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 minimizing configurations along this discontinuity.

1. 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. 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 direction parallel to the axis of the cylinder. For this specific geometry, it is enough to consider the horizontal cross section of the sample, Ω ⊂ R 2 . The phenomenological configuration (ψ, A) is then defined on the domain Ω.
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 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]. In this paper, we focus on the case where the applied magnetic field is a step function, which is not covered in the aforementioned papers. Such magnetic fields are interesting because they give rise to edge currents on the interface separating the distinct values of the magnetic field-the magnetic barrier (see [DHS14,HPRS16,RP00]). Our configuration is illustrated in Figure 2.
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 the superconductivity concentrates at the interface separating the distinct values of the magnetic field.
1.2. The functional and the mathematical set-up. We assume that the domain Ω is open in R 2 , bounded and simply connected. The Ginzburg-Landau (GL) free  (1.1) with ψ ∈ H 1 (Ω; C) and A = (A 1 , A 2 ) ∈ H 1 (Ω; R 2 ). Here, κ > 0 is a large GL parameter, the function B 0 : Ω → [−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 characteristic scale of the sample that measures the size of vortex cores (which is proportional to κ −1 ). Vortex cores are narrow regions in the sample, which corresponds to κ being a large parameter. That is the reason behind our analysis of the asymptotic regime κ → +∞, following many early papers addressing this asymptotic regime (see e.g. [SS07]). We work under the following assumptions on the domain Ω and the magnetic field B 0 (illustrated in Figure 1): Assumption 1.1.
(2) Ω 1 and Ω 2 have a finite number of connected components .
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 (Ω; C) × H 1 (Ω; R 2 )} .
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 of this 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 H = bκ, for some fixed parameter b > 0. Such magnetic field strengths are considered in many papers, [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 > Θ −1 0 , where Θ 0 ≈ 0.59 is a universal constant defined in (2.20) below, the only minimizer of the GL functional (up to change of gauge) is the trivial state (0,F) where curlF = 1 (see [GP99]). 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 to be considered is the second critical field H C 2 , which is much harder to define. When H < H C 2 , then the 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 , then the surface superconductivity regime occurs: the superconductivity disappears from the interior and is localised 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.
1.4. Expected behaviour of magnetic steps. Let us return back to the case where the magnetic field is a step function as in Assumption 1.1. 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 the bulk superconductivity persists, if b > 1/|a| then superconductivity decays exponentially in the bulk of Ω 1 and Ω 2 , and may nucleate in thin layers near Γ ∪ ∂Ω (see Assumption 1.1 and Figure 2). The present contribution affirms the presence of superconductivity in the vicinity of Γ 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 1.5 and 1.7 below.
The aforementioned behaviour of the superconductor in presence of magnetic steps is consistent with the existing literature (for instance see [DHS14,HPRS16,HS15,Iwa85,RP00]). 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 magnetic barrier (Γ in our context). On the other hand, no such snake orbits are formed in the case where 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 [HPRS16,HS15,Iwa85,RP00], and no estimates for the non-linear GL energy in (1.1) were established.
This contribution together with [AK16] provide such estimates. Particularly in the case where a ∈ [−1, 0) and b > 1/|a|, Theorems 1.5 and 1.7 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 ψ. These theorems assert the nucleation of superconductivity near the magnetic barrier Γ (and the surface ∂Ω) when b exceeds the threshold value 1/|a|.
Remark 1.3. Even though the case a ∈ (0, 1) is included in Assumption 1.2, 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 the bulk superconductivity is only restricted to a narrow neighbourhood of the magnetic barrier Γ, and this only occurs when the values of the two magnetic fields interacting near Γ are of opposite signs, that is when a ∈ [−1, 0), (see Figure 2). 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 (7.1). The energy E surf has been studied in many papers [AH07,CR14,Pan02,HFPS11,FK11]. 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.
Theorem 1.5 (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 1.6. In the asymptotics displayed in Theorem 1.5, the term |Γ|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.
Discussion of Theorem 1.5. We will discuss the result in Theorem 1.5 in the interesting case where the magnetic barrier Γ intersects the boundary of Ω. Hence we will assume that ∂Ω j ∩ ∂Ω = ∅ for j ∈ {1, 2}. When this condition is violated, the discussion below can be adjusted easily.
Theorem 1.5 leads to the following observation that mainly relies on Remark 1.4 and the order of the values |a|Θ 0 , Θ 0 , β a and |a|. • For a = −1, we have β a = Θ 0 < |a| (see (2.24)). Consequently, in light of Remark 1.4: -If 1 < b < Θ −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 Figure 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 (3.75). This behaviour is remarkably distinct from the case of a uniform applied magnetic field.
a , then surface superconductivity is confined to the part of the surface near ∂Ω 2 ∩ ∂Ω. At the same time, superconductivity is observed along the magnetic barrier Γ (see Figure 4), its strength is described by the function e a (b). 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]).
0 , then superconductivity only survives along ∂Ω 2 ∩ ∂Ω (see Figure 4). Our results then display the strength of the applied magnetic field responsible for the breakdown of the edge current along the barrier.
0 , then all the energy contributions in Theorem 1.5 vanish. Our next theorem describes the local behaviour of the minimizing order parameter ψ. To that end, we define the following distribution in R 2 , Here ds Γ and ds denote the arc-length measures on Γ and ∂Ω respectively.
and the convergence of T b κ to T is understood in the following sense: 1.6. Notation.
• 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 Ω, and is independent of κ and H . • Let a(κ) and b(κ) be two positive functions, we write : • The quantity o(1) indicates a function of κ, defined by universal quantities, the domain Ω, given functions, etc and such that |o(1)| 1. Any expression o(1) is independent of the minimizer (ψ, A) of (1.1). Similarly, O(1) indicates a function of κ, 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 introduce the space: (1.9) 1.7. Organization of the paper. The rest of the paper is divided into seven sections and one appendix. Section 2 presents some preliminaries, particularly, some a priori estimates, exponential decay results, and a linear 2D operator with a step magnetic field. Theorem 2.8 is an improvement of a result in [HPRS16]. Section 3 introduces a reduced GL energy crucial in the study of superconductivity near the magnetic barrier Γ and we introduce the barrier energy e a (·).
In Section 4, we present the Frenet coordinates defined in a tubular neighbourhood of the curve Γ. These coordinates are frequently used in the study of surface superconductivity (see [FH10, Appendix F]).
Sections 5 and 6 are devoted for the analysis of the local behaviour of the minimizing order parameter near the magnetic barrier Γ, while Section 7 recalls well-known results about the local behaviour the order parameter near the surface ∂Ω.
Collecting the local estimates established in Sections 6 and 7, we prove in Section 8 our main theorems (Theorems 1.5 and 1.7 above).
Finally, in the appendix, we collect some common spectral results used throughout the paper.
One remarkable aspect of our proofs is that we have not used the a priori elliptic L ∞ -estimate (∇ − iκHA)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 (∇ − iκHA)ψ 2 ≤ Cκ and the regularity of the curl-div system (cf. Theorem 2.3). This also spares us the complex derivation of the L ∞ -estimate (see [FH10,Chapter 11]). We have made an effort to keep the proofs reasonably self-contained.

Preliminaries
2.1. 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).
We begin by the following well-known estimate of the order parameter: Recall the magnetic field B 0 introduced in Assumption 1.1. In the next lemma, we will fix the gauge for the magnetic potential generating B 0 (see [AK16, Lemma A.1]): Lemma 2.2. Suppose that the conditions in Assumption 1.1 hold. There exists a unique vector field F ∈ H 1 div (Ω) such that We collect below some useful estimates whose proofs are given in [AK16, Theorem 4.2].

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 Ω 1 and Ω 2 .
2.3. Families of Sturm-Liouville operators on L 2 (R + ). In this section, we will briefly present some spectral properties of the self-adjoint realization on L 2 (R + ) of the Sturm-Liouville operator: defined over the domain: where ξ and γ are two real parameters, and the space B n (R + ) was introduced in (1.9). The quadratic form associated to H[γ, ξ] is Since the embedding of the form domain B 1 (R + ) in L 2 (R + ) is compact, the spectrum of H[γ, ξ] is an increasing sequence of eigenvalues tending to +∞ . Denote by µ(γ, ξ) the first eigenvalue of the operator H[γ, ξ]: and let Θ(γ) = inf ξ∈R µ(γ, ξ) . (2. 3) The Neumann realization. The particular case where γ = 0 corresponds to the Neumann realization, denoted by H N [ξ], with the associated quadratic form (2.4) The Dirichlet realization. Besides the Robin and Neumann realizations, we introduce the Dirichlet realization The associated quadratic form is defined by We introduce the first eigenvalue of H D [ξ] as follows . (2.5) The perturbation theory [Kat66] ensures that the functions are analytic. In addition, recall the following well-known Sturm-Liouville theorems (For instance, see [DH93,RS72]): Theorem 2.6. The following statements hold (1) For all (γ, ξ) ∈ R 2 , the first eigenvalue µ(γ, ξ) of H[γ, ξ] defined in (2.2) is simple, and there exists a unique eigenfunction ϕ γ,ξ satisfying Theorem 2.7. Let Θ(·) be the function defined in (2.3). It holds the following: (1) The function Θ(·) is continuous and increasing .
(2.6) Furthermore, this minimum is non-degenerate, ∂ 2 ξ µ γ, ξ(γ) > 0. 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 (2.8) We introduce the self-adjoint magnetic Hamiltonian in L 2 (R 2 ) We denote the ground state energy of the operator L a by β a = inf sp L a . (2.10) Since the Hamiltonian defined in (2.9) is invariant with respect to translations in the x 1 -direction, we can reduce it 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 domain of h a [ξ] is given by: The spectra of the operators L a and h a [ξ] are linked together as follows (see [FH10,Sec. 4 (2.14) We introduce the first eigenvalue of the fiber operator h a [ξ], µ a (ξ) = inf . (2.15) Consequently, for all a ∈ [−1, 1) \ {0}, we may express the ground state energy in (2.10) by β a = inf ξ∈R µ a (ξ) . (2.16) 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 (2.10) satisfies β a = a . (2.17) The case a = −1. This case is studied in [HPRS16]. Using symmetry arguments, the operator h a [ξ] can be linked to the operator H N [ξ], the Neumann realization on R + of −d 2 /dt 2 + (t − ξ) 2 introduced in Section 2.3. The first eigenvalue µ a (ξ) of h a [ξ] is then simple and satisfies µ a (ξ) = µ N (−ξ) , (2.18) where µ N (·) is introduced in (2.4). By Theorem 2.7, used for γ = 0, we get that (2.20) Furthermore, the minimum at ζ 0 is non-degenerate.
The case −1 < a < 0. See also [HPRS16] for the study of this case. The eigenvalue µ a (ξ) is simple, and there exists ζ a < 0 satisfying |a| ≥ µ a (ζ a ) = min ξ∈R µ a (ξ) . (2.21) Moreover, using the min-max principle, one can easily prove that Combining the foregoing discussion in the case a ∈ [−1, 0), we get β a introduced in (2.10) satisfies |a|Θ 0 ≤ β a ≤ |a| , (2.23) and In the next theorem, we will use a direct approach, different from the one in [HPRS16], to establish the existence of a global minimum ζ a in the case where a ∈ (−1, 0) and to prove that β a < |a|. This slightly improves the estimates in [HPRS16] (see Remark 2.9 below). Theorem 2.8 is necessary to validate Assumption (3.7), under which we work in Section 3.
Remark 2.9. Note that our proof of Theorem 2.8 yields the upper bound which is stronger than β a < |a|.
Collecting (2.17)-(2.22) and the result in Theorem 2.8, we deduce the following facts regarding the bottom of the spectrum of the operator L a introduced in (2.9).
(2) For all a ∈ [−1, 0), |a|Θ 0 ≤ β a < |a| , there exist ζ a < 0 and a function φ a ∈ L 2 (R) such that is a bounded eigenfunction of the operator L a and satisfies L a ψ a = β a ψ a .

Reduced Ginzburg-Landau Energy
3.1. The functional and the main result. Assume that a ∈ [−1, 1) \ {0} is fixed, σ is the step function defined in (2.8) and A 0 is the magnetic potential defined in (2.7). For every R > 0, consider the strip We introduce the space 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 (2.10).
The following holds: The proof of Theorem 3.1 will occupy the rest of this section through a sequence of lemmas.
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 a minimizer of the functional in (3.3).
Proof of Lemma 3.2. We have the obvious upper bound g a (b, R) ≤ G a,b,R (0) = 0.
Next we prove the lower bound g a (b, R) ≥ 0. Pick an arbitrary function u ∈ D R and extend it by zero on R 2 . Using the min-max principle, we get Minimizing over u ∈ D R , we get g a (b, R) ≥ 0.
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 first eigenvalue introduced in (2.10) . Under Assumption (3.7), we will demonstrate the existence of a minimizer of the functional in (3.3) along with an estimate of its decay at infinity. This is the content of Proposition 3.4. Assume that (3.7) holds. For all R > 0, there exists a function ϕ a,b,R ∈ D R such that Here G a,b,R is the functional introduced in (3.3) and g a (b, R) is the ground state energy introduced in (3.4). Furthermore, there exists a universal constant C > 0 such that, for all R > 0, the function ϕ a,b,R satisfies In the proof of Proposition 3.4, we will use the approach in [FKP13, Theorem 3.6] and [Pan02] which 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. In the following, 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 decay properties in Proposition 3.4.
Since the proof of Proposition 3.4 is lengthy, we opt to divide it into several lemmas. First, for every m ∈ N, 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 (2.8). Now we define the ground state energy (3.14) Lemma 3.5. Assume that (3.7) holds. There exists a univeral constant C > 0, and for all R > 0, m ≥ 1, there exists a function ϕ a,b,R,m ∈ D R,m satisfying, and Here G a,b,R,m is the functional introduced in (3.12) and g a (b, R, m) is the ground state energy introduced in (3.14).
Corollary 3.6. There exists a universal constant C > 0 such that, if (3.7) holds, the minimizer ϕ a,b,R,m in Lemma 3.5 satisfies, for all R > 0, m ∈ N, Proof. For the sake of brevity, we will write ϕ m for ϕ a,b,R,m . Using (3.26) and the fact that |x 2 |/ ln |x 2 | 2 ≥ 1, we get On the other hand, using ϕ m ∞ ≤ 1 and b > 1 we get 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 3.5. 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. Let Ω ⊂ R 2 be an open set. If (u m ) m≥1 is a sequence in H k (Ω), then by saying that (u m ) is bounded/convergent in H k loc (Ω), we mean that it is bounded/convergent in H k (K), for every K ⊂ Ω open and relatively compact. A similar terminology applies for boundedness/convergence in C k,α Lemma 3.7. Assume that (3.7) holds. Let R > 0 and α ∈ (0, 1) be fixed. The sequence ϕ a,b,R,m m≥1 defined by Lemma 3.5 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 (3.19) we may write 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 (3.31) and the boundedness of σ and A 0 in K, we get a constant C = C( K, R) such that in light of (3.32). By the interior elliptic estimates (see for instance [FH10, Section E.4.1]), we get that ϕ m ∈ H 2 (K) and 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 , (3.35) Obviously, the functions in (3.34) and (3.35) admit respectively the following weak partial derivatives (3.37) A straightforward computation using (3.32), (3.36) and (3.37) yields in the sense of weak derivatives. By Step Lemma 3.8. Assume that R > 0 and that (3.7) holds. Let ϕ a,b,R,m m≥1 be the sequence defined in Lemma 3.5. There exists 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 Furthermore, for all α ∈ (0, 1), ϕ a,b,R ∈ C 1,α loc (S R ). Proof. We continue writing ϕ m for ϕ a,b,R,m and ϕ for ϕ a,b,R . In the sequel, let α ∈ (0, 1) be fixed.
Let K ⊂ S R be open and relatively compact. By Lemma 3.7, (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). This subsequence and its limit ϕ K are independent of α; we will prove that they are actually independent of the relatively compact set K. This will be done by the standard Cantor's diagonal process that we outline below. For all p ∈ N, set K p = (−R/2, R/2)×(−p, p). Let I 0 = N. The sequence (ϕ m ) m∈I 0 has a subsequence (ϕ m ) m∈I 1 such that it is weakly convergent in H 3 (K 1 ), and strongly convergent in H 2 (K 1 ) and C 0,α (K 1 ). We denote the limit of this sequence by ϕ 1 . Note that ϕ 1 ∈ H 3 (K 1 ). By iteration, we obtain a collection of functions (ϕ p ) p∈N and a collection of subsequences, (ϕ m ) m∈Ip , such that • for every p ∈ N, (ϕ m ) m∈Ip is a subsequence of (ϕ m ) m∈I p−1 .
• for every p ∈ N, the subsequence (ϕ m ) m∈Ip converges weakly to ϕ p in H 3 (K p ) .
• for every p ∈ N, the subsequence (ϕ m ) m∈Ip converges strongly to ϕ p in H 2 (K p ) and C 0,α (K p ) .
The Sobolev embedding of H 3 (K p ) in C 1,α (K p ) yields that ϕ p ∈ C 1,α (K p ). It is useful to note that If p < q, then ϕ p = ϕ q in K p . (3.38) Indeed, the strong convergence of (ϕ m ) m∈Iq to ϕ q in H 2 (K q ) implies the following pointwise convergence of (ϕ m ) m∈Iq in K q (along a subsequence) Similarly, the strong convergence of (ϕ m ) m∈Ip to ϕ p in H 2 (K p ) implies lim m→+∞ ϕ m (x) = ϕ p (x), a.e. in K p .
Since I q ⊂ I p , we get the following pointwise convergence of (ϕ m ) m∈Iq in K p lim m→+∞ ϕ m (x) = ϕ p (x), a.e. in K p . (3.40) Having in hand the continuity of ϕ p and ϕ q , (3.38) follows from (3.39) and (3.40). Now, we are ready to define the limit function ϕ in S R = (−R/2, R/2) × (−∞, +∞) as follows. Let x ∈ S R . There exists p ∈ N such that x ∈ K p . We then define ϕ(x) = ϕ p (x). The function ϕ is well defined by (3.38) and belongs to H 3 loc (S R ), consequently to C 1,α loc (S R ). Next, we will construct a subsequence (ϕ m ) m∈I of (ϕ m ) m∈I 0 (with I ⊂ I 0 ) that converges weakly to the function ϕ in H 3 (K p ), for all p ∈ N. For all p ≥ 1, the set I p ⊂ N consists of a strictly increasing sequence {n 1 (p), n 2 (p), ...}; let n p be the p th element of I p , i.e. n p = n p (p). By induction, we can prove that, for all p, k ∈ N (with k ≥ 2), n k (p + 1) > n k−1 (p + 1) ≥ n k−1 (p). Thus, for all p ∈ N, n p+1 := n p+1 (p + 1) > n p (p) = n p . We define the index set I = {n 1 , n 2 , ...} and note that (ϕ m ) m∈I is a subsequence of (ϕ m ) m≥1 , because n 1 < n 2 < .... Also, it is a subsequence of (ϕ m ) m∈Ip , for every p ∈ N. Consequently, for all p ∈ N, the following strong convergence holds ϕ m −→ m→+∞ m∈I ϕ in H 2 (K p ) and C 0,α (K p ) . (3.41) Finally, if K ⊂ S R is an arbitrary open and relatively compact set, then there exists p ∈ N such that K ⊂ K p . Consequently, we inherit from (3.41) that (ϕ m ) m∈I converges to ϕ in H 2 (K) and C 0,α (K).
Lemma 3.9. Assume that R > 0 and (3.7) holds. Let ϕ a,b,R be the function defined by Lemma 3.8. The following statements hold:

46)
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 3.8. Again, we will use (ϕ m ) and ϕ for (ϕ a,b,R,m ) and ϕ a,b,R respectively. By Lemma 3.5, 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 3.8. By the convergence of ϕ m in H 2 loc (S R ) and C 0,α loc (S R ), we get (3.43) from Now we prove that ϕ ∈ D R . Pick an arbitrary integer m 0 ≥ 1. For all m ≥ m 0 , S R,m 0 ⊂ S R,m . Thus using the decay of ϕ m in (3.31) we have The uniform convergence of ϕ m to ϕ gives us Taking m 0 → +∞, we write by the monotone convergence theorem, This proves that ϕ ∈ L 2 (S R ). Next we will prove that (∇ − iσA 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 (∇ − iσA 0 )ϕ m → (∇ − iσA 0 )ϕ a.e. Furthermore, by Lemma 3.7, 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 (3.31), we may write, for all m 0 ≥ 1, Sending m 0 to +∞ and using the monotone convergence theorem, we get Thus, we have proved that ϕ, (∇ − iσA 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 Lemma 3.10. Assume that (3.7) holds. For all R > 0, the function ϕ a,b,R ∈ D R defined in Lemma 3.8 is a minimizer of G a,b,R , that is 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 (3.4) and (3.14) respectively. In this step, we will prove that lim m→+∞ g a (b, R, m) = g a (b, R) . (3.47) 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 ∈ N. Thus, Next, we will prove that Let ϑ ∈ C ∞ c (R) be a cut-off function satisfying 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 (3.50) By Cauchy's inequality, for all ∈ (0, 1) Thus, using the definition of the ground state energy g a (b, R, m) and the functional G a,b,R in (3.14) and (3.3) respectively, we obtain Taking the successive limits → 0 + then n → +∞, we get (3.49). Combining (3.48) and (3.49), we get (3.47).
Step 2. (The L 4 -norm of the limit function). Let (ϕ m = ϕ a,b,R,m ) be the sequence in Lemma 3.8 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 (3.52) We begin by proving that We introduce lim inf m→+∞ on both sides of (3.54), and we use (3.55) to get This is true for every integer m 0 ≥ 1. Consequently (3.53) simply follows by applying the monotone convergence theorem. Next, we prove that (3.56) Let C be the universal constant in (3.17), > 0 be fixed, and R > 0 be arbitrary. We select an integer m 0 ≥ 1 such that In light of (3.55), there exists m 1 ≥ m 0 such that Noticing that we may write, for all m ≥ m 1 On the other hand, for |x 2 | ≥ m 0 ≥ 1 we have, Thus, the estimate in (3.17) yields for all m ≥ m 0 , . (3.59) Combining (3.58) and (3.59), we get for all m ≥ m 1 ≥ m 0 Taking the successive limits m → +∞ then → 0 + , we get (3.56).
Step 3. (The limit function is a minimizer). The convergence in (3.52) is crucial in establishing that ϕ is a minimizer of G a,b,R . In light of Eq. (3.19), an integration by parts yields, for all m ≥ 1, We take m → +∞, and we use the results in (3.47) and (3.52). We get (3.60) By Lemma 3.9, ϕ ∈ D R and satisfies (3.43), so after integrating by parts, we get Comparing (3.60) and (3.61) yields that G a,b,R (ϕ) = g a (b, R).
Proof of Proposition 3.4. This proposition is simply a convenient collection in one place of already proved facts in Lemma 3.9 and Lemma 3.10.
3.4. 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, translation invariance 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 3.11. Let n ∈ N. Consider the ground state energy g a (b, R) defined in (3.4). It holds g a (b, nR) ≤ ng a (b, R) .
Let u ∈ D R , where D R is the domain defined in (3.2). We define the function v in S R,λ as follows v(x 1 , x 2 ) = u(x 1 − λR, x 2 ), (x 1 , x 2 ) ∈ S R,λ . (3.62) An easy computation shows the invariance of the energy under the aforementioned translation, that is Now, let n ∈ N. Noticing that we define a functionũ in S nR as follows This definition is consistent, since the sets S R,j are disjoint, their closures cover S nR , and u ∈ D R which yields thatũ vanishes on the boundary of every S R,j . Having (3.64), we get consequentlyũ This yields that g a (b, nR) ≤ nG a,b,R (u) .
We choose u ∈ D R to be the minimizer ϕ a,b,R of G a,b,R , defined in Proposition 3.4 and conclude that g a (b, nR) ≤ ng a (b, R) .
Our next result is concerned with the monotonicity of the function R → g a (b, R).
Lemma 3.12. The function R → g a (b, R) defined in (3.4) is monotone non-increasing.
Proof. This follows from the domain monotonicity. Indeed, let r > 0 and u ∈ H 1 0 (S R ) be a minimizer of G a,b,R . Consider the functionũ ∈ H 1 0 (S R+r ) defined as the extension of u by zero on S R+r \ S R . Obviously The existence of the limit of g a (b, R)/R as R → +∞ will follow from a well known abstract result, see Lemma 3.14 below. To apply this abstract result, we need some bounds on the energy g a (b, R). These are given in Lemma 3.13 below.
Lemma 3.13. 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 solely on a and b such that (3.65) Proof.
Let ν a = R φ 4 a (x 2 ) dx 2 . Thus, for t = (1 − bβ a )/ν a we get where C 2 = (1/2)t 2 and C 3 = Cb/ν a depend only on a and b. Lower bound. Let ϕ a,b,R be the minimizer in Proposition 3.4. For simplicity, we will write ϕ = ϕ a,b,R . It follows from the min-max principle that By (3.7), bβ a − 1 < 0. By (3.11), S R |ϕ| 2 dx ≤ CbR, where C > 0 is a universal constant. We choose C 1 = C/β a and get the following inequality Obviously, C 1 depends solely on a.
The next abstract lemma is a key-ingredient in the proof of Theorem 3.1, and more precisely in establishing the existence of the limit energy e a (b) introduced in (3.5). Variants of it were used in many papers, see [FK13,FKP13,Pan02,SS03]. Here we use the version from [FK13, Lemma 2.2].
Suppose that there exists a constant C > 0 such that the estimate holds true for all α ∈ (0, 1), n ∈ N, and ≥ 0 . Then f (l) has a limit A as l → +∞.
Furthermore, for all l ≥ 2l 0 , the following estimate holds We will apply Lemma 3.14 on the function f : R → g a (b, R)/R in order to define e a (b) as lim R→+∞ g a (b, R)/R. To that end, we establish that the above choice of f fulfils the conditions in Lemma 3.14. This is the content of Lemma 3.15. There exists a universal constant C > 0 such that, for all n ∈ N and α ∈ (0, 1), the ground state energy g a (b, R) defined in (3.4) satisfies 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 width of the overlapping region between two strips, when it exists, is α. We consider the partition of unity of R 2 : where C is a universal constant. Define is then a new partition of unity satisfying j |χ R,j | 2 = 1, 0 ≤ χ R,j ≤ 1, j |∇χ R,j | 2 ≤ C α 2 R 2 , supp χ R,j ⊂ S R,j , (3.67) where S R,j = {xR/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 G a,b,n 2 R (χ R,j ϕ a,b,n 2 R ) − C b 2 n 2 α 2 R .
The first inequality above follows from the celebrated IMS localization formula (see [CFKS09,Theorem 3.2]), while the second comes from (3.11) and the properties of (χ R,j ) in (3.67). 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 (see (3.64)), we have G a,b,n 2 R (χ R,j ϕ a,b,n 2 R ) ≥ g a (b, (1 + α)R) . 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 proved 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. For these two items, we handle now the case where 1/|a| ≤ b < 1/β a . In the proof of Item (3), we define 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 3.12 and 3.13, we see that d a,b (·) is non-positive, monotone non-increasing, and that f a,b (·) is bounded. Reformulating (3.66) by taking R = l 2 , we get for ≥ 2 f a,b (nl) ≥ f a,b (1 + α)l − Cb 2 α + 1 α 2 l 2 . Thus, the functions d a,b (l) and f a,b (l) satisfy the assumptions in Lemma 3.14. This assures the existence of a constant e a (b) ≤ 0, depending on a and b, such that Moreover, Lemma 3.13 ensures that e a (b) < 0. The upper bound in Item (5) of Theorem 3.1 follows from Lemma 3.14. It remains to establish a lower bound for g a (b, R)/R. Let n ≥ 1 be an integer. By Lemma 3.11, Dividing both sides by nR and taking n → +∞ yields The monotonicity of the function e a (·) is straightforward and follows from that of g a (·, R). Let > 0, we have g a (b + , R) ≥ g a (b, R). Dividing both sides of this inequality by R then taking R → +∞ gives us e a (b + ) ≥ e a (b). Our final task is to prove the continuity of the function e a (·). Let b ∈ 1/|a|, 1/β a , and > 0. We will prove that e a (·) is right continuous at b. Since e a (·) is monotone non-decreasing, e a (b + ) ≥ e a (b). Consequently, Hence, it is sufficient to prove that lim sup →0 + e a (b + ) ≤ e a (b). We may use the following lower bound from (3.6), (3.68) Let u ∈ D R . We have g a (b + , R) ≤ G a,b+ ,R (u), where the functional G a,·,R is defined in (3.3). We infer from (3.68) that This is true for all u ∈ D R and R ≥ 1. Minimizing over u ∈ D R yields, for all R ≥ 1, Taking R → +∞, we get the desired inequality. Let b ∈ 1/|a|, 1/β a and < 0. Now we prove the left continuity at b. The monotonicity of e a (·) yields that lim sup Let ϕ a,b+ ,R be the minimizer of G a,b+ ,R defined in (3.8). Using the upper bound in (3.6) together with (3.11), we get Taking R → +∞,we get the desired inequality.
3.6. An effective one-dimensional energy. Assume that a ∈ [−1, 1) \ {0} and b > 0. For all ξ ∈ R, consider the functional defined over the space B 1 (R), and let We would like to find a relationship between the 2D-energy in (3.4) and the 1D-energy in (3.70) 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 Under (3.71), the numerical computations indicate that the global minimum β a , defined in (2.10), is attained at a non-degenerate unique point, denoted by ζ a in (2.21) (see [HPRS16, Section 1.3]). To our knowledge, such a uniqueness result has not been analytically proven yet. In the sequel, we will assume that uniqueness of ζ a holds. Under this assumption, Proposition A.4 yields that for each fixed value of b such that 1/|a| < b < 1/β a , there exist two real numbers ξ 1 (a, b) and ξ 2 (a, b) satisfying ξ 1 (a, b) < ζ a < ξ 2 (a, b) , With ξ 1 (a, b) and ξ 2 (a, b) in hand, we can list some elementary properties of the functional E 1D a,b,ξ in (3.69): Theorem 3.16. Let a ∈ [−1, 0) and b ≥ 1/|a|.
(1) The functional E 1D a,b,ξ has a non-trivial minimizer in B 1 (R) if and only if 1/|a| ≤ b < 1/β a . Furthermore, one can find a positive minimizer f ξ , dependent on a and b, such that any minimizer has the form cf ξ where c ∈ C and |c| = 1.
Going back to our step magnetic field problem and the one dimensional energy in (3.69), it is reasonable to make the following conjecture Conjecture 3.17. Assume that −1 ≤ a < 0 and 1/|a| < b < 1/β a , where β a is defined in (2.10). Then, the energy e a (b) introduced in (3.5) satisfies and E 1D a,b (·) is defined in (3.70). By a symmetry argument, Conjecture 3.17 trivially holds in the case a = −1, namely However, there are many points that do not allow us to prove this conjecture in the case where a ∈ (−1, 0). Besides the lack of the uniqueness of the minimum ζ a , the new potential term creates computational difficulties preventing the adoption of the proof in [CR14], (in particular, in the positivity proof of the cost function).

The Frenet Coordinates
In this section, we assume that the set Γ consists of a single smooth curve that may intersect the boundary of Ω transversely in two points. In the general case, Γ consists of a finite number of such curves. By working on each component separately, we reduce to the simple case above.
To study the energy contribution along Γ, we will use the Frenet coordinates which are valid in a tubular neighbourhood of Γ. For more details regarding these coordinates, see e.g. [FH10, Appendix F]. We will list the basic properties of these coordinates here.
For t 0 > 0, we define the open set We introduce the function t : R 2 → R as follows When t 0 is sufficiently small, the transformation is a diffeomorphism whose Jacobian is a(s, t) = det(DΦ) = 1 − tk r (s) .
In Propositions 4.1 and 4.2, we will construct a special gauge transformation that will allow us to express a given vector field in a canonical manner.
Proposition 4.1. For any vector field A = (A 1 , A 2 ) ∈ H 1 (Ω, R 2 ), there exists a H 2 -function ω such that the vector field defined by A new = A − ∇ω satisfies whereÃ new is the vector field associated to A new as in (4.6).

A Local 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).
We start by introducing various (geometric) notations/assumptions. Select a positive number t 0 sufficiently small so that the Frenet coordinates of Section 4 are valid in the tubular neighbourhood Γ(t 0 ) defined in (4.1). 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 . We will assume that κ is sufficiently large so that < t 0 /2. Consider the set and the magnetic potentialF defined in V( ) bỹ Consider the domain D : For u ∈ D , we define the (local) energy |u| 4 a ds dt , (5.5) where a(s, t) = 1 − tk r (s). Now we introduce the following ground state energy (5.6) Using standard variational methods, one can prove the existence of a minimizer u 0 of Our aim is to write matching upper and lower bounds for G 0 as κ → +∞ in the regime H = bκ, a ∈ [−1, 0) and b ≥ 1 |a| .
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 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 the Cauchy's inequality to get Inserting the previous estimate into (5.11) and using the uniform bound |u| ≤ 1, we obtain We introduce the following parameters and define the re-scaled function Recall the parameter b = H/κ in (1.5), in the new scale we may write G a,b,R is the functional introduced in (3.3), andȗ ∈ D R the domain introduced 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 use e a (b) ≤ 0 and the assumptions on κ and to finish the proof of Lemma 5.1.
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 as in (5.1), then where G 0 and e a (b) are defined in (5.6) and (3.5) respectively.
Proof. For R = √ κH, consider ϕ = ϕ a,b,R the minimizer of G a,b,R defined in (3.8). We define the function u in D as follows where χ is a standard smooth cut-off function satisfying Next, we define the following function (with the re-scaled variables) Using the definition of v, the decay of ϕ in (3.11), and the bound of a(s, t) in (5.9), we get where J (u) was defined in (5.11), |v| 4 dγdτ , and = 1/ √ κH. Let χ R (τ ) = χ τ /R = χ t/ . We will estimate now each term of K(v) apart, using mainly the decay of the minimizer ϕ in (3.11) and the properties of the function χ R .
We start with Similarly, we have Next, we may select R 0 sufficiently large so that, for all R ≥ R 0 , we have: The decay of ϕ and (5.17) yield Finally, we write the obvious inequality Gathering the foregoing estimates, we get Invoking Theorem 3.1, we implement (5.19) into (5.16) to get the desired upper bound.

Local Estimates
The aim of this section is to study the concentration of the minimizers (ψ, A) of the functional in (1.1) near the set Γ that separates the values of the applied magnetic field (see Assumption 1.1). This will be displayed by 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 (Ω; C) × H 1 div (Ω) in any open set D ⊂ Ω as (6.1) Choose t 0 > 0 sufficiently small so that the Frenet coordinates of Section 4 are valid in the tubular neighbourhood Γ(t 0 ) defined in (4.1). For all x ∈ Γ(t 0 ), define the point p(x) ∈ Γ as follows dist(x, p(x)) = dist(x, Γ) .
Let ≈ κ −3/4 be a parameter in (5.1) (for some fixed choice of the constants c 1 and c 2 ). Let x 0 ∈ Γ \ ∂Ω that is allowed to vary in such a manner that Consider the following neighbourhood of x 0 , where t(·) is defined in (4.2). For κ sufficiently large (hence sufficiently small), we get that N x 0 ( ) does not intersect the boundary ∂Ω, thanks to (6.2). 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 Γ and the surface ∂Ω (see Theorem 2.4).
The proof of Theorem 6.1 follows by collecting the results of Proposition 6.3 and Proposition 6.4 below, which are derived along the lines of [HK17,Section 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 Section 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 is the function associated to v = e −iκHωx 0 u by the transformation Φ −1 (see (4.5)), and ω x 0 is the gauge transformation function defined in Proposition 4.2.
6.1. 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 ∈ Γ satisfying (6.2), the following is true. If where N x 0 (·) is the neighbourhood defined in (6.3), E 0 is the functional defined 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. Define the function φ 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). Define the function w = e −iκHφx 0 u. Using (6.8) and Cauchy's inequality, we may write 3 |w| 2 . By using that |w| ≤ 1, we get further We may use the relation in (6.7) to write  1). Again, we remind the reader that we assume that (6.4) holds, ≈ κ −3/4 (see (5.1)) and H = bκ.

This yields
Hence, (6.14) The fact that f ψ ∈ H 1 0 N x 0 (ˆ ) and |f ψ| ≤ 1 allows us to use the lower bound result established in Proposition 6.2 for u = f ψ. This yields together with (6.14) This finishes the proof of (6.9), but withˆ appearing instead of . However, this is not harmful, as we could start the argument withˇ = (1 + γ) −1 in place of and then modifyˆ accordingly; in this case we would getˆ = (1 + γ)ˇ = as required.
Proof of (6.10). In light of the first equation in (1.4) satisfied by (ψ, A), we get using integration by parts (see [FK11,(6.2)]) Consequently, We use the previous inequality, (6.16) and the estimate supp |∇f | ≤ Cγ 2 to obtain We insert the lower bound in Proposition 6.2 into (6.17) to get the upper bound of the L 4 -norm in (6.10).
Proof. The proof is divided into five steps.
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). Since (ψ, A) is a minimizer of the functional E κ,H defined in (1.1), we write that is Noticing that E 3 (u, A) = E 3 (ψ, A), we get We use the estimate of E 2 (u, A) in (6.24) to get . We insert the upper bound of E 1 (u, A) in (6.23) in the previous inequality to get Recalling that E 1 (ψ, A) = E 1 ψ, A; N x 0 (ˆ ) , we see that (6.25) is nothing but (6.18) withˆ appearing instead of . Starting the argument with replaced byˇ = (1+γ) −1 , we get (6.25) forˆ = (1 + γ)ˇ = , as required. Therefore, we finished the proof of (6.18).
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 Ω, 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 the identity in (6.16) gives us The previous inequality together with (6.14) and (6.25) establish the lower bound of the L 4 -norm of ψ as κ → +∞.

Surface Superconductivity
In Section 6, we worked under the assumption We investigated the local behaviour of the sample in a tubular neighbourhood of Γ.
In this section, and under the same assumption, we are concerned in the local behaviour of the sample near the boundary of Ω. The analysis of superconductivity near ∂Ω in our case of a step magnetic field (B 0 satisfying 1.1) is essentially the same as that in the uniform field case, since B 0 is constant in each of Ω 1 and Ω 2 . Thereby, the results presented in this section are wellknown in literature since the celebrated work of Saint-James and de Gennes [SJG63]. We refer to [AH07, CR14, FH05, HFPS11, FK11, FKP13, LP99, Pan02] 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 (Ω) for some α ∈ (0, 1). We will adapt these results to our discontinuous magnetic field (see Theorem 7.1 below).
The statement of our main result, Theorem 7.1, involves the surface energy E surf that we introduce in the next section.
The estimates in Theorem 7.1 are already established in [HK17] when B 0 ∈ C 0,α (Ω) for some α ∈ (0, 1). They still hold in our case because B 0 is constant in Ω 1 and Ω 2 (see Assumption 1.1) by repeating the proof given in [HK17].

Proof of Main Results
8.1. Proof of Theorem 1.5. We will work under the assumptions of Theorem 1.5 restricted to the non-trivial case a ∈ [−1, 0) and b > 1/|a| , and we will gather the results of the two previous sections to establish, as κ tends to +∞, asymptotic estimates of the global ground state energy E g.st (κ, H) in (1.3) and of the L 4 -norm of the order parameter ψ, where (ψ, A) is a minimizer of this energy.  where E 0 is defined in (6.1) and B 0 is defined in 1.1. Hence, it suffices to find a relevant lower bound of E 0 ψ, A; Ω . To that end, we will decompose the sample Ω into the sets Γ * ( ), Ω * 1 ( ), Ω * 2 ( ), Ω bulk and T introduced later in this section (see Figure 5) and will establish a lower bound of the energy E 0 ψ, A; ·) in each of the decomposition sets. We assume to be the parameter in (5.1) which satisfies ≈ κ −3/4 . Lower bound in a neighbourhood of the magnetic barrier. We start by introducing the set Γ * ( ) which covers almost all of the set Γ. Recall the assumption that Γ consists of a finite collection of simple smooth curves that may intersect ∂Ω transversely. For the simplicity of the exposition, we will focus on the particular case of a single curve intersecting ∂Ω at two points. The construction below may be adjusted to cover the general case by considering every single component of Γ separately. We may select two constants 0 ∈ (0, 1) and c > 2, and for all ∈ (0, 0 ), a collection of pairwise distinct points ( where N x i ( ) is the set introduced in (6.3). Note that the family (N x i ( )) 1≤i≤N consists of pairwise disjoint sets. Consequently, A uniform lower bound for the local energies E 0 ψ, A; N x i ( ) The last inequality follows from (8.3) and the fact that e a (b) ≤ 0. Lower bound in a neighbourhood of the boundary. Now, we define the two sets Ω * 1 ( ) and Ω * 2 ( ) which cover almost all of the set ∂Ω. In a similar fashion of the definition of Γ * ( ), we fix 0 ∈ (0, 1) and c > 2 and we select collections of points (y j ) N 1 j=1 ⊂ ∂Ω 1 and (z k ) N 2 k=1 ⊂ ∂Ω 2 such that, (y j ) N 1 j=1 ⊂ {u ∈ ∂Ω 1 : dist(u, Γ) > 2 } , (z k ) N 2 k=1 ⊂ {u ∈ ∂Ω 2 : dist(u, Γ) > 2 } , (8.6) ∀ j ∈ {1, ..., N 1 −1}, dist ∂Ω 1 (y j , y j+1 ) = , ∀ k ∈ {1, ..., N 2 −1}, dist ∂Ω 2 (z k , z k+1 ) = , (8.7) where N 1 y j ( ) and N 2 z k ( ) were defined in (7.5). Hence following similar steps as in (8.5), we use the uniform lower bound in Theorem 7.1 together with the estimates in (8.8) to get E 0 ψ, A; Ω * 1 ( ) ≥ |∂Ω 1 ∩ ∂Ω|b − 1 2 κE surf (b) − Cκȓ(κ) , (8.11) and E 0 ψ, A; Ω * 2 ( ) ≥ |∂Ω 2 ∩ ∂Ω|b − 1 2 |a| − 1 2 κE surf b|a| − Cκȓ(κ) .
(8.13) Under our assumptions on b in (6.4) and in (5.1), the exponential decay in Theorem 2.4 allows us to neglect the energy contribution in the bulk, and to particularly write |E 0 ψ, A; Ω bulk | ≤ Cr(κ) .
where f a,ξ is the eigenfunction in Proposition A.1, and γ a (ξ) as in (A.1).
Proof. (Feynman-Hellmann). For simplicity, we write µ, f and h respectively for µ a (ξ), f a,ξ , and h a [ξ] . Differentiating with respect to ξ and integrating by parts in , and recalling that f is normalized, we obtain ∂µ ξ = ∂ ξ hf, f