The Distribution of Superconductivity Near a Magnetic Barrier

We consider the Ginzburg–Landau functional, defined on a two-dimensional simply connected domain with smooth boundary, in the situation when the applied magnetic field is piecewise constant with a jump discontinuity along a smooth curve. In the regime of large Ginzburg–Landau parameter and strong magnetic field, we study the concentration of the minimizing configurations along this discontinuity by computing the energy of the minimizers and their weak limit in the sense of distributions.


Introduction
1.1. Motivation. The Ginzburg-Landau theory, introduced in [LG50], is a phenomenological macroscopic model describing the response of a superconducting sample to an external magnetic field, when the sample is close to its critical temperature T c . The phenomenological quantities associated with a superconductor are the order parameter ψ and the magnetic potential A, where |ψ| 2 measures the density of the superconducting Cooper pairs and curl A represents the induced magnetic field in the sample.
In this paper, the superconducting sample is an infinite cylindrical domain subjected to a magnetic field with a direction parallel to the axis of the cylinder. For this specific geometry, it is enough to consider the horizontal cross section of the sample, ⊂ R 2 . The phenomenological configuration (ψ, A) is then defined on the domain .
The study of the Ginzburg-Landau model in the case of a uniform or a smooth nonuniform applied magnetic field has been the focus of much attention in the literature. We refer to the two monographs [FH10,SS07] for the uniform magnetic field case. Smooth magnetic fields are the subject of the papers [Att15a,Att15b,HK15, LP99,PK02]. Given the current interest in magnetic steps for various physical systems, we focus on the case where the applied magnetic field is a step function, which is not covered in the aforementioned papers.
Nonhomogeneous magnetic fields have been the focus of great amount of research. Current fabrication techniques allow the creation of such magnetic fields [FLBP94,STH+94,GGD+97], something that opens new paths in quantum physics and possible applications [RP98,JBY+97,MJR97]. Indeed, these magnetic fields appear in models involved in nanophysics such as in quantum transport in 2DEG (bidimensional electron gas) (see [PM93,RP00] and references therein) and in the Ginzburg-Landau model in superconductivity [SJST69]. More recently, piecewise constant magnetic fields are considered in the analysis of transport properties in graphene [GDMH+08,ORK+08]. Such magnetic fields are interesting because they induce snake states, carriers of edge currents flowing in the interface separating the distinct values of the magnetic fieldthe magnetic barrier (for instance see [HPRS16,HS15,DHS14,HS08,RP00,PM93]). While such edge currents have been discussed for linear problems in earlier works, the main contribution of this manuscript lies in establishing their existence in the context of the non-linear Ginzburg-Landau functional in superconductivity, by examining the presence of superconductivity along the magnetic barrier. Our configuration is illustrated in Fig. 1.
In an earlier contribution [AK16], we explored the influence of a step magnetic field on the distribution of bulk superconductivity, which highlighted the regime where an edge current might occur near the magnetic barrier. In this contribution, we will demonstrate the existence of such a current by providing examples where superconductivity concentrates at the interface separating the distinct values of the magnetic field.

The functional and the mathematical set-up.
We assume that the domain is open in R 2 , bounded, and simply connected. The Ginzburg-Landau (GL) free energy is given by the functional with ψ ∈ H 1 ( ; C) and A = (A 1 , A 2 ) ∈ H 1 ( ; R 2 ). Here, κ > 0 is a large GL parameter, the function B 0 : → [−1, 1] is the profile of the applied magnetic field, and H > 0 is the intensity of this applied magnetic field. The parameter κ depends on the temperature and the type of the material. It is a physical characteristic scale of the sample, the inverse of the penetration depth, and it measures the size of vortex cores (which is proportional to κ −1 , in some typical situations dependent on the strength of the applied magnetic field). Vortex cores are narrow regions in the sample, which corresponds to κ being a large parameter. That is the main reason behind our analysis of the asymptotic regime κ → +∞, following many early contributions addressing this asymptotic regime (see e.g. [SS07]). We work under the following assumptions on the domain and the magnetic field B 0 , which are quite generic as revealed from the illustration in Fig. 2. = ∂ 1 ∩ ∂ 2 is the union of a finite number of disjoint simple smooth curves { k } k∈K ; we will refer to as the magnetic barrier . (5) = ( 1 ∪ 2 ∪ ) • and ∂ is smooth.
∩ ∂ is either empty or finite . The ground state of the superconductor describes its behaviour at equilibrium. It is obtained by minimizing the GL functional in (1.1) with respect to (ψ, A). The corresponding energy is called the ground state energy, denoted by E g.st (κ, H ), where E g.st (κ, H ) = inf E κ,H (ψ, A) : (ψ, A) ∈ H 1 ( ; C) × H 1 ( ; R 2 ) .
One may restrict the minimization of the GL functional to the space H 1 ( ; C)× H 1 div ( ) where Indeed, the functional in (1.1) enjoys the property of gauge invariance. 1 Consequently, the ground state energy can be written as follows (see [ This restriction allows us to make profit from some well-known regularity properties of vector fields in H 1 div ( ) (see [AK16,Appendix B]).
1.3. Some earlier results for uniform magnetic fields. The value of the ground state energy E g.st (κ, H ) depends on κ and H in a non-trivial fashion. The physical explanation is that a superconductor undergoes phase transitions as the intensity of the applied magnetic field varies.
To illustrate the dependence on the intensity of the applied magnetic field, we assume that κ is large and H = bκ, for some fixed parameter b > 0. Such magnetic field strengths are considered in many papers (for instance see [AH07,LP99,Pan02,SS03]).
Assuming that the applied magnetic field is uniform, which corresponds to taking B 0 = 1 in (1.1), the following scenario takes place. If b > −1 0 , where 0 ≈ 0.59 is a universal constant defined in (2.5) below, the only minimizers of the GL functional are the trivial states (0,F), where curlF = 1 (see [GP99,LP99]). This corresponds in Physics to the destruction of superconductivity when the sample is submitted to a large external magnetic field, and occurs when the intensity H crosses a specific threshold value, the so-called third critical field, denoted by H C 3 .
Another well-known critical field is the second critical field H C 2 , which is much harder to define. When H < H C 2 , superconductivity is uniformly distributed in the interior of the sample (see [SS03]). This is the bulk superconductivity regime. When H C 2 < H < H C 3 , the surface superconductivity regime occurs: superconductivity disappears from the interior and is localized in a thin layer near the boundary of the sample (see [AH07,HFPS11,Pan02,CR14]). The transition from surface to bulk superconductivity takes place when H varies around the critical value κ, which we informally take as the definition of H C 2 (see [FK11]).
One more critical field left is H C 1 . It marks the transition from the pure superconducting phase to the phase with vortices. We refer to [SS07] for its definition.

Expected behaviour under magnetic steps.
Let us return to the case where the magnetic field is a step function as in Assumption 1.2. At some stage, the expected behaviour of the superconductor in question deviates from the one submitted to a uniform magnetic field. Recently, this case was considered in [AK16] and the following was obtained. Suppose that H = bκ and κ is large. If b < 1/|a| then bulk superconductivity persists ; if b > 1/|a| then superconductivity disappears in the bulk of 1 and 2 , and may nucleate in thin layers near ∪ ∂ (see Assumption 1.1 and Fig. 1). The present contribution affirms the presence of superconductivity in the vicinity of when b is greater than, but close to the value 1/|a|, for some negative values of a. The precise statements are given in Theorems 1.7 and 1.11 below.
The aforementioned behaviour of the superconductor in presence of magnetic steps is consistent with the existing literature about the electron motion near the magnetic barrier at which the strength and/or the sign of the magnetic fields change (for instance see [HPRS16,HS15,DHS14,RP00,Iwa85]). Particularly, the case where a ∈ [−1, 0) is called the trapping magnetic step (see [HPRS16]), where the discontinuous magnetic field may create supercurrents (snake orbits) flowing along the discontinuity edge. On the other hand, such supercurrents do not seem detectable in the case when a ∈ (0, 1), which is called the non-trapping magnetic step. However, the approach was generally spectral where some properties of relevant linear models were analysed (see [HPRS16,HS15,Iwa85,RP00]), and no estimates for the non-linear GL energy in (1.1) were established in these cases.
The contribution of this paper together with [AK16] provide such estimates. Particularly in the case when a ∈ [−1, 0) and b > 1/|a|, Theorems 1.7 and 1.11 below establish global and local asymptotic estimates for the ground state energy E g.st (κ, H ), and the L 4 -norm of the minimizing order parameter. These theorems assert the nucleation of superconductivity near the magnetic barrier (and the surface ∂ ) when b crosses the threshold value 1/|a|.
Remark 1.4. Our study does not cover the potentially interesting case a = 0, which deserves to be studied independently in a future work. This case, referred to as magnetic wall, was considered in [RP00,HPRS16].
Remark 1.5. Even though the case a ∈ (0, 1) is included in Assumption 1.3, it will not be central in our study (the reader may notice this in the majority of our theorems statements). The reason is that, our main concern is to analyse the interesting phenomenon happening when bulk superconductivity is only restricted to a narrow neighbourhood of the magnetic edge , and this only occurs when the values of the two magnetic fields interacting near are of opposite signs, that is when a ∈ [−1, 0). This can be seen through the trivial cases in Section 3.2, and is consistent with the aforementioned literature findings (non-trapping magnetic steps). Moreover, the case b < 1/|a| is treated previously in [AK16] and corresponds to the bulk regime.
The statements of the main theorems involve two non-decreasing continuous functions e a : |a| −1 , +∞) → (−∞, 0] and E surf : [1, +∞) → (−∞, 0] , respectively defined in (3.5) and (6.27) below. The energy E surf has been studied in many papers (for instance see [CR14,FKP13,FK11,HFPS11,AH07,Pan02]). We will refer to E surf as the surface energy. The function e a is constructed in this paper, and we will refer to it as the barrier energy. Remark 1.6. It is worthy of mention that e a (b) vanishes if and only if • a ∈ (0, 1) ; or • a ∈ [−1, 0) and b ≥ 1/β a , where β a is defined in (2.11) below and satisfies β a ∈ (0, |a| ) (see Theorem 2.6).
The surface energy E surf (b) vanishes if and only if b ≥ −1 0 , where 0 is the constant defined in (2.5).
The main contribution of this paper is summarized in Theorems 1.7 and 1.11 below.
Theorem 1.7 [Global asymptotics] For all a ∈ [−1, 1)\{0} and b > 1/|a|, the ground state energy E g.st (κ, H ) in (1.3) satisfies, when H = bκ, (1.7) Remark 1.8. In the asymptotics displayed in Theorem 1.7, the term | |b −1/2 e a (b) corresponds to the energy contribution of the magnetic barrier. The rest of the terms indicate the energy contributions of the surface of the sample. In light of Remark 1.6, the critical value b = β −1 a marks the transition between the superconducting and normal states along .
Remark 1.9. The edge creates vertices in the case where ∩ ∂ = ∅ (see Fig. 2) which may have non-trivial energy contributions hidden in the remainder term in (1.6). This case alters the breakdown of superconductivity too and shares some similarities with corner domains [BNF07,CG17,HK18,Ass].
Remark 1.10. Theorem 1.7 does not cover the case when the intensity of the magnetic field satisfies b = 1/|a|. However, we expect that some additional bulk terms will contribute to the estimate of the energy in this case, by analogy with [FK11].
Our next result, Theorem 1.11 below, describes the local behaviour of the minimizing order parameter ψ, thereby enhancing the statement in Theorem 1.7. We define the following distribution in R 2 , Here ds and ds denote the arc-length measures on and ∂ respectively.
Similarly as in [CR16b], we expect that the second correction term in the asymptotics in (1.8) will depend on the surface geometry of and ∂ , and will require a restrictive assumption on the way the support of the test function ϕ meets the edges and ∂ .

Discussion of the main results.
We will discuss the results in Theorems 1.7 and 1.11, in the interesting case where the magnetic barrier intersects the boundary of . Hence, we will assume that ∂ j ∩∂ = ∅ for j ∈ {1, 2}. When this condition is violated, the discussion below can be adjusted easily.
The following observations mainly rely on Remark 1.6 and the order of the values |a| 0 , 0 , β a , and |a|.
• For a = −1, we have β a = 0 < |a| [see (2.26)]. Consequently, in light of Remark 1.6: − If 1 < b < −1 0 , then the surface of the sample carries superconductivity and the entire bulk is in a normal state except for the region near the magnetic barrier (see Fig. 3). Moreover, the energy contributions of the magnetic barrier and the surface of the sample are of the same order and described by the surface energy, since in this case e a (b) = E surf (b), see Remark 3.12. This behaviour is remarkably distinct from the case of a uniform applied magnetic field. − If b ≥ −1 0 , then all the aforementioned energy contributions vanish, E L a (b) = 0. Fig. 4. Superconductivity distribution in the set subjected to a magnetic field B 0 , in the regime where a ∈ (− 0 , 0), H = bκ, and respectively |a| −1 < b < β −1 a and β −1 a ≤ b < |a| −1 −1 0 . The white regions are in a normal state, while the grey regions carry superconductivity • For a ∈ (−1, 0), comparing the values β a , 0 and |a| is more subtle. In (2.18), (2.23) and Theorem 2.6 below, we show that Moreover, numerical results about the variation of β a with respect to a show that β a is strictly decreasing for a ∈ [−1, 0) (see Fig. 5). 2 Having β −1 = 0 [see (2.26)], this suggests that β a < 0 for a ∈ (−1, 0). However, such a result is not rigorously established yet. With (1.9) in hand, Theorem 1.11 and Remark 1.6 indicate the following behaviour for a ∈ (− 0 , 0) and b > |a| −1 : − The part of the sample's surface near ∂ 1 ∩∂ does not carry superconductivity. − If |a| −1 < b < β −1 a , then surface superconductivity is confined to the part of the surface near ∂ 2 ∩ ∂ . At the same time, superconductivity is observed along the magnetic barrier (see Fig. 4). This behaviour is interesting for two reasons. Firstly, it demonstrates the existence of the edge current along the magnetic barrier, which is consistent with physics (see [HPRS16]). Secondly, it marks a distinct behaviour from the one known for uniform applied magnetic fields, in which case the whole surface carries superconductivity evenly (see for instance [HK17,FKP13,Pan02]). − If β −1 a ≤ b < |a| −1 −1 0 , then superconductivity only survives along ∂ 2 ∩ ∂ (see Fig. 4). Our results then display the strength of the applied magnetic field responsible for the breakdown of the edge current along the barrier. − If b ≥ |a| −1 −1 0 , then all energy contributions in Theorem 1.7 disappear. • For a ∈ (0, 1), β a = a [see (2.19)]. When b > a −1 , Theorem 1.11 reveals the absence of superconductivity along the magnetic barrier. As for the distribution of superconductivity along the surface of the sample, we distinguish between two regimes: Regime 1, a ∈ (0, 0 ] The part of the boundary, ∂ 1 ∩ ∂ , does not carry superconductivity. It remains to inspect the energy contribution of ∂ 2 ∩ ∂ . In that respect: 0 , then the entire surface of the sample is in a superconducting state, though the superconductivity distribution is not uniform.
0 , then all the energy contributions in Theorem 1.7 vanish.
• The letter C denotes a positive constant whose value may change from one formula to another. Unless otherwise stated, the constant C depends on the value of a and the domain , and is independent of κ and H . • Let a(κ) and b(κ) be two positive functions. We write a(κ) ≈ b(κ), if there exist constants κ 0 , C 1 and C 2 such that for all κ ≥ κ 0 , C 1 a(κ) ≤ b(κ) ≤ C 2 a(κ). • The quantity o(1) indicates a function of κ, defined by universal quantities, the domain , given functions, etc., and such that |o(1)| → 0 as κ → +∞. Any expression o(1) is independent of the minimizer (ψ, A) of (1.1). Similarly, O(1) indicates a function of κ, absolutely bounded by a constant independent of the minimizers of (1.1). • Let n ∈ N, p ∈ N, N ∈ N, α ∈ (0, 1), K ⊂ R N be an open set. We use the following Hölder space • Let n ∈ N, I ⊂ R be an open interval. We use the space (1.10) 1.7. Heuristics of the proofs. In this section, we present our approach in an informal way, not organized according to the order of appearance of various effective models in the paper, but following a scheme highlighting some important links between these models. We are mainly interested in examining the behaviour of the minimizer of the GL energy in (1.1) near the magnetic barrier . Working under Assumption 1.3, one can use the (Agmon) decay estimates established in [AK16] (see Theorem 2.4) to neglect the bulk energy contribution and restrict the study near the edge and the boundary ∂ .
As the applied magnetic field behaves uniformly near ∂ \ , the study of surface superconductivity is the same as that in the case of uniform fields, frequently encountered in the literature. Therefore in Sect. 6.2, the reader is referred to the existing literature.
The rest of the paper mainly focuses on the study of superconductivity in a tubular neighbourhood of . In Sect. 6, we decompose this neighbourhood into small cells, each of size O(κ −3/2 ), in order to establish the local asymptotics of the minimizer as well as the corresponding energy estimates as κ → +∞. This decomposition aims to reveal the existence of superconductivity in each of these small patches, in a certain regime of the applied magnetic field (i.e. for certain values of the parameter b, as in Assumption 1.3).
Using Frenet coordinates, cut-off functions, a suitable gauge transformation allowing to replace the induced magnetic field A by the applied magnetic field F (curl F = B 0 , see Lemma 2.2), together with a rescaling argument (Sects. 4-6), we may reduce the study of the GL energy in (1.1) into that of the 2D-effective energy G a,b,R defined on Here, x 1 and x 2 are respectively the tangential and the normal coordinates with respect to the magnetic edge. We also consider the ground state energy Hence, we launch an investigation of the new energy model, G a,b,R , with a step magnetic field. It is standard to begin by exploring the linear part of this energy, which leads us to the following linear magnetic Schrödinger operator defined in the plane (Sect. 2.4) The ground state energy corresponding to this operator is denoted by β a . One can easily see that the non-triviality of the energy G a,b,R minimizer (that is when g a (b, R) = 0) is equivalent to 1/|a| < b < 1/β a (under Assumption 1.3). Therefore, to ensure the non-emptiness of the interval (1/|a|, 1/β a ), thus the non-triviality of our study, we shall compare the values |a| and β a . In order to get the aforementioned comparison (of |a| and β a ), we use partial Fourier transform to perform a new reduction, this time of the 2D-operator L a to a 1D-effective operator in R, h a [ξ ], parametrized by ξ ∈ R (Sect. 2.4): and with a lowest eigenvalue denoted by μ a (ξ ). The ground state energy β a satisfies Next, we provide information about this infimum by collecting some spectral properties of the operator h a [ξ ]. This 1D-operator has already been considered in the literature, and some spectral information was established experimentally and rigorously in earlier works (for instance see [HPRS16,HS15,DHS14,RP00,Iwa85]). However, the approach in the aforementioned references was rather complicated, since all energy levels were examined. In addition, some of the spectral results we need in our study were not explicitly stated in these references. Therefore, for the sake of clarity and since we are only interested in the lowest eigenvalue, we opt to use a direct approach to provide such results (see Sect. 2.4). Moreover, our results slightly improve those of the aforementioned works (see Theorem 2.6). Our proofs call some spectral data of well-known effective models in the half-line (Sect. 2.3). From Sect. 2.4, we collect the following useful properties: • β a = a, for a ∈ (0, 1), Here, 0 is the value in (2.5). Now, the comparison of β a and a is in hand and a consequence of this is the following observation: We highlight the contribution of Theorem 2.6 in obtaining the latter property. This gives us the desired information about the values of a and b for which our study is non-trivial. Subsequently, we neglect the case a ∈ (0, 1) and proceed under the more restrictive assumption The main results about the reduced energy G a,b,R are stated in Theorem 3.1. In particular, this theorem introduces the limiting energy e a (b) appearing in our main theorems (Theorems 1.7 and 1.11): R .
In addition, the bounds in the last item of this theorem are important to control the error terms arising while establishing the energy and minimizer estimates in Sect. 6. The proof of Theorem 3.1 occupies Sect. 3. It relies on the approach in [Pan02,FKP13] in the case of uniform fields, with some additional technical difficulties caused by the discontinuity of our magnetic field. For instance, we step carefully while establishing some regularity properties needed in proving the existence of G a,b,R minimizer (see Lemmas B.3-B.6).
Finally, inspired by the recent work of Correggi-Rougerie [CR14] studying the surface superconductivity in the case of constant fields (more precisely by their energy lower bound proof), we interestingly prove that the 2D-limiting energy e a (b) is nothing but a one dimensional energy, E 1D a,b , defined in Sect. 3.6. This reduction serves in providing a more explicit definition of the enregy e a (b) and suggests that the profile of the minimizing order parameter ψ near the edge is as follows (up to a gauge transformation): where ( f 0 ,ξ 0 ) is a minimizing couple of the energy E 1D a,b,ξ defined in (3.16), s is the tangential distance along and t is the normal distance to . Such a profile suggests that the supercurrent along the edge , j = Im ψ(∇ − iκ H A)ψ , behaves to leading order as bκξ 0 f 0 (0) 2 τ , with τ being a unit tangent vector along the edge .
The rigorous derivation of (1.11) is not given in the present paper, but we expect that the analysis in this paper paves the way to a future investigation of the profile of ψ displayed in (1.11). In that respect, a special attention is required due to the nonhomogeneity of the order parameter ψ as revealed in Theorem 1.11; indeed ψ seems to have different profiles along and the parts of ∂ .
One remarkable aspect of our proofs is that we have not used the a priori elliptic L ∞ -estimate (∇ − iκ H A)ψ ∞ ≤ Cκ. Such estimate is not known to hold in our case of discontinuous magnetic field B 0 . Instead, we used the easy energy estimate (∇ − iκ H A)ψ 2 ≤ Cκ and the regularity of the curl-div system (see Theorem 2.3). This also spares us the complex derivation of the L ∞ -estimate (see [FH10,Chapter 11]).
1.8. Organization of the paper. Section 2 presents some preliminaries, particularly, a priori estimates, exponential decay results, and a linear 2D-operator with a step magnetic field. Theorem 2.6 is an improvement of a result in [HPRS16]. Section 3 introduces the 2D-reduced GL energy along with the barrier energy e a (·). In Sect. 4, we present the Frenet coordinates defined in a tubular neighbourhood of the curve . These coordinates are frequently used in the context of surface superconductivity (see [FH10,Appendix F]). In Sect. 5, we introduce a reference energy that describes the local behaviour of the full GL energy in (1.1). Section 6 is devoted for the analysis of the local behaviour of the minimizing order parameter near the edge . Also, we recall well-known results about the local behaviour of the order parameter near the surface ∂ . Finally, collecting all the estimates established in Sect. 6, we complete the proof of our main theorems (Theorems 1.7 and 1.11 above).

A priori estimates.
We present some celebrated estimates needed in the sequel to control the various errors arising while estimating the energy in (1.1). Recall the magnetic field B 0 introduced in Assumption 1.2. In the next lemma, we will fix the gauge for the magnetic potential generating B 0 (see [ We collect below some useful estimates whose proofs are given in [AK16, Theorem 4.2]. Theorem 2.3. Let α ∈ (0, 1) be a constant. Suppose that the conditions in Assumptions 1.1 and 1.2 hold. There exists a constant C > 0 (dependent on b) such that if (1.5) is satisfied and (ψ, 2.2. Exponential decay of the order parameter. The following theorem displays a regime for the intensity of the applied magnetic field where the order parameter and the GL energy are exponentially small in the bulk of the domains 1 and 2 .

A family of Sturm-Liouville operators on L
In this section, we will briefly present some spectral properties of the self-adjoint realization on L 2 (R + ) of the Sturm-Liouville operator: where B 2 (R + ) is the space introduced in (1.10), and ξ and γ are two real parameters. Denote by μ(γ , ξ ) the lowest eigenvalue of the operator The particular case where γ = 0 corresponds to the Neumann realization, and we use the following notation, and 2.4. An operator with a step magnetic field. Let a ∈ [−1, 1)\{0}. We consider the magnetic potential A 0 defined by which satisfies curl A 0 = 1. We define the step function σ as follows. For We introduce the self-adjoint magnetic Hamiltonian The ground state energy of the operator L a is denoted by (2.11) Since the Hamiltonian L a is invariant with respect to translations in the x 1 -direction then, by using the partial Fourier transform with respect to the x 1 -variable, we can reduce L a to a family of Shrödinger operators on L 2 (R), h a [ξ ], parametrized by ξ ∈ R and called fiber operators (see [HPRS16,HS15]). The operator h a [ξ ] is defined by . (2.17) Consequently, for all a ∈ [−1, 1)\{0}, we may express the ground state energy in (2.11) by Below, we collect some properties of the eigenvalue μ a (ξ ).
The case 0 < a < 1 This case is studied in [HS15,Iwa85]. The eigenvalue μ a (ξ ) is simple and is a decreasing function of ξ . The monotonicity of μ a (·) and its asymptotics in Proposition A.4 imply that and that β a , introduced in (2.11), satisfies The case a = −1 This case is studied in [HPRS16]. Using symmetry arguments, μ −1 (ξ ) is simple and satisfies where ξ −1 = −ξ(0) = − √ 0 , 0 and ξ(0) are respectively introduced in (2.5) and (2.7). The case −1 < a < 0 See also [HPRS16] for the study of this case. The eigenvalue μ a (ξ ) is simple, and there exists ξ a < 0 satisfying 3 Combining the foregoing discussion in the case a ∈ [−1, 0), we get that β a , introduced in (2.11), satisfies (2.25) In particular, In the next theorem, we will use a direct approach, different from the one in [HPRS16], to establish the existence of a global minimum ξ a in the case when a ∈ (−1, 0) and to prove that β a < |a|. Theorem 2.6 slightly improves the estimates in [HPRS16], since it provides an upper bound of β a stronger than |a|. This theorem is necessary to validate the hypothesis 1/|a| < 1/β a in (3.7), under which we work in Sect. 3. Indeed, it guarantees the existence of a non-empty b-parameter region where the minimizer of the reduced GL energy G a,b,R , introduced in Sect. 3, is non-trivial, which is key in the study of this energy.
Proof. The proof is inspired by [Kac07]. For all γ ∈ R, let (γ ) and ξ(γ ) be the quantities introduced in (2.4) and (2.6) respectively such that (γ ) = μ γ, ξ(γ ) . Denote by ϕ γ = ϕ γ,ξ(γ ) the positive L 2 -normalized eigenfunction of the operator in (2.2) with eigenvalue (γ ). Define the function where γ and m are two positive constants to be fixed later. One can check that u ∈ Dom q a [ξ ] , hence by the min-max principle, for all ξ ∈ R, . (2.28) Pick ξ ∈ R. We will choose ξ precisely later. The quadratic form q a [ξ ](u) defined in (2.14) can be decomposed as follows: A simple computation gives On the other hand, for t < 0, we do the change of variable y = − √ |a|t, which in turn . That way we get The definition of the function u in (2.27) yields Combining the results in (2.29)-(2.31) and using (2.7), we rewrite (2.28) as follows . Now we choose γ = √ 1/(2|a|(1 − |a|)) and m = √ |a|γ . Using again the fact that (γ ) < 1, we obtain The existence of the global minimum is now standard (it is a consequence of Proposition A.4 in the appendix).
Remark 2.7. Collecting the foregoing results in (2.19)-(2.23) and Theorem 2.6, we deduce the following facts regarding the bottom of the spectrum of the operator L a introduced in (2.10).

The functional and the main result.
Assume that a ∈ [−1, 1)\{0} is fixed, σ is the step function defined in (2.9) and A 0 is the magnetic potential defined in (2.8). For every R > 1, consider the strip We introduce the space Note that the boundary condition in the domain D R is meant in the trace sense. For b > 0, we define the following Ginzburg-Landau energy on D R by along with the ground state energy Our objective is to prove is the ground state energy in (3.4), and β a is defined in (2.11).
The following holds: The proof of Theorem 3.1, along with other properties of e a (b), will occupy the rest of this section.
3.2. The trivial case. We start by handling the trivial situation where the ground state energy vanishes: (1) Under the assumptions in Lemma 3.2, the function u = 0 ∈ D R is the unique minimizer of the functional in (3.3).

Proof of Lemma 3.2. We have the obvious upper bound
For the lower bound, pick an arbitrary function u ∈ D R and extend it by zero on R 2 . Using the min-max principle, we get 3.3. Existence of minimizers. Now we handle the following case (which is complementary to the one in Lemma 3.2) where β a is the lowest eigenvalue introduced in (2.11) . Under the hypothesis in (3.7), we can prove the existence of a non-trivial minimizer of the functional in (3.3) along with decay estimates at infinity.
The proof of Proposition 3.4 relies on the approach in [FKP13, Theorem 3.6] and [Pan02]. It can be described in a heuristic manner as follows. The unboundedness of the set S R makes the existence of the minimizer ϕ a,b,R in (3.8) non-trivial. To overcome this issue, we consider a reduced Ginzburg-Landau energy G a,b,R,m defined on the bounded set S R,m = (−R/2, R/2) × (−m, m), and we establish some decay estimates of its minimizer ϕ a,b,R,m . Later, using a limiting argument on G a,b,R,m and ϕ a,b,R,m for large values of m, we obtain the existence of the minimizer ϕ a,b,R together with the properties in Proposition 3.4. The details are given in Appendix B for the convenience of the reader.

The limit energy.
In this section, we will prove the existence of the limit energy e a (b), defined as the limit of g a (b, R)/R as R → +∞. After that, we will study, when the parameter a is fixed, some properties of the function b → e a (b).
In the sequel, we assume that a, b, R are constants such that R ≥ 1 and (3.7) holds. The next lemma displays some simple, yet very important, property of the energy. This property is mainly needed in Theorem 3.1 to establish an upper bound of the limit energy e a (b).
Lemma 3.5. Let n ∈ N. Consider the ground state energy g a (b, R) defined in (3.4), then Proof. Lemma 3.5 follows from the translation invariance of the integrand of G a,b,R with respect to the variable x 1 and the Dirichlet boundary conditions, where G a,b,R is defined in (3.3).
Our next result easily follows from the property of monotonicity with respect to the domain.
Lemma 3.6. The function R → g a (b, R) defined in (3.4) is monotone non-increasing.
The existence of the limit of g a (b, R)/R as R → +∞ will be derived from a wellknown abstract result (see [FK13, Lemma 2.2]). To apply this abstract result, we need some estimates on the energy g a (b, R), that we give in Lemmas 3.7 and 3.8 below.
Lemma 3.7. Let g a (b, R) be the ground state energy in (3.4). There exist positive constants C 1 , C 2 , and C 3 dependent only on a and b such that Multiplying this equation by ψ a θ 2 R and integrating by parts yield Taking the real part of each side of the equation above, we get Hence, using φ a L 2 (R) = 1 and the properties of θ R in (3.12), we obtain Consequently, for t = √ (1 − bβ a )/ν a and ν a = R |φ a (x 2 )| 4 dx 2 , we get where C 2 = (1/2)t 2 and C 3 = Cb/ν a . Lower bound. Let ϕ = ϕ a,b,R be the minimizer in Proposition 3.4. It follows from the min-max principle that By (3.10), S R |ϕ| 2 dx ≤ CbR, where C > 0 is some constant. Hence, choosing C 1 = C/β a establishes the desired lower bound.
Lemma 3.8. There exists a universal constant C such that, for all n ∈ N and α ∈ (0, 1), Proof. Let n ≥ 1 be a natural number, α ∈ (0, 1) and consider the family of strips Notice that the width of S j is 2(1 + α), and the overlapping occurs only between two adjacent strips (S j and S j−1 , for any j). There exists a universal constantC > 0 and a partition of unity (χ j ) j∈Z of R 2 such that and Since the overlapping is between a finite number of strips, one may further write where C is some universal constant. Define (3.14) where S R, j = {x R/2 : x ∈ S j }. The family of strips (S R, j ) j∈{1,2,...,n 2 } yields a covering of S n 2 R = −n 2 R/2, n 2 R/2 × R by n 2 strips, each of width (1 + α)R. Let ϕ a,b,n 2 R ∈ D n 2 R be the minimizer in Proposition 3.4. We decompose the energy associated to ϕ a,b,n 2 R as follows The first inequality above follows from the celebrated IMS localization formula (see [CFKS09, Theorem 3.2]), while the second comes from (3.10) and the properties of (χ R, j ) in (3.14). Notice that χ R, j ϕ a,b,n 2 R is supported in an infinite strip of width (1 + α)R. By energy translation invariance along the x 1 -direction, we have As a consequence, For R ≥ 1, dividing both sides by n 2 R and using the monotonicity of R → g a (b, R), we get 3.5. Proof of Theorem 3.1. Here we will verify all the statements appearing in Theorem 3.1. Noticing that G a,b,R (0) = 0, we get Item (1). The second item is already proven in Lemma 3.2.
Defining e a (b) = 0 for b ≥ 1/β a , the items (3) and (5) hold trivially since g a (b, R) = 0 in this case. We handle now the case where 1/|a| ≤ b < 1/β a . Define in R the two functions d a,b (l) = g a (b, l 2 ) and f a,b (l) = d a,b (l)/l 2 . Using Lemmas 3.6-3.8, we see that the functions d a,b (l) and f a,b (l) satisfy the following properties: • d a,b (·) is non-positive, monotone non-increasing, and f a,b (·) is bounded.
where C > 0 is a constant dependent on b and independent from l, n and α.
Then, by [FK13, Lemma 2.2], the following limit exists Moreover, for every integer n ≥ 1, Lemma 3.5 asserts that, Dividing both sides by n R and taking n → +∞ yields e a (b) ≤ g a (b, R)/R. By Lemma 3.7, e a (b) < 0 ; that the function e a (·) is monotone non-decreasing follows from the monotonicity of the function b → g a (b, R) ; the continuity of the function e a (·) follows from the estimates in (3.10) and the bounds in (3.6) (see [FKP13,Theorem 3.13]).

An effective one
We would like to find a relationship between the 2D-energy in (3.4) and the 1D-energy in (3.16) for some specific value of ξ . The existing results on the Ginzburg-Landau functional with a uniform magnetic field suggest that we should select ξ so as to minimize the function ξ → E 1D a,b (ξ ), see [AH07,CR14,Pan02]. In light of Remark 3.3, we will assume that a and b satisfy   1/μ a (ξ ). Furthermore, one can find a positive minimizer f a,b,ξ , dependent  on a and b, such that any minimizer has the form c f a,b,ξ where c ∈ C and |c| = 1.
(2) Any minimizer f of E 1D a,b,ξ satisfies f ∞ ≤ 1 and the equation: (3) For 1/|a| < b < 1/β a , there existsξ 0 , dependent on a and b, such that Remark 3.10. Guided by the numerical computations of [HPRS16, Sect. 1.3], we expect that: • the global minimum β a , defined in (2.18), is attained at a unique point ξ a ; • ξ a is the unique critical point of the function ξ → μ a (ξ ) .
However, such results have not been analytically proven yet.
The proof of Proposition 3.9 may be derived as done in [FH10, Sect. 14.2] devoted to the analysis of the following 1D-functional defined over the space B 1 (R + ). We introduce the energies where the remainder term O(1) depends on the geometry and is explicitly computed in [CR16a,CR16b,CDR17]. That has been conjectured by Pan [Pan02], then proven by Almog-Helffer and Helffer-Fournais-Persson [AH07,HFPS11] under a restrictive assumption on b, using a spectral approach. In the whole regime b ∈ (1, −1 0 ), the upper bound part in (3.19) easily holds (see [FH10,Sect. 14.4.2]), while the matching lower bound is more difficult to obtain and has been finally proven by Correggi-Rougerie [CR14]. The proof of Correggi-Rougerie, based on the non-negativity of a certain cost function, was markedly different from the spectral approach of [AH07,HFPS11].
Going back to our step magnetic field problem and the one dimensional energy in (3.15), we prove the following theorem.

20)
and E 1D a,b (·) is defined in (3.16). Remark 3.12. By a symmetry argument, Theorem 3.11 trivially holds in the case a = −1, namely To prove Theorem 3.11, we will adopt the method of [CR14], which relies on remarkable identities, including an energy splitting [LM99], along with the non-positivity of a certain potential function and the non-negativity of another cost function.
We propose the potential and cost functions of our problem. These are defined as follows, and whereξ 0 and f 0 = f a,b,ξ 0 are introduced in Proposition 3.9. We recall the set S R in (3.1) and the energy G a,b,R in (3.3) defined over the space D R in (3.2). Let u ∈ C ∞ 0 (S R ) (note that this space is dense in D R with respect to the norm (∇ −iσ A 0 )u L 2 (S R ) + u L 2 (S R ) ). Since f 0 > 0 on R (see Proposition 3.9), we may introduce the function v via the relation (3.23) Lemma 3.13. It holds
Proof. Note that (3.25) We will compute each term of G a,b,R (u) apart. Starting with (3.26) An integration by parts yields since the functions f 0 and f 0 vanish at ±∞. Plugging (3.27) in (3.26) and using the second item of Proposition 3.9, we find (3.28) Next, we compute the second term of G a,b,R (u) Moreover, by Proposition 3.9 we have We put (3.28)-(3.30) in (3.25) to complete the proof.
Lemma 3.14. Let F 0 and K 0 be the functions defined respectively in (3.21) and (3.22). where Since F 0 (0) = 0 and F 0 (±∞) = 0, we can handle the next term through an integration by parts: Now we handle the integral involving the term in (3.24). An integration by parts yields since u = 0 (and consequently v) for x 1 = ±R/2. We plug (3.32) into (3.31) and we use Cauchy's inequality to get since F 0 ≤ 0. This completes the proof in light of Lemma 3.13 and the definition of the function K 0 .
Now, looking at the expression of E 1 (v) in Lemma 3.14, we obtain Thus, if F 0 ≤ 0, F 0 (±∞) = 0 and K 0 ≥ 0, then we get the lower bound (3.33) Our next task is to verify these conditions. We have the following Feynman-Hellmann equation (see Proposition 3.9): which can be expressed as follows Regarding the function K 0 , we get immediately from (3.22), If we manage to prove that F 0 (±∞) = 0, then by the same argument in [CR14, Lemma 3.2 and Proposition 3.4], we may prove that F 0 ≤ 0 and K 0 ≥ 0. Such information is known in the particular case a = −1, thanks to symmetry considerations and [CR14]; indeed In the asymmetric case when a ∈ (−1, 0), one needs to work a little bit more for obtaining (3.37). The next lemma will be useful for establishing that F 0 (±∞) = 0.
Lemma 3.16 (Alternative expression of F 0 ). It holds Proof. For t ≤ 0 and a < 0, we have Similarly, one proves for t > 0 that Now, we use the Feynman-Hellmann equation in (3.35) and the vanishing of f 0 and f 0 at ∞ to get Since R 1 = 1/|a|R 2 , we conclude that R 1 = R 2 = 0.
Proof. From the definition of F 0 , we have F 0 (0) = 0. In addition, the alternative expression of F 0 in Lemma 3.16 and the decay and vanishing of f 0 and f 0 at ∞ imply that and similarly that F 0 (+∞) = 0. Next, we will study the variations of F 0 . Recall the derivative of F 0 We know that f 0 > 0 on R. Hence, assuming thatξ 0 ≥ 0 yields that F 0 (t) > 0 for all t > 0, which contradicts the fact that F 0 (0) = F 0 (+∞) = 0. This proves thatξ 0 < 0. Consequently, we find that F 0 < 0 in a right-neighbourhood of 0, and F 0 > 0 in a left-neighbourhood of 0. Since F 0 (0) = 0, we find that F 0 ≤ 0 in a neighbourhood of 0.
Remark 3.18. Along the proof of Lemma 3.17, we proved that anyξ 0 minimizing E 1D a,b (·) satisfiesξ 0 < 0. Now, we are ready to prove the non-negativity of the cost function K 0 . Proof. Lemma 3.17 and (3.36) simply imply that K 0 (±∞) = 0. Consequently if K 0 becomes negative at some point t, this definitely means the existence of a global minimum at some point t 0 in R * , since K 0 (0) > 0. We have then K 0 (t 0 ) < 0 and K 0 (t 0 ) = 0, where Since K 0 (t 0 ) = 0 and f 0 (t 0 ) > 0, we get that On the other hand, we may use the alternative expression of F 0 in Lemma 3.16 to write the function K 0 in the following form (3.39) Plug (3.38) into (3.39) to get Since a ∈ [−1, 0), b > 1/|a| and f 0 > 0 everywhere in R, then obviously K 0 (t 0 ) > 0 which is the desired contradiction.
Collecting the aforementioned lemmas, we can now prove Theorem 3.11.
The lower bound e a (b) ≥ E 1D a,b is a consequence of (3.33) after passing to the limit R → +∞.

The Frenet Coordinates
In this section, we assume that the set consists of a single simple smooth curve that may intersect the boundary of transversely in two points. In the general case, consists of a finite number of such (disjoint) curves. By working on each component separately, we reduce to the simple case above.
To study the energy contribution along , we will use the Frenet coordinates which are valid in a tubular neighbourhood of . For more details regarding these coordinates, see e.g. [FH10, Appendix F]. We will list the basic properties of these coordinates here.
Let The curvature k r of is defined by

A Local Effective Energy
In this section, we will introduce a 'local version' of the Ginzburg-Landau functional in (1.1). For this local functional, we will be able to write precise estimates of the ground state energy, which in turn will prove useful in estimating the ground state energy of the full functional in (1.1). Select a positive number t 0 sufficiently small so that the Frenet coordinates of Sect. 4 are valid in the tubular neighbourhood (t 0 ) defined in (4.4). Let 0 < c 1 < c 2 be fixed constants and be a parameter that is allowed to vary in such a manner that We will refer to (5.1) by writing ≈ κ −3/4 . Let s 0 ∈ − | | 2 , | | 2 . After performing a linear change of variable, we may assume that s 0 = 0 (for simplicity). For large values of κ, consider the neighbourhood of s 0 LetF be the magnetic potential defined in V( ) bỹ where σ = σ (s, t) was defined in (2.9). Consider the domain For u ∈ D , we define the (local) energy a −2 (∂ s − iκ HF 1 )u 2 + |∂ t u| 2 − κ 2 |u| 2 + κ 2 2 |u| 4 a ds dt , where a(s, t) = 1 − tk r (s). Now we introduce the following ground state energy Using standard variational methods, one can prove the existence of a minimizer u 0 of G.

Lower bound of G 0 .
Lemma 5.1. Under Assumption (5.7), there exist two constants κ 0 > 1 and C > 0 dependent only on a and b such that, if κ ≥ κ 0 and is as in (5.1), then where G 0 and e a (b) are defined in (5.6) and (3.5) respectively.
Proof. Notice that a(s, t) is bounded in the set V( ) as follows We apply Cauchy's inequality and the uniform bound of u to get where We introduce the parameters R = √ κ H , γ = √ κ Hs, τ = √ κ Ht, and define the re-scaled function In the new scale, we may write where G a,b,R is the functional in (3.3), andȗ ∈ D R the domain in (3.2) (since u ∈ D ). Invoking Theorem 3.1, we conclude that We plug the estimates (5.12) and (5.13) in (5.10), then we use e a (b) ≤ 0 and the assumptions on κ and to complete the proof of Lemma 5.1.

Upper bound of G 0 .
Lemma 5.2. Under Assumption (5.7), there exist two constants κ 0 > 1 and C > 0 dependent only on a and b such that, if κ ≥ κ 0 and is as in (5.1), then where G 0 and e a (b) are defined in (5.6) and (3.5) respectively.
Next, we define the following function (with the re-scaled variables) with γ = √ κ Hs, τ = √ κ Ht. Using (5.9) and (3.10), we get where J (u) was defined in (5.11), |v| 4 dγ dτ , Let χ R (τ ) = χ τ/R = χ t/ . We will estimate now each term of K(v) apart, using mainly the estimates on the minimizer ϕ in (3.10) and the properties of the function χ R . We start with the following two estimates that result from Cauchy-Schwarz inequality, Next, we may select R 0 sufficiently large so that, for all R ≥ R 0 , The decay of ϕ in (3.9), and (5.18) yield Finally, we write the obvious inequality Gathering the foregoing estimates, we get  1.1 and 1.2). This will be displayed by the local estimates of the Ginzburg-Landau energy and the L 4 -norm of the Ginzburg-Landau parameter in Theorem 6.1. We will introduce the necessary notations and assumptions. Starting with the local energy of the configuration (ψ, A) ∈ H 1 ( ; C) × H 1 div ( ), in any open set D ⊂ as follows (6.1) Let ≈ κ −3/4 satisfy (5.1) (for some fixed choice of the constants c 1 and c 2 ). For κ sufficiently large (hence sufficiently small), let x 0 ∈ \∂ be chosen so that Consider the following neighbourhood of x 0 , Thanks to (6.2), we have N x 0 ( ) ⊂ . As a consequence of the assumption in (6.2), all the estimates that we will write will hold uniformly with respect to the point x 0 . We assume that a ∈ [−1, 0) and b > 0 are fixed and satisfy b > 1 |a| .
(6.4) When (6.4) holds, we are able to use the exponential decay of the Ginzburg-Landau parameter away from the set and the surface ∂ (see Theorem 2.4).
where N x 0 (·) is the set in (6.3), E 0 is the local energy in (6.1), and e a (b) is the limiting energy in (3.5). Furthermore, the function r is independent of the point x 0 ∈ .
The proof of Theorem 6.1 follows by combining the results of Proposition 6.3 and Proposition 6.4 below, which are derived along the lines of [HK17,Sect. 4] in the study of local surface superconductivity.
Part of the proof of Theorem 6.1 is based on the following remark. After performing a translation, we may assume that the Frenet coordinates of x 0 are (s = 0, t = 0) (see Sect. 4). Recall the local Ginzburg-Landau energy E 0 introduced in (6.1). Let F be the vector field introduced in Lemma 2.2. We have the following relation where G is defined in (5.5), u ∈ H 1 0 (N x 0 ( )),ṽ = e −iκ H ωũ ,ũ is the function associated to u by the transformation −1 [see (4.5)], and ω = ω − , is the gauge function defined in Lemma 4.1.
6.1.1. Lower bound of the local energy. We start by establishing a lower bound for the local energy E 0 u, A; N x 0 ( ) for an arbitrary function u ∈ H 1 0 (N x 0 ( )) satisfying |u| ≤ 1. We will work under the assumptions made in this section, notably, we assume that (6.4) holds, and ≈ κ −3/4 [see (5.1)], and in the regime where H = bκ. Proposition 6.2. There exist two constants κ 0 > 1 and C > 0 such that, for κ ≥ κ 0 and for all x 0 ∈ satisfying (6.2), the following is true. If where N x 0 (·) is the neighbourhood in (6.3), E 0 is the functional in (6.1), and e a (b) is the limiting energy in (3.5).
Proof. Let α ∈ (0, 1) and F be the vector field introduced in Lemma 2.2. We define the function φ x 0 (x) = A(x 0 ) − F(x 0 ) · x. As a consequence of the fourth item in Theorem 2.3, we get the following useful approximation of the vector potential A We choose α = 2/3 in (6.8). Let h = e −iκ H φ x 0 u. Using (6.8), Cauchy's inequality, and the uniform bound |h| ≤ 1, we may write Finally, the lower bound in Lemma 5.1, together with the inequality e a (b) ≤ 0, yield the claim of the inequality.

Sharp upper bound on the L 4 -norm.
We will derive a lower bound of the local energy E 0 ψ, A; N x 0 ( ) and an upper bound of the L 4 -norm of ψ, valid for any critical point (ψ, A) of the functional in (1.1). Again, we remind the reader that we assume that (6.4) holds, ≈ κ −3/4 [see (5.1)] and H = bκ. Proposition 6.3. There exist two constants κ 0 > 1 and C > 0 such that, for all x 0 ∈ satisfying (6.2), the following is true.

Sharp lower bound on the L 4 -norm.
Complementary to Proposition 6.3, we will prove Proposition 6.4 below, whose conclusion holds for minimizing configurations only. We continue working under the assumption that (6.4) holds, ≈ κ −3/4 [see (5.1)] and H = bκ.
Proof. The proof is divided into five steps.
Step 1. Construction of a test function and decomposition of the energy. The construction of the test function is inspired from that by Sandier and Serfaty, in their study of bulk superconductivity in [SS03]. For γ = κ −3/16 andˆ = (1 + γ ) , we define the function , φ x 0 is the gauge function introduced in (6.8), ω = ω s 0 ,s 1 is the function introduced in Lemma 4.1 for s 0 = −ˆ and s 1 =ˆ , is the coordinate transformation in (4.3), u 0 is a minimizer of the functional G ·, V(ˆ ) defined in (5.5), and η is a smooth function satisfying (6.21) Recall the energies defined in (1.1) and (6.1). Let us write the obvious decomposition Adding the magnetic energy term κ 2 H 2 curl A − B 0 2 L 2 ( ) on both sides, we obtain the following identity, since the same magnetic energy term is present in both energies E κ,H (·, A) and E ·, A; N x 0 (ˆ ) . We denote by 2γ ) . Hence, we get the following decomposition of the functional in (1.1), Step 2. Estimating E 1 (u, A). Using (6.8) for α = 2/3, |v 0 | ≤ 1 and the Cauchy-Schwarz inequality, we write But by (6.7), we have E 0 v 0 , F; N x 0 (ˆ ) = G u 0 , V(ˆ ) . Hence, Lemma 5.2 and (6.22) imply Step 3. Estimating E 2 (u, A).
Then, we do a straightforward computation, similar to the one done in the proof of (6.14), replacing f by η and N x 0 (ˆ ) by N x 0 (˜ )\N x 0 (ˆ ). This gives the following relation between E 2 (u, A) and E 2 (ψ, A) Step 4. Estimating E 1 (ψ, A).
Step 5. Lower bound of the L 4 -norm of ψ. Consider the function f defined in (6.11).
We use the properties of this function, mainly that f = 1 in N x 0 ( ) and 0 ≤ f ≤ 1 in , to obtain Following an argument similar to the one for (6.13), we divide the set γ }, and we use this time the exponential decay of |ψ| 4 , deduced from Theorem 2.4, to get Inserting (6.26) into (6.16) gives The previous inequality together with (6.14) and (6.25) establish the lower bound in (6.19).

Surface superconductivity.
In this section, we are concerned in the local behaviour of the sample near the boundary of , under the assumption The analysis of superconductivity near ∂ in our case of a step magnetic field (B 0 satisfying 1.2) is essentially the same as that in the uniform field case, since B 0 is constant in each of 1 and 2 . Thereby, the results presented in this section are well-known in the literature since the celebrated work of Saint-James and de Gennes [SJG63]. We refer to [CG17,CR16a,CR16b,CR14,FKP13,FK11,HFPS11,AH07,FH05,Pan02,LP99] for rigorous results in general 2D and 3D samples subjected to a constant magnetic field, and to [NSG+09] for recent experimental results. Particularly, local surface estimates were recently established in [HK17], when B 0 ∈ C 0,α ( ) for some α ∈ (0, 1). We will adapt these results to our discontinuous magnetic field (see Theorem 6.5 below).
The statement of Theorem 6.5 involves the surface energy E surf , that we introduce in the next section.
where (γ , τ ) ∈ R 2 , A 0 (γ , τ ) = (−τ, 0), U R = (−R/2, R/2) × (0, +∞), and The boundary condition in the domain W(U R ) is meant in the trace sense. Let d(b, R) be the ground state energy defined by Pan proved in [Pan02] the existence of a non-decreasing continuous function E surf : Later, it was proven that (see e.g. [CR14]) . One important property of the function E surf (·) is (see [FH05]) This property allows us to extend the function E surf (·) continuously to [1, +∞), by setting it to zero on [ −1 0 , +∞). This extension of the surface energy is still denoted by E surf for simplicity.

Local surface superconductivity.
Let t 0 > 0 and j ∈ {1, 2}. We define the following set Assume that t 0 is sufficiently small, then for any x ∈ j (t 0 ), there exists a unique point , p(x)) .
Let ≈ κ −3/4 be the parameter in (5.1). Assume that κ is sufficiently large and choose We introduce the following neighbourhood of x 0 The assumption on x 0 in (6.30) guarantees that N j x 0 ( ) ⊂ j . Consequently, the estimates in Theorem 6.5 below hold uniformly with respect to the point x 0 .
where N j x 0 (·) is the set in (6.31), and E 0 is the local energy in (6.1). Furthermore, the functionȓ is independent of the point x 0 .
The estimates in Theorem 6.5 are established in [HK17], when the function B 0 is smooth. Since B 0 is constant in the neighbourhood N j x 0 ( ), the proof in [HK17] still holds in our case.

Proof of main results.
In this section, we work under the conditions of Theorems 1.7 and 1.11. We will gather the results of the two previous sections to establish the two aforementioned theorems.
Proof of Theorem 1.11. We will decompose the sample into the sets * ( ), * 1 ( ), analyse the behaviour of the minimizer in each of these sets. We assume to be the parameter in (5.1) which satisfies ≈ κ −3/4 . In a neighbourhood of the magnetic barrier We start by introducing the set * = * ( ) which covers almost all of the set . Recall the assumption that consists of a finite collection of simple disjoint smooth curves that may intersect ∂ transversely. For the simplicity of the exposition, we will focus on the particular case of a single curve intersecting ∂ at two points. The construction below may be adjusted to cover the general case by considering every single component of separately. We may select two constants 0 ∈ (0, 1) and c > 2, and for all ∈ (0, 0 ), a collection of pairwise distinct where N x i ( ) is the set introduced in (6.3). The family N x i ( ) 1≤i≤N consists of pairwise disjoint sets. The number N depends on as follows In a neighbourhood of the boundary Now, we define the two sets * 1 = * 1 ( ) and * 2 = * 2 ( ) which cover almost all of the set ∂ . In a similar fashion to the definition of * ( ), we fix 0 ∈ (0, 1) and c > 2 and we select two collections of points for 1 ≤ j ≤ N 1 − 1 and 1 ≤ k ≤ N 2 − 1. Furthermore, where N 1 y j ( ) and N 2 z k ( ) are defined in (6.31). The numbers N 1 and N 2 depend on as follows The bulk set Next, we introduce the set bulk = bulk ( ) representing the bulk of the sample: In a neighbourhood of the T -zone We finally introduce the remaining set in the decomposition of , the neighbourhood T = T ( ) of ∩ ∂ The definition of the sets * , * 1 , * 2 and bulk in (6.32), (6.34), (6.35) and (6.36) ensures that |T | = O( 2 ) as → 0. Behaviour of the minimizer Now, we are ready to prove the convergence of |ψ| 4 in the sense of distributions, claimed in Theorem 1.11.
Let ϕ ∈ C ∞ c (R 2 ). We have We will estimate each of these right hand side integrals. Starting with C is a constant independent of κ.
Next, we have [see (6.32)] For i ∈ {1, . . . , N }, let p i and q i be two points of such that We may write We estimate |ϕ(x) − ϕ( p i )| in N x i ( ) by the mean value theorem. Using the size of N x i ( ) and the bound ψ L ∞ ( ) ≤ 1, we get for some C independent of κ. Hence, by (5.1) and (6.33) On the other hand, using the uniform bounds in (6.10) and (6.19), we get where C is a constant independent of κ. We use further that and N = O(1) by (6.33). We get that whereC is a new constant independent of κ. Combining (6.40)-(6.43) yields (6.45) We combine (6.44) and (6.45) to obtain But by (6.33) Hence, One can proceed similarly to prove that and κ * 2 Gathering pieces in (6.38), (6.39), (6.46) and (6.47), we establish Theorem 1.11.
Proof of Theorem 1.7 We apply Theorem 1.11 for ϕ ∈ C ∞ c (R 2 ) such that ϕ = 1 in a neighbourhood of to get (1.7).
Multiplying both sides of the first equation in (1.4) by ψ then integrating by parts give where E(ψ, A; ·) and E 0 (ψ, A; ·) are the energies in (6.1). Using (6.48) and (1.7), we get the lower bound of E g.st (κ, H ) in (1.6). The upper bound of E g.st (κ, H ) can be derived by the help of a suitable trial configuration. We are still considering the parameter as in (5.1). Let F be the magnetic potential introduced in Lemma 2.2. We define the function h ∈ H 1 ( ; where * ( ) and N x i ( ) are respectively the sets in (6.32) and (6.3), v i (x) = e iκ H ω u i • −1 (x), ω = ω − , is the gauge function in Lemma 4.1, is the coordinate transformation in (4.3), u i is defined by u i (s, t) = u 0 (s − s i , t) for (s i , t i ) = −1 (x i ), and u 0 is the minimizer of G(·, V( )) defined in (5.5). From the definition of v i , we derive the following [see (6.7)] The previous identity together with Lemma 5.2, (6.33) and ( ≈ κ −3/4 ) give (6.49) Similarly, for j ∈ {1, 2}, using the results of Theorem 6.5, one may define a function where * j ( ) is defined in (6.34) and (6.35). Now, we define the trial function Noticing that E g.st (κ, H ) ≤ E h, F; = E 0 h, F; [see (6.1)], we gather the results in (6.49) and (6.50) to derive the upper bound in (1.6). and lowest eigenvalue The perturbation theory [Kat66] ensures that the functions are analytic, where μ(γ , ξ ) and μ N (ξ ) are respectively defined in (2.3) and (2.5). We list the following well-known spectral properties (for instance see [DH93,RS72,Kac06]): Proposition A.2. The lowest eigenvalue μ a (ξ ) of h a [ξ ] is simple. Furthermore, there exists a positive eigenfunction g a,ξ normalized with respect to the norm · L 2 (R) . g a,ξ is the unique function satisfying such properties.
The bounds in Lemma A.3 are useful for establishing Proposition A.4, which is crucial in our study of the eigenvalue μ a (ξ ) (see Sect. 2.4).
• If a ∈ [−1, 0), then Proof. We will prove the lemma in the case a ∈ (−1, 0). The proof follows similarly in the case a ∈ (0, 1). We start by establishing the upper bound in (A.3). Let ξ ∈ R. Consider u = u D −ξ the normalized eigenfunction of the operator H D [−ξ ] defined in (A.1), corresponding to the lowest eigenvalue μ D (−ξ). Then Noticing that u ∈ H 1 0 (R + ), we extend it by zero on R − (the extension is still denoted by u for simplicity). Hence, we have q a [ξ ](u) = μ D (−ξ), where q a [ξ ] is the quadratic form in (2.14). Using the min-max principle, we get .
sponding to the lowest eigenvalue μ D (−ξ/ √ |a|), we can prove that by the min-max principle, after employing the change of variable x = −t/ √ |a| and extending the resulting function by 0 on R + .
Next, we establish the lower bound in (A.3). We consider g = g a,ξ the normalized eigenfunction of the operator h a [ξ ] corresponding to the lowest eigenvalue μ a (ξ ) (see Proposition A.2). We have Using the min-max principle, we write a lower bound for each integral appearing in (A.4). Indeed, where g a,ξ is the eigenfunction in Proposition A.2.
Proof. (Feynman-Hellmann) For simplicity, we write μ, g and h respectively for μ a (ξ ), g a,ξ , and h a [ξ ] . Differentiating with respect to ξ and integrating by parts in Hence using and recalling that g is normalized, we obtain Integrating by parts the right hand side of (A.9), and using (A.8) establish the result.
Proof. Let ξ a be such that β a = μ a (ξ a ) (see [HPRS16]). We use the lower bound proof of Lemma A.3, with g = g a,ξ a the positive normalized eigenfunction of the operator h a [ξ a ] corresponding to μ a (ξ a ) (see Proposition A.2). We get Since g is normalized and positive, and |a| 0 < 0 for a ∈ (−1, 0), the proof is completed.
Proof. Suppose that ξ a ≥ 0. Let g a,ξ a be the positive normalized eigenfunction of the operator h a [ξ a ] corresponding to the lowest eigenvalue μ a (ξ a ) (see Proposition A.2).

Appendix B. Decay Estimates for the 2D-Effective Model
The aim of this appendix is to prove Proposition 3.4. Recall that we work under (3.7), namely, where β a is the lowest eigenvalue introduced in (2.11) . For every m ∈ N and R > 1, we introduce the set S R,m = (−R/2, R/2) × (−m, m) and the functional defined over the space Here σ was defined in (2.9). Now we define the ground state energy  (ϕ a,b,R,m ) = g a (b, R, m) .
Here G a,b,R,m is the functional introduced in (B.1) and g a (b, R, m) is the ground state energy introduced in (B.3).
Proof. The boundedness and the regularity of the domain S R,m guarantee the existence of a minimizer ϕ m := ϕ a,b,R,m of G a,b,R,m in D R,m , satisfying Multiply (B.8) by χ 2 ϕ m and integrate by parts, (B.10) It follows from (B.9) and (B.10) Using Hölder's inequality, (B.12) Now, using Cauchy-Schwarz inequality together with (B.11) and (B.12), we obtain Consequently, under the assumption 1 ≤ 1/|a| ≤ b < 1/β a , we get (B.6). Inserting (B.6) into (B.12), we get We still need to establish To that end, we select η ∈ C ∞ (R) such that η(x 2 ) = 0 if |x 2 | ≤ 1, and η(x 2 ) = √ |x 2 |/ ln |x 2 | if |x 2 | ≥ 4. Multiplying the equation in (B.8) by η 2 ϕ m and integrating It is easy to check by a straightforward computation, and using Cauchy's inequality, that Proof. For the sake of brevity, we will write ϕ m for ϕ a,b,R,m . Using (B.14) and the fact that |x 2 |/ ln |x 2 | 2 ≥ 1 for |x 2 | ≥ 4, we get On the other hand, using ϕ m ∞ ≤ 1 and b > 1 we get Next, since ϕ m satisfies a simple integration by parts over S R,m yields Now, we will investigate the regularity of the minimizer ϕ a,b,R,m in Lemma B.1. We have to be careful at this point since the magnetic field is a step function and therefore has singularities. As a byproduct, we will extract a convergent subsequence of (ϕ a,b,R,m ) m≥1 .
We will use the following terminology.  1 and α ∈ (0, 1) be fixed. The sequence ϕ a,b,R,m m≥1 defined by Lemma B.1 is bounded in H 3 loc (S R ) and consequently in C 1,α loc (S R ). Proof. For simplicity, we will write ϕ m = ϕ a,b,R,m . The proof is split into three steps.
Step 1. We first prove the boundedness of ϕ m in H 2 loc (S R ). Using (B.8) we may write Let K ⊂ S R be open and relatively compact. Choose an open and bounded set K such that K ⊂ K ⊂ S R . There exists m 0 ∈ N such that for all m ≥ m 0 , K ⊂ S R,m and by Cauchy's inequality, Using |ϕ m | ≤ 1, the decay estimate in (B.19) and the boundedness of σ and A 0 in K , we get a constant C = C( K , R) such that where C is a constant independent from m. This proves that (ϕ m ) m≥1 is bounded in H 2 loc (S R ).
Step 2. Here we will improve the result in Step 1 and prove that (ϕ m ) m≥1 is bounded in H 3 loc (S R ). It is enough to prove that the sequence ∇ϕ m m≥1 is bounded in H 2 loc (S R ). Let ς m = ∂ x 2 ϕ m . We will prove that ς m m≥1 is bounded in L 2 loc (S R ). Recall that, for all x = (x 1 , x 2 ) ∈ R 2 , A 0 (x) = (−x 2 , 0) and σ (x) = 1 R + (x 2 ) + a1 R − (x 2 ) , hence, in the sense of weak derivatives. By Step 1, the sequence (ϕ m ) is bounded in H 2 loc (S R ). Consequently, since |ϕ m | ≤ 1, it is clear that ( ς m ) m≥1 is bounded in L 2 loc (S R ). By the interior elliptic estimates, we get that (ς m = ∂ x 2 ϕ m ) m≥1 is bounded in H 2 loc (S R ). In a similar fashion, we prove that (∂ x 1 ϕ m ) m≥1 is bounded in H 2 loc (S R ).
Step 3. Finally, for every relatively compact open set K ⊂ , the space H 3 (K ) is embedded in C 1,α (K ). Consequently, ϕ m is bounded in C 1,α loc (S R ).
Lemma B.4. Assume that R > 1 and that (3.7) holds. Let ϕ a,b,R,m m≥1 be the sequence defined in Lemma B.1. There exist a function ϕ a,b,R ∈ H 3 loc (S R ) and a subsequence, denoted by ϕ a,b,R,m m≥1 , such that ϕ a,b,R,m −→ ϕ a,b,R in H 2 loc (S R ) and ϕ a,b,R,m −→ ϕ a,b,R in C 0,α loc (S R ) α ∈ (0, 1) .
Proof. We continue writing ϕ m for ϕ a,b,R,m . Let K ⊂ S R be open and relatively compact. By Lemma B.3, (ϕ m ) m≥1 is bounded in H 3 (K ), hence it has a weakly convergent subsequence by the Banach-Alaoglu theorem. By the compact embedding of H 3 (K ) in H 2 (K ), and of H 2 (K ) in C 0,α (K ), we may extract a subsequence, that we denote by (ϕ m ), such that it is strongly convergent in H 2 (K ) and C 0,α (K ). The subsequence in Lemma B.4 and its limit are then constructed via the standard Cantor's diagonal process.
Lemma B.5. Let R > 1 and ϕ a,b,R be the function defined by Lemma B.4. The following statements hold: where C > 0 is a universal constant, and D R is the space introduced in (3.2).
Proof. Let (ϕ a,b,R,m ) be the subsequence in Lemma B.4. Again, we will use (ϕ m ) and ϕ for (ϕ a,b,R,m ) and ϕ a,b,R respectively.
By Lemma B.1, the inequality |ϕ m | ≤ 1 holds for all m. The inequality |ϕ| ≤ 1 then follows from the uniform convergence of (ϕ m ) stated in Lemma B.4. By the convergence of ϕ m in H 2 loc (S R ) and C 0,α loc (S R ), we get (B.27) from −b(∇ − iσ A 0 ) 2 ϕ m = (1 − |ϕ m | 2 )ϕ m . Taking m 0 → +∞, we write by the monotone convergence theorem, This proves that ϕ ∈ L 2 (S R ). Next we will prove that (∇ − iσ A 0 )ϕ ∈ L 2 (S R ). In light of the convergence of (ϕ m ) in H 1 loc (S R ), we can refine the subsequence (ϕ m ) so that (∇ − iσ A 0 )ϕ m → (∇ − iσ A 0 )ϕ a.e. Furthermore, by Lemma B.3, ϕ m is bounded in C 1 loc (S R ), hence in C 1 (S R,m 0 ), for all m 0 ≥ 1. Using the dominated convergence theorem and the estimate in (B.19), we may write, for all m 0 ≥ 1, Sending m 0 to +∞ and using the monotone convergence theorem, we get Thus, we have proven that ϕ, (∇ −iσ A 0 )ϕ ∈ L 2 (S R ). It remains to prove that ϕ satisfies the boundary condition To see this, let x 2 ∈ R. There exists m 0 such that x 2 ∈ (−m 0 , m 0 ). By the convergence of (ϕ m ) to ϕ in C 0,α (S R,m 0 ), we get Finally, we may use similar limiting arguments to pass from the decay estimates of ϕ m in (B.5) and (B.6) to the decay estimates of ϕ in (B.28) and (B.29). Now, we are ready to establish the existence of a minimizer of the Ginzburg-Landau energy G(a, b, R) defined in the unbounded set S R . Lemma B.6. Let R > 1. The function ϕ a,b,R ∈ D R defined in Lemma B.4 is a minimizer  of G a,b,R , that is   G a,b,R (ϕ a,b,R ) = g a (b, R).
Here G a,b,R is the functional introduced in (3.3) and g a (b, R) is the ground state energy defined in (3.4).
Proof. The proof is divided into three steps.
Step 1. (Convergence of the ground state energy) . Let g a (b, R, m) and g a (b, R) be the energies defined in (B.3) and (3.4) respectively. In this step, we will prove that lim m→+∞ g a (b, R, m) = g a (b, R) . (B.31) Let u ∈ D R,m . We can extend u by 0 to a functionũ ∈ D R . As an immediate consequence, we get g a (b, R, m) ≥ g a (b, R), for all m ≥ 1. Thus, lim inf m→+∞ g a (b, R, m) ≥ g a (b, R). Next, we will prove that lim sup m→+∞ g a (b, R, m) ≤ g a (b, R) . (B.32) Consider (ϕ n ) ⊂ D R a minimizing sequence of G a,b,R , that is g a (b, R) = lim n→+∞ G a,b,R (ϕ n ).
Consider the re-scaled function ϑ m (x 2 ) = ϑ(x 2 /m). The function ϑ m (x 2 )ϕ n (x) restricted to S R,m belongs to D R,m and consequently Taking the successive limits → 0 + and n → +∞, we get (B.32).
Step 2. (The L 4 -norm of the limit function). Let (ϕ m = ϕ a,b,R,m ) be the sequence in Lemma B.4 which converges to the function ϕ = ϕ a,b,R . We would like to verify that the limit function ϕ is a minimizer of the functional G a,b,R . To that end, we will prove first that Taking the successive limits m → +∞ and → 0 + , we get (B.39).
Step Proof of Proposition 3.4. All the properties stated in Proposition 3.4 (except the nontriviality of the minimizer) are simply a convenient collection in one place of already proven facts in Lemmas B.5 and B.6. With these properties in hand, the non-triviality of ϕ a,b,R follows from Lemma 3.7.