Boundary Behavior of the Ginzburg-Landau Order Parameter in the Surface Superconductivity Regime
BBoundary Behavior of the Ginzburg-Landau OrderParameter in the Surface Superconductivity Regime
M. Correggi a , N. Rougerie b a Dipartimento di Matematica e Fisica, Universit`a degli Studi Roma Tre,L.go San Leonardo Murialdo, 1, 00146, Rome, Italy. b Universit´e de Grenoble 1 & CNRS, LPMMCMaison des Magist`eres CNRS, BP166, 38042 Grenoble Cedex, France.
January 11th, 2015
Abstract
We study the 2D Ginzburg-Landau theory for a type-II superconductor in an appliedmagnetic field varying between the second and third critical value. In this regime the orderparameter minimizing the GL energy is concentrated along the boundary of the sample andis well approximated to leading order (in L norm) by a simplified 1D profile in the directionperpendicular to the boundary. Motivated by a conjecture of Xing-Bin Pan, we address thequestion of whether this approximation can hold uniformly in the boundary region. We provethat this is indeed the case as a corollary of a refined, second order energy expansion includingcontributions due to the curvature of the sample. Local variations of the GL order parameterare controlled by the second order term of this energy expansion, which allows us to prove thedesired uniformity of the surface superconductivity layer. Contents a r X i v : . [ m a t h - ph ] J a n orreggi, Rougerie – Surface Superconductivity A Useful Estimates on 1D Functionals 40
A.1 Properties of optimal phases and densities . . . . . . . . . . . . . . . . . . . . . . . 41A.2 Positivity of the cost function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
The Ginzburg-Landau (GL) theory of superconductivity, originating in [GL], provides a phe-nomenological, macroscopic, description of the response of a superconductor to an applied magneticfield. Several years after it was introduced, it turned out that it could be derived from the micro-scopic BCS theory [BCS, Gor] and should thus be seen as a mean-field/semiclassical approximationof many-body quantum mechanics. A mathematically rigorous derivation starting from BCS theoryhas been provided recently [FHSS].Within GL theory, the state of a superconductor is described by an order parameter Ψ : R → C and an induced magnetic vector potential κσ A : R → R generating an induced magnetic field h = κσ curl A . The ground state of the theory is found by minimizing the energy functional G GL κ,σ [Ψ , A ] = (cid:90) Ω d r (cid:110) | ( ∇ + iκσ A ) Ψ | − κ | Ψ | + κ | Ψ | + ( κσ ) | curl A − | (cid:111) , (1.1)where κ > κσ measures the intensity of the external magnetic field, that we assume to be constant throughoutthe sample. We consider a model for an infinitely long cylinder of cross-section Ω ⊂ R , a compactsimply connected set with regular boundary.Note the invariance of the functional under the gauge transformationΨ → Ψ e − iκσϕ , A → A + ∇ ϕ, (1.2)which implies that the only physically relevant quantities are the gauge invariant ones such as theinduced magnetic field h and the density | Ψ | . The latter gives the local relative density of electronsbound in Cooper pairs. It is well-known that a minimizing Ψ must satisfy | Ψ | ≤
1. A value | Ψ | = 1(respectively, | Ψ | = 0) corresponds to the superconducting (respectively, normal) phase where all(respectively, none) of the electrons form Cooper pairs. The perfectly superconducting state with | Ψ | = 1 everywhere is an approximate ground state of the functional for small applied field and thenormal state where Ψ vanishes identically is the ground state for large magnetic field. In betweenthese two extremes, different mixed phases can occur, with normal and superconducting regionsvarying in proportion and organization.A vast mathematical literature has been devoted to the study of these mixed phases in type-IIsuperconductors (characterized by κ > / √ κ → ∞ (extreme type-II).Reviews and extensive lists of references may be found in [FH3, SS2, Sig]. Two main phenomenaattracted much attention: • The formation of hexagonal vortex lattices when the applied magnetic field varies betweenthe first and second critical field, first predicted by Abrikosov [Abr], and later experimentallyobserved (see, e.g., [H et al ]). In this phase, vortices (zeros of the order parameter withquantized phase circulation) sit in small normal regions included in the superconductingphase and form regular patterns. Here we use the units of [FH3], other choices are possible, see, e.g., [SS2]. orreggi, Rougerie – Surface Superconductivity • The occurrence of a surface superconductivity regime when the applied magnetic fields variesbetween the second and third critical fields. In this case, superconductivity is completelydestroyed in the bulk of the sample and survives only at the boundary, as predicted in [SJdG].We refer to [N et al ] for experimental observations.We refer to [CR] for a more thorough discussion of the context. We shall be concerned with thesurface superconductivity regime, which in the above units translates into the assumption σ = bκ (1.3)for some fixed parameter b satisfying the conditions1 < b < Θ − (1.4)where Θ is a spectral parameter (minimal ground state energy of the shifted harmonic oscillatoron the half-line, see [FH3, Chapter 3]):Θ := inf α ∈ R inf (cid:26)(cid:90) R + d t (cid:0) | ∂ t u | + ( t + α ) | u | (cid:1) , (cid:107) u (cid:107) L ( R + ) = 1 (cid:27) . (1.5)From now on we introduce more convenient units to deal with the surface superconductivity phe-nomenon: we define the small parameter ε = 1 √ σκ = 1 b / κ (cid:28) ε → G GL ε [Ψ , A ] = (cid:90) Ω d r (cid:26)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) ∇ + i A ε (cid:19) Ψ (cid:12)(cid:12)(cid:12)(cid:12) − bε (cid:0) | Ψ | − | Ψ | (cid:1) + bε | curl A − | (cid:27) . (1.7)We shall denote E GL ε := min (Ψ , A ) ∈ D GL G GL ε [Ψ , A ] , (1.8)with D GL := (cid:8) (Ψ , A ) ∈ H (Ω; C ) × H (Ω; R ) (cid:9) , (1.9)and denote by (Ψ GL , A GL ) a minimizing pair (known to exist by standard methods [FH3, SS2]).The salient features of the surface superconductivity phase are as follows: • The GL order parameter is concentrated in a thin boundary layer of thickness ∼ ε = ( κσ ) − / .It decays exponentially to zero as a function of the distance from the boundary. • The applied magnetic field is very close to the induced magnetic field, curl A ≈ • Up to an appropriate choice of gauge and a mapping to boundary coordinates, the groundstate of the theory is essentially governed by the minimization of a 1D energy functional inthe direction perpendicular to the boundary.A review of rigorous statements corresponding to these physical facts may be found in [FH3]. Oneof their consequences is the energy asymptotics E GL ε = | ∂ Ω | E ε + O (1) , (1.10) orreggi, Rougerie – Surface Superconductivity | ∂ Ω | is the length of the boundary of Ω, and E is obtained by minimizing the functional E ,α [ f ] := (cid:90) + ∞ d t (cid:26) | ∂ t f | + ( t + α ) f − b (cid:0) f − f (cid:1)(cid:27) , (1.11)both with respect to the function f and the real number α . We proved recently [CR] that (1.10)holds in the full surface superconductivity regime, i.e. for 1 < b < Θ − . This followed a series ofpartial results due to several authors [Alm1, AH, FH1, FH2, FHP, LP, Pan], summarized in [FH3,Theorem 14.1.1]. Some of these also concern the limiting regime b (cid:37) Θ − . The other limitingcase b (cid:38) b (cid:37) GL for (1.1) has the structureΨ GL ( r ) ≈ f (cid:0) τε (cid:1) exp (cid:0) − iα sε (cid:1) exp { iφ ε ( s, t ) } (1.12)where ( f , α ) is a minimizing pair for (1.11), ( s, τ ) = (tangent coordinate, normal coordinate) areboundary coordinates defined in a tubular neighborhood of ∂ Ω with τ = dist( r , ∂ Ω) for any point r there and φ ε is a gauge phase factor (see (5.4)), which plays a role in the change to boundarycoordinates. Results in the direction of (1.12) may be found in the following references: • [Pan] contains a result of uniform distribution of the energy density at the domain’s boundaryfor any 1 ≤ b < Θ − ; • [FH1] gives fine energy estimates compatible with (1.12) when b (cid:37) Θ − ; • [AH] and then [FHP] prove that (1.12) holds at the level of the density, in the L sense, for1 . ≤ b < Θ − ; • [FK] and then [Kac] investigate the concentration of the energy density when b is close to 1; • [FKP] contains results about the energy concentration phenomenon in the 3D case.In [CR, Theorem 2.1] we proved that (cid:13)(cid:13) | Ψ GL | − f (cid:0) τε (cid:1)(cid:13)(cid:13) L (Ω) ≤ Cε (cid:28) (cid:13)(cid:13) f (cid:0) τε (cid:1)(cid:13)(cid:13) L (Ω) (1.13)for any 1 < b < Θ − in the limit ε →
0. A very natural question is whether the above estimate maybe improved to a uniform control (in L ∞ norm) of the local discrepancy between the modulus of thetrue GL minimizer and the simplified normal profile f (cid:0) τε (cid:1) . Indeed, (1.13) is still compatible withthe vanishing of Ψ GL in small regions, e.g., vortices, inside of the boundary layer. Proving thatsuch local deviations from the normal profile do not occur would explain the observed uniformityof the surface superconducting layer (see again [N et al ] for experimental pictures). Interest inthis problem (stated as Open Problem number 4 in the list in [FH3, Page 267]) originates from aconjecture of X.B. Pan [Pan, Conjecture 1] and an affirmative solution has been provided in [CR]for the particular case of a disc sample. The purpose of this paper is to extend the result to generalsamples with regular boundary (the case with corners is known to require a different analysis [FH3,Chapter 15]).Local variations (on a scale O ( ε )) in the tangential variable are compatible with the energyestimate (1.10), and thus the uniform estimate obtained for disc samples in [CR] is based on anexpansion of the energy to the next order: E GL ε = 2 πE (cid:63) ( k ) ε + O ( ε | log ε | ) , (1.14) orreggi, Rougerie – Surface Superconductivity E (cid:63) ( k ) is the minimum (with respect to both the real number α and the function f ) of the ε -dependent functional E k,α [ f ] := (cid:90) c | log ε | d t (1 − εkt ) (cid:26) | ∂ t f | + ( t + α − εkt ) (1 − εkt ) f − b (cid:0) f − f (cid:1)(cid:27) , (1.15)where the constant c has to be chosen large enough and k = R − is the curvature of the discunder consideration, whose radius we denote by R . Of course, (1.11) is simply the above functionalwhere one sets k = 0, ε = 0, which amounts to neglect the curvature of the boundary. When thecurvature is constant, (1.14) in fact follows from a next order expansion of the GL order parameterbeyond (1.12): Ψ GL ( r ) ≈ f k (cid:0) τε (cid:1) exp (cid:0) − iα ( k ) sε (cid:1) exp { iφ ε ( s, t ) } (1.16)where ( α ( k ) , f k ) is a minimizing pair for (1.15). Note that for any fixed kf k = f (1 + O ( ε )) , α ( k ) = α (1 + O ( ε )) , (1.17)so that (1.16) is a slight refinement of (1.12) but the O ( ε ) correction corresponds to a contributionof order 1 beyond (1.10) in (1.14), which turns out to be the order that controls local densityvariations.As suggested by the previous results in the disc case, the corrections to the energy asymp-totics (1.10) must be curvature-dependent. The case of a general sample where the curvature ofthe boundary is not constant is then obviously harder to treat than the case of a disc, where oneobtains (1.14) by a simple variant of the proof of (1.10), as explained in our previous paper [CR].In fact, we shall obtain below the desired uniformity result for the order parameter in generaldomains as a corollary of the energy expansion ( γ is a fixed constant) E GL ε = 1 ε (cid:90) | ∂ Ω | d s E (cid:63) ( k ( s )) + O ( ε | log ε | γ ) (1.18)where the integral runs over the boundary of the sample, k ( s ) being the curvature of the boundaryas a function of the tangential coordinate s . Just as the particular case (1.14), (1.18) contains theleading order (1.10), but O (1) corrections are also evaluated precisely. As suggested by the energyformula, the GL order parameter has in fact small but fast variations in the tangential variablewhich contribute to the subleading order of the energy. More precisely, one should think of theorder parameter as having the approximate formΨ GL ( r ) = Ψ GL ( s, τ ) ≈ f k ( s ) (cid:0) τε (cid:1) exp (cid:0) − iα ( k ( s )) sε (cid:1) exp { iφ ε ( s, t ) } (1.19)with f k ( s ) , α ( k ( s )) a minimizing pair for the energy functional (1.15) at curvature k = k ( s ). Themain difficulty we encounter in the present paper is to precisely capture the subtle curvature depen-dent variations encoded in (1.19). What our new result (we give a rigorous statement below) (1.19)shows is that curvature-dependent deviations to (1.12) do exist but are of limited amplitude andcan be completely understood via the minimization of the family of 1D functionals (1.15). A cru-cial input of our analysis is therefore a detailed inspection of the k -dependence of the ground stateof (1.15).We can deduce from (1.18) a uniform density estimate settling the general case of [Pan, Con-jecture 1] and [FH3, Open Problem 4, page 267]. We believe that the energy estimate (1.18) is ofindependent interest since it helps in clarifying the role of domain curvature in surface supercon-ductivity physics. It was previously known (see [FH3, Chapters 8 and 13] and references therein)that corrections to the value of the third critical field depend on the domain’s curvature, but appli-cations of these results are limited to the regime where b → Θ − when ε →
0. The present paper orreggi, Rougerie – Surface Superconductivity < b < Θ − .This role may seem rather limited since it only concerns the second order in the energy asymptoticsbut it is in fact crucial in controlling local variations of the order parameter and allowing to provea strong form of uniformity for the surface superconductivity layer.Our main results are rigorously stated and further discussed in Section 2, their proofs occupythe rest of the paper. Some material from [CR] is recalled in Appendix A for convenience. Notation.
In the whole paper, C denotes a generic fixed positive constant independent of ε whose value changes from formula to formula. A O ( δ ) is always meant to be a quantity whoseabsolute value is bounded by δ = δ ( ε ) in the limit ε →
0. We use O ( ε ∞ ) to denote a quantity (likeexp( − ε − )) going to 0 faster than any power of ε and | log ε | ∞ to denote | log ε | a where a > ε , e.g., ε | log ε | ∞ which is a O ( ε − c ) for any 0 < c <
1, and hence we usually do notspecify the precise power a . Acknowledgments.
M.C. acknowledges the support of MIUR through the FIR grant 2013 “Con-densed Matter in Mathematical Physics (Cond-Math)” (code RBFR13WAET). N.R. acknowledgesthe support of the ANR project Mathostaq (ANR-13-JS01-0005-01). We also acknowledge the hos-pitality of the
Institut Henri Poincar´e , Paris. We are indebted to one of the anonymous refereesfor the content of Remarks 2.2 and 2.3.
We first state the refined energy and density estimates that reveal the contributions of the domain’sboundary. As suggested by (1.19), we now introduce a reference profile that includes these varia-tions. A piecewise constant function in the tangential direction is sufficient for our purpose and wethus first introduce a decomposition of the superconducting boundary layer that will be used inall the paper. The thickness of this layer in the normal direction should roughly be of order ε , butto fully capture the phenomenon at hand we need to consider a layer of size c ε | log ε | where c isa fixed, large enough constant. By a passage to boundary coordinates and dilation of the normalvariable on scale ε (see [FH3, Appendix F] or Section 4 below), the surface superconducting layer˜ A ε := { r ∈ Ω | τ ≤ c ε | log ε |} , (2.1)where τ := dist( r , ∂ Ω) , (2.2)can be mapped to A ε := { ( s, t ) ∈ [0 , | ∂ Ω | ] × [0 , c | log ε | ] } . (2.3)We split this domain into N ε = O ( ε − ) rectangular cells {C n } n =1 ,...,N ε of constant side length (cid:96) ε ∝ ε in the s direction. We denote s n , s n +1 = s n + (cid:96) ε the s coordinates of the boundaries of thecell C n : C n = [ s n , s n +1 ] × [0 , c | log ε | ]and we may clearly choose (cid:96) ε = ε | ∂ Ω | (1 + O ( ε ))for definiteness. We will approximate the curvature k ( s ) by its mean value k n in each cell: k n := (cid:96) − ε (cid:90) s n +1 s n d s k ( s ) . orreggi, Rougerie – Surface Superconductivity f n := f k n , α n := α ( k n )respectively the optimal profile and phase associated to k n , obtained by minimizing (1.15) firstwith respect to f and then to α .The reference profile is then the piecewise continuous function g ref ( s, t ) := f n ( t ) , for s ∈ [ s n , s n +1 ] and ( s, t ) ∈ A ε , (2.4)that can be extended to the whole domain Ω by setting it equal to 0 for dist( r , ∂ Ω) ≥ c ε | log ε | .We compare the density of the full GL order parameter to g ref in the next theorem. Note thatbecause of the gauge invariance of the energy functional, the phase of the order parameter is notan observable quantity, so the next statement is only about the density | Ψ GL | . Theorem 2.1 ( Refined energy and density asymptotics ) . Let Ω ⊂ R be any smooth, bounded and simply connected domain. For any fixed < b < Θ − , inthe limit ε → , it holds E GL ε = 1 ε (cid:90) | ∂ Ω | d s E (cid:63) ( k ( s )) + O ( ε | log ε | ∞ ) . (2.5) and (cid:13)(cid:13) | Ψ GL | − g (cid:0) s, ε − τ (cid:1)(cid:13)(cid:13) L (Ω) = O ( ε / | log ε | ∞ ) (cid:28) (cid:13)(cid:13) g (cid:0) s, ε − t (cid:1)(cid:13)(cid:13) L (Ω) . (2.6) Remark 2.1 [The energy to subleading order]The most precise result prior to the above is [CR, Theorem 2.1] where the leading order is computedand the remainder is shown to be at most of order 1. Such a result had been obtained beforein [FHP] for a smaller range of parameters, namely for 1 . ≤ b < Θ − , see also [FH3, Chapter 14]and references therein. The above theorem evaluates precisely the O (1) term, which is betterappreciated in light of the following comments:1. In the effective 1D functional (1.15), the parameter k that corresponds to the local curvatureof the sample appears with an ε prefactor. As a consequence, one may show (see Section 3.1below) that for all s ∈ [0 , | ∂ Ω | ] E (cid:63) ( k ( s )) = E (cid:63) (0) + O ( ε ) (2.7)so that (2.5) contains the previously known results. More generally we prove below that (cid:12)(cid:12) E (cid:63) ( k ( s )) − E (cid:63) ( k ( s (cid:48) )) (cid:12)(cid:12) ≤ Cε | s − s (cid:48) | so that E (cid:63) ( k ( s )) has variations of order ε on the scale of the boundary layer. Thesecontribute to a term of order 1 that is included in (2.5). Actually one could investigate theasymptotics (2.7) further, aiming at evaluating explicitly the error O ( ε ) and therefore thecurvature contribution to the energy. This would in particular be crucial in the analysisdescribed in Remark 2.3 below, but we do not pursue it here for the sake of brevity.2. Undoing the mapping to boundary coordinates, one should note that g ref ( s, ε − t ) has fastvariations (at scale ε ) in both the t direction and s directions. The latter are of limitedamplitude however, which explains that they enter the energy only at subleading order, andwhy a piecewise constant profile is sufficient to capture the physics.3. We had previously proved the density estimate (1.13), which is less precise than (2.6). Notein particular that (2.6) does not hold at this level of precision if one replaces g (cid:0) s, ε − t (cid:1) bythe simpler profile f ( ε − t ). We are free to impose f n ≥
0, which we always do in the sequel. orreggi, Rougerie – Surface Superconductivity
84. Strictly speaking the function g ref is defined only in the boundary layer ˜ A ε , so that (2.6)should be interpreted as if g ref would vanish outside ˜ A ε . However the estimate there isobviously true thanks to the exponential decay of Ψ GL . Remark 2.2 [Regime b → b →
1, where surface superconductivity is also present (see [Alm1, Pan, FK]). The main reason forassuming b > b → b > b →
1, its proof requiressome non-trivial modifications. Moreover while for b ≥ b (cid:37)
1, a bulk term appears inthe energy asymptotics [FK] and the problem becomes much more subtle.We now turn to the uniform density estimates that follow from the above theorem. Here we canbe less precise than before. Indeed, as suggested by the previous discussion, a density deviation oforder ε on a length scale of order ε only produces a O ( ε ) error in the energy. Thus, using (2.5)we may only rule out local variations of a smaller order than the tangential variations includedin (2.4), and for this reason we will compare | Ψ GL | in L ∞ norm only to the simplified profile f ( ε − τ ), since by (1.17) f ( t ) − f k ( t ) = O ( ε ). Also, the result may be proved only in a regionwhere the density is relatively large , namely in A bl := (cid:8) r ∈ Ω : f (cid:0) ε − τ (cid:1) ≥ γ ε (cid:9) ⊂ (cid:110) dist( r , ∂ Ω) ≤ ε (cid:112) | log γ ε | (cid:111) , (2.8)where bl stands for “boundary layer” and 0 < γ ε (cid:28) γ ε (cid:29) ε / | log ε | a , (2.9)where a > to the power of | log ε | appearing in (2.5). Theinclusion in (2.8) follows from (A.6) below and ensures we are really considering a significantboundary layer: recall that the physically relevant region has a thickness roughly of order ε | log ε | . Theorem 2.2 ( Uniform density estimates and Pan’s conjecture ) . Under the assumptions of the previous theorem, it holds (cid:13)(cid:13)(cid:12)(cid:12) Ψ GL ( r ) (cid:12)(cid:12) − f (cid:0) ε − τ (cid:1)(cid:13)(cid:13) L ∞ ( A bl ) ≤ Cγ − / ε ε / | log ε | ∞ (cid:28) . (2.10) In particular for any r ∈ ∂ Ω we have (cid:12)(cid:12)(cid:12)(cid:12) Ψ GL ( r ) (cid:12)(cid:12) − f (0) (cid:12)(cid:12) ≤ Cε / || log ε | ∞ | (cid:28) , (2.11) where C does not depend on r . Estimate (2.11) solves the original form of Pan’s conjecture [Pan, Conjecture 1]. In addition,since f is strictly positive, the stronger estimate (2.10) ensures that Ψ GL does not vanish in theboundary layer (2.8). A physical consequence of the theorem is thus that normal inclusions suchas vortices in the surface superconductivity phase may not occur. This is very natural in view ofthe existing knowledge on type-II superconductors but had not been proved previously. Recall that it decays exponentially far from the boundary. Assuming that (2.5) holds true with an error of order ε | log ε | γ , for some given γ >
0, the constant a can be anynumber satisfying a > ( γ + 3). orreggi, Rougerie – Surface Superconductivity GL around the boundary ∂ Ω defined as2 π deg (Ψ , ∂ Ω) := − i (cid:90) ∂ Ω d s | Ψ | Ψ ∂ s (cid:18) Ψ | Ψ | (cid:19) , (2.12) ∂ s standing for the tangential derivative. Theorem 2.2 ensures that deg (Ψ , ∂ B R ) ∈ Z is well-defined. Roughly, this quantity measures the number of quantized phase singularities (vortices)that Ψ GL has inside Ω. Our estimate is as follows: Theorem 2.3 ( Winding number of Ψ GL on the boundary). Under the previous assumptions, any GL minimizer Ψ GL satisfies deg (cid:0) Ψ GL , ∂ Ω (cid:1) = | Ω | ε + | α | ε + O ( ε − / | log ε | ∞ ) (2.13) in the limit ε → . Note that the remainder term in (2.13) is much larger than ε − | α ( k ) − α | = O (1) so that theabove result does not allow to estimate corrections due to curvature. We believe that, just as wehad to expand the energy to second order to obtain the refined first order results Theorems 2.2and 2.3, obtaining uniform density estimates and degree estimates at the second order wouldrequire to expand the energy to the third order, which goes beyond the scope of the present paper.We had proved Theorems 2.2 and 2.3 before in the particular, significantly easier, case where Ω isa disc. The next subsection contains a sketch of the proof of the general case, where new ingredientsenter, due to the necessity to take into account the non-trivial curvature of the boundary. Beforeproceeding, we make a last remark in this direction: Remark 2.3 [Curvature dependence of the order parameter]In view of previous results [FH1] in the regime b (cid:37) Θ − , a larger curvature should imply alarger local value of the order parameter. In the regime of interest to this paper, this will onlybe a subleading order effect, but it would be interesting to capture it by a rigorous asymptoticestimate.It has been proved before [Pan, FK] that in the surface superconductivity regime (1.4)1 b / ε | Ψ GL | d r −→ ε → C ( b )d s ( r ) (2.14)as measures, with d r the Lebesgue measure and d s ( r ) the 1D Hausdorff measure along the bound-ary. Here C ( b ) > b . A natural conjecture is that one canderive a result revealing the next-order behavior, of the form1 ε (cid:18) b / ε | Ψ GL | d r − C ( b )d s ( r ) (cid:19) −→ ε → C ( b ) k ( s )d s ( r ) (2.15)with C ( b ) > b . The form of the right-hand side is motivated by two consid-erations: • In view of [FH1] we should expect that increasing k increases the local value of | Ψ GL | , whencethe sign of the correction; • Since the curvature appears only at subleading order in this regime, perturbation theorysuggests that the correction should be linear in the curvature.We plan to substantiate this picture further in a later work. orreggi, Rougerie – Surface Superconductivity In the regime of interest to this paper, the GL order parameter is concentrated along the boundaryof the sample and the induced magnetic field is extremely close to the applied one. The toolsallowing to prove these facts are well-known and described at length in the monograph [FH3]. Weshall thus not elaborate on this and the formal considerations presented in this subsection take asstarting point the following effective functional G A ε [ ψ ] := (cid:90) | ∂ Ω | d s (cid:90) c | log ε | d t (1 − εk ( s ) t ) (cid:26) | ∂ t ψ | + 1(1 − εk ( s ) t ) | ( ε∂ s + ia ε ( s, t )) ψ | − b (cid:2) | ψ | − | ψ | (cid:3)(cid:27) , (2.16)where ( s, t ) represent boundary coordinates in the original domain Ω, the normal coordinate t having been dilated on scale ε , and ψ can be thought of as Ψ GL ( r ( s, εt )), i.e., the order parameterrestricted to the boundary layer. We denote k ( s ) the curvature of the original domain and haveset a ε ( s, t ) := − t + εk ( s ) t + εδ ε , (2.17)with δ ε := γ ε − (cid:106) γ ε (cid:107) , γ := 1 | ∂ Ω | (cid:90) Ω d r curl A GL , (2.18) (cid:98) · (cid:99) standing for the integer part. Note that a specific choice of gauge has been made to ob-tain (2.16).Thanks to the methods exposed in [FH3], one can show that the minimization of the abovefunctional gives the full GL energy in units of ε − , up to extremely small remainder terms, provided c is chosen lare enough. To keep track of the fact that the domain A ε = [0 , | ∂ Ω | ] × [0 , c | log ε | ]corresponds to the unfolded boundary layer of the original domain and ψ to the GL order parameterin boundary coordinates, one should impose periodicity of ψ in the s direction.Here we shall informally explain the main steps of the proof that G A ε = (cid:90) | ∂ Ω | d s E (cid:63) ( k ( s )) + O ( ε | log ε | ∞ ) . (2.19)where G A ε is the ground state energy associated to (2.16). When k ( s ) ≡ k is constant (the disccase), one may use the ansatz ψ ( s, t ) = f ( t ) e − i ( ε − αs − εδ ε s ) . (2.20)and recover the functional (1.15). It is then shown in [CR] that the above ansatz is essentiallyoptimal if one chooses α = α ( k ) and f = f k . An informal sketch of the proof in the case k = 0is given in Section 3.2 therein. The main insight in the general case is to realize that the aboveansatz stays valid locally in s . Indeed, since the terms involving k ( s ) in (2.16) come multiplied byan ε factor, it is natural to expect variations in s to be weak and the state of the system to beroughly of the form (1.19), directly inspired by (2.20).As usual the upper and lower bound inequalities in (2.19) are proved separately. Upper bound.
To recover the integral in the energy estimate (2.19), we use a Riemann sumover the cell decomposition A ε = (cid:83) N ε n =1 C n introduced at the beginning of Section 2.1. Indeed, asalready suggested in (2.4), a piecewise constant approximation in the s -direction will be sufficient.Our trial state roughly has the form ψ ( s, t ) = f n ( t ) e − i ( ε − α n s − εδ ε s ) , for s n ≤ s ≤ s n +1 . (2.21) orreggi, Rougerie – Surface Superconductivity ε per cell. Evaluatingthe energy of the trial state in this way we obtain an upper bound of the form G A ε ≤ N ε (cid:88) n =1 | s n +1 − s n | E (cid:63) ( k n )(1 + o (1)) + O ( ε ) (2.22)where the o (1) error is due to the necessary modifications to (2.21) to make it continuous. Thecrucial point is to be able to control this error by showing that the modification needs not be alarge one. This requires a detailed analysis of the k dependence of the relevant quantities E (cid:63) ( k ), α ( k ) and f k obtained by minimizing (1.15). Indeed, we prove in Section 3.1 below that (cid:12)(cid:12) E (cid:63) ( k ) − E (cid:63) ( k (cid:48) ) (cid:12)(cid:12) ≤ Cε | log ε | ∞ | k − k (cid:48) | , | α ( k ) − α ( k (cid:48) ) | ≤ Cε / | log ε | ∞ | k − k (cid:48) | / and, in a suitable norm, f k (cid:48) = f k + O (cid:16) ε / | log ε | ∞ | k − k (cid:48) | / (cid:17) , which will allow to obtain the desired control of the o (1) in (2.22) and conclude the proof by aRiemann sum argument. Lower bound.
In view of the argument we use for the upper bound, the natural idea to obtainthe corresponding lower bound is to use the strategy for the disc case we developed in [CR] locallyin each cell. In the disc case, a classical method of energy decoupling and Stokes’ formula lead tothe lower bound G A ε [ ψ ] (cid:39) E (cid:63) ( k ) + (cid:90) A ε d s d t (1 − εkt ) K k ( t ) (cid:16) | ∂ t v | + ε (1 − εkt ) | ∂ s v | (cid:17) (2.23)where we have used the strict positivity of f k to write ψ ( s, t ) = f k ( t ) e − i ( ε − α ( k ) s − εδ ε s ) v ( s, t ) (2.24)and the “cost function” is K k ( t ) = f k ( t ) + F k ( t ) ,F k ( t ) = 2 (cid:90) t dη η + α ( k ) − εkη − εkη f k ( η ) . This method is inspired from our previous works on the related Gross-Pitaevskii theory of rotatingBose-Einstein condensates [CRY, CPRY1, CPRY2] (informal summaries may be found in [CPRY3,CPRY4]). Some of the steps leading to (2.23) have also been used before in this context [AH].The desired lower bound in the disc case follows from (2.23) and the fact that K k is essentially positive for any k . This is proved by carefully exploiting special properties of f k and α ( k ).To deal with the general case where the curvature is not constant, we again split the domain A ε into small cells, approximate the curvature by a constant in each cell and use the above strategylocally. A serious new difficulty however comes from the use of Stokes’ formula in the derivationof (2.23). We need to reduce the terms produced by Stokes’ formula to expressions involvingonly first order derivatives of the order parameter, using further integration by parts. In the disccase, boundary terms associated with this operation vanish due to the periodicity of ψ in the s We simplify the argument for pedagogical purposes. More precisely it is positive except possibly for large t , a region that can be handled using the exponential decayof GL minimizers (Agmon estimates). orreggi, Rougerie – Surface Superconductivity f k and α ( k ) in (2.24),the boundary terms do not vanish since we artificially introduce some (small) discontinuity bychoosing a cell-dependent profile f k n as reference.To estimate these boundary terms we proceed as follows: the term at s = s n +1 , made of onepart coming from the cell C n and one from the cell C n +1 is integrated by parts back to become abulk term in the cell C n . In this sketch we ignore a rather large amount of technical complicationsand state what is essentially the conclusion of this procedure: G A ε [ ψ ] (cid:39) N ε (cid:88) n =1 (cid:20) | s n +1 − s n | E (cid:63) ( k n ) + (cid:90) C n d s d t (1 − εk n t ) ˜ K n (cid:18) | ∂ t u n | + ε (1 − εk n t ) | ∂ s u n | (cid:19)(cid:21) (2.25)where u n ( s, t ) = f − k n ( t ) e i ( ε − α ( k ) s + εδ ε s ) ψ ( s, t ) (2.26)and the “modified cost function” is˜ K n ( s, t ) = K k n ( t ) − | ∂ s χ n ( s ) || I n,n +1 ( t ) | − | χ n ( s ) | | ∂ t I n,n +1 ( t ) | ,I n,n +1 ( t ) = F k n ( t ) − F k n +1 ( t ) f k n ( t ) f k n +1 ( t ) , and χ n is a suitable localization function supported in C n with χ n ( s n +1 ) = 1 that we use to performthe integration by parts in C n . Note that the dependence of the new cost function on both k n and k n +1 is due to the fact that the original boundary terms at s n +1 that we transform into bulk termsin C n involved both u n and u n +1 .The last step is to prove a bound of the form | I n,n +1 ( t ) | + | ∂ t I n,n +1 ( t ) | ≤ Cε | log ε | ∞ f k n ( t ) (2.27)on the “correction function” I n,n +1 , so that˜ K n ( t ) ≥ (1 − Cε | log ε | ∞ ) f k n ( t ) + F k n ( t ) . This allows us to conclude that (essentially) ˜ K n ≥ K k n in [CR] and thus concludes the lower bound proof modulo the same Riemann sum argument asin the upper bound part. Note the important fact that the quantity in the l.h.s. of (2.27) is provedto be small relatively to f k n ( t ), including in a region where the latter function is exponentiallydecaying. This bound requires a thorough analysis of auxiliary functions linked to (1.15) and isin fact a rather strong manifestation of the continuity of this minimization problem as a functionof k .The rest of the paper is organized as follows: Section 3 contains the detailed analysis of theeffective, curvature-dependent, 1D problem. The necessary continuity properties as function ofthe curvature are given in Subsection 3.1 and the analysis of the associated auxiliary functions inSubsection 3.2. The details of the energy upper bound are then presented in Section 4 and theenergy lower bound is proved in Section 5. We deduce our other main results in Section 6. Ap-pendix A recalls for the convenience of the reader some material from [CR] that we use throughoutthe paper. This section is devoted to the analysis of the 1D curvature-dependent reduced functionals whoseminimization allows us to reconstruct the leading and sub-leading order of the full GL energy. Weshall prove results in two directions: orreggi, Rougerie – Surface Superconductivity • We carefully analyse the dependence of the 1D variational problems as a function of curvaturein Subsection 3.1. Our analysis, in particular the estimate of the subleading order of the GLenergy, requires some quantitative control on the variations of the optimal 1D energy, phaseand density when the curvature parameter is varied, that is when we move along the boundarylayer of the original sample along the transverse direction. • In our previous paper [CR] we have proved the positivity property of the cost function whichis the main ingredient in the proof of the energy lower bound in the case of a disc (constantcurvature). As mentioned above, the study of general domains with smooth curvature thatwe perform here will require to estimate more auxiliary functions, which is the subject ofSubsection 3.2.We shall use as input some key properties of the 1D problem at fixed k that we proved in [CR].These are recalled in Appendix A below for the convenience of the reader. We take for granted the three crucial but standard steps of reduction to the boundary layer,replacement of the vector potential and mapping to boundary coordinates. Our considerationsthus start from the following reduced GL functional giving the original energy in units of ε − , upto negligible remainders: G A ε [ ψ ] := (cid:90) | ∂ Ω | d s (cid:90) c | log ε | d t (1 − εk ( s ) t ) (cid:26) | ∂ t ψ | + 1(1 − εk ( s ) t ) | ( ε∂ s + ia ε ( s, t )) ψ | − b (cid:2) | ψ | − | ψ | (cid:3)(cid:27) , (3.1)where k ( s ) is the curvature of the original domain. We have set a ε ( s, t ) := − t + εk ( s ) t + εδ ε , (3.2)and δ ε := γ ε − (cid:106) γ ε (cid:107) , γ := 1 | ∂ Ω | (cid:90) Ω d r curl A GL , (3.3) (cid:98) · (cid:99) standing for the integer part. The boundary layer in rescaled coordinates is denoted by A ε := { r ∈ Ω | dist( r , ∂ Ω) ≤ c ε | log ε |} . (3.4)The effective functionals that we shall be concerned with in this section are obtained by com-puting the energy (3.1) of certain special states. In particular we have to go beyond the simpleans¨atze considered so far in the literature, e.g., in [FH3, CR], and obtain the following effectiveenergies: •
2D functional with definite phase . Inserting the ansatz ψ ( s, t ) = g ( s, t ) e − i ( ε − S ( s ) − εδ ε s ) (3.5)in (3.1), with g and S respectively real valued density and phase, we obtain E S [ g ] := (cid:90) c | log ε | d t (cid:90) | ∂ Ω | d s (1 − εk ( s ) t ) (cid:26) | ∂ t g | + ε (1 − εk ( s ) t ) | ∂ s g | + (cid:0) t + ∂ s S − εt k ( s ) (cid:1) (1 − εtk ( s )) g − b (cid:0) g − g (cid:1) (cid:27) . (3.6) orreggi, Rougerie – Surface Superconductivity ∂ s S = α ∈ π Z we may obtain a simpler functional of thedensity alone E α [ g ] := (cid:90) c | log ε | d t (cid:90) | ∂ Ω | d s (1 − εk ( s ) t ) (cid:26) | ∂ t g | + ε (1 − εk ( s ) t ) | ∂ s g | + W α ( s, t ) g − b (cid:0) g − g (cid:1)(cid:27) , (3.7)where W α ( s, t ) = (cid:0) t + α − k ( s ) εt (cid:1) (1 − k ( s ) εt ) . (3.8)However to capture the next to leading order of (3.1) we do consider a non-constant ∂ s S toaccommodate curvature variations, which is in some sense the main novelty of the presentpaper. In particular, (3.7) does not provide the O ( ε ) correction to the full GL energy. Onthe opposite (3.6) does, once minimized over the phase factor S as well as the density g . Wewill not prove this directly although it follows rather easily from our analysis. •
1D functional with given curvature and phase . If the curvature k ( s ) ≡ k is constant (the disccase), the minimization of (3.7) reduces to the 1D problem E k,α [ f ] := (cid:90) c | log ε | d t (1 − εkt ) (cid:110) | ∂ t f | + V k,α ( t ) f − b (cid:0) f − f (cid:1)(cid:111) , (3.9)with V k,α ( t ) := ( t + α − εkt ) (1 − εkt ) . (3.10)In the sequel we shall denote I ε = [0 , c | log ε | ] =: [0 , t ε ] . (3.11)Note that (3.9) includes O ( ε ) corrections due to curvature. As explained above our approachis to approximate the curvature of the domain as a piecewise constant function and hence animportant ingredient is to study the above 1D problem for different values of k , and provesome continuity properties when k is varied. For k = 0 (the half-plane case, sometimesreferred to as the half-cylinder case) we recover the familiar E ,α [ f ] := (cid:90) c | log ε | d t (cid:110) | ∂ t f | + ( t + α ) f − b (cid:0) f − f (cid:1)(cid:111) , (3.12)which has been known to play a crucial role in surface superconductivity physics for a longtime (see [FH3, Chapter 14] and references therein).In this section we provide details about the minimization of (3.9) that go beyond our previousstudy [CR, Section 3.1]. We will use the following notation: • Minimizing (3.9) with respect to f at fixed α we get a minimizer f k,α and an energy E ( k, α ). • Minimizing the latter with respect to α we get some α ( k ) and some energy E (cid:63) ( k ). It followsfrom (3.14) below that α ( k ) is uniquely defined. • Corresponding to E (cid:63) ( k ) := E ( k, α ( k )) we have an optimal density f k , which minimizes E ( k, α ( k )), and a potential V k ( t ) := V k,α ( k ) ( t ) . orreggi, Rougerie – Surface Superconductivity k ) of theseobjects: Proposition 3.1 ( Dependence on curvature of the 1D minimization problem ) . Let k, k (cid:48) ∈ R be bounded independently of ε and < b < Θ − , then the following holds: (cid:12)(cid:12) E (cid:63) ( k ) − E (cid:63) ( k (cid:48) ) (cid:12)(cid:12) ≤ Cε | k − k (cid:48) || log ε | ∞ (3.13) and | α ( k ) − α ( k (cid:48) ) | ≤ C ( ε | k − k (cid:48) | ) / | log ε | ∞ . (3.14) Finally, for all n ∈ N , (cid:13)(cid:13)(cid:13) f ( n ) k − f ( n ) k (cid:48) (cid:13)(cid:13)(cid:13) L ∞ ( I ε ) ≤ C ( ε | k − k (cid:48) | ) / | log ε | ∞ . (3.15)We first prove (3.13) and (3.14) and explain that these estimates imply the following lemma: Lemma 3.1 ( Preliminary estimate on density variations ) . Under the assumptions of Proposition 3.1 it holds (cid:13)(cid:13) f k − f k (cid:48) (cid:13)(cid:13) L ( I ε ) ≤ C ( ε | k − k (cid:48) | ) / | log ε | ∞ . (3.16) Proof of Lemma 3.1.
We proceed in three steps:
Step 1. Energy decoupling.
We use the strict positivity of f k recalled in the appendix towrite any function f on I ε as f = f k v. We can then use the variational equation (A.1) satisfied by f k to decouple the α (cid:48) , k (cid:48) functional inthe usual way, originating in [LM]. Namely, we integrate by parts and use the fact that f k satisfiesNeumann boundary conditions to write (cid:90) c | log ε | d t (1 − εk (cid:48) t )( ∂ t f ) = (cid:90) c | log ε | d t (1 − εk (cid:48) t ) (cid:2) v ( ∂ t f k ) + f k ( ∂ t v ) + 2 f k ∂ t f k v∂ t v (cid:3) = (cid:90) c | log ε | d t (1 − εk (cid:48) t ) (cid:104) f k ( ∂ t v ) + (cid:16) εk (cid:48) − εk (cid:48) t − εk − εkt (cid:17) v f k ∂ t f k − f k v (cid:0) V k + b ( f k − (cid:1)(cid:105) . Inserting this into the definition of E k (cid:48) ,α (cid:48) and using (A.3), we obtain for any f E k (cid:48) ,α (cid:48) [ f ] = E (cid:63) ( k ) + F red [ v ]+ (cid:90) c | log ε | d t (1 − εk (cid:48) t ) ( V k (cid:48) ,α (cid:48) ( t ) − V k ( t )) f k v + 1 b ε ( k (cid:48) − k ) (cid:90) c | log ε | d t tf k + ε (cid:90) c | log ε | d t (cid:16) k (cid:48) − k − εk (cid:48) t − εkt (cid:17) | v | f k ∂ t f k (3.17)with F red [ v ] = (cid:90) c | log ε | d t (1 − εk (cid:48) t ) (cid:110) f k ( ∂ t v ) + b f k (cid:0) − v (cid:1) (cid:111) . (3.18)In the case α (cid:48) = α ( k ) we can insert the trial state v ≡ E (cid:63) ( k (cid:48) ) ≤ E k (cid:48) ,α ( k ) ≤ E (cid:63) ( k ) + Cε | k − k (cid:48) || log ε | ∞ (3.19) orreggi, Rougerie – Surface Superconductivity f k recalled in Appendix A and the easy estimate (cid:12)(cid:12) V k (cid:48) ,α ( k ) ( t ) − V k ( t ) (cid:12)(cid:12) ≤ Cε | k − k (cid:48) || log ε | ∞ for any t ∈ I ε . Changing the role of k and k (cid:48) in (3.19) we obtain the reverse inequality E (cid:63) ( k ) ≤ E (cid:63) ( k (cid:48) ) + Cε | k − k (cid:48) || log ε | ∞ , and hence (3.13) is proved. Step 2. Use of the cost function.
We now consider the case α (cid:48) = α ( k (cid:48) ) , f = f k (cid:48) and boundfrom below the term on the second line of (3.17). A simple computation gives (cid:90) c | log ε | d t (1 − εk (cid:48) t ) (cid:0) V k (cid:48) ,α ( k (cid:48) ) − V k,α ( k ) (cid:1) f k v = (cid:90) c | log ε | d t (1 − εkt ) − ( α ( k (cid:48) ) − α ( k )) (cid:0) t + α ( k ) + α ( k (cid:48) ) − εkt (cid:1) f k v + O ( ε | k − k (cid:48) | )= ( α ( k (cid:48) ) − α ( k )) (cid:90) c | log ε | d t (1 − εk (cid:48) t ) − f k v + 2( α ( k (cid:48) ) − α ( k )) (cid:90) c | log ε | d t t + α ( k ) − εkt − εkt f k v + O ( ε | k − k (cid:48) | ) . (3.20)We may now follow closely the procedure of [CR, Section 5.2]: with the potential function F k defined in (A.8) below we have 2 t + α ( k ) − εkt − εkt f k = ∂ t F k ( t )and hence an integration by parts yields (boundary terms vanish thanks to Lemma A.3)2 (cid:90) c | log ε | d t t + α ( k ) − εkt − εkt f k v = − (cid:90) c | log ε | d t F k v∂ t v. (3.21)We now split the integral into one part running from 0 to ¯ t k,ε and a boundary part running from¯ t k,ε to c | log ε | , where ¯ t k,ε is defined in (A.12) and (A.13) below. For the second part, it followsfrom the decay estimates of Lemma A.2 that (cid:90) c | log ε | ¯ t k,ε d t F k v∂ t v = O ( ε ∞ ) . (3.22)To see this, one can simply adapt the procedure in [CR, Eqs. (5.21) – (5.28)]. The bound (3.22)is in fact easier to derive than the corresponding estimate in [CR] because the decay estimates inLemma A.2 are stronger than the Agmon estimates we had to use in that case. Details are thusomitted.We turn to the main part of the integral (3.21), which lives in [0 , ¯ t k,ε ] . Since F k is negative wehave, using Lemma A.4 and Cauchy-Schwarz, (cid:12)(cid:12)(cid:12)(cid:12) α ( k (cid:48) ) − α ( k )) (cid:90) ¯ t k,ε d t F k v∂ t v (cid:12)(cid:12)(cid:12)(cid:12) ≤ ( α ( k (cid:48) ) − α ( k )) (cid:90) ¯ t k,ε d t (1 − εk (cid:48) t ) − | F k | v + (cid:90) ¯ t k,ε d t (1 − εk (cid:48) t ) | F k | ( ∂ t v ) ≤ (1 − d ε )( α ( k (cid:48) ) − α ( k )) (cid:90) ¯ t k,ε d t (1 − εk (cid:48) t ) − f k v + (1 − d ε ) (cid:90) ¯ t k,ε d t (1 − εk (cid:48) t ) f k ( ∂ t v ) orreggi, Rougerie – Surface Superconductivity < d ε ≤ C | log ε | − . Inserting this bound and (3.22) in (3.17), using (3.20) and (3.21),yields the lower bound E (cid:63) ( k (cid:48) ) ≥ E (cid:63) ( k ) + (cid:90) c | log ε | d t (1 − εtk (cid:48) ) (cid:26) d ε f k ( ∂ t v ) + d ε ( α (cid:48) − α ( k )) (1 − εtk (cid:48) ) f k v + f k b (cid:0) − v (cid:1) (cid:27) + ε (cid:90) c | log ε | d t v f k ∂ t f k (cid:18) k (cid:48) − k − εtk (cid:48) − εtk (cid:19) − Cε | k − k (cid:48) || log ε | ∞ (3.23)where v = f k (cid:48) /f k and we also used the uniform bound (A.2) to estimate the fourth term of ther.h.s. of (3.17). Step 3. Conclusion.
We still have to bound the first term in the second line of (3.23): ε (cid:90) c | log ε | d t v f k ∂ t f k (cid:18) k (cid:48) − k − εk (cid:48) t − εkt (cid:19) = 12 (cid:20) v f k (cid:18) εk (cid:48) − εk − εk (cid:48) t − εkt (cid:19)(cid:21) c | log ε | + (cid:90) c | log ε | d t v f k εk ( k (cid:48) − k )(1 − εkt ) − (cid:90) c | log ε | d t v∂ t vf k (cid:18) εk (cid:48) − εk − εk (cid:48) t − εkt (cid:19) . The first two terms are both O ( ε | k − k (cid:48) || log ε | ∞ ) thanks to (A.2) applied to f k (cid:48) = f k v . For thethird one we write (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) c | log ε | d t v∂ t vf k (cid:18) εk (cid:48) − εk − εk (cid:48) t − εkt (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cε | k − k (cid:48) || log ε | ∞ (cid:90) c | log ε | d t v | ∂ t v | f k ≤ Cε | k − k (cid:48) || log ε | ∞ (cid:20) (cid:90) c | log ε | d t f k v + (cid:90) c | log ε | d t f k ( ∂ t v ) (cid:21) . Inserting this in (3.23), using again (A.2) and dropping a positive term, we finally get E (cid:63) ( k (cid:48) ) ≥ E (cid:63) ( k ) + | log ε | − ( α ( k (cid:48) ) − α ( k )) (cid:90) c | log ε | d t (1 − εk (cid:48) t ) f k (cid:48) + 12 b (cid:90) c | log ε | d t (1 − εk (cid:48) t ) (cid:0) f k − f k (cid:48) (cid:1) − Cε | k − k (cid:48) || log ε | ∞ (3.24)where we have chosen d ε = | log ε | − , which is compatible with the requirement 0 < d ε ≤ C | log ε | − . Combining with the estimate (3.13) that we proved in Step 1 concludes the proofof (3.14). To get (3.16) one has to use in addition (A.6), which guarantees that under the assump-tions 1 < b < Θ − (cid:107) f k (cid:48) (cid:107) L ( I ε ) ≥ C > C independent of ε .To conclude the proof of Proposition 3.1 there only remains to discuss (3.15). We shall upgradethe estimate (3.16) to better norms, taking advantage of the 1D nature of the problem and usinga standard bootstrap argument. Proof of Proposition 3.1.
We write f k = f k (cid:48) + ( f k − f k (cid:48) ) and expand the energy E (cid:63) ( k ) = E k [ f k ], orreggi, Rougerie – Surface Superconductivity f k (cid:48) : E (cid:63) ( k ) ≥ E (cid:63) ( k (cid:48) ) + (cid:90) I ε d t (1 − εkt ) | ∂ t ( f k − f k (cid:48) ) | + (cid:90) I ε d t (1 − εkt ) V k ( f k − f k (cid:48) ) + (cid:90) I ε d t (1 − εkt )( V k − V k (cid:48) ) f k (cid:48) + 2 (cid:90) I ε d t (1 − εkt ) f k (cid:48) ( f k − f k (cid:48) )( V k − V k (cid:48) )+ 12 b (cid:90) I ε d t (1 − εkt ) (cid:2) f k (cid:48) ( f k − f k (cid:48) ) + 4 f k (cid:48) ( f k − f k (cid:48) ) + ( f k − f k (cid:48) ) − f k − f k (cid:48) ) (cid:3) − Cε | k − k (cid:48) || log ε | ∞ where the O ( ε | k − k (cid:48) || log ε | ∞ ) is as before due to the replacement of the curvature k ↔ k (cid:48) . Usingthe same procedure to expand E (cid:63) ( k (cid:48) ) = E k (cid:48) [ f k (cid:48) ] and combining the result with the above weobtain E (cid:63) ( k ) ≥ E (cid:63) ( k ) + 2 (cid:90) I ε d t (1 − εkt ) | ∂ t ( f k − f k (cid:48) ) | + (cid:90) I ε d t (1 − εkt )( V k + V k (cid:48) )( f k − f k (cid:48) ) + (cid:90) I ε d t (1 − εkt )( V k − V k (cid:48) )( f k (cid:48) − f k )+ 2 (cid:90) I ε d t (1 − εkt )( f k (cid:48) ( f k − f k (cid:48) ) − f k ( f k (cid:48) − f k ))( V k − V k (cid:48) )+ 12 b (cid:90) I ε d t (1 − εkt )( f k − f k (cid:48) ) (cid:2) f k (cid:48) + 4 f k + 4 f k (cid:48) f k − (cid:3) − Cε | k − k (cid:48) || log ε | ∞ . Hence it holds Cε | k − k (cid:48) || log ε | ∞ ≥ (cid:90) I ε d t (1 − εkt ) | ∂ t ( f k − f k (cid:48) ) | + (cid:90) I ε d t (1 − εkt )( V k − V k (cid:48) )( f k − f k (cid:48) )+ (cid:90) I ε d t (1 − εkt )( f k − f k (cid:48) ) (cid:20) V k + V k (cid:48) + 2 b (cid:0) f k (cid:48) + f k + f k (cid:48) f k − (cid:1)(cid:21) . (3.25)Next we note that thanks to (3.14)sup I ε | V k − V k (cid:48) | ≤ C ( | α ( k ) − α ( k (cid:48) ) | + ε | k − k (cid:48) | ) | log ε | ∞ ≤ C ( ε | k − k (cid:48) | ) / | log ε | ∞ as revealed by an easy computation starting from the expression (3.10). Thus, using (3.16) andthe Cauchy-Schwartz inequality, (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) I ε d t (1 − εkt )( V k − V k (cid:48) )( f k − f k (cid:48) ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C | log ε | / sup I ε | V k − V k (cid:48) | (cid:13)(cid:13) f k − f k (cid:48) (cid:13)(cid:13) L ( I ε ) ≤ Cε | k − k (cid:48) || log ε | ∞ . (3.26)For the term on the third line of (3.25) we notice that, using the growth of the potentials V k and V k (cid:48) for large t , the integrand is positive in˜ I ε := (cid:104) c (log | log ε | ) / , c | log ε | (cid:105) orreggi, Rougerie – Surface Superconductivity c and ε small enough. On the other hand, combining (3.16) and the pointwiselower bound in (A.6) we have (cid:107) f k − f k (cid:48) (cid:107) L (˜ I ε ) ≤ C ( ε | k − k (cid:48) | ) / | log ε | ∞ . Splitting the integral into two pieces we thus have (cid:90) I ε d t (1 − εkt )( f k − f k (cid:48) ) (cid:2) V k + V k (cid:48) + b (cid:0) f k (cid:48) + f k + f k (cid:48) f k − (cid:1)(cid:3) ≥ − Cε | k − k (cid:48) || log ε | ∞ . Using this and (3.26) we deduce from (3.25) that (cid:90) I ε d t (1 − εkt ) | ∂ t ( f k − f k (cid:48) ) | ≤ Cε | k − k (cid:48) || log ε | ∞ (3.27)and combining with the previous L bound this gives (cid:107) f k − f k (cid:48) (cid:107) H (˜ I ε ) ≤ C ( ε | k − k (cid:48) | ) / | log ε | ∞ . Since we work on a 1D interval, the Sobolev inequality implies (cid:107) f k − f k (cid:48) (cid:107) L ∞ (˜ I ε ) ≤ C ( ε | k − k (cid:48) | ) / | log ε | ∞ . (3.28)In particular (cid:12)(cid:12)(cid:12) f k ( c (log | log ε | ) / ) − f k (cid:48) ( c (log | log ε | ) / ) (cid:12)(cid:12)(cid:12) ≤ C ( ε | k − k (cid:48) | ) / | log ε | ∞ . Then, integrating the bound (3.27) from c (log | log ε | ) / to c | log ε | we can extend (3.28) to thewhole interval I ε : (cid:107) f k − f k (cid:48) (cid:107) L ∞ ( I ε ) ≤ C ( ε | k − k (cid:48) | ) / | log ε | ∞ , which is (3.15) for n = 0. The bounds on the derivatives follow by a standard bootstrap argument,inserting the L ∞ bound in the variational equations. In this Section we collect some useful estimates of other quantities involving the 1D densities aswell as the optimal phases. It turns out that we need an estimate of the k -dependence of ∂ t log( f k ),provided in the following Proposition 3.2 ( Estimate of logarithmic derivatives ) . Let k, k (cid:48) ∈ R be bounded independently of ε and < b < Θ − , then the following holds: (cid:13)(cid:13)(cid:13)(cid:13) f (cid:48) k f k − f (cid:48) k (cid:48) f k (cid:48) (cid:13)(cid:13)(cid:13)(cid:13) L ∞ ( I ε ) ≤ C ( ε | k − k (cid:48) | ) / | log ε | ∞ . (3.29) Proof.
Let us denote for short g ( t ) := f (cid:48) k ( t ) f k ( t ) − f (cid:48) k (cid:48) ( t ) f k (cid:48) ( t ) . (3.30)We first notice that the estimate is obviously true in the region where f k ≥ | log ε | − M for any M > | g ( t ) | ≤ | f (cid:48) k − f (cid:48) k (cid:48) | f k + | f (cid:48) k (cid:48) | | f k − f k (cid:48) | f k f k (cid:48) ≤ | log ε | M | f (cid:48) k − f (cid:48) k (cid:48) | + | log ε | M +3 | f k − f k (cid:48) |≤ C ( ε | k − k (cid:48) | ) / | log ε | ∞ . orreggi, Rougerie – Surface Superconductivity t ∗ be the unique solution to f k ( t ∗ ) = | log ε | − M (uniqueness follows from the properties of f k discussed in Proposition A.1). To complete the proof it thus suffices to prove the estimate in theregion [ t ∗ , c | log ε | ]. Notice also that thanks to (A.6), it must be that t ∗ → ∞ when ε → t ∗ , t ε ] (recall (3.11)), one has g ( t ∗ ) = O (cid:16) ( ε | k − k (cid:48) | ) / | log ε | M (cid:17) , g ( t ε ) = 0 , (3.31)because of Neumann boundary conditions. Hence if the supremum of | g | is reached at the boundarythere is nothing to prove. Let us then assume that sup t ∈ [ t ∗ ,t ε ] | g | = | g ( t ) | , for some t ∗ < t < t ε ,such that g (cid:48) ( t ) = 0, i.e., f (cid:48)(cid:48) k ( t ) f k ( t ) − f (cid:48)(cid:48) k (cid:48) ( t ) f k (cid:48) ( t ) + ( f (cid:48) k ( t )) f k ( t ) − ( f (cid:48) k (cid:48) ( t )) f k (cid:48) ( t ) = 0 . (3.32)Since f k and f k (cid:48) are both decreasing in [ t ∗ , t ε ] (see again Proposition A.1) we also have( f (cid:48) k ( t )) f k ( t ) − ( f (cid:48) k (cid:48) ( t )) f k (cid:48) ( t ) = (cid:20) | f (cid:48) k ( t ) | f k ( t ) + | f (cid:48) k (cid:48) ( t ) | f k (cid:48) ( t ) (cid:21) g ( t ) . (3.33)The variational equations satisfied by f k and f k (cid:48) on the other hand imply (cid:12)(cid:12)(cid:12)(cid:12) f (cid:48)(cid:48) k ( t ) f k ( t ) − f (cid:48)(cid:48) k (cid:48) ( t ) f k (cid:48) ( t ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) εkf (cid:48) k ( t )(1 − εkt ) f k ( t ) − εk (cid:48) f (cid:48) k (cid:48) ( t )(1 − εk (cid:48) t ) f k (cid:48) ( t ) + V k ( t ) − V k (cid:48) ( t ) − b (cid:0) f k ( t ) − f k (cid:48) ( t ) (cid:1) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:104) ( ε | k − k (cid:48) | ) / | log ε | ∞ + ε | g ( t ) | (cid:105) , (3.34)thanks to (3.14) and (3.15). For the first two terms the estimate (A.7) has also been used for thederivatives f (cid:48) k and f (cid:48) k (cid:48) : εkf (cid:48) k ( t )(1 − εkt ) f k ( t ) − εk (cid:48) f (cid:48) k (cid:48) ( t )(1 − εk (cid:48) t ) f k (cid:48) ( t ) = O ( ε ) g ( t ) + f (cid:48) k (cid:48) ( t ) f k (cid:48) ( t ) (cid:18) εk − εkt − εk (cid:48) − εk (cid:48) t (cid:19) = O ( ε ) g ( t ) + O ( ε | k − k (cid:48) | ) . Plugging (3.33) and (3.34) into (3.32), we get the estimate (cid:20) | f (cid:48) k ( t ) | f k ( t ) + | f (cid:48) k (cid:48) ( t ) | f k (cid:48) ( t ) + O ( ε ) (cid:21) g ( t ) = O (cid:16) ( ε | k − k (cid:48) | ) / | log ε | ∞ (cid:17) . (3.35)Now if | f (cid:48) k ( t ) | f k ( t ) + | f (cid:48) k (cid:48) ( t ) | f k (cid:48) ( t ) ≥ | log ε | − , the result follows immediately. Therefore we can assume that | f (cid:48) k ( t ) | f k ( t ) + | f (cid:48) k (cid:48) ( t ) | f k (cid:48) ( t ) ≤ | log ε | − , (3.36)but we claim that this also implies | f (cid:48) k ( t ) | f k ( t ) + | f (cid:48) k (cid:48) ( t ) | f k (cid:48) ( t ) ≤ | log ε | − for any t ∈ [ t , t ε ] . (3.37)Indeed, setting h k ( t ) := − f (cid:48) k ( t ) /f k ( t ) , orreggi, Rougerie – Surface Superconductivity
21a simple computation involving the variational equation (A.1) yields h (cid:48) k ( t ) = − εkf (cid:48) k ( t )(1 − εkt ) f k ( t ) − V k ( t ) + 1 b (cid:0) − f k ( t ) (cid:1) + h k ( t ) = − V k ( t ) + h k ( t ) + O (1) , using (A.7) again. Hence h (cid:48) k ( t ) <
0, since V k ( t ) (cid:29)
1, which follows from t > t ∗ (cid:29)
1, andtherefore (3.37) holds. An identical argument applies to h k (cid:48) and thus to the sum h k + h k (cid:48) =: h. Finally, the explicit expression of g (cid:48) ( t ) in combination with (3.37) gives for t ≥ t | g ( t ) | = (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) t ε t d η g (cid:48) ( η ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:90) t ε t d η (cid:104) ( h ( η ) + O ( ε )) | g ( η ) | + O (cid:16) ( ε | k − k (cid:48) | ) / | log ε | ∞ (cid:17)(cid:105) ≤ C | log ε | − sup t ∈ [ t ,t ε ] | g ( t ) | + O (cid:16) ( ε | k − k (cid:48) | ) / | log ε | ∞ (cid:17) , (3.38)which implies the result.The above estimate is mainly useful in providing bounds on quantities of the form I k,k (cid:48) ( t ) := F k ( t ) − F k (cid:48) ( t ) f k ( t ) f k (cid:48) ( t ) , (3.39)alluded to in Subsection 2.2. As announced there, the main difficulty is that we need to show that I k,k (cid:48) is small relatively to f k , which is the content of the following corollary. We need the followingnotation [0 , ¯ t k,ε ] := (cid:8) t : f k ( t ) ≥ | log ε | f k ( t ε ) (cid:9) . (3.40)Note that the monotonicity for large t of f k guarantees that the above set is indeed an intervaland that ¯ t k,ε = t ε + O (log | log ε | ) . (3.41) Corollary 3.1 ( Estimates on the correction function ) . Under the assumptions of Proposition 3.2, it holds sup t ∈ [0 ,t ε ] (cid:12)(cid:12)(cid:12)(cid:12) I k,k (cid:48) f k (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ( ε | k − k (cid:48) | ) / | log ε | ∞ (3.42) and, setting ¯ t ε := min { ¯ t k,ε , ¯ t k (cid:48) ,ε } , sup t ∈ [0 , ¯ t ε ] (cid:12)(cid:12)(cid:12)(cid:12) ∂ t I k,k (cid:48) f k (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ( ε | k − k (cid:48) | ) / | log ε | ∞ . (3.43) Proof.
We write I k,k (cid:48) ( t ) f k ( t ) = F k ( t ) f k ( t ) − F k (cid:48) ( t ) f k (cid:48) ( t )Using the definition of the potential function (A.8) and its properties (A.9), we can rewrite F k ( t ) f k ( t ) − F k (cid:48) ( t ) f k (cid:48) ( t ) = − (cid:90) t ε t d η (cid:20) b k ( η ) f k ( η ) f k ( t ) − b k (cid:48) ( η ) f k (cid:48) ( η ) f k (cid:48) ( t ) (cid:21) = (cid:90) t ε t d η (cid:20) b k ( η ) (cid:18) f k (cid:48) ( η ) f k (cid:48) ( t ) − f k ( η ) f k ( t ) (cid:19) + ( b k (cid:48) ( η ) − b k ( η )) f k (cid:48) ( η ) f k (cid:48) ( t ) (cid:21) . (3.44) orreggi, Rougerie – Surface Superconductivity η ≥ t f k (cid:48) ( η ) f k (cid:48) ( t ) ≤ C, (3.45)as it easily follows by combining the monotonicity of f k for t large with its strict positivity closeto the origin (see Proposition A.1 and Lemma A.2 for the details). Hence we can bound the lastterm on the r.h.s. of (3.44) as (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) t ε t d η ( b k (cid:48) ( η ) − b k ( η )) f k (cid:48) ( η ) f k (cid:48) ( t ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C | log ε | (cid:107) b k (cid:48) − b k (cid:107) L ∞ ( I ε ) = O (cid:16) ( ε | k − k (cid:48) | ) / | log ε | ∞ (cid:17) , (3.46)since by (3.14) b k (cid:48) ( t ) − b k ( t ) = (1 + O ( ε )) (cid:0) O ( ε | k − k (cid:48) | t ) + α ( k ) − α ( k (cid:48) ) (cid:1) = O (cid:16) ( ε | k − k (cid:48) | ) / | log ε | ∞ (cid:17) . For the first term on the r.h.s. of (3.44) we exploit the estimate f k (cid:48) ( η ) f k (cid:48) ( t ) − f k ( η ) f k ( t ) = O (cid:16) ( ε | k − k (cid:48) | ) / | log ε | ∞ (cid:17) , which can be proven by writing f k ( η ) f k ( t ) = exp (cid:26)(cid:90) ηt d τ f (cid:48) k ( τ ) f k ( τ ) (cid:27) , which implies (cid:12)(cid:12)(cid:12)(cid:12) f k (cid:48) ( η ) f k (cid:48) ( t ) − f k ( η ) f k ( t ) (cid:12)(cid:12)(cid:12)(cid:12) = f k (cid:48) ( η ) f k (cid:48) ( t ) (cid:12)(cid:12)(cid:12)(cid:12) − exp (cid:26)(cid:90) ηt d τ (cid:20) f (cid:48) k ( τ ) f k ( τ ) − f (cid:48) k (cid:48) ( τ ) f k (cid:48) ( τ ) (cid:21)(cid:27)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:90) ηt d τ (cid:12)(cid:12)(cid:12)(cid:12) f (cid:48) k ( τ ) f k ( τ ) − f (cid:48) k (cid:48) ( τ ) f k (cid:48) ( τ ) (cid:12)(cid:12)(cid:12)(cid:12) exp (cid:26)(cid:90) ηt d τ (cid:12)(cid:12)(cid:12)(cid:12) f (cid:48) k ( τ ) f k ( τ ) − f (cid:48) k (cid:48) ( τ ) f k (cid:48) ( τ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:27) ≤ C ( ε | k − k (cid:48) | ) / | log ε | ∞ , (3.47)where we have used (3.45), the estimate | − e δ | ≤ | δ | e | δ | , δ ∈ R , and (3.29). Putting together(3.44) with (3.46) and (3.47), we conclude the proof of (3.42).To obtain (3.29) we first note that since F (cid:48) k ( t ) ≤
0, the positivity of K k in [0 , ¯ t k,ε ] recalled inLemma A.4 ensures that (cid:12)(cid:12)(cid:12)(cid:12) F k ( t ) f k ( t ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ , ¯ t k,ε ]. Then we may use (3.29) again to estimatesup t ∈ [0 , ¯ t ε ] (cid:12)(cid:12)(cid:12)(cid:12) ∂ t I k,k (cid:48) f k (cid:12)(cid:12)(cid:12)(cid:12) = sup t ∈ [0 , ¯ t ε ] (cid:20) | (1 − εkt ) b k − (1 − εk (cid:48) t )) b k (cid:48) | + 2 (cid:12)(cid:12)(cid:12)(cid:12) F k (cid:48) f k (cid:48) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f (cid:48) k f k − f (cid:48) k (cid:48) f k (cid:48) (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) ≤ C ( ε | k − k (cid:48) | ) / | log ε | ∞ , and the proof is complete. We now turn to the proof of the energy upper bound corresponding to (2.5), namely we prove thefollowing: orreggi, Rougerie – Surface Superconductivity Proposition 4.1 ( Upper bound to the full GL energy ) . Let < b < Θ − and ε be small enough. Then it holds E GL ε ≤ ε (cid:90) | ∂ Ω | d s E (cid:63) ( k ( s )) + Cε | log ε | ∞ (4.1) where s (cid:55)→ k ( s ) is the curvature function of the boundary ∂ Ω as a function of the tangentialcoordinate. This result is proven as usual by evaluating the GL energy of a trial state having the expectedphysical features. As is well-known [FH3], such a trial state should be concentrated along theboundary of the sample, and the induced magnetic field should be chosen close to the appliedone. Before entering the heart of the proof, we briefly explain how these considerations allow usto reduce to the proof of an upper bound to the reduced functional (3.1). We define G A ε := inf {G A ε [ ψ ] , ψ (0 , t ) = ψ ( | ∂ Ω | , t ) } , (4.2)the infimum of the reduced functional under periodic boundary conditions in the tangential direc-tion and prove Lemma 4.1 ( Reduction to the boundary functional ) . Under the assumptions of Proposition 4.1, it holds E GL ε ≤ ε G A ε + Cε ∞ . (4.3) Proof.
This is a standard reduction for which more details may be found in [FH3, Section 14.4.2]and references therein. See also [CR, Sections 4.1 and 5.1]. We provide a sketch of the proof forcompleteness.We first pick the trial vector potential as A trial = F where F is the induced vector potential written in a gauge where div F = 0, namely the uniquesolution of div F = 0 , in Ω , curl F = 1 , in Ω , F · ν = 0 , on ∂ Ω . Next we introduce boundary coordinates as described in [FH3, Appendix F]: let γ ( ξ ) : R \ ( | ∂ Ω | Z ) → ∂ Ωbe a counterclockwise parametrization of the boundary ∂ Ω such that | γ (cid:48) ( ξ ) | = 1. The unit vectordirected along the inward normal to the boundary at a point γ ( ξ ) will be denoted by ν ( ξ ). Thecurvature k ( ξ ) is then defined through the identity γ (cid:48)(cid:48) ( ξ ) = k ( ξ ) ν ( ξ ) . Our trial state will essentially live in the region˜ A ε := { r ∈ Ω | dist( r , ∂ Ω) ≤ c ε | log ε |} , (4.4)and in such a region we can introduce tubular coordinates ( s, εt ) (note the rescaling of the normalvariable) such that, for any given r ∈ ˜ A ε , εt = dist( r , ∂ Ω), i.e., r ( s, εt ) = γ (cid:48) ( s ) + εt ν ( s ) , (4.5) orreggi, Rougerie – Surface Superconductivity ε small enough. Hence the boundary layerbecomes in the new coordinates ( s, t ) A ε := { ( s, t ) ∈ [0 , | ∂ Ω | ] × [0 , c | log ε | ] } . (4.6)We now pick a function ψ ( s, t ) defined on A ε , satisfying periodic boundary conditions in the s variable. Using a smooth cut-off function χ ( t ) with χ ( t ) ≡ t ∈ [0 , c | log ε | ] and χ ( t )exponentially decreasing for t > c | log ε | , we associate to ψ the GL trial stateΨ trial ( r ) := ψ ( s, t ) χ ( t ) exp { iφ trial ( s, t ) } , where φ trial is a gauge phase (analogue of (5.4)) depending on A trial , i.e., φ trial ( s, t ) := − ε (cid:90) t d η ν ( s ) · A trial ( r ( s, εη )) + 1 ε (cid:90) s d ξ γ (cid:48) ( ξ ) · A trial ( r ( ξ, − (cid:18) | Ω || ∂ Ω | ε − (cid:22) | Ω || ∂ Ω | ε (cid:23)(cid:19) s. (4.7)Then, with the definition of G A ε as in (3.1), a relatively straightforward computation gives E GL [Ψ trial , A trial ] ≤ ε G A ε [ ψ ] + Cε ∞ , and the desired result follows immediately. Note that this computation uses the gauge invarianceof the GL functional, e.g., through [FH3, Lemma F.1.1].The problem is now reduced to the construction of a proper trial state for G A ε . To capture the O ( ε ) correction (which depends on curvature) to the leading order of the GL energy (which doesnot depend explicitly on curvature), we need a more elaborate function than has been consideredso far. The construction is detailed in Subsection 4.1 and the computation completing the proofof Proposition 4.1 is given in Subsection 4.2. We start by recalling the splitting of the domain A ε defined in (3.4) into N ε ∝ ε − rectangularcells {C n } n =1 ...N ε with boundaries s n , s n +1 in the s -coordinate such that s n +1 − s n = (cid:96) ε ∝ ε, so that N ε = | ∂ Ω | (cid:96) ε . We denote C n = [ s n , s n +1 ] × [0 , c | log ε | ] , (4.8)with the convention that s = 0, for simplicity. We will approximate the curvature k ( s ) inside eachcell by its mean value and set k n := (cid:96) − ε (cid:90) s n +1 s n d s k ( s ) . (4.9)We also denote by α n = α ( k n ) (4.10)the optimal phase associated to k n , obtained by minimizing E ( α, k n ) with respect to α as inSection 3.1. orreggi, Rougerie – Surface Superconductivity k n − k n +1 = O ( ε ) . (4.11)Indeed if we assume that sup s ∈ [0 , π ] | ∂ s k ( s ) | ≤ C < ∞ (independent of ε ), one gets (cid:96) − ε (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) k n +1 k n d sk ( s ) − (cid:90) k n +2 k n +1 d sk ( s ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:96) − ε (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) k n +1 k n d s (cid:90) k n +1 s d η ∂ η k ( η )+ (cid:90) k n +2 k n +1 d s (cid:90) sk n +1 d η ∂ η k ( η ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C(cid:96) ε = O ( ε ) . We can then apply Proposition 3.1 to obtain α n − α n +1 = O ( ε | log ε | ∞ ) , (4.12) (cid:13)(cid:13)(cid:13) f ( m ) n − f ( m ) n +1 (cid:13)(cid:13)(cid:13) L ∞ ( I ε ) = O ( ε | log ε | ∞ ) , (4.13)for any finite m ∈ N .Our trial state has the form ψ trial ( s, t ) = g ( s, t ) exp (cid:8) − i (cid:0) ε − S ( s ) − εδ ε s (cid:1)(cid:9) (4.14)where δ ε is the number (3.3). The density g and phase factor S are defined as follows: • The density. The modulus of our wave function is constructed to be essentially piecewiseconstant in the s -direction, with the form f k n ( t ) in the cell C n . The admissibility of the trialstate requires that g be continuous and we thus set: g ( s, t ) := f k n + χ n , (4.15)where the function χ n satisfies χ n ( s, t ) = (cid:40) , at s = s n ,f k n +1 ( t ) − f k n ( t ) , at s = s n +1 , (4.16)the continuity at the s n boundary being ensured by χ n − . A simple choice is given by χ n ( s, t ) = (cid:0) f k n +1 ( t ) − f k n ( t ) (cid:1) (cid:18) − s − s n +1 s n − s n +1 (cid:19) . (4.17)Note that | k n − k n +1 | ≤ C | s n − s n +1 | ≤ Cε since the curvature is assumed to be a smoothfunction of s . Clearly, in view of Proposition 3.1 we can impose the following bounds on χ n : | χ n | ≤ Cε | log ε | ∞ , | ∂ t χ n | ≤ Cε | log ε | ∞ , | ∂ s χ n | ≤ C | log ε | ∞ , (4.18)so that χ n is indeed only a small correction to the desired density f k n in C n . • The phase. The phase of the trial function is dictated by the refined ansatz (1.19): withinthe cell C n it must be approximately equal to α n and globally it must define an admissiblephase factor, i.e., vary of a multiple of 2 π after one loop. We then let S = S ( s ) = S loc ( s ) + S glo ( s )where S loc varies locally (on the scale of a cell) and S glo varies globally (on the scale of thefull interval [0 , | ∂ Ω | ]) and is chosen to enforce the periodicity on the boundary of the trialstate. The term S loc is the main one, and its s derivative should be equal to α n in each cell orreggi, Rougerie – Surface Superconductivity C n in order that the evaluation of the energy be naturally connected to the 1D functional westudied before, as explained in Section 3.1. We define S loc recursively by setting: S loc ( s ) = (cid:40) α s, in C ,α n ( s − s n ) + S loc ( s n ) , in C n , n ≥ , (4.19)which in particular guarantees the continuity of S loc on [ s , s N ε +1 [. Moreover we easilycompute (recall that s = 0) S loc ( s n ) = n − (cid:88) m =2 α m ( s m +1 − s m ) + α s = (cid:90) s n d s α ( s ) + O ( ε | log ε | ∞ ) . (4.20)The factor S glo ensures that S ( s N ε +1 ) − S ( s ) = S ( s N ε +1 ) ∈ πε Z , which is required for (4.14) to be periodic in the s -direction and hence to correspond to asingle-valued wave function in the original variables. The conditions we impose on S glo arethus S glo ( s ) = 0 (4.21) S glo ( s N ε +1 ) = 2 πε ( α N ε ( s N ε +1 − s N ε ) + S loc ( s N ε ) − (cid:98) α N ε ( s N ε +1 − s N ε ) + S loc ( s N ε ) (cid:99) )with (cid:98) . (cid:99) standing for the integer value. Thanks to (4.20), we have α N ε ( s N ε +1 − s N ε ) + S loc ( s N ε ) = O (1)and we can thus clearly impose that S glo be regular and | S glo | ≤ Cε, | ∂ s S glo | ≤ Cε. (4.22)
Remark 4.1 ( s -dependence of the trial state)The main novelty here is the fact that the density and phase of the trial state have (small)variations on the scale of the cells which are of size O ( ε ) in the s -variable. A noteworthypoint is that the phase needs not have a t -dependence to evaluate the energy at the level ofprecision we require. Basically this is associated with the fact that the t term in (3.2) comesmultiplied with an ε factor. The main point that renders the computation of the energydoable is (4.18) and this is where the analysis of Subsection 3.1 enters heavily. We may now complete the proof of Proposition 4.1 by proving
Lemma 4.2 ( Upper bound for the boundary functional ) . With ψ trial given by the preceding construction, it holds G A ε [ ψ trial ] ≤ (cid:90) | ∂ Ω | d sE (cid:63) ( k ( s )) + O ( ε | log ε | ∞ ) . (4.23)The upper bound (4.1) follows from Lemmas 4.1 and 4.2 since ψ trial is periodic in the s -variableand hence an admissible trial state for G A ε . Proof.
As explained in Subsection 3.1, inserting (4.14) into (3.1) yields G A ε [ ψ trial ] = E S [ g ] (4.24)where E S [ g ] is defined in (3.6). For clarity we split the estimate of the r.h.s. of the above equationinto several steps. We use the shorter notation f n for f k n when this generates no confusion. orreggi, Rougerie – Surface Superconductivity Step 1. Approximating the curvature.
In view of the continuity of the trial function, theenergy is the sum of the energies restricted to each cell. We approximate k ( s ) by k n in C n asannounced, and note that since k is regular we have | k ( s ) − k ( s n ) | ≤ Cε in each cell, with aconstant C independent of j . We thus have E S [ g ] ≤ N ε (cid:88) n =1 (cid:90) C n d t d s (1 − εk n t ) (cid:26) | ∂ t g | + ε (1 − εk n t ) | ∂ s g | + (cid:0) t + ∂ s S − εt k n (cid:1) (1 − εk n t ) g − b (cid:0) g − g (cid:1)(cid:41) (cid:0) O ( ε ) (cid:1) (4.25)since each k -dependent term comes multiplied with an ε factor. Step 2. Approximating the phase. In C n we have ∂ s S = α n + ∂ s S glo = α n + O ( ε ) . We can thus expand the potential term: (cid:90) C n d t d s (cid:0) t + ∂ s S − εt k n (cid:1) − εk n t g = (cid:90) C n d t d s (cid:0) t + α n − εt k n (cid:1) − εk n t g + 2 (cid:90) C n d t d s ∂ s S glo t + α n − εt k n − εk n t g + (cid:90) C n d t d s ( ∂ s S glo ) − εk n t g (4.26)and obviously (cid:90) C n d t d s ( ∂ s S glo ) − εk n t g ≤ Cε | log ε | ∞ , because of (4.22) and the size of C n in the s direction. Next we note that in C n g = f n + 2 f n χ n + χ n so that, using (A.5) and the fact that ∂ s S glo only depends on s we have (cid:90) C n d t d s ∂ s S glo t + α n − εt k n − εk n t g = (cid:90) C n d t d s ∂ s S glo t + α n − εt k n − εk n t (cid:0) f n χ n + χ n (cid:1) , which is easily bounded by Cε | log ε | ∞ using (4.18), (4.22) and the fact that | s n +1 − s n | ≤ Cε .All in all: (cid:90) C n d t d s (cid:0) t + ∂ s S − εt k n (cid:1) − εk n t g = (cid:90) C n d t d s (cid:0) t + α n − εt k n (cid:1) − εk n t g + O ( ε | log ε | ∞ ) . (4.27) Step 3. The 1D functional inside each cell.
We now have to estimate an essentially 1Dfunctional in each cell, closely related to (3.9): (cid:90) C n d t d s (1 − εk n t ) (cid:26) | ∂ t g | + ε (1 − εk n t ) | ∂ s g | + (cid:0) t + α n − εt k n (cid:1) (1 − εk n t ) g − b (cid:0) g − g (cid:1) (cid:27) . (4.28) orreggi, Rougerie – Surface Superconductivity g according to (4.15) in the above expression and use the variational equa-tion (A.1) to cancel the first order terms in χ n . This yields (cid:90) C n d s d t (1 − εk n t ) (cid:110) | ∂ t g | + ε (1 − εk n t ) | ∂ s g | + V k n ( t ) g − b (cid:0) g − g (cid:1)(cid:111) = (cid:96) ε E (cid:63) ( k n )+ (cid:90) C n d s d t (1 − εk n t ) (cid:110) | ∂ t χ n | + ε (1 − εtk n ) | ∂ s χ n | + V k n χ n + b (cid:0) χ n f n + 4 χ n f n + χ n − χ n (cid:1)(cid:111) = (cid:96) ε E (cid:63) ( k n ) + O ( ε | log ε | ∞ ) , (4.29)where we only have to use (4.18) to obtain the final estimate. Step 4, Riemann sum approximation.
Gathering all the above estimates we obtain E S [ g ] ≤ (cid:96) ε N ε (cid:88) n =1 E (cid:63) ( k n ) (cid:0) O ( ε ) (cid:1) + O ( ε | log ε | ∞ ) = (cid:90) | ∂ Ω | d s E (cid:63) ( k ( s )) + O ( ε | log ε | ∞ ) . (4.30)Indeed, (3.13) implies that inside C n (cid:12)(cid:12) E (cid:63) ( k n ) − E (cid:63) ( k ( s )) (cid:12)(cid:12) ≤ Cε(cid:96) ε | log ε | ∞ ≤ Cε | log ε | ∞ . (4.31)Recognizing a Riemann sum of N ε ∝ ε − terms in (4.30) and recalling that E (cid:63) ( k n ) is of order 1,irrespective of n , thus leads to (4.30). Combining (4.24) and (4.30) we obtain (4.23) which concludesthe proof of Lemma 4.2 and hence that of Proposition 4.1, via Lemma 4.1. The main result proven in this section is the following
Proposition 5.1 ( Energy lower bound ) . Let Ω ⊂ R be any smooth simply connected domain. For any fixed < b < Θ − , in the limit ε → , it holds E GL ε ≥ ε (cid:90) | ∂ Ω | d s E (cid:63) ( k ( s )) − Cε | log ε | ∞ . (5.1)We first reduce the problem to the study of decoupled functionals in the boundary layer inSubsection 5.1 and then provide lower bounds to these in Subsection 5.2, which contains the mainnew ideas of our proof. As in Section 4, the starting point is a restriction to the boundary layer together with a replacementof the vector potential. We refer to the proof of Lemma 4.1 and in particular (4.5) for the definitionof the boundary coordinates.
Lemma 5.1 ( Reduction to the boundary functional ) . Under the assumptions of Proposition 5.1, it holds E GL ε ≥ ε G A ε [ ψ ] − Cε | log ε | , (5.2) with ψ ( s, t ) = Ψ GL ( r ( s, εt )) e − iφ ε ( s,t ) in A ε , φ ε ( s, t ) is a global phase defined in (5.4) below and G A ε is the boundary functional defined in (3.1) orreggi, Rougerie – Surface Superconductivity Proof.
A simplified version of the result for disc samples is proven in [CR, Proposition 4.1], wherea rougher lower bound is also derived for general domains. This latter result is obtained bydropping the curvature dependent terms from the energy, which was sufficient for the analysiscontained there. Here we need more precision in order to obtain a remainder term of order o ( ε ).We highlight here the main steps and skip most of the technical details.A suitable partition of unity together with the standard Agmon estimates (see [FH1, Section14.4]) allow to restrict the integration to the boundary layer: E GL ε ≥ (cid:90) ˜ A ε d r (cid:26)(cid:12)(cid:12)(cid:12)(cid:16) ∇ + i A GL ε (cid:17) Ψ (cid:12)(cid:12)(cid:12) − bε (cid:2) | Ψ | − | Ψ | (cid:3)(cid:27) + O ( ε ∞ ) . (5.3)where Ψ is given in terms of Ψ GL in the form Ψ = f Ψ GL for some function 0 ≤ f ≤ t , with support containing the set ˜ A ε defined by (4.4)and contained in { r ∈ Ω | dist( r , ∂ Ω) ≤ Cε | log ε |} for a possibly large constant C . The constant c in the definition (4.4) of the boundary layer has tobe chosen large enough, but the choice of the support of f remains to any other extent arbitraryand one can clearly pick f in such a way that f = 1 in ˜ A ε and going smoothly to 0 outside of it.The second ingredient of the proof is the replacement of the magnetic potential A GL but thiscan be done following the same strategy applied to disc samples in [CR, Eqs. (4.18) – (4.26)],whose estimates are not affected by the dependence of the curvature on s . The crucial propertiesused there are indeed provided by the Agmon estimates, see below. The phase factor involved inthe gauge transformation is explicitly given by φ ε ( s, t ) := − ε (cid:90) t d η ν ( s ) · A GL ( r ( s, εη )) + 1 ε (cid:90) s d ξ γ (cid:48) ( ξ ) · A GL ( r ( ξ, − δ ε s. (5.4)The overall prefactor ε − in the energy is then inherited from the rescaling of the normalcoordinate τ = εt in the tubular neighborhood of the boundary. Note here the use of a differentconvention with respect to both [CR, FH1], where the tangential coordinate s was rescaled too.We need to rephrase some well-known decay estimates in a form suited to our needs. TheAgmon estimates proven in [FH2, Eq. (12.9)] can be translated into analogous bounds applyingto ψ ( s, t ) = Ψ GL ( r ( s, εt )) e − iφ ε ( s,t ) in A ε : for some constant A > (cid:90) A ε d s d t (1 − εk ( s ) t ) e At (cid:26) | ψ ( s, t ) | + (cid:12)(cid:12)(cid:12)(cid:16) ( ε∂ s , ∂ t ) + i ˜ A ( s,t ) ε (cid:17) ψ ( s, t ) (cid:12)(cid:12)(cid:12) (cid:27) = O (1) , (5.5)with (see, e.g., [CR, Eqs. (4.19) – (4.20)])˜ A ( s, t ) := (cid:0) (1 − εk ( s ) t ) γ (cid:48) ( s ) · A GL ( r ( s, εt )) + ε ∂ s φ ε (cid:1) e s . (5.6)In addition we are going to use two additional bounds proven in [FH2, Eq. (10.21) and (11.50)]: (cid:107) ψ (cid:107) L ∞ ( A ε ) ≤ , (cid:107) ( ε∂ s , ∂ t ) ψ (cid:107) L ∞ ( A ε ) ≤ C. (5.7)These bounds imply the following Lemma 5.2 ( Useful consequences of Agmon estimates ) . Let ¯ t = c | log ε | (1 + o (1)) for some c large enough, then for any a, b, s ∈ [0 , π ) , (cid:90) ba d s | ψ ( s, ¯ t ) | = O ( ε ∞ ) , (cid:90) c | log ε | ¯ t d t | ψ ( s , t ) | = O ( ε ∞ ) . (5.8) orreggi, Rougerie – Surface Superconductivity Proof.
We start by considering the first estimate: let χ ( t ) be a suitable smooth function withsupport in [ t , ¯ t ], with t = ¯ t − c , for some c >
0, and such that 0 ≤ χ ≤ χ (¯ t ) = 1 and | ∂ t χ | ≤ C .Then one has (cid:90) ba d s | ψ ( s, ¯ t ) | = (cid:90) ba d s χ (¯ t ) | ψ ( s, ¯ t ) | = (cid:90) ba d s (cid:90) ¯ tt d t [ χ ( t ) ∂ t | ψ ( s, t ) | + | ψ ( s, t ) | ∂ t χ ( t )] ≤ Ce − At (cid:26)(cid:20) (cid:90) A ε d s d t e At | ∂ t | ψ ( s, t ) || (cid:21) / + (cid:20) (cid:90) A ε d s d t e At | ψ ( s, t ) | (cid:21) / (cid:27) = O ( ε ∞ ) , (5.9)by (5.5), the diamagnetic inequality and the assumption on t and ¯ t . Indeed the factor e − At = ε Ac (1+ o (1)) can be made smaller than any power of ε by taking c large enough.For the second estimate we use a tangential cut-off function, i.e., a smooth monotone function χ ( s ) with support in [ s , π ], such that 0 ≤ χ ≤ χ ( s ) = 1, χ (2 π ) = 0, and | ∂ s χ | ≤ C . Then asin the estimate above (recall that t ε := c | log ε | ) (cid:90) t ε ¯ t d t | ψ ( s , t ) | = (cid:90) t ε ¯ t d t χ ( s ) | ψ ( s , t ) | = − (cid:90) πs d s (cid:90) t ε ¯ t d t [ χ ( s ) ∂ s | ψ ( s, t ) | + | ψ ( s, t ) | ∂ s χ ( s )] ≤ Ce − A ¯ t (cid:26) ε − (cid:20) (cid:90) A ε d s d t e At | ε∂ s | ψ ( s, t ) || (cid:21) / + (cid:20) (cid:90) A ε d s d t e At | ψ ( s, t ) | (cid:21) / (cid:27) = O ( ε ∞ ) , (5.10)where the main ingredients are again (5.5), the diamagnetic inequality and the assumption on ¯ t .We now introduce some reduced energy functionals defined over the cells we have introducedbefore, see Subsection 4.1 for the notation. We are going to perform an energy decoupling `a laLassoued-Mironescu [LM] in each cell: we write ψ ( s, t ) =: u n ( s, t ) f n ( t ) exp (cid:8) − i (cid:0) α n ε + δ ε (cid:1) s (cid:9) , (5.11)and introduce the reduced functionals E n [ u ] := (cid:90) C n d s d t (1 − εk n t ) f n (cid:110) | ∂ t u | + − εk n t ) | ε∂ s u | − εb n ( t ) J s [ u ] + b f n (cid:0) − | u | (cid:1) (cid:111) , (5.12)with b n ( t ) := t + α n − εk n t (1 − εk n t ) , (5.13)and J s [ u ] := ( iu, ∂ s u ) = i ( u ∗ ∂ s u − u∂ s u ∗ ) . (5.14)Note that in (5.12) the curvature is approximated by its mean value in the cell C n . Theseobjects play a crucial role in the sequel, as per Lemma 5.3 ( Lower bound in terms of the reduced functionals ) . With the previous notation G A ε [ ψ ] ≥ (cid:90) | ∂ Ω | d s E (cid:63) ( k ( s )) + N ε (cid:88) n =1 E n [ u n ] − Cε | log ε | ∞ (5.15) Let us assume that s − π > C >
0, otherwise one can take as a support for χ the complement set, i.e., [0 , s ]. orreggi, Rougerie – Surface Superconductivity Proof.
With the above cell decomposition, we can estimate G A ε [ ψ ] ≥ N ε (cid:88) n =1 E GL n [ ψ ] − Cε | log ε | ∞ , (5.16)where E GL n [ ψ ] := (cid:90) C n d s d t (1 − εk n t ) (cid:110) | ∂ t ψ | + − εk n t ) | ( ε∂ s + ia n ( t )) ψ | − b (cid:2) | ψ | − | ψ | (cid:3)(cid:111) , (5.17)and a n ( t ) := − t + εk n t + εδ ε . (5.18)The remainder term has been estimated as follows: the replacement of k ( s ) by k n produces twodifferent rests which can be estimated separately, i.e., (cid:90) C n d s d t ( k ( s ) − k n ) t (cid:110) | ∂ t ψ | − b (cid:2) | ψ | − | ψ | (cid:3)(cid:111) = O ( ε | log ε | ∞ ) , (5.19)1 ε (cid:90) C n d s d t (cid:110) − εk ( s ) t | ( ε∂ s + ia ε ( s, t )) ψ | − − εk n t | ( ε∂ s + ia n ( t )) ψ | (cid:111) = O ( ε | log ε | ∞ ) . (5.20)In estimating the first error term (5.19), we use the fact that k ( s ) − k n = O ( ε )and the bounds (5.7) together with the cell size. For the second estimate the same ingredients aresufficient as well, in addition to the simple boundsup ( s,t ) ∈C n | a ε ( s, t ) − a n ( t ) | ≤ Cε sup ( s,t ) ∈C n | k ( s ) − k n | | log ε | = O ( ε | log ε | ) . Inside any given cell C n we can then decouple the functional in the usual way (see [CR,Lemma 5.2] for a statement in this context) to obtain E GL n [ ψ ] = E (cid:63) ( k n ) (cid:96) ε + E n [ u n ] . (5.21)The first term in (5.21) is a Riemann sum approximation of the leading order term in (5.1): using(4.31), we immediately get N ε (cid:88) n =1 E (cid:63) ( k n ) (cid:96) ε = N ε (cid:88) n =1 E (cid:63) ( k n )( s n +1 − s n )= N ε (cid:88) n =1 (cid:90) s n +1 s n d s (cid:2) E (cid:63) ( k ( s )) + O ( ε | log ε | ∞ ) (cid:3) = (cid:90) | ∂ Ω | d s E (cid:63) ( k ( s )) + O ( ε | log ε | ∞ ) , (5.22)which concludes the proof. In view of our previous reductions in Lemma 5.3, the final lower bound (5.1) is a consequence ofthe following lemma orreggi, Rougerie – Surface Superconductivity Lemma 5.4 ( Lower bound on the reduced functionals ) . With the previous notation, we have N ε (cid:88) n =1 E n [ u n ] ≥ | log ε | − N ε (cid:88) n =1 (cid:90) C n d s d t (1 − εk n t ) f n (cid:104) | ∂ t u n | + − εk n t ) | ε∂ s u n | (cid:105) + 12 bε N ε (cid:88) n =1 (cid:90) C n d s d t (1 − εk n t ) f n (cid:0) − | u n | (cid:1) − Cε | log ε | ∞ (5.23)Proposition 5.1 now follows by a combination of Lemmas 5.1, 5.3 and 5.4 because the two sumsin the right-hand side of (5.23) are positive. These terms will prove useful to obtain our densityand degree estimates in Section 6.We can now focus on the proof of Lemma 5.4, which is the core argument of the proof ofProposition 5.1. Proof of Lemma 5.4.
The proof is split into two rather different steps. In the first one we essentiallyfollow the strategy of [CR, Section 5.2] to control the main part of the only potentially negative termin (5.12). This is done locally inside each cell and uses mainly the positivity of the cost function,Lemma A.4. This strategy however involves an application of Stokes’ formula and subsequentfurther integrations by parts to put the so obtained terms in such a form (involving only first orderderivatives, see (5.28)) that they can be compared with the kinetic one. This produces unphysicalsurface terms located on the boundaries of the (rather artificial) cells we have introduced. Thesecond step of the proof consists in controlling those, which requires to sum them all and reorganizethe sum in a convenient manner. It is in this step only that we cease working locally inside eachcell.
Step 1. Lower bound inside each cell.
First, we split the integration over two regions, onewhere a suitable lower bound to the density f n holds true and another one yielding only a verysmall contribution. More precisely we set R n := (cid:8) ( s, t ) ∈ C n : f n ( t ) ≥ | log ε | f n ( t ε ) (cid:9) . (5.24)Note that the monotonicity for large t of f n (see Proposition A.1) guarantees that R n := [ s n , s n +1 ] × [¯ t n,ε , t ε ] , ¯ t n,ε = t ε + O (log | log ε | ) . (5.25)Now we use the potential function F n ( t ) defined as F n ( t ) := 2 (cid:90) t d η (1 − εk n η ) f n ( η ) b n ( η ) = 2 (cid:90) t d η f n ( η ) η + α n − εk n η − εk n η , (5.26)and compute − ε (cid:90) C n d s d t (1 − εk n t ) f n ( t ) b k ( t ) J s [ u n ] = ε (cid:90) C n d s d t F n ( t ) ∂ t J s [ u n ] , (5.27)where we have exploited the vanishing of F n at t = 0 and t = t ε . Now we split the r.h.s. of theabove expression into an integral over D n := C n \ R n and a rest. In order to compare the first partwith the kinetic energy and show that the sum is positive, we have to perform another integrationby parts: ε (cid:90) D n d s d t F n ( t ) ∂ t J s [ u n ] = 2 ε (cid:90) ¯ t n,ε d t (cid:90) s n +1 s n d s F n ( t ) ( i∂ t u n , ∂ s u n )+ ε (cid:90) ¯ t n,ε d tF n ( t ) [ J t [ u n ]( s n +1 , t ) − J t [ u n ]( s n , t )] . (5.28) orreggi, Rougerie – Surface Superconductivity ε (cid:90) D n d t d s F n ( t ) ( i∂ t u n , ∂ s u n ) ≥ − (cid:90) D n d s d t | F n ( t ) | | ∂ t u n | | ε∂ s u n |≥ − (cid:90) D n d s d t (1 − εk n t ) F n ( t ) (cid:104) | ∂ t u n | + − εk n t ) | ε∂ s u n | (cid:105) , (5.29)where we have used the inequality ab ≤ ( δa + δ − b ) and the negativity of F n ( t ) (see Lemma A.3).Combining the above lower bound with (5.12) and (5.16) and dropping the part of the kinetic energylocated in R n , we get E n [ u n ] ≥ (cid:90) D n d s d t (1 − εk n t ) K n ( t ) (cid:20) | ∂ t u n | + − εk n t ) | ε∂ s u n | (cid:21) + ε (cid:90) ¯ t n,ε d tF n ( t ) [ J t [ u n ]( s n +1 , t ) − J t [ u n ]( s n , t )] + ε (cid:90) R n d s d t F n ( t ) ∂ t J s [ u n ]+ d ε (cid:90) C n d s d t (1 − εk n t ) f n (cid:104) | ∂ t u n | + − εk n t ) | ε∂ s u n | (cid:105) + 12 b (cid:90) C n d s d t (1 − εk n t ) f n (cid:0) − | u n | (cid:1) , (5.30)where K n ( t ) := K k n ( t ) , (5.31)is the cost function defined in (A.10), for some given d ε , satisfying (A.11). The third term in(5.30) is bounded from below by a quantity smaller than any power of ε , provided c is chosenlarge enough. This is shown using the same strategy as in [CR, Eq. (5.21) and following discussion]and we skip the details for the sake of brevity. For the first term we use the positivity of K n providedby Lemma A.4. We then conclude E n [ u n ] ≥ ε (cid:90) ¯ t n,ε d tF n ( t ) [ J t [ u n ]( s n +1 , t ) − J t [ u n ]( s n , t )]+ d ε (cid:90) C n d s d t (1 − εk n t ) f n (cid:104) | ∂ t u n | + − εk n t ) | ε∂ s u n | (cid:105) + 12 b (cid:90) C n d s d t (1 − εk n t ) f n (cid:0) − | u n | (cid:1) + O ( ε ∞ ) , (5.32)and there only remains to bound the first term on the r.h.s. from below. We are not actually ableto bound the term coming from cell n separately, so in the next step we put back the sum overcells. Step 2. Summing and controlling boundary terms.
We now conclude the proof of (5.23)by proving the following inequality: ε N ε (cid:88) n =1 (cid:90) ¯ t n,ε d tF n ( t ) [ J t [ u n ]( s n +1 , t ) − J t [ u n ]( s n , t )] ≥− C | log ε | − N ε (cid:88) n =1 (cid:90) C n d s d t (1 − εk n t ) f n (cid:104) | ∂ t u n | + − εk n t ) | ε∂ s u n | (cid:105) − Cε | log ε | ∞ . (5.33)Grouping (5.32) and (5.33), choosing d ε = 2 | log ε | − (which we are free to do) concludes the proof. orreggi, Rougerie – Surface Superconductivity ε N ε (cid:88) n =1 (cid:90) ¯ t n,ε d tF n ( t ) [ J t [ u n ]( s n +1 , t ) − J t [ u n ]( s n , t )]= ε N ε (cid:88) n =1 (cid:20) (cid:90) ¯ t n,ε d t [ F n ( t ) J t [ u n ]( s n , t ) − F n +1 ( t ) J t [ u n +1 ]( s n , t )] + R n (cid:21) , (5.34)where, assuming without loss of generality that ¯ t n,ε < ¯ t n +1 ,ε , R n := − (cid:90) ¯ t n +1 ,ε ¯ t n,ε d t F n +1 ( t ) J t [ u n +1 ]( s n +1 , t ) , (5.35)If on the other hand ¯ t n,ε > ¯ t n +1 ,ε , in (5.34) ¯ t n,ε should be replaced with ¯ t n +1 ,ε and in place of R n one would find (cid:90) ¯ t n,ε ¯ t n +1 ,ε d t F j ( t ) J t [ u j ]( s n +1 , t ) . In other words the remainder R n is inherited from the fact that the decomposition C n = D n ∪ R n clearly depends on n and the boundary terms in (5.34) do not compensate exactly. However it isclear from what follows that the estimate of such a boundary term is the same in both cases andessentially relies on the second inequality in (5.8): recalling that f n +1 ( t ) + F n +1 ( t ) ≥ t ≤ ¯ t n +1 ,ε , we have | R n | = (cid:90) ¯ t n +1 ,ε ¯ t n,ε d t | F n +1 ( t ) | | J t [ u n +1 ]( s n +1 , t ) | ≤ (cid:90) ¯ t n +1 ,ε ¯ t n,ε d tf n +1 ( t ) | u n +1 ( s n +1 , t ) | | ∂ t u n +1 ( s n +1 , t ) |≤ (cid:90) ¯ t n +1 ,ε ¯ t n,ε d t | ψ ( s n +1 , t ) | [ | ∂ t ψ ( s n +1 , t ) | + | u n +1 ( s n +1 , t ) | | ∂ t f n +1 ( t ) | ] ≤ C | log ε | (cid:90) ¯ t n +1 ,ε ¯ t n,ε d t | ψ ( s n +1 , t ) | = O ( ε ∞ ) , (5.36)where we have used the bounds (5.7) and (A.7), i.e., | f (cid:48) n +1 | ≤ | log ε | f n +1 ( t ). The identity (5.34)hence yields ε N ε (cid:88) n =1 (cid:90) ¯ t n,ε d tF n ( t ) [ J t [ u n ]( s n +1 , t ) − J t [ u n ]( s n , t )]= ε N ε (cid:88) n =1 (cid:90) ¯ t n,ε d t [ F n ( t ) J t [ u n ]( s n , t ) − F n +1 ( t ) J t [ u n +1 ]( s n , t )] + O ( ε ∞ ) . (5.37)Using now the definitions (5.11) of u n and u n +1 , we get u n +1 ( s, t ) = f n ( t ) f n +1 ( t ) e i ( α n +1 − α n ) s u n ( s, t ) , (5.38)so that J t [ u n +1 ]( s n , t ) = iG n,n +1 ( t ) G (cid:48) n,n +1 ( t ) | u n ( s n , t ) | + G n,n +1 ( t ) J t [ u n ]( s n , t ) , (5.39) orreggi, Rougerie – Surface Superconductivity G n,n +1 ( t ) := f n ( t ) f n +1 ( t ) . (5.40)Then we can compute ε (cid:90) ¯ t n,ε d t [ F n J t [ u n ]( s n , t ) − F n +1 J t [ u n +1 ]( s n , t )]= ε (cid:90) ¯ t n,ε d t (cid:2) F n ( t ) − F n +1 ( t ) G n,n +1 ( t ) (cid:3) J t [ u n ]( s n , t ) − iε (cid:90) ¯ t n,ε d t F n +1 ( t ) ∂ t (cid:0) G n,n +1 ( t ) (cid:1) | u n ( s n , t ) | , but we know that the l.h.s. of the above expression is real, so that we can take the real part of theidentity above obtaining ε (cid:90) ¯ t n,ε d t [ F n J t [ u n ]( s n , t ) − F n +1 J t [ u n +1 ]( s n , t )] = ε (cid:90) ¯ t n,ε d t (cid:2) F n − F n +1 G n,n +1 (cid:3) J t [ u n ]( s n , t ) . (5.41)To estimate the r.h.s. we integrate by parts back by introducing a suitable cut-off function. Let,for any given n = 1 , . . . , N ε , χ n ( s ) be a suitable smooth function, such that χ n ( s n ) = 1 , χ (cid:0) ( s n + s n +1 ) (cid:1) = 0and (cid:2) s n , ( s n + s n +1 ) (cid:1) ⊂ supp( χ n ) , | ∂ s χ n | ≤ Cε − . (5.42)We can rewrite ε (cid:90) ¯ t n,ε d t (cid:2) F n − F n +1 G n,n +1 (cid:3) J t [ u n ]( s n , t ) = ε (cid:90) ¯ t n,ε d t χ n ( s n ) (cid:2) F n − F n +1 G n,n +1 (cid:3) J t [ u n ]( s n , t )= ε (cid:90) ¯ t n,ε d t (cid:90) ( s n + s n +1 ) s n d s (cid:110) χ n ( s ) I n,n +1 ( t ) ∂ s ( J t [ u n ]) + ∂ s ( χ n ( s )) I n,n +1 ( t ) J t [ u n ] (cid:111) , (5.43)where we have set for short (compare with (3.39)) I n,n +1 ( t ) := F n ( t ) − F n +1 ( t ) G n,n +1 ( t ) = F n ( t ) − F n +1 ( t ) f n ( t ) f n +1 ( t ) . (5.44)The first contribution to (5.43) can be cast in a form analogous to (5.29): ε (cid:90) ¯ t n,ε d t (cid:90) ( s n + s n +1 ) s n d s χ n ( s ) I n,n +1 ( t ) ∂ s ( J t [ u n ])= ε (cid:90) ¯ t n,ε d t (cid:90) ( s n + s n +1 ) s n d s χ n ( s ) { I n,n +1 ( t ) ( i∂ s u n , ∂ t u n ) − ∂ t ( I n,n +1 ( t )) J s [ u n ] } + ε (cid:90) ( s n + s n +1 ) s n d s χ n ( s ) I n,n +1 (¯ t n,ε ) J s [ u n ]( s, ¯ t n,ε ) . (5.45) orreggi, Rougerie – Surface Superconductivity ε (cid:90) ¯ t n,ε d t (cid:90) ( s n + s n +1 ) s n d s χ n ( s ) I n,n +1 ( t ) ( i∂ s u n , ∂ t u n ) ≥ − (cid:90) D n d s d t | I n,n +1 ( t ) | | ε∂ s u n | | ∂ t u n |≥ − Cε | log ε | ∞ (cid:90) D n d s d t (1 − εk n t ) f n (cid:104) | ∂ t u n | + − εk n t ) | ε∂ s u n | (cid:105) , (5.46)where we have used (3.42) with k = k n , k (cid:48) = k n +1 and recalled that | k n − k n +1 | ≤ Cε to bound I n,n +1 . The last term in (5.45) can be easily shown to provide a small correction: using (3.42)again yields | I n,n +1 (¯ t n,ε ) | ≤ Cε | log ε | ∞ f n (¯ t n,ε ) , so that by (5.8) and (5.25) (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) ( s n + s n +1 ) s n d s χ n ( s ) I n,n +1 (¯ t n,ε ) J s [ u n ]( s, ¯ t n,ε ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:90) ( s n + s n +1 ) s n d s | I n,n +1 (¯ t n,ε ) | | J s [ u n ] |≤ Cε | log ε | ∞ (cid:90) ( s n + s n +1 ) s n d s f n (¯ t n,ε ) | u n ( s, ¯ t n,ε ) | | ∂ s u n ( s, ¯ t n,ε ) |≤ C | log ε | ∞ (cid:107) ε∂ s ψ (cid:107) ∞ (cid:90) ( s n + s n +1 ) s n d s | ψ ( s, ¯ t n,ε ) | = O ( ε ∞ ) , (5.47)where we have estimated the s -derivative of ψ by means of (5.7). Hence, combining (5.45) with(5.46) and (5.47), we can bound from below (5.43) as ε (cid:90) ¯ t n,ε d t (cid:2) F n − F n +1 G n,n +1 (cid:3) J t [ u n ]( s n , t ) ≥ ε (cid:90) ¯ t n,ε d t (cid:90) ( s n + s n +1 ) s n d s {− ∂ t I n,n +1 J s [ u n ] + ∂ s χ n I n,n +1 J t [ u n ] }− Cε | log ε | ∞ (cid:90) D n d s d t (1 − εk n t ) f n (cid:104) | ∂ t u n | + − εk n t ) | ε∂ s u n | (cid:105) + O ( ε ∞ ) . (5.48)To complete the proof it only remains to estimate the first two terms on the r.h.s. of the expressionabove, which again requires to borrow a bit of the kinetic energy. Using (3.29) we havesup t ∈ [0 , ¯ t n,ε ] (cid:12)(cid:12)(cid:12)(cid:12) ∂ t I n,n +1 f n (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cε | log ε | ∞ , so that ε (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) ¯ t n,ε d t (cid:90) ( s n + s n +1 ) s n d s ∂ t I n,n +1 J s [ u n ] (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cε | log ε | ∞ (cid:90) D n d s d t f n | u n | | ε∂ s u n |≤ Cε | log ε | ∞ (cid:90) D n d s d t (cid:104) δ − εk n t f n | ε∂ s u n | + δ | ψ | (cid:105) ≤ C | log ε | − (cid:90) D n d s d t − εk n t f n | ε∂ s u n | + O ( ε | log ε | ∞ ) , (5.49)where we have chosen δ = ε | log ε | a , for some suitably large a > | log ε | prefactor (this generates the coefficient | log ε | − ), and used (5.7) to estimate the remaining term. orreggi, Rougerie – Surface Superconductivity | ∂ s χ | ≤ Cε − , to get ε (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) ¯ t n,ε d t (cid:90) ( s n + s n +1 ) s n d s ∂ s χ n I n,n +1 J t [ u n ] (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cε | log ε | ∞ (cid:90) D n d s d t f n | u n | | ∂ t u n |≤ Cε | log ε | ∞ (cid:90) D n d s d t (cid:104) δ f n | ∂ t u n | + δ | ψ | (cid:105) ≤ C | log ε | − (cid:90) D n d s d t f n | ∂ t u n | + O ( ε | log ε | ∞ ) , (5.50)where we have made the same choice of δ as in (5.49).Collecting all the previous estimates yields our claim (5.33) (recall that there are N ε ∝ ε − terms to be summed, whence the final error of order ε | log ε | ∞ ). In this section we prove the main results about the behavior of | Ψ GL | close to the boundary of thesample ∂ Ω and an estimate of its degree at ∂ Ω.We first notice that the L estimate stated in (2.6) is in fact a trivial consequence of the energyasymptotics (2.5): putting together the lower bounds (5.2), (5.15) and (5.23) with the upper bound(4.1), we obtain 12 εb N ε (cid:88) n =1 (cid:90) C n d s d t (1 − εk n t ) f n (cid:0) − | u n | (cid:1) ≤ Cε | log ε | γ , (6.1)for some power γ large enough (recall the meaning of the notation | log ε | ∞ ). Now, using the factthat k n = k ( s ) (1 + O ( ε )) inside C n , we can easily reconstruct (2.6), once everything has beenexpressed in the original unscaled variables and the definitions (2.4) and (5.11) has been exploited(recall also that ψ ( s, t ) = Ψ GL ( r ( s, εt )). See also [CR, Section 4.2] for further details.We now focus on the refined density estimate discussed in Theorem 2.2 and the proof of Pan’sconjecture. The result is obtained via an adaptation of the arguments used in [CR, Section 5.3],originating in [BBH1]. The general idea is now rather standard so we will mainly comment on thechanges needed to make those argument work in the present setting. Proof of Theorem 2.2.
The two main ingredients of the proof are the above estimate (6.1) anda pointwise bound on the gradient of u n . Once combined, the two estimates imply that thefunction | u n | cannot be too far from 1 anywhere in the boundary layer A bl (see (2.8) for its precisedefinition). Step 1, gradient estimate.
A minor difference with the setting in [CR, Section 5.3] is due tothe convention we used to avoid a scaling of the tangential coordinate s . This is just a matter ofnotation and by following [CR, Proof of Lemma 5.3], we can show that, for any n = 1 , . . . , N ε , | ∂ t | u n || ≤ Cf − n ( t ) | log ε | , | ∂ s | u n || ≤ Cf − n ( t ) ε − . (6.2)Notice the second estimate above, which is a consequence of not scaling the coordinate s .We now prove (6.2). From the definitions of ψ and u n we immediately have | ∂ t | u n | ( s, t ) | ≤ f − n ( t ) | f (cid:48) n ( t ) | | ψ ( s, t ) | + f − n ( t ) | ∂ t | ψ ( s, t ) ||≤ Cf − n ( t ) (cid:2) | log ε | + | ∂ t | ψ ( s, t ) || (cid:3) , (6.3) orreggi, Rougerie – Surface Superconductivity | ∂ s | u n | ( s, t ) | ≤ f − n ( t ) | ∂ s | ψ ( s, t ) || where we have used [CR, Equation (A.28)]. The result is then a consequence of [Alm1, Theorem 2.1]or [AH, Equation (4.9)] in combination with the diamagnetic inequality (see [LL]), which yield (cid:12)(cid:12) ∇ (cid:12)(cid:12) Ψ GL (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:16) ∇ + i A GL ε (cid:17) Ψ GL (cid:12)(cid:12)(cid:12) ≤ Cε − = ⇒ | ∂ t | ψ ( s, t ) || + ε | ∂ s | ψ ( s, t ) || ≤ C. (6.4) Step 2, uniform bound on u n . We first observe that the estimate (cid:107) f n − f (cid:107) ∞ = O ( ε ) provenin (3.15) guarantees that f n ( t ) ≥ γ ε , for any ( s, t ) ∈ C n ∩ A bl and ∀ n = 1 , . . . , N ε . (6.5)Now we can apply a standard argument to show that | u n | can not differ too much from 1 in A bl .The proof is done by contradiction. We choose some 0 < c < a and define σ ε := ε / γ − / ε | log ε | c (cid:28) | log ε | c − a/ (cid:28) . (6.6)Suppose for contradiction that there exists a point ( s , t ) in C n ∩ A bl such that | − | u n ( s , t ) || ≥ σ ε . Then by (6.2) we can construct a rectangle-like region R ε ⊂ C n ∩ A bl of tangential length εγ ε σ ε (cid:28) ε and normal length (cid:37) ε with (cid:37) ε := γ ε σ ε | log ε | − (cid:28) ε / | log ε | c − − a/ (cid:28) | log ε | / , (6.7)where | − | u n ( s, t ) || ≥ σ ε . To complete the proof it suffices to estimate from below1 ε N ε (cid:88) n =1 (cid:90) C n d s d t (1 − εk n t ) f n (cid:0) − | u n | (cid:1) ≥ ε (cid:90) R ε d s d t (1 − εk n t ) f n (cid:0) − | u n | (cid:1) ≥ γ ε σ ε (cid:37) ε = γ ε σ ε | log ε | − = ε | log ε | c − (cid:29) ε | log ε | γ , (6.8)where γ is the power of | log ε | appearing in the r.h.s. of (6.1) and we have chosen c so that c ≥ ( γ + 3). Recalling the condition a > c we also have a > ( γ + 3), which coincides withthe assumption on γ ε (see (2.9)). Under such conditions the estimate above contradicts the upperbound (6.1) and the result is proven. Step 3, conclusion.
Now we know that in A bl ∩ C n || u n | − | ≤ σ ε , and it is easy to translate this estimate in an analogous one for | ψ ( s, t ) | and therefore | Ψ GL | .Indeed, in the cell C n | ψ | = | Ψ GL | = f n | u n | modulo a change of variables. The final estimate on | Ψ GL | then involves the reference profile g ref but the bound (cid:107) f n − f (cid:107) ∞ = O ( ε ) again allows the replacement of g ref with f .We can now turn to the proof of the estimate of the winding number of Ψ GL along ∂ Ω. orreggi, Rougerie – Surface Superconductivity Proof of Theorem 2.3.
Thanks to the positivity of g ref at t = 0 (see Lemma A.2) and the resultdiscussed above, Ψ GL never vanishes on ∂ Ω and therefore its winding number is well defined. Therest of the proof follows the lines of [CR, Proof of Theorem 2.4].The first part is the estimate of the winding number contribution of the phase φ ε involved inthe change of gauge ψ ( s, t ) = Ψ GL ( r ( s, εt )) e − iφ ε ( s,t ) but this can be done exaclty as in [CR, Proofof Lemma 5.4]:2 π deg (cid:0) Ψ GL , ∂ Ω (cid:1) − π deg ( ψ, ∂ Ω) = (cid:90) | ∂ Ω | d s γ (cid:48) ( s ) · ∇ φ ε ( s, t ) = (cid:90) | ∂ Ω | d s ∂ s φ ε ( s, φ ε (2 π, − φ ε (0 ,
0) = 1 ε (cid:90) | ∂ Ω | d s γ (cid:48) ( s ) · A GL ( r ( s, − | ∂ Ω | δ ε = 1 ε (cid:90) Ω d r curl A GL − | ∂ Ω | δ ε . (6.9)Now by the elliptic estimate [FH3, Eq. (11.51)] (cid:13)(cid:13) curl A GL − (cid:13)(cid:13) C (Ω) = O ( ε ) , and the Agmon estimate [FH1, Eq. (12.10)] (cid:13)(cid:13) ∇ (curl A GL − (cid:13)(cid:13) L (Ω \A ε ) = O ( ε ∞ ) , we get (cid:13)(cid:13) curl A GL − (cid:13)(cid:13) L ( A ε ) ≤ Cε | log ε | (cid:13)(cid:13) ∇ (cid:0) curl A GL − (cid:1)(cid:13)(cid:13) L ∞ (Ω) = O ( ε | log ε | ) , (cid:13)(cid:13) curl A GL − (cid:13)(cid:13) L (Ω \A ε ) ≤ C (cid:13)(cid:13) curl A GL − (cid:13)(cid:13) L (Ω \A ε ) ≤ C (cid:13)(cid:13) ∇ (curl A GL − (cid:13)(cid:13) L (Ω \A ε ) = O ( ε ∞ ) , (6.10)via Sobolev inequality. Altogether we can thus replace curl A GL with 1 in (6.9), so obtaining2 π deg (cid:0) Ψ GL , ∂ Ω (cid:1) − π deg ( ψ, ∂ Ω) = | Ω | ε + O ( | log ε | ) . (6.11)A minor modification in the proof is then due to the cell decomposition and the use of a differentdecoupling in each cell: the analogue of [CR, Lemma 5.4] is the following N ε (cid:88) n =1 (cid:90) s n +1 s n d s J s [ u n ]( s,
0) = O ( | log ε | ∞ ) . (6.12)To see that, we introduce a tangential cut-off function χ ( t ) with support contained in [0 , | log ε | − ]and such that 0 ≤ χ ≤ χ (0) = 1 and | ∂ t χ | = O ( | log ε | ). Then we compute (cid:90) s n +1 s n d s J s [ u n ]( s,
0) = (cid:90) s n +1 s n d s (cid:90) | log ε | d t [ ∂ t χJ s [ u n ]( s, t ) + χ∂ t J s [ u n ]( s, t )] = (cid:90) | log ε | d t (cid:26) (cid:90) s n +1 s n d s [ ∂ t χJ s [ u n ]( s, t ) + 2 χ ( i∂ t u n , ∂ s u n )] + J t [ u n ]( s n +1 , t ) − J t [ u n ]( s n , t ) (cid:27) (6.13) orreggi, Rougerie – Surface Superconductivity N ε (cid:88) n =1 (cid:90) s n +1 s n d s J s [ u n ]( s,
0) = N ε (cid:88) n =1 (cid:90) | log ε | − d t (cid:90) s n +1 s n d s [( ∂ t χ ) J s [ u n ]( s, t ) + 2 χ ( i∂ t u n , ∂ s u n )] − N ε (cid:88) n =1 (cid:90) | log ε | − d t [ J t [ u n +1 ]( s n +1 , t ) − J t [ u n ]( s n +1 , t )] . (6.14)The three terms on the r.h.s. of the above expression are going to be bounded independently. Wefirst observe that, exactly like we derived (6.1), one can also extract from the comparison betweenthe energy upper and lower bounds (see (5.23)) the following estimate: N ε (cid:88) n =1 (cid:90) C n d s d t (1 − εk n t ) f n (cid:110) | ∂ t u n | + − εk n t ) | ε∂ s u n | (cid:111) ≤ Cε | log ε | ∞ . (6.15)Then we can estimate the absolute value of the first two terms on the r.h.s. of (6.14) by using theCauchy-Schwarz inequality N ε (cid:88) n =1 (cid:90) | log ε | − d t (cid:90) s n +1 s n d s [ C | log ε || u n | | ∂ s u n | + 2 | ∂ t u n | | ∂ s u n | ] ≤ C N ε (cid:88) n =1 (cid:90) C n d s d t (1 − εk n t ) f n (cid:104) | log ε | | u n | + − εk n t ) | ∂ s u n | + | ∂ t u n | (cid:105) , (6.16)where we have exploited the pointwise lower bound (A.6), which implies f n ( t ) ≥ C > t ∈ [0 , | log ε | − ] and n = 1 , . . . , N ε , to put back the density f n in the expression. Now the bound f n | u n | = | ψ | ≤ N ε (cid:88) n =1 (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) | log ε | − d t (cid:90) s n +1 s n d s [( ∂ t χ ) J s [ u n ]( s, t ) + 2 χ ( i∂ t u n , ∂ s u n )] (cid:12)(cid:12)(cid:12)(cid:12) ≤ C | log ε | ∞ . (6.17)On the other hand the definition (5.11) of u n implies that (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) | log ε | − d t J t [ u n +1 ]( s n +1 , t ) − J t [ u n ]( s n +1 , t ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) | log ε | − d t (cid:18) f n +1 − f n (cid:19) J t [ | ψ | ]( s n +1 , t ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cε | log ε | ∞ (cid:90) | log ε | − d t | ψ || ∂ s | ψ || ≤ C | log ε | ∞ , (6.18)thanks to (3.15), the already mentioned lower bound on f n in [0 , | log ε | − ] and the standard bound (cid:107)∇ ψ (cid:107) ∞ ≤ Cε − (see, e.g., [FH3, Eq. (11.50)]).Hence (6.12) is proven and the rest of the proof is just a repetition of the estimates in [CR,Proof of Theorem 2.4]. Note that, as already anticipated in the comments after Theorem 2.3 α n = α (1 + O ( ε )), so that the optimal phases α n can all be replaced with α . A Useful Estimates on 1D Functionals
Here we recall some preliminary results obtained in [CR]. We start in Subsection A.1 with ele-mentary properties of the minimizing 1D profiles and carry on in Subsection A.2 by recalling thecrucial positivity property of the cost function we mentioned in Subsection 2.2. orreggi, Rougerie – Surface Superconductivity A.1 Properties of optimal phases and densities
This subsection contains a summary of results on the 1D minimization problem that follow fromrelatively standard methods. We start with the well-posedness of the minimization problem atfixed α . The following is [CR, Proposition 3.1]. Proposition A.1 ( Optimal density f k,α ) . For any given α ∈ R , k ≥ and ε small enough, there exists a minimizer f k,α to E k,α , unique upto sign, which we choose to be non-negative. It solves the variational equation − f (cid:48)(cid:48) k,α + εk − εkt f (cid:48) k,α + V k,α ( t ) f k,α = b (cid:16) − f k,α (cid:17) f k,α (A.1) with boundary conditions f (cid:48) k,α (0) = f (cid:48) k,α ( c | log ε | ) = 0 . Moreover f k,α satisfies the estimate (cid:107) f k,α (cid:107) L ∞ ( I ε ) ≤ and it is monotonically decreasing for t ≥ max (cid:104) , − α + √ b − Cε (cid:105) . In addition E k,α is a smoothfunction of α ∈ R and E k,α = − b (cid:90) I ε d t (1 − εkt ) f k,α ( t ) . (A.3)Next we consider the minimization problem as a function of the phase α , dealt with in [CR,Lemma 3.1]. Here Θ − is defined as in (1.5). Lemma A.1 ( Optimal phase α ( k )) . For any < b < Θ − , k ≥ and ε small enough, there exists at least one α ( k ) minimizing E k,α : inf α ∈ R E k,α = E k,α ( k ) =: E (cid:63) ( k ) . (A.4) Setting f k := f k,α ( k ) we have that f k > everywhere and (cid:90) I ε d t t + α ( k ) − εkt − εkt f k ( t ) = 0 . (A.5)We also use some decay and gradient estimates for the minimizing density. The following is acombination of [CR, Proposition 3.3 and Lemma A.1] Lemma A.2 ( Useful bounds on f k,α ) . For any < b < Θ − , k ∈ R and ε sufficiently small, there exist two positive constants c, C > independent of ε such that c exp (cid:110) − (cid:0) t + √ (cid:1) (cid:111) ≤ f k ( t ) ≤ C exp (cid:110) − ( t + α ) (cid:111) , (A.6) for any t ∈ I ε .Moreover there exists a finite constant C such that | f (cid:48) k ( t ) | ≤ C , for t ∈ (cid:104) , | α | + √ b (cid:105) , | log ε | f k ( t ) , for t ∈ (cid:104) | α | + √ b , c | log ε | (cid:105) . (A.7) orreggi, Rougerie – Surface Superconductivity A.2 Positivity of the cost function
A less standard part of our analysis in [CR] is the introduction of a cost function K k whosepositivity is one of the crucial ingredients of the energy lower bounds in the present paper.Let us first recall the definition of the potential function associated with f k : F k ( t ) := 2 (cid:90) t d η (1 − εkη ) f k ( η ) η + α ( k ) − εkη (1 − εkη ) , (A.8)which has the following properties [CR, Lemma 3.2]: Lemma A.3 ( Properties of the potential function F k ) . For any < b < Θ − , k ∈ R and ε sufficiently small, we have F k ( t ) ≤ , in I ε , F k (0) = F k ( t ε ) = 0 . (A.9)The cost function that naturally enters the analysis is then K k ( t ) = (1 − d ε ) f k ( t ) + F k ( t ) (A.10)where d ε is any parameter satisfying0 < d ε ≤ C | log ε | − , as ε → . (A.11)The positivity property we exploit is proved in [CR, Proposition 3.5]. Let¯ I k,ε := (cid:8) t ∈ I ε : f k ( t ) ≥ | log ε | f k ( t ε ) (cid:9) , (A.12)which is an interval in the t variable, i.e., ¯ I k,ε = [0 , ¯ t k,ε ] , (A.13)with ¯ t k,ε ≥ t ε − C log | log ε | = c | log ε | (cid:16) − O (cid:16) log | log ε || log ε | (cid:17)(cid:17) . (A.14)We then have Lemma A.4 ( Positivity of the cost function ) . For any d ε ∈ R + satisfying (A.11) , < b < Θ − , k > and ε sufficiently small, we have K k ( t ) ≥ , for any t ∈ ¯ I k,ε . (A.15) References [Abr]
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