into unseen domains. \\citet{d-reptile} introduces meta-learning and uses source domains to meta-learn the parameters of the model used to initialize the fine-tuning process of the target domain. One constraint of such methods is that they rely on domain similarity for transfer, and therefore cannot be applied to general domains.\n\nAnother thread of approaches tries to exploit external knowledge. \\citet{DBLP:conf\/asru\/ChenWR13} and \\citet{ddswws} utilize FrameNet-style \\cite{framenet} semantic frames and named entity recognition (NER) as the weak supervision for slot candidates. \\citet{rc2},\\citet{rc1},  \\citet{zero-dst-qa} and \\citet{transferQA} reformulate DST into a Reading Comprehension (RC) task and make use of the abundant RC data and frameworks to overcome the data scarcity issue in the DST task. \\citet{self-sup} investigate two self-supervised objectives: preserving latent consistency and modeling conversational behavior. However, they have limited performance owing to the limited common knowledge. \n\n\n\n\n\\subsection{Prompt Learning}\nWith the rapid development of large-scale pre-trained language models (PLM), a new paradigm arise public's attention: ``\\textit{pre-train, prompt, and predict} \\cite{prompt-survey}''. Instead of adapting PLM to downstream tasks via objective engineering, prompt learning reformulates downstream tasks to look more like those solved during the original PLM training with the help of a textual prompt. GPT-3 model \\cite{gpt3} achieves remarkable few-shot performance solely by leveraging a few task demonstrations as input context (e.g., \\textit{``Translate English into French''}) and a natural-language prompt (e.g., \\textit{``cheese ==> ''}). However, training such a huge model (175B parameters) is difficult. A more usual prompt learning method is ``prompt-based fine-tune'': utilize a moderately-sized PLM for which fine-tuning is computationally efficient and fine-tune it with the task-related prompts. It shows good performance in many few-shot scenarios. \\citet{DBLP:conf\/acl\/GaoFC20} use RoBERT-large and design automatic prompt generation for text classification. \\citet{DBLP:conf\/acl\/LiL20} add continuous task-specific vector as prompt to each transformer layer and achieve improvements in low-resource text summarization. For DST task, \\citet{sup-dst-prompt} use slots as prompt directly and generate the corresponding values, which needs a lot of labeled training data for fine-tuning PLM. For few-shot DST, the prompt learning-based methods are still under-explored.\n\n\n\\section{Conclusion}\nFor the lack of labeled data in practical DST tasks, we design a prompt learning framework, which consists of two main components (value-based prompt and inverse prompt mechanism). Our model can effectively probe DST-related knowledge from pre-trained language models and utilize it for DST task. Experiments show that our model outperforms existing state-of-the-art methods under different levels of resources. It achieves an improvement of 7.3\\% joint goal accuracy under the extreme low-resource settings (only 0.1\\% training data). In addition, this framework doesn't rely on the known ontology of slot types. With extensive experiments, we find that it can generate slots that are not seen in source domains and are not pre-defined as well with high probabilities. In the future, we'll focus on improving the performance of extracting value candidates.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Acknowledgement}\nWe thank M. V. Berry and J. H. Eberly for discussions, and\nacknowledge funding by Projects No. GIU07\/40, No. FIS2009-12773-C02-01,\nNo. NSFC 60806041, No. 08QA14030, No. 2007CG52, No. S30105,\nNo. ANR-09-BLAN-0134-01, and Juan de la Cierva Program.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{s_intro}\n\nLet $\\mathcal{M}$ be a compact surface homeomorphic to $\\mathbb{S}^2$, embedded in\n$\\mathbb{R}^3.$ For $\\kappa, h>0$ and $\\mathbf{A}$ a vector field on\n$\\mathcal{M},$ we consider the Ginzburg-Landau functional\n$\\mathcal{G}_{\\mathcal{M},\\kappa}:H^1(\\mathcal{M};\\mathbb{C})\\to\\mathbb{R}_{+},$\n\\begin{equation}\\label{GLS}\n\\mathcal{G}_{\\mathcal{M},\\kappa}(\\psi)=\\int_\\mathcal{M}\\(\\abs{\\nabla_\\mathcal{M}-ih\\mathbf{A}\\psi}^2+\\frac{\\kappa^2}{2}\\(\\abs{\\psi}^2-1\\)^2\\)d\\mathcal{H}_{\\mathcal{M}}^2(x).\n\\end{equation}\nThe functional $\\mathcal{G}_{\\mathcal{M},\\kappa}$ arises as the $\\Gamma$-limit (see \\cite{CS}) of the full 3d  Ginzburg-Landau energy\n\n\\begin{equation}\\label{GL3D}\n\\begin{split}\nG_{\\varepsilon,\\kappa}(\\psi,A)& =\\frac{1}{\\varepsilon}\\Bigg[\\int_{\\Omega_\\varepsilon}\\(\\abs{(\\nabla-iA)\\psi}^2+\\frac{\\kappa^2}{2}\\(\\abs{\\psi}^2-1\\)^2\\)dx \\\\\n&\\quad +\\int_{{\\mathbb R}^3}\\abs{\\nabla\\times A-\\mathbf{H}_{ext}}^2 dx\\Bigg].\n\\end{split}\n\\end{equation}\nwhere for all $\\varepsilon>0$ sufficiently small, $\\Omega_\\varepsilon$ corresponds to a uniform tubular neighborhood of $\\mathcal{M}.$\nIn \\eqref{GL3D} $\\mathbf{H}_{ext}$ is the external magnetic field. As $\\varepsilon \\to 0 ,$ the field completely penetrates the sample which then implies that in the $\\Gamma$-limit $A$ is prescribed to be equal to $\\mathbf{A},$ the tangential component of a divergence free vector field $\\mathbf{A}^e$ such that $\\nabla\\times h \\mathbf{A}^e=\\mathbf{H}_{ext}.$\n\nA central question in Ginzburg-Landau theory is the determination of the so-called {\\it critical fields}.\nThe first critical field corresponds to the appearance of zeros of $\\psi$ carrying non-trivial degree -- called vortices in this context -- in minimizers of the energy.\n\n\nThe analysis in \\cite{CS} includes the computation of the first critical field of a thin shell of a surface of revolution subject to a constant vertical field which turns out to be surprisingly simple and depending only on an intrinsic quantity, in the $ \\kappa\\to\\infty$ limit:\n$$\nH_{c_1}\\sim\\(\\frac{4\\pi}{\\mbox{Area of }\\mathcal{M}}\\)\\ln \\kappa.\n$$ \nThis result is extended in \\cite{C}, to general surfaces and magnetic fields. For a fixed field $\\mathbf{H}^e$, an external magnetic field of the form $\\mathbf{H}_{ext}=h( \\kappa)\\mathbf{H}^e=h( \\kappa)\\nabla\\times \\mathbf{A}^e$ is considered. Then the first critical field is\n$$\nH_{c_1}\\sim\\frac{1}{\\max_\\mathcal{M} *F-\\min_\\mathcal{M} *F}\\ln  \\kappa,\n$$\nwhere $d^{*}F=*d*F=\\mathbf{A}$ and $*$ denotes the Hodge star-operator.\nIn fact, the study shows also that, somewhat remarkably, not all fields $\\mathbf{H}^e$ give rise to a first critical field. This phenomenon is related to the geometry and relative location of $\\mathcal{M}$ with respect to $\\mathbf{H}^e.$ For $\\mathbf{H}^e$ that yield a finite $H_{c_1},$ the topological obstruction imposed by $\\mathcal{M}$ implying that the total degree of $\\frac{\\psi}{\\abs{\\psi}}$ is zero is used in \\cite {C} to show that there is an even number of vortices in minimizers of $\\mathcal{G}_{\\mathcal{M},\\kappa},$ half with positive degree, half with negative degree concentrating respectively on the set where $*F$ achieves its minimum and maximum.  The optimal number $2n$ and location of vortices and anti-vortices in $\\mathcal{M}$ is established in \\cite{C} for values of $h(k)$ slightly above $H_{c_1}$ and in addition it is shown that if the minimum and maximum of $*F$ is attained at finitely many points then the two sets of vortices  minimize, independently, a renormalized energy.\n\n\nThe results in \\cite{C} and \\cite{CS} cover only a moderate regime; in these works the intensity of the applied  field  is $H_{c_1} +\\mathcal{O}(\\ln\\ln \\kappa )$ and thus the number of vortices remains bounded as $\\kappa$ goes to infinity.\n\nOnce the value of $h$ becomes much larger than $H_{c_1},$ that is there is a constant $C>0$ such that $h-H_{c_1}\\geq C\\ln\\kappa,$ then the number of vortices in minimizers diverges as $\\kappa\\to\\infty$. For even larger $h$,  superconductivity persists  only in a narrow region in the sample.\n\n\nIn the case of an infinite cylinder whose cross section is a domain $\\Omega\\subseteq{\\mathbb R}^2$ and for constant applied fields parallel to the axis of the cylinder a reduction to a two-dimensional problem is possible.\nIn this case it is known that as the intensity increases superconductivity is lost  in the bulk and only a thin superconductivity region near $\\partial\\Omega$ persists (see Chapter 7 in \\cite{SS2}).\nFor much higher values still, superconductivity is completely lost: this value is known as $H_{c_3}$ and is estimated by a delicate spectral analysis of the magnetic Laplacian operator as in the monograph \\cite{fournaishelffer10}.\n\nIn our setting, corresponding to the above functional $\\mathcal{G}_{\\mathcal{M},\\kappa}$ \\eqref{GLS} on the compact surface $\\mathcal M$, there is no boundary, so what happens to the superconductivity region is not obvious. Another  crucial difference lies in the behaviour of the (normalized) magnetic field $H$ induced on $\\mathcal M$, which is the normal component of $\\mathbf H^e$, or equivalently $H\\, d\\mathcal H^2_{\\mathcal M}=d\\mathbf A$ (viewing $\\mathbf A$ as a 1-form). Namely, in our case, $H$ vanishes and changes sign. The spectral analysis in \\cite{montgomery95} therefore suggests that superconductivity should persist near the set $\\lbrace H=0\\rbrace$, where the external magnetic field is tangent to the surface $\\mathcal M$. In \\cite{pankwek02} the authors study the case of a vanishing magnetic field in the infinite cylinder model, and observe indeed nucleation of superconductivity near the zero locus of the magnetic field, for very high values of the applied field (near the putative $H_{c_3}$) under the condition that the gradient of the magnetic field does not vanish on its zero locus. The problem of the determination of the upper critical field for vanishing fields remains largely open otherwise.  Here, we are concerned with much lower values of the applied field: a main motivation of this work is to understand the transition from the vortexless to normal state regimes.\n\nAnother interesting difference is the fact that in the infinite cylinder model only positive  vortices exist  and so the location and growth of the vortex region is always ruled by the competing effects of  mutual repulsion, and confinement provided by the external field. In the present setting, this is no longer the case. Vortices of positive and negative degree must coexist and so repulsion and attraction are common features  of the relative placement of vortices in $\\mathcal{M}$, this without taking into account the external field.\n\n In this way,  the shrinking of the superconductivity region is a multifaceted phenomenon. Moreover, the problems mentioned in the characterization of this region are present even in the most emblematic case of a constant external field $\\mathbf H^e$: the region of persistence of superconductivity does not only depend on the field and on the topology of $\\mathcal{M},$ but also on extrinsic geometric properties of the surface; the relative position of $\\mathcal{M}$ with respect to $\\mathbf H^e$ affects $H$ and therefore the zero locus of the induced field. \n\n\nIn the present work we address the question of identifying the region where superconductivity persists in the $\\kappa\\to\\infty$ limit, when \n$$\\frac{H_{c1}}{h}$$ is small; we show that as this quantity gets small superconductivity persists in a small neighborhood of the place where the applied field is tangential to the sample, provided the field satisfies a generic non-degeneracy condition(see \\eqref{Hnondegen} below). \nAnother thrust of this work is aimed at uncovering some new intermediate regimes only present in this setting, when the normal component of the external field changes sign multiples times. In the model problem of a surface of revolution and constant vertical field, we  identify several structural transitions undergone by the superconductivity region. Furthermore, we observe a new phenomenon which we refer to as {\\it freezing of the boundary}, where a component of the vortex region stops growing even after increasing the intensity of the external field. This phenomenon holds in great generality (not only in the surface of revolution case), as is shown at the end of section \\ref{s_interm}.\n\n\n\nTo carry out our analysis  we start by using a reduction to a mean field model, first derived rigorously in \\cite{SS1}. More precisely, if we write a critical point $\\psi$ of $\\mathcal{G}_{\\mathcal{M},\\kappa}$ in polar form $\\psi=\\rho e^{i\\phi},$ variations of the phase yield $d(\\rho^2(d\\phi-hd^{*}F))=0,$ and because $H_{dr}^1(\\mathcal{M})=0$ this implies there is a $V$ such that $*dV=\\rho^2(d\\phi-hd^{*}F).$\nTaking $V=hW$, the function $W$ is expected to minimize\n\\begin{equation}\\label{mf}\n\\int_\\mathcal{M}\\abs{\\nabla_\\mathcal{M} W}^2 d\\mathcal{H}_\\mathcal{M}^2\n+\\frac{\\ln  \\kappa}{h}\\int_\\mathcal{M}\\abs{-\\Delta_\\mathcal{M} W+\\Delta_\\mathcal{M} *F}d\\mathcal{H}_\\mathcal{M}^2.\n\\end{equation}\n\nThe details of this mean field reduction can be found in \\cite{SS1} in the case of a positive external field applied in a bounded planar domain. However, the analysis in \\cite{SS1} does not handle the additional restriction of total zero mass which affects the construction of an upper bound in this setting.  The steps needed to extend the proof to the present case are included in Appendix~\\ref{a_meanfield}. \n\n\nThe measure $-\\Delta_{\\mathcal{M}} V +\\Delta_{\\mathcal{M}} *F$ can be interpreted as the normalized measure generated by the vortices. \nOn the other hand,\nwe observe that  \n\\begin{equation*}\n\\Delta_{\\mathcal{M}} *F \\, d\\mathcal H^2_{\\mathcal M} = d*d*F =d\\mathbf A= Hd\\mathcal H^2_{\\mathcal M},\n\\end{equation*}\nwhere the function $H$ is the normal component of the external magnetic field $\\mathbf H^e$ relative to $\\mathcal{M}$. In what follows we refer to $H$ simply as \\textit{the} magnetic field, and we assume that $H\\in C^1(\\mathcal M)$. Moreover, we drop the explicit dependence on $\\mathcal{M}$ in expressions like $\\Delta_{\\mathcal M}$, $\\nabla_{\\mathcal M}$. \n\n\nBefore we  state our main result we make the following assumption: there exists $\\beta>0$ such that\n\\begin{equation} \\label{defbeta}\\lim_{\\kappa\\to \\infty }\\frac{\\ln \\kappa}{h}=\\beta.\\end{equation}\n Once the connection to the mean field problem \\eqref{mf} is established we proceed to locate very precisely the region of persistence of superconductivity, that is, the region $SC_\\beta$ where the vorticity measure $-\\Delta V + H$ vanishes. We find that this region corresponds to a $\\beta^{\\frac{1}{3}}$ neighborhood of the set where $H$ vanishes, in the $\\beta\\to 0$ limit. More precisely,\n\n\\begin{theorem}\\label{simplifiedmain} Under the nondegeneracy assumption that $\\nabla H$ is nowhere vanishing on $\\{H=0\\}, $ there exists $C>0$ independent of $\\beta$ such that the superconductivity region $SC_\\beta$ is contained in $\\{x\\in \\mathcal{M}: d(x, \\{H=0\\})<C\\beta^{\\frac{1}{3}}\\}$, and contains  $\\{x\\in \\mathcal{M}: d(x, \\{H=0\\})<C^{-1}\\beta^{\\frac{1}{3}}\\}$,\nfor $\\beta$ sufficiently small.\n\\end{theorem}   \n\n\n\nThe nondegeneracy assumption on $H$ implies that the set $\\lbrace H=0\\rbrace$ is a finite union of smooth closed curves. It is the same assumption as the one made in \\cite{montgomery95,pankwek02} for the study of the third critical field $H_{c_3}$.\n\nTo prove Theorem \\ref{simplifiedmain} we reformulate the mean field approximation as an obstacle problem, and construct comparison functions. We note that a construction in the same spirit was carried out in \\cite[Appendix~A]{VorLatt} for the planar Ginzburg-Landau model in a different context. In our case however the construction is not immediate, because our obstacle problem is two-sided and our magnetic field $H$ changes sign. Indeed, our proof makes use of a comparison principle for two sided obstacle problems proved in \\cite{dalmasomoscovivaldi} which allows to compare solutions to obstacle problems corresponding to different data $H$. Hence the comparison functions will not be merely ``super- or sub-solutions'' of our problem, but actual solutions of modified problems. In particular they have to be quite regular. As a consequence, we cannot use functions of the  distance to $\\lbrace H=0\\rbrace$ as comparison functions. We have to use a particular coordinate system near each component of $\\{H=0\\}$ and explicitly build local functions satisfying local obstacle problems with appropriate modifications of $H$. Pasting these constructions we are able to  appeal to \\cite{dalmasomoscovivaldi} to obtain the desired estimates. In so doing we note a key feature of the proof, related to the fact that the obstacle problem is two-sided: the barriers thus obtained cannot be used independently to get neither the inner nor the outer bound separately, but together they yield the conclusion of the theorem. This is explained in more detail in section \\ref{s_smallbeta}.\n\nThanks to Theorem~\\ref{simplifiedmain}, we have  a clear picture of the superconductivity region for $\\beta\\to 0$: it is a union of tubular neigborhoods of the connected components of $\\lbrace H=0\\rbrace$. In particular, the superconductivity region has at least as many connected components as $\\lbrace H=0\\rbrace$. On the other hand, we also have a clear picture of the superconductivity region as $\\beta\\to\\beta_c$, where positive (resp. negative) vortices are concentrated near the points where $*F$ achieves its maximum (resp. minimum). In particular, the superconductivity region has, generically, one connected component. In the last part of this work, we investigate the intermediate regimes. If $\\lbrace H=0\\rbrace$ has more than one connected component, transitions \\textit{have} to occur: when $\\beta$ crosses some critical value, the number of connected components of $SC_\\beta$ changes. \n\nStudying such transitions, and determining the values of $\\beta$ at which they occur, seems out of our reach in all generality. That is why we concentrate first on a simple model problem. We consider a surface of revolution around the vertical axis $\\mathbf{e_z}$, and assume that the external magnetic field $\\mathbf H^e = \\mathbf{e_z}$ is vertical and constant. (In fact in Section~\\ref{ss_symcase}, more general magnetic fields are considered.) In that case, the induced field $H$ on $\\mathcal M$ is just $H=\\mathbf{e_z}\\cdot \\nu$, where $\\nu$ is an outward normal vector on $\\mathcal M$. The set $\\lbrace H=0\\rbrace$ consists exactly of the points where $\\mathbf{e_z}$ is tangent to $\\mathcal M$, and it is a union of circles. Note that $H$ has to change sign an odd number of times, since $H=-1$ at the `south pole' and $+1$ at the `north pole', thus there are an odd number of those circles. As explained above, interesting transitions happen when $\\lbrace H=0\\rbrace$ has more than one connected component. Therefore we focus on the simplest non-trivial situation, which corresponds to $\\lbrace H=0\\rbrace$ consisting of three circles.\nWe state loosely here the result that we obtain for that simple model problem in Section~\\ref{ss_symcase} (see Figure~\\ref{figreg}).\n\n\\begin{prop}\\label{loose1d}\nThere exists $\\beta_c>\\beta_1^* \\geq \\beta_2^* >0$ such that\n\\begin{itemize}\n\\item for $\\beta\\in (\\beta^*_1,\\beta_c)$, $SC_\\beta$ has one connected component,\n\\item for $\\beta\\in (\\beta^*_2,\\beta^*_1)$, $SC_\\beta$ has two connected components,\n\\item for $\\beta\\in (0,\\beta^*_2)$, $SC_\\beta$ has three connected components.\n\\end{itemize}\nMoreover, for $\\beta\\in(\\beta^*_2,\\beta^*_1)$, one connected component of $SC_\\beta$ remains constant.\n\\end{prop}\n\n\\begin{figure}[ht]\n\\begin{center}\n\\begin{subfigure}{.32\\textwidth}\n\\includegraphics[width=\\textwidth]{reg1.pdf}\n\\caption{$\\beta\\in (\\beta_1^*,\\beta_c)$}\n\\label{figreg1}\n\\end{subfigure}\n\\hspace{.05\\textwidth}\n\\begin{subfigure}{.24\\textwidth}\n\\includegraphics[width=\\textwidth]{reg2.pdf}\n\\caption{$\\beta\\in (\\beta_2^*,\\beta_1^*)$}\n\\label{figreg2}\n\\end{subfigure}\n\\hspace{.05\\textwidth}\n\\begin{subfigure}{.24\\textwidth}\n\\includegraphics[width=\\textwidth]{reg3.pdf}\n\\caption{$\\beta\\in (0,\\beta_2^*)$}\n\\label{figreg3}\n\\end{subfigure}\n\\caption{The region $SC_\\beta$ in the three regimes of Proposition~\\ref{loose1d}}\n\\label{figreg}\n\\end{center}\n\\end{figure}\n\nThe most striking part of Proposition~\\ref{loose1d} is the appearance of an intermediate regime in which one connected component of $SC_\\beta$ remains constant: one part of the free boundary is \\textit{frozen}. In Section~\\ref{ss_freez} we identify the features responsible for such `freezing' of the boundary and prove a similar `freezing property' in a general (non-symmetric) setting (see Proposition~\\ref{prop_freez}).\n\n\nAn other interesting outcome of the precise version of Proposition~\\ref{loose1d} (Proposition~\\ref{prop_regimes1d} in Section~\\ref{ss_symcase}) are the expressions of the critical values $\\beta^*_1$ and $\\beta^*_2$, in terms of integral quantities involving $\\mathbf A$ and the parametrization of $\\mathcal M$. Transfering these conditions to a general non-symmetric setting seems far from obvious and constitutes an interesting challenge. \n\n\nThe plan of the paper is as follows.\nIn the next section we collect some basic properties of solutions to an obstacle problem that serves as the starting point in our analysis. In section \\ref{s_smallbeta} we identify the thin region of superconductivity when $\\beta$ is small. In section \\ref{s_interm} we turn to the symmetric situation and identify in Proposition~\\ref{prop_regimes1d} the further transitions as $\\beta$ decreases to zero from $\\beta_c=\\max(*F)-\\min(*F).$ We also prove the `freezing of the boundary' property at the end of  section \\ref{s_interm}. \n\n\\subsubsection*{Acknowledgements}\n\nA.C. is\nsupported by the Fields postdoctoral fellowship. He wishes to thank\nRobert L. Jerrard for his incredible support. X.L. is supported by a Labex Milyon doctoral mobility scholarship for his stay at McMaster University. He wishes to thank his Ph.D. advisor Petru Mironescu and his McMaster supervisors Stanley Alama and Lia Bronsard for their great support.\n\n\\section{The obstacle problem}\\label{s_obstacle}\nThis preamble is devoted to the derivation of the obstacle problem dual to the mean field approximation. We also prove some basic results we will need later on. We think it is worthwhile recording these properties because in our setting, even in the by now classical application of the duality theorem which allows for the obstacle problem formulation, there is an inherent degeneracy we have to account for which is not present in other similar results in the literature. \n\nIn the first part of this section we show that -- as in \\cite[Chapter~7]{SS2} -- the minimizer of \n\\begin{equation}\\label{Ebeta}\nE_\\beta (V)=\\int_{\\mathcal M} \\abs{\\nabla V}^2  +\\beta \n \\int_{\\mathcal M} \\abs{ -\\Delta V + H} \n\\end{equation}\n is the solution of an obstacle problem, and then we study general properties of the contact set. There are two main differences with the obstacle problem arising in \\cite[Chapter~7]{SS2}.\n \\begin{itemize}\n \\item In our case there are no boundary conditions and the minimizer is well-defined only up to a constant. We need to deal with this degeneracy.\n \\item While in \\cite[Chapter~7]{SS2} the obstacle problem is one-sided, we have to consider a two-sided obstacle problem. This is due to the fact that, in our case, the magnetic field $H$ changes sign.\n \\end{itemize}\n \nThe functional $E_\\beta$ is, under assumption \\eqref{defbeta}, the limit of the sequence of energies considered in \\eqref{mf}. The link between $E_\\beta$ and the superconductivity region is, as mentioned in the introduction, proved in appendix A. \n \n\n\\subsection{Derivation of the obstacle problem}\\label{ss_derivobstacle}\n\n\\begin{prop}\\label{prop_obstacle}\nLet $\\beta>0$. A function $V_0\\in H^1(\\mathcal M)$ minimizes $E_\\beta$ \\eqref{Ebeta} if and only if $V_0$ minimizes\n\\begin{equation}\\label{F(V)}\n\\mathcal F (V) = \\int_{\\mathcal M}\\!\\left(\\abs{\\nabla V}^2 + 2  HV\\right)\n\\end{equation}\namong all $V\\in H^1(\\mathcal M)$ such that $(\\esssup V-\\essinf V)\\leq \\beta$.\n\\end{prop}\n\n\\begin{remark}\\label{rem_obstacle}\nSince the functional $\\mathcal F(V)$ is translation invariant, $V_0$ coincides, up to a constant, with any minimizer of the two-sided obstacle problem\n\\begin{equation*}\n\\min \\left\\lbrace \\int_{\\mathcal M}\\!\\left(\\abs{\\nabla V}^2 + 2 HV\\right) \\colon V\\in H^1(\\mathcal M),\\: \\abs{V}\\leq\\beta \/2 \\right\\rbrace.\n\\end{equation*}\nMoreover, recalling that $H=\\Delta *F$, this obstacle problem can also be rephrased as\n\\begin{equation}\\label{obstacle*F}\n\\min \\left\\lbrace \\int_{\\mathcal M}\\!\\abs{\\nabla(V-*F)}^2 \\colon V\\in H^1(\\mathcal M),\\: \\abs{V}\n \\leq \\beta\/2 \\right\\rbrace.\n\\end{equation}\nThe fact that minimizers coincide only up to a constant does not matter, since the physically relevant object is the vorticity measure $-\\Delta V +H$. Moreover, it is easy to check that, if the obstacle problem \\eqref{obstacle*F} admits a solution $V$ that `touches' the obstacles, i.e. satisfies $\\max V -\\min V=\\beta$, then this solution is unique because any other solution differs from it by a constant, which has to be zero. On the other hand, a solution satisfying $\\max V-\\min V <\\beta$ would have to be $V=*F+\\alpha$ for some constant $\\alpha$. Therefore, for $\\beta\\leq \\max *F-\\min *F$ the solution is unique.\n\\end{remark}\n\nThe proof of Proposition~\\ref{prop_obstacle} relies on the following classical result of convex analysis (easily deduced from \\cite{rockafellar66} or \\cite[Theorem~1.12]{B}). \n\n\\begin{lemma}\\label{lem_dual}\nLet $\\mathcal H$ be a Hilbert space and $\\varphi: \\mathcal H\\to{\\mathbb R}\\cup\\{+\\infty\\}$ be a convex lower semi-continuous function.\nThen the minimizers of the problems \n\\begin{equation*}\\min_{x\\in \\mathcal H}\\(\\frac{1}{2}\\norm{x}^2_{\\mathcal H}+\\varphi(x)\\)\\quad \\mbox{ and } \\quad\n\\min_{y\\in \\mathcal H}\\(\\frac{1}{2}\\norm{y}^2_{\\mathcal H}+\\varphi^{*}(-y)\\)\n\\end{equation*}\ncoincide, where $\\varphi^{*}$ denotes the Fenchel conjugate of $\\varphi$, \n\\begin{equation*}\n\\varphi^{*}(y):=\\sup_{z\\in \\mathcal H}\\langle y,z\\rangle_{\\mathcal H}-\\varphi(z).\n\\end{equation*}\n\\end{lemma}\n\n\\begin{proof}[Proof of Proposition~\\ref{prop_obstacle}:]\nWe apply Lemma~\\ref{lem_dual} in the Hilbert space\n\\begin{equation*}\n\\mathcal H:=\\dot H^1(\\mathcal M)=\\left\\lbrace V\\in H^1(\\mathcal M)\\colon \\int_{\\mathcal M} V =0\\right\\rbrace,\n\\end{equation*}\nendowed with the norm $\\norm{V}^2=\\int \\abs{\\nabla V}^2$, to the function\n\\begin{equation}\\label{phibeta}\n\\varphi(V)=\\varphi_\\beta(V)=\\frac\\beta 2 \\int_{\\mathcal M} \\abs{-\\Delta V +H }.\n\\end{equation}\nIn formula \\eqref{phibeta}, it is implicit that $\\varphi(V)=+\\infty$ if $\\mu = -\\Delta V + H$ is not a Radon measure. Note that, when $\\mu$ \\textit{is} a Radon measure, it must have zero average $\\int\\mu=0$, since $\\mu=\\Delta (*F-V)$.\n\nWe compute the Fenchel conjugate of $\\varphi$. It holds\n\\begin{align*}\n\\varphi^*(V) & \n= \\sup_{U\\in \\mathcal H } \\left\\lbrace \\int_{\\mathcal M}\\! \\nabla V \\cdot \\nabla U -\\frac\\beta 2 \\int_{\\mathcal M}\\! \\abs{-\\Delta U +H} \\right\\rbrace \\\\\n& = -\\int_{\\mathcal M} HV \n + \\sup_{U\\in \\mathcal H} \\left\\lbrace \\int_{\\mathcal M}\\! (-\\Delta U + H)V -\\frac\\beta 2 \\int_{\\mathcal M}\\! \\abs{-\\Delta U +H} \\right\\rbrace \\\\\n& = -\\int_{\\mathcal M} H V + \\sup_{ \\int P =0} \\left\\lbrace \\int_{\\mathcal M}\\left( P V-\\frac\\beta 2 \\abs{P}\\right) \\right\\rbrace.\n\\end{align*}\nIn the last equality, the supremum may -- by a density argument -- be taken over all $L^2$ functions $P$ with zero average.\n\nIf $(\\esssup V-\\essinf V)\\leq\\beta$, then $\\abs{V+\\alpha}\\leq \\beta\/2$ for some $\\alpha\\in{\\mathbb R}$, so that\n\\begin{equation*}\n\\int_{\\mathcal M}\\!\\left( P V-\\frac\\beta 2 \\abs{P}\\right) = \\int_{\\mathcal M}\\! \\left( (V+\\alpha)P -\\frac\\beta 2 |P| \\right) \\leq 0,\n\\end{equation*}\nand in that case\n\\begin{equation*}\n\\varphi^*(V)=-\\int_{\\mathcal M}\\! HV.\n\\end{equation*}\nOn the other hand, if $(\\esssup V-\\essinf V)>\\beta$, then up to translating $V$ we may assume that $\\lbrace V>\\beta\/2\\rbrace$ and $\\lbrace V<-\\beta\/2\\rbrace$ have positive measures. It is then easy to construct a function $P$ supported in those sets, such that $\\int P=0$, $\\int |P|=1$, and $\\int PV>\\beta\/2$. Using $\\lambda P$ as a test function for arbitrary $\\lambda>0$, we deduce that $\\varphi^*(V)=+\\infty$.\n\nFrom Lemma~\\ref{lem_dual} it follows that $V_0\\in \\dot H^1(\\mathcal M)$ minimizes $E_\\beta$ if and only if $V_0$ minimizes \n\\begin{equation*}\n\\frac 12 \\int _{\\mathcal M}\\! \\abs{\\nabla V}^2 + \\int_{\\mathcal M}\\! HV \n\\end{equation*}\namong $V\\in \\dot H^1(\\mathcal M)$ such that $\\esssup V-\\essinf V\\leq \\beta$. Since both problems are invariant under addition of a constant, the restriction to the space $\\dot H^1(\\mathcal M)$ can be relaxed to obtain Proposition~\\ref{prop_obstacle}.\n\\end{proof}\n\n\\subsection{Basic properties}\\label{ss_basicprop}\n\nIn this section we concentrate on the obstacle problem\n\\begin{equation}\\label{obstacleH}\n\\min \\left\\lbrace \\int_{\\mathcal M}\\!\\left( \\abs{\\nabla V}^2 + 2HV\\right)\\colon V\\in H^1(\\mathcal M),\\: |V|\\leq \\beta\/2 \\right\\rbrace.\n\\end{equation}\nWe recall the classical interpretation of \\eqref{obstacleH} as a free boundary problem, and establish a monotonicity property of the free boundary.\n\nThe first step to these basic properties is the reformulation of the obstacle problem \\eqref{obstacleH} as a variational inequality: a function $V\\in H^1(\\mathcal M)$ solves \\eqref{obstacleH} if and only if $\\abs{V}\\leq \\beta\/2$ and\n\\begin{equation}\\label{varineq}\n\\int_{\\mathcal M}\\! \\nabla V \\cdot \\nabla (W-V) \\geq -\\int_{\\mathcal M}\\! H(W-V) \\qquad\\forall W\\in H^1(\\mathcal M),\\: |W|\\leq \\beta\/2.\n\\end{equation}\nThe proof of this weak formulation is elementary and can be found in many textbooks on convex analysis. See for instance \\cite{rodrigues}.\n\nNext we recall the standard reformulation of \\eqref{varineq} as a free boundary problem.\n\\begin{lemma}\\label{lem_freebound}\nA function $V\\in H^1(\\mathcal M)$ with $\\abs{V}\\leq \\beta\/2$ solves \\eqref{obstacleH} or equivalently \\eqref{varineq} if and only if \n\\begin{equation}\\label{freebound}\n\\left\\lbrace\n\\begin{aligned}\n V&\\in W^{2,p}(\\mathcal M),\\quad 1<p<\\infty,\\\\\n \\Delta V & = H \\quad \\text{in }\\lbrace \\abs{V}<\\beta\/2\\rbrace,\\\\\n 0 & \\geq H \\quad \\text{in }\\lbrace V=\\beta\/2 \\rbrace,\\\\\n 0 & \\leq H \\quad \\text{in }\\lbrace V=-\\beta\/2 \\rbrace.\n\\end{aligned}\n\\right.\n\\end{equation}\nIn particular $V\\in C^{1,\\alpha}(\\mathcal M)$, so that at every regular point of the free boundaries $\\partial \\lbrace V=\\pm\\beta\/2\\rbrace$, the function $V$ satisfies the overdetermining boundary conditions $V=\\pm\\beta\/2$ and $\\partial V \/ \\partial\\nu =0$.\n\\end{lemma}\n\nThe only non-elementary part of Lemma~\\ref{lem_freebound}  is the $W^{2,p}$ regularity of the solution. There are many textbooks on the subject, at least for the one-sided obstacle problem. See for instance \\cite{petrosyan}. For the two-sided obstacle problem we refer to \\cite{dalmasomoscovivaldi}, in which the authors study very fine properties of two-sided obstacle problems. For the convenience of the reader, since most of the literature  only deals with a one-sided obstacle (and \\cite{dalmasomoscovivaldi} reaches far beyond this quite elementary result), we provide a sketch of the proof in Appendix~\\ref{a_prooffreebound}.\n\n\n\n\nRecall that in our case, $\\mu =-\\Delta V + H$ represents the vorticity measure. In light of Lemma~\\ref{lem_freebound}, this measure is supported in $\\lbrace V=\\pm\\beta\/2\\rbrace$. In that region, vortices are distributed with density $H$. \n\nFor $\\beta > \\beta_c$, where\n\\begin{equation}\\label{beta_c}\n\\beta_c := \\max (*F) - \\min (*F),\n\\end{equation}\nthe function $*F + \\alpha$ solves the obstacle problem \\eqref{obstacleH}, as long as the constant $\\alpha$ satisfies $\\max (*F) -\\beta\/2 \\leq \\alpha \\leq \\min (*F) +\\beta\/2$, and the vorticity measure $-\\Delta V+H$ is identically zero.\n\nFor $\\beta\\leq \\beta_c$, the solution $V=V_\\beta$ of the obstacle problem~\\eqref{obstacleH} must satisfy\n\\begin{equation*}\n\\max V_\\beta -\\min V_\\beta =\\beta,\n\\end{equation*}\nand therefore is unique (see Remark~\\ref{rem_obstacle}). Recall that the superconductivity region $SC_\\beta$ is defined as the set where the vorticity measure $-\\Delta V +H$ vanishes. According to Lemma~\\ref{lem_freebound}, that region is exactly\n\\begin{equation}\\label{SCbeta}\nSC_\\beta =\\lbrace \\abs{V_\\beta}<\\beta\/2 \\rbrace.\n\\end{equation}\n\n\n\nA first basic property of the superconductivity region $SC_\\beta$ is its monotonicity.\n\\begin{prop}\\label{prop_monot}\nFor any $0<\\beta_1 < \\beta_2\\leq \\beta_c$, it holds\n\\begin{equation*}\nSC_{\\beta_1} \\subset SC_{\\beta_2}.\n\\end{equation*}\n\\end{prop}\nIn other words, increasing the intensity of the applied magnetic field shrinks the region of persisting superconductivity, which consistant with physical intuition.\nSince we have to deal with a two-sided obstacle problem, this monotonicity property is not as obvious as in \\cite[Chapter~7]{SS2}. To prove it, we use a comparison principle for two-sided obstacle problems \\cite[Lemma~2.1]{dalmasomoscovivaldi}. We state and prove here a particular form that will also be useful later on.\n\n\\begin{lemma}\\label{lem_comparison}\nLet $H_1\\geq H_2$ be bounded, real-valued functions on $\\mathcal M$. Let also $\\alpha_1\\leq\\alpha_2$ and $\\beta_1\\leq\\beta_2$ be real numbers. For $j=1,2$, let $V_j\\in H^1(\\mathcal M)$ solve respectively the obstacle problems\n\\begin{equation*}\n\\min \\left\\lbrace \\int_{\\mathcal M}\\! \\left(\\abs{\\nabla V}^2 + 2H_jV\\right)\\colon \\alpha_j\\leq V\\leq \\beta_j\\right\\rbrace.\n\\end{equation*}\nThen either $V_1-V_2$ is constant, or $V_1\\leq V_2$.\n\\end{lemma}\n\\begin{proof}\nFor the convenience of the reader, we provide here the elementary proof, which consists in remarking that\n\\begin{equation*}\nW_1 = \\min(V_1,V_2)\\quad\\text{and }\nW_2 = \\max(V_1,V_2)\n\\end{equation*}\nare admissible test functions in the variational inequalities \n\\begin{equation*}\n \\int_{\\mathcal M}\\! \\nabla V_j \\cdot \\nabla (W_j-V_j)\\geq -\\int_{\\mathcal M}\\! H_j(W_j-V_j),\\qquad\\forall W_j\\in H^1,\\: \\alpha_j\\leq W_j\\leq \\beta_j.\n\\end{equation*}\n Substracting the resulting inequalities, we obtain\n\\begin{equation*}\n\\int_{\\mathcal M}\\abs{\\nabla (V_1-V_2)_+}^2 \\leq \\int_{\\mathcal M}(H_2-H_1)(V_1-V_2)_+ \\leq 0,\n\\end{equation*}\nwhere $(V_1-V_2)_+=\\max(V_1-V_2,0)$. We conclude that $(V_1-V_2)_+$ is a constant function.\n\\end{proof}\n\nWith Lemma~\\ref{lem_comparison} at hand, we may prove the monotonicity of the superconductivity region.\n\n\\begin{proof}[Proof of Proposition~\\ref{prop_monot}:]\nLet $V_1$ and $V_2$ denote the solution of the obstacle problem \\eqref{obstacleH} corresponding respectively to $\\beta=\\beta_1$ and $\\beta=\\beta_2$. Let \n\\begin{equation*}\n\\widetilde V_1 =V_1 +\\beta_1\/2,\\quad\\text{and}\\quad \\widetilde V_2=V_2+\\beta_2\/2,\n\\end{equation*}\nso that for $j=1,2$, $\\widetilde V_j$ solves the obstacle problem\n\\begin{equation*}\n\\min\\left\\lbrace \\int_{\\mathcal M}\\!\\left(\\abs{\\nabla V}^2 + 2HV\\right)\\colon 0\\leq V\\leq \\beta_j\\right\\rbrace.\n\\end{equation*}\nTherefore, applying Lemma~\\ref{lem_comparison} with $H_1=H_2=H$, $\\alpha_1=\\alpha_2=0$ and $\\beta_1\\leq \\beta_2$, we deduce that\n\\begin{equation*}\nV_1+\\beta_1\/2\\leq V_2 +\\beta_2\/2.\n\\end{equation*}\n(If $\\widetilde V_1- \\widetilde V_2$ is constant, then $\\beta_2 = \\max V_1-\\min V_1 = \\beta_1$.)\nIn particular, we obtain that\n\\begin{equation*}\n\\lbrace V_1 >-\\beta_1\/2 \\rbrace \\subset \\lbrace V_2 > -\\beta_2\/2\\rbrace.\n\\end{equation*}\nBy a similar argument, we show that\n\\begin{equation*}\n\\lbrace V_1 <\\beta_1\/2 \\rbrace \\subset \\lbrace V_2 < \\beta_2\/2\\rbrace,\n\\end{equation*}\nand conclude that $SC_{\\beta_1}\\subset SC_{\\beta_2}$.\n\\end{proof}\n\n\\begin{remark}\\label{rem_cont}\nIt follows from the above proof  that\n\\begin{equation*}\n|V_1-V_2|\\leq (\\beta_2-\\beta_1)\/2,\n\\end{equation*}\nthus proving the continuity of $\\beta\\mapsto V_\\beta$ for $0\\leq \\beta\\leq\\beta_c$.\n\\end{remark}\n\n\\section{The small $\\beta$ limit}\\label{s_smallbeta}\nIn this section we study what happens to the superconductivity set when the intensity of the field is high enough to confine it in a narrow region. \nWe make the (generic) non degeneracy assumption that\n\\begin{equation}\\label{Hnondegen}\n|H|+|\\nabla H| > 0\\quad\\text{in }\\mathcal M.\n\\end{equation}\nIn other words, $\\nabla H\\neq 0$ in $\\lbrace H=0\\rbrace$. This implies in particular that the set $\\Sigma:=\\lbrace H=0\\rbrace$ where the magnetic field vanishes is a finite disjoint union of smooth closed curves. We also note that condition \\eqref{Hnondegen} also implies that we are not in the situation where not even the first critical field is defined(see \\cite{C}, Theorem 3.1).\n\nLet us say a few words here about the nondegeneracy assumption \\eqref{Hnondegen}. This is the  same nondegeneracy assumption that has been considered in works on the spectral analysis of the magnetic Laplacian \\cite{montgomery95} and on higher applied magnetic fields in Ginzburg-Landau \\cite{pankwek02,attar14}.\nMoreover, we emphasize that \\eqref{Hnondegen} is a generic assumption, in the following sense.\n\\begin{lem}\\label{generic}\nThe set of $H$ satisfying \\eqref{Hnondegen} is open and dense in $C^1(\\mathcal M)$.\n\\end{lem}\n\\begin{proof}\nThe fact that \\eqref{Hnondegen} is an open condition in $C^1(\\mathcal M)$ is clear. The density follows from a transversality theorem by Quinn \\cite[Theorem~3]{quinn70}, applied to the $C^1$ map\n\\begin{equation*}\n\\Phi\\colon C^1(\\mathcal M) \\times \\mathcal M\\to\\mathbb R,\\quad (H,x)\\mapsto H(x).\n\\end{equation*}\nFor a function $H\\in C^1(\\mathcal M)$, \\eqref{Hnondegen} is equivalent to $\\Phi(H,\\cdot)$ being transverse to $\\lbrace 0\\rbrace$. Clearly, $D_H\\Phi(H,x)=I_{C^1(\\mathcal M)}$ is Fredholm, and $\\Phi$ is transverse to $\\lbrace 0\\rbrace$. Therefore, the set of $H$ such that $\\Phi(H,\\cdot)$ is transverse to $\\lbrace 0\\rbrace$ is dense in $C^1(\\mathcal M)$. \n\\end{proof}\n\n\n\nWe are interested in the behavior, as $\\beta\\to 0$, of the superconductivity region $SC_\\beta$ \\eqref{SCbeta}.\n\nWe let $d\\colon\\mathcal M\\to\\mathbb R_+$ denote the distance function to the set $\\Sigma=\\lbrace H=0\\rbrace$, that is\n\\begin{equation}\\label{defd}\nd(x)=\\mathrm{dist}(x,\\lbrace H=0\\rbrace).\n\\end{equation}\nIn this context we characterize the behavior of $SC_\\beta$ in terms of the function $d$, as follows(this is a more explicit version of Theorem \\ref{simplifiedmain}).\n\n\\begin{theorem}\\label{thm_scbeta}\nUnder the non-degeneracy assumption \\eqref{Hnondegen} on the magnetic field, there exists $\\beta_0>0$ and $C>0$ such that, for $\\beta\\in(0,\\beta_0)$, \\begin{equation}\\label{boundsscbeta}\n\\left\\lbrace d\\leq \\frac 1C \\beta^{1\/3}\\right\\rbrace \\subset SC_\\beta \\subset \\left\\lbrace d\\leq C\\beta^{1\/3} \\right\\rbrace,\n\\end{equation}\nwhere $SC_\\beta$ is the superconductivity region  \\eqref{SCbeta}, and $d$ denotes the distance to the zero locus of the magnetic field \\eqref{defd}.\n\\end{theorem}\n\nIn the proof we construct explicit solutions to modified obstacle problems, in order to apply the comparison principle Lemma~\\ref{lem_comparison}. The comparison functions are constructed locally near each component $\\Gamma$ of $\\lbrace H=0\\rbrace$, and then we need to extend and paste these functions and the associated modified obstacle problem data. Although the construction looks local, it is worth noting that we really need to make it near \\textit{every} component $\\Gamma$ of $\\lbrace H=0\\rbrace$. Otherwise the pasting would not provide us with obstacle problems comparable to the original one, because a solution has to change sign near \\textit{every} curve $\\Gamma$.\n\n\n\n\\begin{remark}\\label{rem_outer+thick}\nAnother natural approach to proving Theorem~\\ref{thm_scbeta} would be to construct separate comparison functions in $\\lbrace H>0\\rbrace$ and $\\lbrace H<0\\rbrace$. In those regions, the obstacle problem becomes one-sided, so that more standard constructions with a classical comparison principle can be made. On the other hand, there is no boundary conditions in those regions, so that such a construction would only provide us with the outer bound \n\\begin{equation}\\label{outer}\nSC_\\beta \\subset \\lbrace d\\leq C \\beta^{1\/3}\\rbrace.\n\\end{equation}\n To obtain the bounds \\eqref{boundsscbeta} which show that\nthe superconductivity set extends to \\underline{both} sides of the zero locus of $H$ by a $\\beta^{\\frac{1}{3}}$ margin, it seems that we really have to appeal to the comparison principle for two-sided obstacle problems. \nHowever, if we would just content ourselves with showing that the superconductivity set had `thickness' proportional to $\\beta^{\\frac{1}{3}}$, namely\n\\begin{equation}\\label{thickness}\n\\mathrm{dist}(\\lbrace V=\\beta\/2\\rbrace,\\lbrace V=-\\beta\/2\\rbrace)\\geq c\\beta^{1\/3},\n\\end{equation}\nthere would be a simpler way. In fact \\eqref{thickness} can be directly inferred from \\eqref{outer}. This is a simple consequence of the interpolated elliptic estimate (see \\cite[Appendix~A]{BBH1})\n\\begin{equation}\\label{interpolestim}\n\\norm{\\nabla V}^2_\\infty \\leq C \\norm{\\Delta V}_\\infty \\norm{V}_\\infty,\n\\end{equation}\nwhich implies, since $|V|\\leq\\beta$ and \n$|\\Delta V|=|H\\ensuremath{\\mathonebb{1}}_{SC_\\beta}| \\leq C \\beta^{1\/3},$\nthat\n\\begin{equation}\\label{gradientbound}\n|\\nabla V|\\leq C \\beta^{2\/3} \\quad\\text{in }\\mathcal M.\n\\end{equation}\nHence, for any $x_\\pm\\in \\lbrace V=\\pm\\beta\/2\\rbrace$ and any arc-length parametrized curve $\\gamma(s)$, ($0\\leq s\\leq \\ell$) going from $x_-$ to $x_+$, it holds\n\\begin{equation*}\n\\beta =V(x_+)-V(x_-) = \\int_0^\\ell \\nabla V(\\gamma(s))\\cdot\\gamma'(s) \\, ds \\leq  C \\beta^{2\/3}\\ell,\n\\end{equation*}\nso that the length of $\\gamma$ satisfies $\\ell\\geq c\\beta^{1\/3}$, which proves \\eqref{thickness}.\n\\end{remark}\n\nNext we turn to the proof of Theorem~\\ref{thm_scbeta}.\n\n\\begin{proof}[Proof of Theorem~\\ref{thm_scbeta}:]\nWe will construct, for small enough $\\beta$, bounded functions $H_1\\leq H\\leq H_2$, and comparison functions $V_1$ and $V_2$ of regularity $W^{2,\\infty}$, satisfying for $j=1,2$,\n\\begin{equation}\\label{eqVj}\n\\begin{gathered}\n\\Delta V_j=H_j \\ensuremath{\\mathonebb{1}}_{\\abs{V_j}<\\beta\/2},\\\\\n \\abs{V_j}\\leq \\beta\/2,\\quad H_j\\geq 0 \\text{ in }\\lbrace V_j=-\\beta\/2\\rbrace,\\quad H_j\\leq 0\\text{ in }\\lbrace V_j=\\beta\/2\\rbrace,\n \\end{gathered}\n\\end{equation}\nand the bounds\n\\begin{equation}\\label{boundsVj}\n\\left\\lbrace d\\leq \\frac 1C \\beta^{1\/3}\\right\\rbrace \\subset \\left\\lbrace \\abs{V_j}<\\beta\/2\\right\\rbrace \\subset \\left\\lbrace d\\leq C\\beta^{1\/3} \\right\\rbrace.\n\\end{equation}\nBy Lemma~\\ref{lem_freebound}, \\eqref{eqVj} implies that $V_j$ solves the obstacle problem \\eqref{obstacleH} with $H=H_j$. Therefore we may apply the comparison principle for two-sided obstacle problems (Lemma~\\ref{lem_comparison}) to conclude that $V_1\\geq V \\geq V_2$. In view of the bounds \\eqref{boundsVj} satisfied by $V_1$ and $V_2$, this obviously implies that the superconductivity region satisfies the bounds \\eqref{boundsscbeta}.\n\nThe rest of the proof is devoted to constructing $V_1$ and $V_2$. To this end we introduce good local coordinates in a neighborhood of $\\Sigma=\\lbrace H=0\\rbrace$. Recall that, thanks to the nondegeneracy assumption \\eqref{Hnondegen}, $\\Sigma$ is a finite union of closed smooth curves. Let us fix one of them, $\\Gamma$, together with an arc-length parametrization of it:\n\\begin{equation*}\n\\Gamma =\\left\\lbrace \\gamma(x) \\colon x\\in{\\mathbb R}\/\\ell\\mathbb Z\\right\\rbrace,\\quad \\abs{\\gamma'(x)}=1.\n\\end{equation*}\nLet us also fix a smooth normal vector $\\nu(x)$ to $\\Gamma$ on $\\mathcal M$, that is\n\\begin{equation*}\n\\nu(x)\\in T_{\\gamma(x)}\\mathcal M,\\quad \\abs{\\nu}=1,\\quad \\nu\\cdot\\gamma'=0,\n\\end{equation*}\nand impose that $\\nu(x)$ points in the direction of $\\lbrace H>0\\rbrace$ (since $H<0$ on one side of $\\Gamma$ and $H<0$ on the other side). We introduce Fermi coordinates along $\\Gamma$: for small enough $\\delta$, the map\n\\begin{equation*}\n{\\mathbb R}\/\\ell\\mathbb Z \\times (-\\delta,\\delta)\\to\\mathcal M,\\quad (x,y)\\mapsto \\exp_{\\gamma(x)}(y\\nu(x)),\n\\end{equation*}\nis a diffeomorphism. It defines local coordinates $(x,y)$ on $\\mathcal M$ in a neighborhood of $\\Gamma$, in which the Laplace operator has the form\n\\begin{equation}\\label{laplacefermi}\n\\Delta =\\frac 1f\\left(\\partial_y f\\partial_y +\\partial_x f^{-1} \\partial_x\\right),\n\\end{equation}\nwhere $f(x,y)=1-y\\kappa(x,y)$ for some smooth function $\\kappa$. Note that $y$ is nothing else than the signed distance to $\\Gamma$, and in particular $|y|=d$ in a neighborhood of $\\Gamma$. While this is a coordinate system that follows well the geometry of a neighborhood of $\\gamma , $ we actually need one  where the Laplacian allows us to reduce our construction to a $1d$ problem. To that end let $(x,z)$ be the local coordinates where\n\\begin{equation}\\label{zcoord}\nz=y+\\frac 12 y^2\\kappa(x,y).\n\\end{equation}\nClearly the map $(x,y)\\mapsto (x,z)$ is a diffeomorphism for small enough $y$, so that $(x,z)$ define indeed local coordinates on $\\mathcal M$.  The reason for using the coordinates $(x,z)$ is that the Laplace operator is then approximately\n\\begin{equation*}\n\\Delta \\approx \\partial_x^2 + \\partial_z^2,\n\\end{equation*}\nwhich will allow us to obtain nice bounds for functions depending only on $z$.\n\nNote that, since we choose the normal vector $\\nu$ to point in the direction of  $\\lbrace H>0\\rbrace$, and since $\\abs{\\nabla H}\\geq c>0$ in a neighborhood of $\\Gamma$ thanks to the nondegeneracy assumption \\eqref{Hnondegen}, it holds\n\\begin{equation*}\n\\partial_z H \\geq c>0,\\quad \\abs{z}<\\delta.\n\\end{equation*}\nOn the other hand, $\\nabla H$ is bounded, so that there exist $C\\geq c>0$ such that\n\\begin{equation}\\label{Hz}\nCz\\ensuremath{\\mathonebb{1}}_{z<0} + cz\\ensuremath{\\mathonebb{1}}_{z>0} \\leq H \\leq cz\\ensuremath{\\mathonebb{1}}_{z<0} + Cz\\ensuremath{\\mathonebb{1}}_{z>0},\\qquad \\abs{z}<\\delta.\n\\end{equation}\n\nNext we concentrate on the construction of $V_1$ ($H_1$ will be defined accordingly). Away from the set $\\Sigma$, we simply define\n\\begin{equation}\\label{V1away}\nV_1=-\\mathrm{sign}(H)\\beta\/2 \\quad\\text{in }\\lbrace d>\\delta\/2\\rbrace.\n\\end{equation}\nThe interesting part is of course what happens near $\\Sigma$. Near each of the smooth curves $\\Gamma\\subset\\Sigma$, we will look for $V_1$ in the form $V_1=v(z)$, where $v$ is a $W^{2,\\infty}$ function satisfying\n\\begin{equation}\\label{V1near}\nv(z)=\\begin{cases}\n\\beta\/2 & \\text{ for }z<-\\eta_-,\\\\\n-\\beta\/2 & \\text{ for }z>\\eta_+,\n\\end{cases}\n\\end{equation}\nfor some parameters $\\eta_\\pm>0$ that will depend on $\\beta$. A straightforward computation using \\eqref{laplacefermi} and \\eqref{zcoord} shows that\n\\begin{equation}\\label{DeltaV1}\n\\Delta V_1 =v''(z) + z \\left(g_1(x,z) v''(z) + g_2(x,z) v'(z)\\right),\n\\end{equation}\nwhere $g_1$ and $g_2$ are bounded functions. We are going to define in $(-\\eta_-,\\eta_+)$ the function $v$ so that \n\\begin{equation}\\label{boundsv}\nv''\\leq 2Cz\\ensuremath{\\mathonebb{1}}_{z<0} + \\frac{c}{2}z\\ensuremath{\\mathonebb{1}}_{z>0},\\quad |v'|= o(\\beta),\\; |v''|= o(\\beta).\n\\end{equation}\nWe then define $H_1$ in $(-\\eta_{-} , \\eta_+)$ simply as $\\Delta V_1 .$\nThus, recalling \n\\eqref{Hz}, we will have, for small enough $\\beta>0$,\n\\begin{equation}\n\\Delta V_1 = H_1\\ensuremath{\\mathonebb{1}}_{|V_1|<\\beta\/2}\\qquad\\text{with }H_1\\leq H\\;\\text{in }\\lbrace -\\eta_-<z<\\eta_+\\rbrace.\n\\end{equation}\nIt is then straightforward to extend $H_1$ to a function defined on $\\mathcal M$, such that $H_1\\leq H$, and having the same sign as $H$ outside of $\\lbrace -\\eta_-<z<\\eta_+\\rbrace$. The resulting $H_1$ and $V_1$ satisfy \\eqref{eqVj}. \n\nThus it remains to show that we can indeed define $v(z)$ in $\\lbrace -\\eta_-<z<\\eta_+\\rbrace$, satisfying the bounds \\eqref{boundsv}. We look for $v$ in the form\n\\begin{equation}\\label{formv}\nv(z)=\\begin{cases}\nv_-(z) &\\text{ for }-\\eta_-<z<0,\\\\\nv_+(z) &\\text{ for }0<z<\\eta_+,\n\\end{cases}\n\\quad\\text{with }v_\\pm(z)\\text{ polynomial.}\n\\end{equation}\nFirst of all, for $v$ to be of class $W^{2,\\infty}$ around the points $\\pm\\eta_\\pm$, we should impose\n\\begin{equation}\\label{regv}\nv_-(-\\eta_-)=\\beta\/2,\\quad v_+(\\eta_+)=-\\beta\/2,\\quad v_-'(-\\eta_-)=v_+'(\\eta_+)=0.\n\\end{equation}\nThus we take $v_\\pm$ to be of the form\n\\begin{equation}\\label{v+v-}\n\\begin{aligned}\nv_-(z)&=(z + \\eta_-)^2(A_-z+B_-) +\\frac{\\beta}{2} \\\\\n&= A_-z^3 + (B_-+2\\eta_-A_-)z^2 + (2\\eta_-B_- +\\eta_-^2A_-)z +\\eta_-^2B_-+\\frac{\\beta}{2} ,\\\\\nv_+(z)&=(z-\\eta_+)^2 (A_+z +B_+) -\\frac\\beta 2 \\\\\n&= A_+z^3 + (B_+-2\\eta_+A_+)z^2 +(-2\\eta_+B_+ +\\eta_+^2 A_+)z +\\eta_+^2 B_+ -\\frac\\beta 2.\n\\end{aligned}\n\\end{equation}\nFor $v$ to be of class $W^{2,\\infty}$ around $z=0$, we have to impose\n\\begin{equation}\\label{condv0}\n\\begin{gathered}\n\\eta_-^2B_-+\\frac{\\beta}{2} = \\eta_+^2 B_+ -\\frac\\beta 2,  \\\\\n2\\eta_-B_- +\\eta_-^2A_- =-2\\eta_+B_+ +\\eta_+^2 A_+. \n\\end{gathered}\n\\end{equation}\nWe also need to ensure that\n\\begin{equation}\\label{v''}\nv''\\leq 2Cz\\ensuremath{\\mathonebb{1}}_{z<0} + \\frac{c}{2}z\\ensuremath{\\mathonebb{1}}_{z>0},\n\\end{equation}\nso we impose\n\\begin{equation}\\label{condv''}\n6A_- = 2C,\\quad 6A_+ = \\frac c2,\\quad B_-+2\\eta_-A_- = B_+-2\\eta_+A_+ =0,\n\\end{equation}\nso that we even have an equality in \\eqref{v''}.\nPlugging \\eqref{condv''} into \\eqref{condv0}, we find\n\\begin{equation}\\label{condeta}\n\\frac c6 \\eta_+^3 + \\frac{2C}{3}\\eta_-^3 =\\beta,\\quad 4C\\eta_-^2 = c\\eta_+^2,\n\\end{equation}\nwhich leads us to choose\n\\begin{equation}\\label{eta}\n\\eta_\\pm = \\alpha_\\pm\\beta^{1\/3},\n\\end{equation}\nwhere $\\alpha_\\pm>0$ are the solutions of\n\\begin{equation*}\n4C\\alpha_-^2 = c\\alpha_+^2,\\quad \\frac c6 \\alpha_+^3 + \\frac{2C}{3}\\alpha_-^3=1.\n\\end{equation*}\nWith $A_\\pm$, $B_\\pm$ and $\\eta_\\pm$ chosen as in \\eqref{condv''}-\\eqref{eta}, the function $v$ is of class $W^{2,\\infty}$ and satisfies \\eqref{v''}. Moreover, it is straightforward to check that\n\\begin{equation*}\n|v'|+|v''|\\leq C\\beta^{1\/3} \\quad\\text{ in }(-\\eta_-,\\eta_+),\n\\end{equation*}\nso that \\eqref{boundsv} is satisfied, which concludes the construction of $V_1$ satisfying \\eqref{eqVj}. On the other hand $V_1$ obviously satisfies \\eqref{boundsVj} since\n\\begin{equation*}\n\\lbrace \\abs{V_1}<\\beta\/2\\rbrace = \\lbrace -\\eta_- < z<\\eta_+ \\rbrace.\n\\end{equation*}\n\nWe omit the construction of $V_2$, which is completely similar to the one just performed.\n\\end{proof}\n\n\n\n\n\n\\section{Intermediate regimes}\\label{s_interm}\n\nAs discussed in the introduction (Section~\\ref{s_intro}), in the present section we want to understand the transitions occurring as $\\beta$ decreases from $\\beta_c$ to 0, when the set $\\lbrace H=0\\rbrace$ has more than one connected component.\n\n\nIn Section~\\ref{ss_symcase} we study in detail a special case with rotational symmetry along a vertical axis, to provide some insight into the transition from the vortexless state to the zero solution. The reason to restrict to this setting is that it encapsules, what we believe are, the most interesting changes in the superconducting set that can occur. \n\nOn the one hand, once we drop the assumption of rotational symmetry, changes in $H$ inside the sample could lead to arbitrarily intricate solutions to the obstacle problem for different values of $\\beta,$ so a general theorem is not available. On the other hand the symmetries we consider highlight many model situations with remarkable properties.  One of these is the striking phenomenon that some parts of the free boundary may freeze: that is, remain constant with respect to $\\beta$, for $\\beta$ in some interval. In Section~\\ref{ss_freez} we generalize this observation to the general, non-symmetric case. \n\nAs mentioned earlier, a generalization of the other properties is precluded due to the wide variety of solutions one could construct, having the freedom to choose both $H$ and $\\mathcal{M}.$ Nevertheless, we believe that under some more restrictive assumptions, in particular fixing the topology of the level sets of $H,$ one could extend the result on existence of the transitions observed in Proposition~\\ref{prop_regimes1d}, however the role of the integral conditions on $I_{\\pm}, J$ is not so easily transferable or even identifiable anymore. \n\n\n\n\n\\subsection{Detailed study of a symmetric case}\\label{ss_symcase}\n\nHere we consider a surface of revolution of the form\n\\begin{equation*}\n\\mathcal M =\\left\\lbrace (\\rho(\\phi)\\cos \\theta,\\rho(\\phi)\\sin\\theta,z(\\phi))\\colon \\phi\\in [0,\\pi],\\,\\theta\\in[0,2\\pi] \\right\\rbrace,\n\\end{equation*}\nwhere $\\rho$ and $z$ are smooth functions linked by the relation \n\\begin{equation*}\nz(\\phi)\\tan\\phi=\\rho(\\phi),\n\\end{equation*}\nand satisfying $\\rho(0)=\\rho(\\pi)=0$, $\\rho>0$ in $(0,\\pi)$,  $z'(0)=z'(\\pi)=0$, and\n\\begin{equation*}\n\\gamma:=\\sqrt{(\\rho')^2+(z')^2}\\geq c>0.\n\\end{equation*}\nThe volume form on such $\\mathcal M$ is $d\\mathcal H^2_{\\mathcal M}=\\rho\\gamma d\\theta d\\phi$.\n\nThe induced magnetic potential $\\mathbf A$ on $\\mathcal M$ is also assumed to be symmetric, of the form\n\\begin{equation*}\n\\mathbf A = a(\\phi) d\\theta = \\frac{a(\\phi)}{u(\\phi)} \\hat e_\\theta,\n\\end{equation*}\nand we make the following assumptions on the functions $a$:\n\\begin{itemize}\n\\item[(a1):] $a(0)=a(\\pi)=0$, and $a>0$ in $(0,\\pi)$.\n\\item[(a2):] $a'>0$ in $(0,\\phi_1)$ and $(\\phi_2,\\phi_3)$ and $a'<0$ in $(\\phi_1,\\phi_2)$ and $(\\phi_3,\\pi)$, for some $0<\\phi_1<\\phi_2<\\phi_3<\\pi$.\n\\end{itemize}\nThe function $a(\\phi)$ hast two local maxima $a_1=a(\\phi_1)$ and $a_3=a(\\phi_3)$, and one local minimum $a_2=a(\\phi_2)$. To simplify notations to come, we assume in addition that $a_1<a_3$. See Figure~\\ref{figa1}.\n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[width=6cm]{a1.pdf}\n\\caption{The shape of $a(\\phi)$.}\n\\label{figa1}\n\\end{center}\n\\end{figure}\n\n\\begin{remark}\\label{rem_unifhex}\nThe case, presented in the introduction (Section~\\ref{s_intro}), of a uniform external magnetic field $\\mathbf H^e =\\mathbf{e_z}$ corresponds to $a=u^2\/2$.\n\\end{remark}\n\nIn that setting, the functions $H$ and $*F$ are also axially symmetric: they depend only on $\\phi$ , and are given by\n\\begin{equation*}\nH=\\frac{a'}{\\rho\\gamma},\\quad  (*F)'= a\\frac{\\gamma}{\\rho}.\n\\end{equation*}\n\nBy uniqueness (up to a possible additive constant), the solution of the obstacle problem \\eqref{obstacleH} is also rotationally symmetric: it holds $V=v(\\phi)$. Since $V\\in C^1(\\mathcal M)$, the function $v$ should satisfy\n\\begin{equation*}\nv\\in C^1([0,\\pi]),\\quad v'(0)=v'(\\pi)=0.\n\\end{equation*} \nMoreover, the free boundary problem \\eqref{freebound} becomes\n\\begin{equation}\\label{1dfreebound}\n\\left\\lbrace\n\\begin{aligned}\n\\abs{v}&\\leq \\beta\/2 &\\text{in }[0,\\pi],\\\\\n\\left(\\rho\\gamma^{-1} v'-a\\right)'&=0 & \\text{in }\\lbrace |u|<\\beta\/2\\rbrace, \\\\\na'&\\geq 0 & \\text{in }\\lbrace v=-\\beta\/2\\rbrace,\\\\\na'&\\leq 0 &\\text{in }\\lbrace v=\\beta\/2\\rbrace.\n\\end{aligned}\n\\right.\n\\end{equation}\n\nWe investigate, for $\\beta<\\beta_c$, the changes in the shape of the superconducting set $SC_\\beta=\\lbrace \\abs{v}<\\beta\/2\\rbrace$.\n The critical values at which that shape changes depend on the values of integrals $\\int a\\,\\gamma\\rho^{-1} d\\phi$ on some intervals related to the level sets of $a(\\phi)$.\nThat is why we start by fixing some notations concerning the level sets of $a(\\phi)$. There are three different cases, depicted in Figure~\\ref{figlevela}:\n\\begin{itemize}\n\\item For $\\alpha\\in (0,a_2)$, $\\lbrace a=\\alpha\\rbrace =\\lbrace \\phi_- < \\phi_+\\rbrace$.\n\\item For $\\alpha\\in (a_2,a_1)$, $\\lbrace a=\\alpha\\rbrace =\\lbrace \\phi_- < \\psi_+ < \\psi_- < \\phi_+\\rbrace$.\n\\item For $\\alpha\\in (a_1,a_3)$, $\\lbrace a=\\alpha\\rbrace = \\lbrace \\psi_- < \\phi_+\\rbrace$. \n\\end{itemize}\nThe functions $\\phi_\\pm(\\alpha)$, $\\psi_\\pm(\\alpha)$ are continuous on their intervals of definition.\n\n\n\nFor $\\alpha\\in (a_2,a_1)$, we define\n\\begin{equation}\\label{defIJ}\n\\begin{gathered}\nI_-(\\alpha)=\\int_{\\phi_-}^{\\psi_+} (a-\\alpha)\\frac\\gamma\\rho\\, d\\phi,\\quad I_+(\\alpha)=\\int_{\\psi_-}^{\\phi_+} (a-\\alpha)\\frac\\gamma\\rho\\, d\\phi,\\\\\nJ(\\alpha)=-\\int_{\\psi_+}^{\\psi_-} (a-\\alpha)\\frac\\gamma\\rho\\, d\\phi.\n\\end{gathered}\n\\end{equation}\n\nThose integrals corresponds to ``weighted'' areas of the regions depicted in Figure~\\ref{figIJ}, with respect to the measure $\\gamma\\rho^{-1} d\\phi$. Note that both the integrands and the intervals of integration depend on $\\alpha$.\n\n\n\nWe identify a critical value of $\\alpha$ with respect to these integrals.\n\\begin{lemma}\\label{critalpha}\nThere exists $\\alpha_*\\in (a_2,a_1)$ such that:\n\\begin{itemize}\n\\item for $a_2<\\alpha<\\alpha_*$, $J<\\min(I_\\pm)$.\n\n\\item for $\\alpha_*<\\alpha<a_1$, $\\min(I_\\pm) < J$.\n\\end{itemize}\n\\end{lemma}\n\n\\begin{proof}\nIt follows from the obvious facts that $J$ is increasing, $I_\\pm$ are decreasing, $J(a_2)=0$, $I_-(a_1)=0$, and the functions are continuous.\n\\end{proof}\n\n\\begin{figure}[ht]\n\\begin{center}\n\\begin{subfigure}{.4\\textwidth}\n\\includegraphics[width=\\textwidth]{a2.pdf}\n\\caption{$\\alpha\\in (0,a_2)$}\n\\label{figa2}\n\\end{subfigure}\n\\hspace{.05\\textwidth}\n\\begin{subfigure}{.4\\textwidth}\n\\includegraphics[width=\\textwidth]{a3.pdf}\n\\caption{$\\alpha\\in (a_2,a_1)$}\n\\label{figa3}\n\\end{subfigure}\n\n\\vspace{1em}\n\\begin{subfigure}{.4\\textwidth}\n\\includegraphics[width=\\textwidth]{a4.pdf}\n\\caption{$\\alpha\\in (a_1,a_3)$}\n\\label{figa4}\n\\end{subfigure}\n\\caption{Level sets $\\lbrace a=\\alpha\\rbrace$}\n\\label{figlevela}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[width=6cm]{IJ.pdf}\n\\caption{The integrals $I_\\pm$ and $J$.}\n\\label{figIJ}\n\\end{center}\n\\end{figure}\n\nNow we may give the precise version of Proposition~\\ref{loose1d}.\n\\begin{prop}\\label{prop_regimes1d}\nLet $\\beta_c>\\beta^*_1 \\geq \\beta^*_2 >0$ be defined by\n\\begin{equation*}\n\\beta^*_1:=\\max(I_\\pm(\\alpha^*)),\\quad \\beta^*_2:=\\min (I_\\pm(\\alpha^*)).\n\\end{equation*}\nThen the conclusion of Propostion~\\ref{loose1d} holds:\n\\begin{itemize}\n\\item For $\\beta_c >\\beta >\\beta^*_1$,  $SC_\\beta$ is an interval.\n\\item For $\\beta^*_1>\\beta>\\beta^*_2$, $SC_\\beta$ is the union of two disjoint intervals, one of them independent of $\\beta$. \n\\item For $\\beta^*_2>\\beta>0$, $SC_\\beta$ is the union of three disjoint intervals.\n\\end{itemize}\n\\end{prop}\n\n\\begin{remark}\nIt may happen that $I_-(\\alpha^*)=I_+(\\alpha^*)$. In that case, $\\beta^*_1=\\beta^*_2$ and the second regime predicted by Proposition~\\ref{prop_regimes1d} never happens.\n\\end{remark}\n\n\n\n\\begin{proof}[Proof of Proposition~\\ref{prop_regimes1d}:]\nBy uniqueness (see Remark~\\ref{rem_obstacle}), it suffices to exhibit, for each regime listed in Proposition~\\ref{prop_regimes1d}, a solution of \\eqref{1dfreebound} satisfying the desired properties.\n\n\\textbf{Case 1:} $\\beta\\in (\\beta^*_1,\\beta_c)$. The function\n\\begin{equation*}\nI(\\alpha):=\\int_{\\phi_-}^{\\phi_+} (a-\\alpha)\\frac\\gamma\\rho\\, d\\phi,\\quad\\alpha\\in (0,a_1),\n\\end{equation*}\nis continuous, decreasing and satisfies $I(0)=\\beta_c$ and $I(\\alpha^*)=\\beta^*_1$. Therefore there exists a unique $\\alpha\\in (0,\\alpha^*)$ such that $I(\\alpha)=\\beta$. We define\n\\begin{equation*}\nv(\\phi)=\n\\begin{cases}\n-\\beta\/2 & \\text{ for }\\phi\\in (0,\\phi_-),\\\\\n-\\beta\/2 + \\int_{\\phi_-}^{\\phi} (a-\\alpha)\\frac\\gamma\\rho\\, d\\tilde\\phi & \\text{ for }\\phi\\in (\\phi_-,\\phi_+),\\\\\n\\beta\/2 & \\text{ for }\\phi\\in (\\phi_+,\\pi).\n\\end{cases}\n\\end{equation*}\nThe shape of the function $v$ is sketched in Figure~\\ref{figu1}.\n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[width=8cm]{u1.pdf}\n\\caption{The shape of $v$ for $\\beta\\in(\\beta^*_1,\\beta_c)$}\n\\label{figu1}\n\\end{center}\n\\end{figure}\n\nThe function $v$ is clearly continuous since $\\beta$ has been chosen accordingly. Moreover, it holds\n\\begin{equation*}\nv'(\\phi_+)=(a(\\phi_+)-\\alpha)\\frac\\gamma\\rho = v'(\\phi_-)=0,\n\\end{equation*}\nsince by definition $a(\\phi_+)=a(\\phi_-)=\\alpha$. Hence $v$ is in fact $C^1$ in $[0,\\pi]$. Also by definition, $a'\\geq 0$ in $(0,\\phi_-)$ and $a'\\leq 0$ in $(\\phi_+,\\pi)$. In addition, we clearly have $(\\rho\\gamma^{-1}v'-a)'=0$ in $(\\phi_-,\\phi_+)$. To prove that $v$ solves \\eqref{1dfreebound}, it only remains to show that $|v|<\\beta\/2$ in $(\\phi_+,\\phi_-)$. We consider two different cases, depending on whether $\\alpha\\in (0,a_2]$ or $\\alpha\\in (a_2,\\alpha^*)$.\n\nIf $\\alpha\\in (0,a_2)$, then (see Figure~\\ref{figa2})\n\\begin{equation*}\nv'=(a-\\alpha)\\frac\\gamma\\rho>0\\quad\\text{in }(\\phi_-,\\phi_+),\n\\end{equation*}\nso that $v$ is increasing on $(\\phi_-,\\phi_+)$ and it clearly holds $|v|<\\beta\/2$. For $\\alpha=a_2$ the derivative $v'$ only vanishes at one point and the same conclusion is valid.\n\nIf, on the other hand $\\alpha\\in (a_2,\\alpha^*)$, then (see Figure~\\ref{figu1})\n\\begin{equation*}\nv'=(a-\\alpha)\\frac\\gamma\\rho\\begin{cases}\n>0 &\\text{ in }(\\phi_-,\\psi_+),\\\\\n<0 & \\text{ in }(\\psi_+,\\psi_-),\\\\\n>0 & \\text{ in }(\\psi_-,\\phi_+).\n\\end{cases}\n\\end{equation*}\nTherefore it suffices to check that $v(\\psi_+)<\\beta\/2$ and $v(\\psi_-)>-\\beta\/2$. We have, since $I(\\alpha)=\\beta$ and by definition of $I_\\pm$ and $J$ (see Figure~\\ref{figIJ}),\n\\begin{equation*}\n\\begin{split}\nv(\\psi_+)-\\beta\/2 & = I_-(\\alpha)-\\beta = I_-(\\alpha)-I(\\alpha)=J(\\alpha)-I_+(\\alpha),\\\\\nv(\\psi_-)+\\beta\/2 & = I_-(\\alpha)-J(\\alpha).\n\\end{split}\\end{equation*}\nSince $\\alpha<\\alpha^*$ we find indeed (by definition of $\\alpha^*$) that $v(\\psi_+)<\\beta\/2$ and $v(\\psi_-)>-\\beta\/2$, and in that case also we conclude that $v$ solves the free boundary problem\n\\eqref{1dfreebound}.\n\n\\textbf{Case 2:} $\\beta\\in (\\beta^*_2,\\beta^*_1)$. We treat the case where $\\min (I_\\pm(\\alpha^*))=I_-(\\alpha^*)$. Thus $\\beta^*_1=I_+(\\alpha^*)$ and $\\beta^*_2=I_-(\\alpha^*)$. The other case can be dealt with similarly. \n\nThe function $I_+(\\alpha)$ is continuous and decreasing on $(a_2,a_3)$ and satisfies $I_+(\\alpha^*)=\\beta^*_1$ and $I_+(a_3)=0<\\beta^*_2$ (see Figure~\\ref{figIJ}). Therefore there exists $\\alpha>\\alpha^*$ such that $I_+(\\beta)=\\alpha$. We denote by $\\psi_-$ and $\\phi_+$ the two points of $\\lbrace a=\\alpha \\rbrace \\cap (\\phi_2,\\pi)$, and by $\\phi_-^*<\\psi_+^*<\\psi_-^*$ the three  points of $\\lbrace a=\\alpha^*\\rbrace \\cap (0,\\phi_3)$ (as in Figure~\\ref{figu2} below). Note that, since $\\alpha>\\alpha^*$, $\\psi_-^*<\\psi_-$. Next we define\n\\begin{equation*}\nv(\\phi)=\n\\begin{cases}\n-\\beta\/2 &\\text{ for }\\phi\\in (0,\\phi_-^*),\\\\\n-\\beta\/2 +\\int_{\\phi_-^*}^{\\phi}(a-\\alpha^*)\\frac\\gamma\\rho\\, d\\tilde\\phi &\\text{ for }\\phi\\in (\\phi_-^*,\\psi_-^*),\\\\\n-\\beta\/2 & \\text{ for }\\phi\\in (\\psi_-^*,\\psi_-),\\\\\n-\\beta\/2 +\\int_{\\psi_-}^{\\phi}(a-\\alpha)\\frac\\gamma\\rho\\, d\\tilde\\phi & \\text{ for }\\phi\\in (\\psi_-,\\phi_+),\\\\\n\\beta\/2 & \\text{ for }\\phi\\in(\\phi_+,\\pi).\n\\end{cases}\n\\end{equation*}\nThe shape of the function $v$ is sketched in Figure~\\ref{figu2}.\n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[width=8cm]{u2.pdf}\n\\caption{The shape of $v$ for $\\beta\\in(\\beta^*_2,\\beta^*_1)$}\n\\label{figu2}\n\\end{center}\n\\end{figure}\n\nContinuity of $v$ at $\\psi_-^*$ is ensured by the fact that $I_-(\\alpha^*)=J(\\alpha^*)$. Continuity at $\\phi_+$ by $I_+(\\alpha)=\\beta$. The function $v$ is $C^1$ because the facts that $a(\\phi_-^*)=a(\\psi_-^*)=\\alpha^*$ and $a(\\psi_-)=a(\\phi_+)=\\alpha$ guarantee that $v'(\\phi_-^*)=v'(\\psi_-^*)=v'(\\psi_-)=v'(\\phi_+)=0$. The sign of $a'$ is positive in $(0,\\phi_-^*)$ and $(\\psi_-^*,\\psi_-)$ and negative in $(\\phi_+,\\pi)$.  In the two intervals $(\\phi_-^*,\\psi_-^*)$ and $(\\psi_-,\\phi_+)$, the equation $(\\rho\\gamma^{-1}v'-a)'=0$ is obviously satisfied, and it remains to check that $|v|<\\beta\/2$ in those intervals.\n\nSince $v'=(a-\\alpha)\\gamma\\rho^{-1} >0$ in $(\\psi_-,\\phi_+)$, it clearly holds $|v|<\\beta\/2$ in $(\\psi_-,\\phi_+)$.\n\nIn the interval $(\\phi_-^*,\\psi_-^*)$, the sign of $v'$ shows that $v$ attains its minimum at the boundary and its maximum at $\\psi_+^*$, and it holds\n\\begin{equation*}\nv(\\psi_+^*)-\\beta\/2=-\\beta + I_-(\\alpha^*) =-\\beta + \\beta^*_2 <0.\n\\end{equation*}\nWe conclude that $v$ solves the free boundary problem \\eqref{1dfreebound}. Moreover, the interval $(\\phi_-^*,\\psi_-^*)$ clearly does not depend on $\\beta$.\n\n\\textbf{Case 3:} $\\beta\\in (0,\\beta^*_2)$. \nSince $I_-$ is continuous and decreasing, $I_-(\\alpha^*)>\\beta^*_2$ and $I_-(a_1)=0$, there exists $\\alpha_1>\\alpha^*$ such that $I_-(\\alpha_1)=\\beta$. Similarly, there exist $\\alpha_2<\\alpha^*$ and $\\alpha_3>\\alpha^*$ such that $J(\\alpha_2)=I_+(\\alpha_3)=\\beta$. We denote by\n\\begin{equation*}\n0<\\phi_-^1<\\psi_+^1 <\\psi_+^2 <\\psi_-^2 <\\psi_-^3 <\\phi_+^3<\\pi\n\\end{equation*}\nthe points such that (see Figure~\\ref{figu3})\n\\begin{equation*}\n\\begin{gathered}\n\\lbrace a=\\alpha_1\\rbrace\\cap (0,\\phi_2)=\\lbrace \\phi_-^1,\\psi_+^1\\rbrace,\\\\\n\\lbrace a=\\alpha_2\\rbrace\\cap (\\phi_1,\\phi_3) =\\lbrace \\psi_+^2,\\psi_-^2\\rbrace,\\\\\n\\lbrace a=\\alpha_3\\rbrace\\cap (\\phi_2,\\pi)=\\lbrace \\psi_-^3,\\phi_+^3\\rbrace.\n\\end{gathered}\\end{equation*}\n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[width=8cm]{u3.pdf}\n\\caption{The shape of $v$ for $\\beta\\in(0,\\beta^*_2)$}\n\\label{figu3}\n\\end{center}\n\\end{figure}\n\nThen we define\n\\begin{equation*}\nv(\\phi)=\\begin{cases}\n-\\beta\/2 &\\text{ for }\\phi\\in (0,\\phi_-^1)\\text{ or }\\phi\\in (\\psi_-^2,\\psi_-^3)\\\\\n-\\beta\/2 +\\int_{\\phi_-^1}^{\\phi}(a-\\alpha_1)\\frac\\gamma\\rho\\, d\\tilde\\phi &\\text{ for }\\phi\\in (\\phi_-^1,\\psi_+^1),\\\\\n\\beta\/2 & \\text{ for }\\phi\\in (\\psi_+^1,\\psi_+^2)\\text{ or }\\phi\\in (\\phi_+^3,\\pi),\\\\\n\\beta\/2 +\\int_{\\psi_+^2}^{\\phi}(a-\\alpha_2)\\frac\\gamma\\rho\\, d\\tilde\\phi & \\text{ for }\\phi\\in (\\psi_+^2,\\psi_-^2)\\\\\n-\\beta\/2 +\\int_{\\psi_-^3}^{\\phi}(a-\\alpha_3)\\frac\\gamma\\rho\\, d\\tilde\\phi & \\text{ for }\\phi\\in (\\psi_-^3,\\phi_+^3).\n\\end{cases}\n\\end{equation*}\nThe shape of the function $v$ is sketched in Figure~\\ref{figu3}.\n\n\n\nAs above the $C^1$ regularity of $v$ follows from the definitions of $\\alpha_1$, $\\alpha_2$ and $\\alpha_3$. The sign of $a'$ is positive in $(0,\\phi_-^1)\\cup (\\psi_-^2,\\psi_-^3)$ and negative in $(\\psi_+^1,\\psi_+^2)\\cup (\\phi_+^3,\\pi) $. The equation $(\\rho\\gamma^{-1}v'-a)'=0$ is satisfied in the three intervals $(\\phi_-^1,\\psi_+^1)$,  $(\\psi_+^2,\\psi_-^2)$ and $(\\psi_-^3,\\phi_+^3)$. Moreover in those intervals, the function $v$ is monotone, hence $|v|<\\beta\/2$. Therefore $v$ solves the free boundary problem \\eqref{1dfreebound}.\n\\end{proof}\n\n\n\n\n\n\n\\subsection{`Freezing' of the free boundary}\\label{ss_freez}\n\n\\begin{prop}\\label{prop_freez}\nAssume that, for some $\\beta_0\\in (0,\\beta_c)$, one connected component $\\omega$ of the superconductivity set $SC_{\\beta_0}$ is such that $V_{\\beta_0}$ takes the same value on each connected component of $\\partial\\omega$. Then there exists $\\delta>0$ such that\n\\begin{equation}\\label{freez}\nSC_\\beta \\cap\\overline \\omega  = SC_{\\beta_0}\\cap\\overline\\omega =\\omega,\n\\end{equation}\nfor all $\\beta\\in (\\beta_0-\\delta,\\beta_0]$.\n\\end{prop}\n\nIn Figure~\\ref{figfreez} we show a situation corresponding to Proposition~\\ref{prop_freez}, with $V=-\\beta\/2$ on every connected component of $\\partial\\omega$.\n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[width=4.5cm]{freez.pdf}\n\\caption{An example of the situation of Proposition~\\ref{prop_freez}}\n\\label{figfreez}\n\\end{center}\n\\end{figure}\n\n\\begin{remark}\nThe assumption on $\\beta_0$ in Proposition~\\ref{prop_freez} corresponds exactly to what happens in the symmetric case (Proposition~\\ref{prop_regimes1d}) in the regime $\\beta_1^*>\\beta>\\beta_2^*$, where $u(\\phi_-^*)=u(\\psi_-^*)=-\\beta\/2$ (Figure~\\ref{figu2}).\n\\end{remark}\n\n\\begin{proof}[Proof of Proposition~\\ref{prop_freez}:]\nWe present the proof in the case where $V=-\\beta_0\/2$ on every connected component of $\\partial\\omega$. The case $V=\\beta_0\/2$ on $\\partial\\omega$ can be dealt with similarly.\n\nSince $V<\\beta_0\/2$ in $\\omega$ and $V=-\\beta_0\/2$ on $\\partial\\omega$, it holds\n\\begin{equation*}\nm:=\\max_{\\overline\\omega} V <\\beta_0\/2,\n\\end{equation*}\nand we define\n\\begin{equation*}\n\\delta:=  \\frac 12 {\\beta_0}-m >0.\n\\end{equation*}\n\nLet $\\beta\\in (\\beta_0-\\delta,\\beta_0]$, and define\n\\begin{equation}\\label{tildeV0}\n\\widetilde V_0 := V_{\\beta_0} + \\frac{1}{2}(\\beta_0-\\beta).\n\\end{equation} \nThe definitions of $m$ and $\\delta$ ensure that it holds\n\\begin{equation}\\label{ineqtildeV0}\n-\\beta\/2 \\leq \\widetilde V_0 \\leq \\frac 12 {\\beta_0} -\\delta + \\frac{1}{2}(\\beta_0-\\beta) < \\beta\/2\\quad\\text{in }\\overline\\omega.\n\\end{equation}\nWe claim that \n\\begin{equation}\\label{freezV}\nV_\\beta =\\widetilde V_0 \\quad\\text{in }\\overline\\omega,\n\\end{equation}\nwhich obviously implies \\eqref{freez}. \n\nNote that the proof of Proposition~\\ref{prop_monot} implies that it always holds\n\\begin{equation}\\label{VleqtildeV}\nV_\\beta \\leq \\widetilde V_0 \\quad\\text{in }\\mathcal M.\n\\end{equation}\nLet $\\omega_\\beta = SC_\\beta\\cap \\omega$, and \n\\begin{equation}\\label{Unonneg}\nU:=\\widetilde V_0 - V_\\beta \\geq 0.\n\\end{equation}\nNote that $U\\in C^{1,\\alpha}(\\overline\\omega)$, and $U=0$ on $\\partial\\omega$ (since, by definition of $\\omega$, $\\widetilde V_0=0$ on $\\partial\\omega$). \n\nLet $\\omega':=\\omega\\cap SC_\\beta$. It holds\n\\begin{equation}\\label{DeltaU}\n\\Delta U = H\\ensuremath{\\mathonebb{1}}_{\\omega\\setminus\\omega'}\\quad\\text{ in }\\omega.\n\\end{equation}\nFrom \\eqref{ineqtildeV0} and \\eqref{VleqtildeV} it follows that\n\\begin{equation*}\nV_\\beta <\\beta\/2 \\quad\\text{in }\\overline \\omega.\n\\end{equation*}  \nTherefore, recalling the free boundary formulation \\eqref{freebound}, we have $H\\geq 0$ in $\\omega\\setminus\\omega'$. In particular \\eqref{DeltaU} implies that\n\\begin{equation*}\n\\Delta U\\geq 0 \\quad \\text{ in }\\omega.\n\\end{equation*}\nLet $\\varepsilon>0$ and consider\n\\begin{equation*}\n\\varphi:=\\max(U-\\varepsilon,0)\\in H^1(\\omega).\n\\end{equation*}\nRecalling that $U\\in C(\\overline \\omega)$ and $U=0$ on $\\partial\\omega$, we know that  $\\varphi$ has compact support inside $\\omega$. Thus we may integrate by part (without knowing anything about the regularity of $\\partial\\omega$) to obtain\n\\begin{equation*}\n\\int_\\omega |\\nabla \\varphi|^2 =\\int_\\omega \\nabla\\varphi\\cdot\\nabla U = -\\int_\\omega \\varphi \\Delta U \\leq 0,\n\\end{equation*}\nand we deduce that $\\varphi\\equiv 0$ in $\\omega$, which implies that $U\\leq\\varepsilon$ in $\\omega$. Letting $\\varepsilon\\to 0$, we conclude that $U\\leq 0$ in $\\omega$, which, together with \\eqref{Unonneg}, shows that \\eqref{freezV} holds.\n\\end{proof}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}