Decay of superconductivity away from the magnetic zero set

Abstract

We establish exponential bounds on the Ginzburg–Landau order parameter away from the curve where the applied magnetic field vanishes. In the units used in this paper, the estimates are valid when the parameter measuring the strength of the applied magnetic field is comparable with the Ginzburg–Landau parameter. This completes a previous work by the authors analyzing the case when this strength was much higher. Our results display the distribution of surface and bulk superconductivity and are valid under the assumption that the magnetic field is Hölder continuous.

1 Introduction

1.1 The functional

In non-dimensional units, the Ginzburg–Landau functional is defined as follows,

$$\begin{aligned} {\mathcal {E}}(\psi ,{\mathbf {A}})=\int _\Omega \left( |(\nabla -i\kappa H{\mathbf {A}})\psi |^2-\kappa ^2|\psi |^2+\frac{\kappa ^2}{2}|\psi |^4+(\kappa H)^2|{{\mathrm{curl}}}{\mathbf {A}}-B_0|^2\right) dx, \end{aligned}$$

(1.1)

where:

• $\Omega \subset \mathbb R^2$ is an open, bounded and simply connected set with a $C^\infty$ boundary;

• $(\psi ,{\mathbf {A}})\in H^1(\Omega ;{\mathbb {C}})\times H^1(\Omega ;\mathbb R^2)$;

• $\kappa >0$ and $H>0$ are two parameters;

• $B_0$ is a real-valued function in $L^2(\Omega )$.

The superconducting sample is supposed to occupy a long cylinder with vertical axis and horizontal cross section $\Omega$. The parameter $\kappa$ is the Ginzburg–Landau parameter that expresses the properties of the superconducting material. The applied magnetic field is $\kappa HB_0 \vec {e}$, where $\vec { e}=(0,0,1)$. The configuration pair $(\psi ,{\mathbf {A}})$ describes the state of superconductivity as follows: $|\psi |^2$ measures the density of the superconducting Cooper pairs, ${{\mathrm{curl}}}{\mathbf {A}}$ measures the induced magnetic field in the sample and $j:=(i\psi ,\nabla \psi -i\kappa H{\mathbf {A}}\psi )$ measures the induced super-current. Here $(\cdot ,\cdot )$ denotes the inner product in $\mathbb C$ defined as follows, $(u,v)=u_1v_1+u_2v_2$ where $u=u_1+iu_2$ and $v=v_1+iv_2$.

At equilibrium, the state of the superconductor is described by the (minimizing) configurations $(\psi ,{\mathbf {A}})$ that realize the following ground state energy

$$\begin{aligned} \mathrm {E}_{\mathrm{gs}}(\kappa , H)=\inf \left\{ {\mathcal {E}}(\psi ,{\mathbf {A}})~:~(\psi ,{\mathbf {A}})\in H^1(\Omega ;\mathbb C)\times H^1(\Omega ;{\mathbb {R}}^2)\right\} . \end{aligned}$$

(1.2)

Such configurations are critical points of the functional introduced in (1.1), that is they solve the following system of Euler-Lagrange equations ($\nu$ is the unit inward normal on the boundary)

$$\begin{aligned} \left\{ \begin{array}{lll} -\big (\nabla -i\kappa H{\mathbf {A}}\big )^2\psi &{}=\kappa ^2(1-|\psi |^2)\psi &{}\mathrm{in}\ \Omega ,\\ -\nabla ^{\perp } \big ({{\mathrm{curl}}}{\mathbf {A}}-B_0\big )&{}= (\kappa H)^{-1}\mathrm{Im}\big (\overline{\psi }(\nabla -i\kappa H \mathbf{A})\psi \big ) &{} \mathrm{in}\ \Omega ,\\ \nu \cdot (\nabla -i\kappa H{\mathbf {A}})\psi &{}=0 &{} \mathrm{on}\ \partial \Omega ,\\ \mathrm{curl}{} \mathbf{A}&{}=B_0 &{} \mathrm{on}\ \partial \Omega . \end{array} \right. \end{aligned}$$

(1.3)

Once a choice of $(\kappa ,H)$ is fixed, the notation $(\psi ,{\mathbf {A}})_{\kappa ,H}$ stands for a solution of (1.3). When $B_0$ belongs to $C^0(\overline{\Omega })$, we introduce two constants $\beta _0$ and $\beta _1$ that will play a central role in this paper:

$$\begin{aligned} \beta _0:=\sup _{x\in \overline{\Omega }}|B_0(x)|\quad \mathrm{and}\quad \beta _1:=\sup _{x\in \partial \Omega }|B_0(x)|. \end{aligned}$$

(1.4)

1.2 The case with a constant magnetic field

A huge mathematical literature is devoted to the analysis of the functional in (1.1) when the magnetic field is constant. This corresponds to taking $B_0=1$ in (1.1). The two monographs [15, 40] and the references therein are mainly devoted to this subject. One important situation is the transition from bulk to surface superconductivity. This happens when the parameter H increases between two critical values $H_{C_2}$ and $H_{C_3}$ called the second and third critical fields respectively.

In this analysis the de Gennes constant plays a central role. This constant is universal and defined as follows

$$\begin{aligned} \Theta _0=\inf _{\xi \in \mathbb R}\Big \{\inf _{\Vert u\Vert _2=1}\Big (\int _0^\infty \big (|u'(t)|^2+(t-\xi )^2|u(t)|^2\big )dt\Big )\Big \}. \end{aligned}$$

(1.5)

Furthermore, it is known (cf. [15]) that

$$\begin{aligned} \frac{1}{2}<\Theta _0<1. \end{aligned}$$

(1.6)

The de Gennes constant appears indeed in the asymptotics of $H_{C_3}$ for $\kappa$ large

$$\begin{aligned} H_{C_3}\sim \Theta _0^{-1}\kappa , \end{aligned}$$

while we have for the second critical field

$$\begin{aligned} H_{C_2}\sim \kappa . \end{aligned}$$

To be more specific, if $b>0$ is a constant and $(\psi ,{\mathbf {A}})_{\kappa , H}$ is a minimizer of the functional in (1.1) for $H=b\kappa$ (and $B_0=1$), the concentration of $\psi$ in the limit $\kappa \rightarrow \infty$ depends strongly on b.

If $0<b<1$, then $\psi$ is uniformly distributed in the domain $\Omega$ (cf. [28, 41]). If $1<b<\Theta _0^{-1}$, then $\psi$ is concentrated on the surface and decays exponentially in the bulk (cf. [12, 35]). If $b>\Theta _0^{-1}$, then $\psi =0$ (cf. [25, 32]). The critical cases when b is close to 1 or $\Theta _0^{-1}$ are thoroughly analyzed in [14, 16].

1.3 The case with a non-vanishing magnetic field

The case of a non-constant magnetic field $B_0$ satisfying the assumptions

$$\begin{aligned} B_0\in C^0({\overline{\Omega }})\quad \mathrm{and}\quad \inf _{x\in {\overline{\Omega }}}B_0(x)>0, \end{aligned}$$

is qualitatively similar to the constant magnetic field case. This situation is reviewed in [22, Sec. 2.2]. Surface superconductivity is studied in [15], while the transition to the normal solution is discussed in [37].

1.4 The case with a vanishing magnetic field

The results in this paper are valid for a large class of applied magnetic fields, see Assumption 1.2 below. However, one interesting situation covered by our results is the case where the applied magnetic field has a non-trivial zero set. In the presence of such an applied magnetic field, we will study the concentration of the minimizers $(\psi ,{\mathbf {A}})_{\kappa ,H}$ of (1.1) in the asymptotic limit $\kappa \rightarrow +\infty$ and $H\approx \kappa$. Unlike the results in [15, 37] that only investigate surface superconductivity, the situation discussed here includes bulk superconductivity as well.

The discussion in this subsection is focusing on magnetic fields that satisfy:

Assumption 1.1

(On the applied magnetic field) 

• (1) The function $B_0$ is in $C^{1}(\overline{\Omega })$.

• (2) The set $\Gamma :=\{x\in {\overline{\Omega }}~:~B_0(x)=0\}$ is non-empty and consists of a finite disjoint union of simple smooth curves.

• (3) $\Gamma \cap \partial \Omega$ is either empty or a finite set.

• (4) For all $x\in {\overline{\Omega }}, |B_0(x)|+|\nabla B_0(x)|\ne 0$.

• (5) The set $\Gamma$ is allowed to intersect $\partial \Omega$ transversely. More precisely, if $\Gamma \cap \partial \Omega \not =\emptyset$, then on this set, $\nu \times \nabla B_0\not =0$, where $\nu$ is the normal vector field along $\partial \Omega$.

A much weaker assumption will be described later (cf. Assumption 1.2). Under Assumption 1.1, we may introduce the following two non-empty open sets

$$\begin{aligned} \Omega _+=\{x\in \Omega ~:~B_0(x)>0\}\quad \mathrm{and}\quad \Omega _-=\{x\in \Omega ~:~B_0(x)<0\}. \end{aligned}$$

(1.7)

The boundaries of $\Omega _\pm$ are given as follows

$$\begin{aligned} \partial \Omega _\pm = \Gamma \cup (\overline{\Omega }_\pm \cap \partial \Omega ). \end{aligned}$$

Magnetic fields satisfying Assumption 1.1 are discussed in many contexts:

• In geometry, the magnetic Laplacian arises naturally as a natural Laplacian associated with a given connection on a U(1)-bundle (cf. [30]). Magnetic fields with a non-trivial zero set appear in [33] under the appealing question: can we hear the zero locus of a magnetic field?

• In the semi-classical analysis of the spectrum of Schrödinger operators with magnetic fields satisfying Assumption 1.1 (and $\Gamma \subset \Omega$). These operators are extensively studied in [13, 20, 24].

• In the study of the time-dependent Ginzburg–Landau equations [4, 5], applied magnetic fields as in Assumption 1.1 naturally appear in the presence of applied electric currents. Actually, for specific samples, there are electrical currents that produce sign-changing induced magnetic fields.

• For superconducting surfaces submitted to constant magnetic fields [11], the constant magnetic field may induce a smooth sign-changing magnetic field on the surface.

• In the transition from normal to superconducting configurations [36], one meets the problem of determining H such that the ground state energy in (1.2) vanishes on a curve meeting transversally the boundary. The results in [36] are sharpened in [9, 34].

• The asymptotics of the ground state energy in (1.2) and the concentration of the corresponding minimizers for large values of $\kappa$ and H is analyzed in [7, 8, 22, 23].

Of particular importance to us are the results of Attar in [7]. These results hold under Assumption 1.1, for $H=b\kappa$ with $b>0$ constant. One of the results in [7] is that the ground state energy in (1.2) satisfies, as $\kappa \rightarrow +\infty$,

$$\begin{aligned} \mathrm {E}_{\mathrm{gs}}(\kappa , H)=\kappa ^2\int _\Omega g(b|B_0(x)|)dx+o(\kappa ^2). \end{aligned}$$

(1.8)

Here the function $g(\cdot )$, which was introduced by Sandier–Serfaty in [41], is a continuous non-decreasing function defined on $[0,\infty )$ and vanishes on $[1,\infty )$ (cf. (2.5) for more details).

K. Attar also obtained an interesting formula displaying the local distribution of the minimizing order parameter $\psi$. If $(\psi ,{\mathbf {A}})_{\kappa ,H}$ is a minimizer of the functional in (1.1) for

$$\begin{aligned} H=b\kappa , \end{aligned}$$

and if ${\mathcal {D}}$ is an open set in $\Omega$ with a smooth boundary, then, as $\kappa \rightarrow +\infty$,

$$\begin{aligned} \int _{{\mathcal {D}}}|\psi (x)|^4dx=-2\int _{{\mathcal {D}}} g(b|B_0(x)|)dx+o(1). \end{aligned}$$

(1.9)

The interest for an $L^4$ control of the order parameter dates back to Y. Almog (see [2] and the discussion in the book [15, Ch. 12, Sec. 12.6]). Using the Ginzburg–Landau equations, the $L^4$-norm of the order parameter is directly connected with the Ginzburg–Landau energy of the corresponding minimizer.

The formula in (1.9) shows that $\psi$ is weakly localized in the neighborhood of $\Gamma$, ${\mathcal {V}} \left( \frac{1}{b}\right)$, where:

$$\begin{aligned} {\mathcal {V}} \left( \epsilon \right) :=\Big \{ x \in \Omega , |B_0(x)| \le \epsilon \Big \}. \end{aligned}$$

(1.10)

For taking account of the boundary effects (the surface superconductivity should play a role like in the constant magnetic field case) we also introduce in $\partial \Omega$ the subset

$$\begin{aligned} {\mathcal {V}}^{\mathrm{bnd}}\left( \epsilon \right) := \big \{ x\in \partial \Omega ,\Theta _0 |B_0(x)| \le \epsilon \big \}. \end{aligned}$$

(1.11)

We would like to measure the strength of the (exponential) decay of the minimizing order parameter $\psi$ in the domains

$$\begin{aligned} \omega \left( \frac{1}{b}\right) := \Omega {\setminus }{\mathcal {V}}\left( \frac{1}{b}\right) . \end{aligned}$$

(1.12)

Note the role played by the two constants introduced in (1.4). If $\frac{1}{b}\ge \beta _0$, then ${\mathcal {V}}(\frac{1}{b})=\Omega$. For this reason we will focus on the values of b above $\beta _0^{-1}$. We also observe that, if $\frac{1}{b}\ge \Theta _0\beta _1$, then ${\mathcal {V}}^{\mathrm{bnd}}(\frac{1}{b})=\partial \Omega$. Hence, boundary effects are expected to appear when $b<\frac{1}{\Theta _0\beta _1}$.

Loosely speaking, we would like to prove that, for all values of $b\ge \beta _0^{-1}$, the density $|\psi |^2$ is exponentially small (in the $L^2$-sense) outside the set ${\mathcal {V}}(\frac{1}{b})\cup {\mathcal {V}}^{\mathrm{bnd}}(\frac{1}{b})$. This will lead us to two distinct regimes:

Regime I: For $\beta _0^{-1}< b\le (\Theta _0\beta _1)^{-1}, {\mathcal {V}}^{\mathrm{bnd}}(\frac{1}{b})=\partial \Omega$ and $\partial \Omega$ carries surface superconductivity everywhere. This is illustrated in Fig. 1.

Fig. 1

Illustration of Regime I for $H=b\kappa$ and $b=1/\varepsilon$: Superconductivity is destroyed in the dark regions and survived on the entire boundary

Regime II: For $b>(\Theta _0\beta _1)^{-1}$, we will get that $\psi$ is exponentially small outside the set ${\mathcal {V}}^{\mathrm{bnd}}(\frac{1}{b})$. Here we have two cases:

• As b increases, surface superconductivity shrinks to the points of $\{x\in \partial \Omega ,~B_0(x)=0\}$, provided that this set is non-empty (cf. Fig. 2).

• If $\{B_0(x)=0\}\cap \partial \Omega =\emptyset$, then, for sufficiently large values of b, no surface superconductivity is left (cf. Fig. 3).

Fig. 2

Illustration of Regime II for $H=b\kappa$ and $b=1/\varepsilon$: Superconductivity is also destroyed on the boundary parts $\{\Theta _0|B_0(x)|>\varepsilon \}\cap \partial \Omega$

Fig. 3

Illustration of Regime II when $\{B_0=0\}\cap \partial \Omega =\emptyset , H=b\kappa , b=1/\varepsilon$ and $\varepsilon$ is small: Superconductivity is destroyed on the entire boundary and is concentrated in the set $\{|B_0|<\varepsilon \}$

Regime II is consistent with the results of [22, Thm. 3.6] devoted to the complementary regime where $b\gg 1$ as $\kappa \rightarrow +\infty$.

The results in this paper confirm the behavior described in these two regimes and are valid under a much weaker assumption than Assumption 1.1 (cf. Assumption 1.2 below).

The transition to the normal state is studied in [9, 34, 36]. This happens, when $\kappa$ is large, for $H\sim c_*\kappa ^2$ (equivalently $b \sim c_*\kappa$), where $c_*>0$ is a constant explicitly defined by the domain $\Omega$ and the function $B_0$.

1.5 Main results

In this paper, we will first work under the following assumption:

Assumption 1.2

 

• The function $B_0$ is in $C^{0,\alpha }({\overline{\Omega }})$ for some $\alpha \in (0,1)$;

• The constants $\beta _0$ and $\beta _1$ in (1.4) satisfy $\beta _1\ge \beta _0>0$.

Note that this assumption is much weaker than Assumption 1.1. With the previous notation our main theorem is:

Theorem 1.3

(Exponential decay outside the superconductivity region) Suppose that Assumption 1.2 holds, that $b>\beta _0^{-1}$ and let O be an open set such that $\overline{O}\subset \omega \big (\frac{1}{b}\big )$, where $\omega (\frac{1}{b})$ is the domain introduced in (1.12)

There exist $\kappa _0>0$, $C>0$ and $\alpha _0>0$ such that, if $\kappa \ge \kappa _0$ and $(\psi ,{\mathbf {A}})_{\kappa ,H}$ is a solution of (1.3) for $H=b\kappa$, then the following inequality holds

$$\begin{aligned} \Vert \psi \Vert _{H^1(O)}\le C e^{-\alpha _0\kappa }. \end{aligned}$$

(1.13)

Furthermore, if $b>(\Theta _0\beta _1)^{-1}$, then the estimate in (1.13) holds when the open set O satisfies

$$\begin{aligned} \overline{O}\subset \left\{ x\in \partial \Omega ,~\Theta _0|B_0(x)|>\frac{1}{b}\right\} \cup {\omega }\left( \frac{1}{b}\right) . \end{aligned}$$

The proof of Theorem 1.3 follows from the stronger conclusion of Theorem 3.1, establishing Agmon like estimates.

Remark 1.4

(Sign-changing magnetic fields) In addition to Assumption 1.2, suppose that $\Omega _+$ and $\Omega _-$ are non-empty. The constant $\beta _0$ in (1.4) can be expressed as follows

$$\begin{aligned} \beta _0=\max (\beta _0^+,\beta _0^-)\quad \mathrm{where~}\beta _0^\pm =\sup _{x\in \overline{\Omega _\pm }}|B_0(x)|. \end{aligned}$$

We will discuss the conclusion of Theorem 1.3 when $\beta _0^+<\beta _0^-$. We have:

• If $(\beta _0)^{-1}<b<(\beta _0^+)^{-1}$, then $\omega (\frac{1}{b})\cap \Omega _+=\emptyset$. Consequently, the exponential decay occurs in $\omega (\frac{1}{b})\cap \Omega _-$.

• If $(\beta _0^+)^{-1}\le b$, then the exponential decay occurs in both $\omega (\frac{1}{b})\cap \Omega _+$ and $\omega (\frac{1}{b})\cap \Omega _-$.

The situation when $\beta _0^-<\beta _0^+$ can be discussed similarly. Next, we suppose that the two sets

$$\begin{aligned} (\partial \Omega )_+:=\{x\in \partial \Omega ~,~B_0(x)>0\}\quad \mathrm{and} \quad (\partial \Omega )_-:=\{x\in \partial \Omega ~,~B_0(x)<0\} \end{aligned}$$

are non-empty, and we express the constant $\beta _1$ in (1.4) as follows

$$\begin{aligned} \beta _1=\max (\beta _1^+,\beta _1^-)\quad \mathrm{where~}\beta _1^\pm =\sup _{x\in \overline{(\partial \Omega )_\pm }}|B_0(x)|. \end{aligned}$$

According to Theorem 1.3, when $\beta _1^+<\beta _1^-$ and $(\beta _1)^{-1}<b<(\beta _1^+)^{-1}$, then the exponential decay occurs on $\{x\in \partial \Omega ,~\Theta _0b|B_0(x)|>1\}\cap (\partial \Omega )_-$, since $\{x\in \partial \Omega ,~\Theta _0 b|B_0(x)|>1\}\cap (\partial \Omega )_+=\emptyset$.

Our next result discusses the optimality of Theorem 1.3. This theorem determines a part of the boundary where the order parameter (the first component $\psi$ of the minimizer) is exponentially small. Outside this part of the boundary, we will prove that the $L^4$ norm of the order parameter is not exponentially small. In physical terms, superconductivity is present there.

The statement of Theorem 1.5 involves the following notation:

• For all $t>0, {\widetilde{\Omega }}(t)=\{x\in \mathbb R^2~:~\mathrm{dist}(x,\partial \Omega )<t\}$.

• By smoothness of $\partial \Omega$, there exists a geometric constant $t_0$ such that, for all $x\in {\widetilde{\Omega }}(t_0)$, we may assign a unique point $p(x)\in \partial \Omega$ such that $\mathrm{dist}(p(x),x)= \mathrm{dist}(x,\partial \Omega )$.

• If $b>0$, we define the open subset in $\mathbb R^2$

  $$\begin{aligned} {\widetilde{\Omega }}(t_0,b)=\left\{ x\in {\widetilde{\Omega }}(t_0)~:~ 1< b|B_0(p(x))|<\Theta _0^{-1}\right\} . \end{aligned}$$

  (1.14)

• $E_{\mathrm{surf}}:[1,\Theta _0^{-1})\rightarrow (-\infty ,0)$ is the surface energy function which will be defined in (4.5) later. This function is continuous and non-decreasing.

• If ${\widetilde{\Omega }}(t_0,b)\not =\emptyset$, we define the following distribution in ${\mathcal {D}}'\big ({\widetilde{\Omega }}(t_0,b)\big )$:

  $$\begin{aligned} C_c^\infty \big ({\widetilde{\Omega }}(t_0,b)\big )\ni \varphi \mapsto {\mathcal {T}}_b(\varphi )=-2\int _{{\widetilde{\Omega }}(t_0,b)\cap \partial \Omega }\sqrt{\frac{1}{b|B_0(x)|}}E_{\mathrm{surf}}\big (b|B_0(x)|\big )\varphi (x)ds(x), \end{aligned}$$

  (1.15)

  where ds is the surface measure on $\partial \Omega$.

• If $D\subset {\overline{\Omega }}$, we introduce the local Ginzburg–Landau energy in D as follows

  $$\begin{aligned} {\mathcal {E}}(\psi ,{\mathbf {A}};D)=\displaystyle \int _{D} \left( |(\nabla -i\kappa H{\mathbf {A}})\psi |^2-\kappa ^2|\psi |^2+\frac{\kappa ^2}{2} |\psi (x)|^4\right) dx. \end{aligned}$$

  (1.16)

• ${\mathbf {1}}_\Omega$ denotes the characteristic function of the set $\Omega$.

Theorem 1.5

(Existence of surface superconductivity)  Suppose that Assumption 1.2 holds, that $b>\beta _0^{-1}$ and that ${\widetilde{\Omega }}(t_0,b)\not =\emptyset$, where $\beta _0$ is the constant introduced in (1.4). If $(\psi ,{\mathbf {A}})_{\kappa ,H}$ is a minimizer of the functional in (1.1) for $H=b\kappa$, then as $\kappa \rightarrow \infty$, we have the following weak convergence¹

$$\begin{aligned} \kappa {\mathbf {1}}_\Omega |\psi _{\kappa ,H}|^4\rightharpoonup {\mathcal {T}}_b\quad \mathrm{in~}{\mathcal {D}}'\big ({\widetilde{\Omega }}(t_0,b)\big ). \end{aligned}$$

(1.17)

Remark 1.6

Theorem 1.5 demonstrates the existence of surface superconductivity. We can interpret the assumption in Theorem 1.5 in two different ways.

• If $H=b\kappa , b>0$ is fixed and $x_0\in \partial \Omega$, then to find superconductivity near $x_0$, this point should satisfy $1< b|B_0(x_0)|<\Theta _0^{-1}$.

• If $x_0\in \partial \Omega$ is fixed and $|B_0(x_0)|$ is small, then to find superconductivity near $x_0$, the intensity of the applied magnetic field should be increased in such a manner that $H=b\kappa$ and $1< b|B_0(x_0)|<\Theta _0^{-1}$.

Our last result confirms that the region $\{B_0(x)<\frac{\kappa }{H}\}$ carries superconductivity everywhere. To state it, we will use the following notation:

• If $p,q\in \partial \Omega , \mathrm{dist}_{\partial \Omega }(p,q)$ denotes the (arc-length) distance in $\partial \Omega$ between p and q.

• For $x_0\in \mathbb R^2$ and $r>0$, we denote by $Q_r(x_0) = x_0 + (-r/2,r/2)^2$ the interior of the square of center $x_0$ and side r. When $x_0=0$, we write $Q_r=Q_r(0)$.

• For $(x,\ell )\in {\overline{\Omega }}\times (0,t_0/2)$, we will use the following notation:

  $$\begin{aligned} {\mathcal {W}}(x_0,\ell )= \left\{ \begin{array}{ll} \{x\in {\overline{\Omega }}~:~\mathrm{dist}_{\partial \Omega }(p(x),x_0)<\ell ~ \text{ and } ~\mathrm{dist}(x,\partial \Omega )<2\ell \}&{}\mathrm{~if~}x_0\in \partial \Omega ,\\ Q_{2\ell }(x_0)&{}\mathrm{~if~}x_0\in \Omega . \end{array}\right. \end{aligned}$$

  (1.18)

Theorem 1.7

(The bulk superconductivity region)  Suppose that Assumption 1.2 holds for some $\alpha \in (0,1), b>0$ and $\frac{2}{2+\alpha }<\rho <1$ be two constants. Let $x_0\in {\overline{\Omega }}$ such that $|B_0(x_0)|<\frac{1}{b}$.

There exist $\kappa _0>0$, a function $\mathrm{r}:[\kappa _0,+\infty )\rightarrow \mathbb R_+$ such that $\lim _{\kappa \rightarrow +\infty }\mathrm{r}(\kappa )=0$ and, for all $\kappa \ge \kappa _0$ and for all critical point $(\psi ,{\mathbf {A}})_{\kappa ,H}$ of the functional in (1.1) with $H=b\kappa$, the following two inequalities hold,

$$\begin{aligned} \left| \frac{1}{|{\mathcal {W}}(x_0,\kappa ^{-\rho })|}\int _{{\mathcal {W}}_{x_0}(\kappa ^{-\rho })}|\psi (x)|^4dx+2g\big (b|B_0(x_0)|\big )\right| \le \mathrm{r}(\kappa ) \end{aligned}$$

and

$$\begin{aligned} \Big |{\mathcal {E}}\Big (\psi ,{\mathbf {A}};{\mathcal {W}}(x_0,\kappa ^{-\rho })\Big )-\kappa ^2g\big (b|B_0(x_0)|\big )\Big |\le \kappa ^2\mathrm{r}(\kappa ). \end{aligned}$$

Here $g(\cdot )$ is the continuous function appearing in (1.8) and (1.9) (see Sect. 2.1 for its definition and properties).

The result in Theorem 1.7 is a variant of the formula in (1.9) valid for applied magnetic fields which are only Hölder continuous, thereby generalizing the results by Attar [7] and Sandier–Serfaty [41]. This will be clarified further in Remark 1.9.

In the bulk superconductivity region displayed in Theorem 1.7, vortices are expected to exist, since the energy of minimizers is strictly lower that that of the normal or perfectly superconducting states. Even in the case of a uniform magnetic field, their detection remains an open problem (cf. [41]). However, if the magnetic field is not constant, the vortices will not arrange on a lattice and will have a non-uniform distribution. We refer to [7] for existing results about the non-uniform distributions of vortices, but for a different regime of the applied magnetic field.

Remark 1.8

Let us choose fixed constants $\gamma$ and $\rho$ such that $\frac{2}{2+\alpha }<\rho <1$ and $0<\gamma <1-\rho$. Our proof of Theorem 1.7 yields that the constant $\kappa _0$ and the function $r(\kappa )$ in Theorem 1.7 can be selected independently of the point $x_0$ provided that

• $\kappa ^{-2\gamma } \le b|B_0(x_0)|<1$;

• $x_0\in \partial \Omega$ or $\mathrm{dist}(x_0,\partial \Omega )\ge 4\kappa ^{-\rho }$.

The condition $\mathrm{dist}(x_0,\partial \Omega )\ge 4\kappa ^{-\rho }$ ensures that $Q_{2\kappa ^{-\rho }}(x_0)\subset \overline{\Omega }$, which is needed in the proof of Theorem 1.7.

Remark 1.9

Let $\gamma \in (0, \frac{\alpha }{2+ \alpha })$. If we assume furthermore the following geometric condition

$$\begin{aligned} \big |\{x\in {\overline{\Omega }}~,~|B_0(x)|\le \kappa ^{-2\gamma }\} \big |=o(1)\quad (\kappa \rightarrow \infty ), \end{aligned}$$

(1.19)

then Theorem 1.7 implies the weak convergence

$$\begin{aligned} |\psi _{\kappa ,H} (\cdot ) |^4\rightharpoonup -2g\big (b|B_0(\cdot )|\big )\quad \mathrm{in~}{\mathcal {D}}'(\Omega ). \end{aligned}$$

In (1.19), we have used the following notation. If $E\subset \mathbb R^2, |E|$ denotes the Lebesgue (area) measure of E. Note that the condition in (1.19) holds under Assumption 1.1 considered in [7].

The rest of the paper is organized as follows. In Sect. 2, we collect various results that will be used throughout the paper. Section 3 is devoted to the proof of Theorem 1.3. In Sect. 4, we present the proof of Theorem 1.5. Finally, we prove Theorem 1.7 in Sect. 5.

In the proofs, we avoid the use of the a priori elliptic $L^\infty$-estimates, whose derivation is quite complicated (cf. [15, Ch. 11]), thereby providing new proofs for the results in [35, 41]. To our knowledge, these $L^\infty$-estimates have not been established when the magnetic field $B_0$ is only Hölder continuous.

2 Preliminaries

2.1 The bulk energy function

The energy function $g(\cdot )$, hereafter called the bulk energy, has been constructed in [41]. We will recall its construction here. It plays a central role in the study of ‘bulk’ superconductivity, both for two and three dimensional problems (cf. [17, 19]). Furthermore, it is related to the periodic solutions of (1.3) and the Abrikosov energy (cf. [1, 16]).

For $b\in (0,+\infty ), r>0$, and $Q_r=(-r/2,r/2)\times (-r/2,r/2)$, we define the functional,

$$\begin{aligned} F_{b,Q_r}(u)=\int _{Q_r}\left( b|(\nabla -i{\mathbf {A}}_0)u|^2-|u|^2+\frac{1}{2}|u|^4\right) dx, \quad \text{ for } u\in H^1(Q_r). \end{aligned}$$

(2.1)

Here, the magnetic potential ${\mathbf {A}}_0$ is expressed using the symmetric gauge,

$$\begin{aligned} {\mathbf {A}}_0(x)=\frac{1}{2}(-x_2,x_1),\quad \text{ for } x=(x_1,x_2)\in \mathbb R^2, \end{aligned}$$

(2.2)

and gives rise to the unit constant magnetic field.

We define the Dirichlet and Neumann ground state energies by

$$\begin{aligned}&e_D(b,r)=\inf \{F_{b,Q_r}(u)~:~u\in H^1_0(Q_r)\}, \end{aligned}$$

(2.3)

$$\begin{aligned}&e_N(b,r)=\inf \left\{ F_{b,Q_r}(u)~:~u\in H^1(Q_r)\right\} . \end{aligned}$$

(2.4)

We can define $g(\cdot )$ as follows (cf. [7, 17, 41])

$$\begin{aligned} \forall ~b>0,\quad g(b)=\lim _{r\rightarrow \infty }\frac{e_D(b,r)}{|Q_r|}=\lim _{r\rightarrow \infty }\frac{e_N(b,r)}{|Q_r|}, \end{aligned}$$

(2.5)

where $|Q_r| =r^2$ denotes the area of $Q_r$.

Furthermore, there exists a universal constant $C>0$ such that

$$\begin{aligned} \forall ~b>0,\quad \forall ~r>1,\quad g(b)\le \frac{e_D(b,r)}{|Q_r|}\le \frac{e_N(b,r)}{|Q_r|}+\frac{C}{r}\le g(b)+\frac{2C}{r}. \end{aligned}$$

(2.6)

One can show that the function $g(\cdot )$ is a non decreasing continuous function such that

$$\begin{aligned} g(0)=-\frac{1}{2},\quad g(b)< 0 \text{ when } b <1, \text{ and } \quad g(b)=0 \text{ when } b\ge 1. \end{aligned}$$

(2.7)

2.2 The magnetic Laplacian

We need two results about the magnetic Laplacian. The first result concerns the Dirichlet magnetic Laplace operator in a bounded set $\Omega$ with a strong constant magnetic field B, that is

$$\begin{aligned} -(\nabla -iB{\mathbf {A}}_0)^2\quad \mathrm{in~}L^2(\Omega ), \end{aligned}$$

with the Dirichlet condition

$$\begin{aligned} u=0~\mathrm{on~}\partial \Omega . \end{aligned}$$

Here ${\mathbf {A}}_0$ is the vector field introduced in (2.2), with ${{\mathrm{curl}}}{\mathbf {A}}_0=1$. It is based on the elementary spectral inequality (cf. [15, Lem. 1.4.1]):

Lemma 2.1

For all $B\in \mathbb R$ and $\phi \in H^1_0(\Omega )$, it holds

$$\begin{aligned} \int _\Omega |(\nabla -iB{\mathbf {A}}_0)\phi |^2dx\ge |B|\int _\Omega |\phi (x)|^2dx. \end{aligned}$$

The second result concerns the Neumann magnetic Laplace operator in a bounded set $\Omega$ with a strong constant magnetic field B, that is

$$\begin{aligned} -(\nabla -iB{\mathbf {A}}_0)^2\quad \mathrm{in~}L^2(\Omega ), \end{aligned}$$

with the (magnetic) Neumann condition

$$\begin{aligned} \nu \cdot (\nabla -iB{\mathbf {A}}_0)u=0~\mathrm{on~}\partial \Omega . \end{aligned}$$

Here $\nu$ is the unit inward normal vector on $\partial \Omega$. The asymptotic behavior of the groundstate energy as $|B|\rightarrow \infty$ is well known (cf. [21, 31] and [15, Prop. 8.2.2]):

Lemma 2.2

There exist ${\hat{\beta }}_0>0$ and $C>0$ such that, if $|B|\ge {\hat{\beta }}_0$ and $\phi \in H^1(\Omega )$,

$$\begin{aligned} \int _\Omega |(\nabla -iB{\mathbf {A}}_0)\phi |^2dx\ge \left( \Theta _0|B|-C|B|^{3/4}\right) \int _\Omega |\phi |^2dx. \end{aligned}$$

2.3 Universal bound on the order parameter

If $(\psi ,{\mathbf {A}})$ is a solution of (1.3), then $\psi$ satisfies in $\Omega$ (cf. [15, Prop. 10.3.1])

$$\begin{aligned} |\psi (x) |\le 1. \end{aligned}$$

(2.8)

2.4 The magnetic energy

Let us introduce the space of vector fields

$$\begin{aligned} H^1_\mathrm{div}(\Omega )=\{{\mathbf {A}}\in H^1(\Omega ;\mathbb R^2)~:~\mathrm{div}{\mathbf {A}}=0\mathrm{~in~}\Omega \quad \mathrm{and}\quad \nu \cdot {\mathbf {A}}=0~\mathrm{on~}\partial \Omega \}. \end{aligned}$$

(2.9)

The functional in (1.1) is invariant under the gauge transformations $(\psi ,{\mathbf {A}})\mapsto (e^{i\phi }\psi ,{\mathbf {A}}+\nabla \phi )$. Consequently, if $(\psi ,{\mathbf {A}})$ solves (1.3), we may apply a gauge transformation such that the new configuration $({\widetilde{\psi }} = e^{i\phi }\psi , {\widetilde{A}}=A+\nabla \phi )$ is a solution of (1.3) and furthermore ${\widetilde{{\mathbf {A}}}}\in H^1_\mathrm{div}(\Omega )$. Having this in hand, we always assume that every critical/minimizing configuration $(\psi ,{\mathbf {A}})$ satisfies ${\mathbf {A}}\in H^1_\mathrm{div}(\Omega )$ which simply amounts to a gauge transformation.

Since $\Omega$ For given $B_0 \in L^2(\Omega )$, there exists a unique vector field satisfying

$$\begin{aligned} {\mathbf {F}}\in H^1_\mathrm{div}(\Omega ) \quad \mathrm{and}\quad {{\mathrm{curl}}}{\mathbf {F}}=B_0. \end{aligned}$$

(2.10)

Actually, ${\mathbf {F}}=\nabla ^\bot f$ where $f\in H^2(\Omega )\cap H^1_0(\Omega )$ is the unique solution of $-\Delta f=B_0$. The uniqueness of ${\mathbf {F}}$ is a consequence of the simple connectedness of $\Omega$.

Remark 2.3

By the elliptic Schauder Hölder estimates (see for example Appendix E.3 in [15]), if in addition $B_0 \in C^{0,\alpha }(\overline{\Omega })$ for some $\alpha >0$, then the vector field ${\mathbf {F}}$ is smooth of class $C^{1,\alpha }(\overline{\Omega })$.

We recall the following result from [7]:

Proposition 2.4

Let $\gamma \in (0,1)$ and $0<c_1<c_2$ be fixed constants. Suppose that $B_0\in L^2(\Omega )$. There exist $\kappa _0>0$ and $C>0$ such that, if $\kappa \ge \kappa _0, c_1\kappa \le H \le c_2\kappa$ and if $(\psi ,{\mathbf {A}})_{\kappa ,H}\in H^1(\Omega )\times H^1_\mathrm{div}(\Omega )$ is a minimizer of (1.2), then

$$\begin{aligned} \Vert {\mathbf {A}}-{\mathbf {F}}\Vert _{C^{0,\gamma }(\overline{\Omega })}\le \frac{C}{\kappa }. \end{aligned}$$

The proof of Proposition 2.4 given in [7] is made under the assumption $B_0\in C^\infty ({\overline{\Omega }})$, but it still holds under the weaker assumption $B_0\in L^2(\Omega )$.

The next result gives the existence of a useful gauge transformation that allows us to approximate the vector field ${\mathbf {F}}$ locally by a vector field generating a constant magnetic field. It is similar to the result in [7, Lem. A.3], but the difference here is that we only assume ${\mathbf {F}}\in C^{1,\alpha }(\overline{\Omega })$ instead of $C^2$.

Lemma 2.5

Let $\alpha \in (0,1), r_0>0$ and $B_0\in C^{0,\alpha }({\overline{\Omega }})$. There exists $C>0$ and for any $a\in \overline{\Omega }$ a function $\varphi _a \in C^{2,\alpha } ( {\mathbb {R}}^2)$ such that, if $r\in (0,r_0]$ and $\ B(a,r)\cap \Omega \not =\emptyset ,$ then

$$\begin{aligned} \forall ~x\in \overline{B(a,r)\cap \Omega },\quad |{\mathbf {F}}(x)-B_0(a){\mathbf {A}}_0(x-a)-\nabla \varphi _a(x)|\le C r^{1+\alpha }. \end{aligned}$$

Here ${\mathbf {F}}$ is the vector field satisfying (2.10).

Proof of Lemma 2.5

Since the boundary of $\Omega$ is smooth and ${\mathbf {F}}\in C^{1,\alpha }(\overline{\Omega };\mathbb R^2)$, the vector field ${\mathbf {F}}$ admits an extension ${\widehat{{\mathbf {F}}}}$ in $C^{1,\alpha }(\mathbb R^2;\mathbb R^2)$. We get in this way an extension ${\widehat{B}}_0 = {{\mathrm{curl}}}{\widehat{{\mathbf {F}}}}$ of $B_0$ in $C^{0,\alpha } ({\mathbb {R}}^2)$. We now define in ${\mathbb {R}}^2$, the two vector fields

$$\begin{aligned} {\widetilde{{\mathbf {F}}}}(y)={\widehat{{\mathbf {F}}}}(a+y),\quad \mathbf {{\widetilde{A}}}(y)=\left( \int _0^1 s{\widehat{B}}_0(a+sy)ds\right) (-y_2,y_1). \end{aligned}$$

Clearly, ${{\mathrm{curl}}}{\widetilde{{\mathbf {F}}}}={{\mathrm{curl}}}\mathbf {{\widetilde{A}}}={\widehat{B}}_0(a+y)$. Consequently, by integrating the closed 1-form associated with ${\widetilde{F}} -\mathbf {{\widetilde{A}}}$, there exists a function ${\widetilde{\varphi }}\in C^{2,\alpha }({\mathbb {R}}^2)$ such that

$$\begin{aligned} {\widetilde{{\mathbf {F}}}}-\nabla {\widetilde{\varphi }}=\mathbf {{\widetilde{A}}}, {\widetilde{\phi }} (0)=0. \end{aligned}$$

Since ${\widehat{B}}_0\in C^{0,\alpha }({\mathbb {R}}^2), \mathbf {{\widetilde{A}}} (y)=B_0(a)(-y_2,y_1)+{\mathcal {O}}(r^{1+\alpha })$ in $\overline{B(0,r)}$. We then define the function $\varphi _a$ by $\varphi _a(x)={\widetilde{\varphi }}(x-a)+B_0(a)\Big (a_2x_1-a_1x_2\Big )$. This implies

$$\begin{aligned} \forall ~x\in \overline{B(a,r)},\quad |\mathbf{{\widehat{F}}} (x)-B_0(a){\mathbf {A}}_0(x-a)-\nabla \varphi _a(x)|\le C r^{1+\alpha }, \end{aligned}$$

and the lemma by restriction to $\overline{\Omega }$. $\square$

2.5 Lower bound of the kinetic energy term

The main result in this subsection is:

Proposition 2.6

Let $0<c_1<c_2$ be fixed constants. Suppose that $\alpha \in (0,1]$ and $B_0\in C^{0,\alpha }({\overline{\Omega }})$. There exist $\kappa _0>0$ and $C>0$ such that the following is true, with

$$\begin{aligned} \sigma (\alpha ) = \frac{2\alpha }{3+\alpha }. \end{aligned}$$

(2.11)

• (1) For

  • $\kappa \ge \kappa _0, c_1 \kappa \le H\le c_2 \kappa$;

  • $(\psi ,{\mathbf {A}})_{\kappa ,H}$ a solution of (1.3);

  • $\phi \in H^1(\Omega )$ satisfies $\mathrm{supp}\phi \subset \{x\in {\overline{\Omega }},~|B_0(x)|>0\}$,

  we have

  $$\begin{aligned} \int _\Omega |(\nabla -i\kappa H{\mathbf {A}})\phi (x)|^2dx\ge \Theta _0\kappa H\int _\Omega \big (|B_0(x)|-C \kappa ^{-\sigma (\alpha )}\big ) |\phi (x)|^2dx. \end{aligned}$$

• (2) If in addition $\phi =0$ on $\partial \Omega$, then

  $$\begin{aligned} \int _\Omega |(\nabla -i\kappa H{\mathbf {A}})\phi (x)|^2dx\ge \kappa H\int _\Omega \big (|B_0(x)|-C\kappa ^{-\sigma (\alpha )}\big )|\phi (x)|^2dx. \end{aligned}$$

The estimates in Items (1) and (2) in this proposition are known when the vector field ${\mathbf {A}}$ is $C^2$, independent of $(\kappa ,H), {{\mathrm{curl}}}{\mathbf {A}}\not =0$ and $B_0$ is replaced by ${{\mathrm{curl}}}{\mathbf {A}}$ (cf. Lemma 2.2 and [20]).

For $\alpha =1$ (i.e. $B_0$ is Lipschitz) the errors in Proposition 2.6 and Lemma 2.2 are of the same order.

Proof of Proposition 2.6

Let us choose an arbitrary $\phi \in H^1(\Omega )$. All constants below are independent of $\phi$. For the sake of simplicity, we will work under the additional assumption that $\mathrm{supp}\phi \subset \{B_0>0\}$.

Step 1. Decomposition of the energy via a partition of unity. 

For $\ell >0$ we consider the partition of unity in $\mathbb R^2$

$$\begin{aligned} \sum _{j}\chi _j^2=1,\quad \sum _{j}|\nabla \chi _j|^2\le C \ell ^{-2}\quad \mathrm{in}\mathbb R^2,\quad \mathrm{and}\quad \mathrm{supp}\chi _j\subset B(a_j^\ell ,\ell ). \end{aligned}$$

Here the construction is first done for $\ell =1$ and then for general $\ell >0$ by dilation. Hence the constant C is independent of $\ell$. Although the points $(a_j^\ell )$ depend on $\ell$, we omit below the reference to $\ell$ and write $a_j$ for $a_j^\ell$.

In what follows, we will use this partition of unity with

$$\begin{aligned} \ell =\kappa ^{-\rho },\quad \quad 0<\rho <1\quad \mathrm{and}\quad \kappa \mathrm{~ large~ enough}. \end{aligned}$$

Using this partition of unity, we may estimate from below the kinetic energy term as follows

$$\begin{aligned} \int _\Omega |(\nabla -i\kappa H{\mathbf {A}})\phi |^2dx\ge \sum _{j} \left( \int _\Omega |(\nabla -i\kappa H{\mathbf {A}})(\chi _j\phi )|^2dx-C\ell ^{-2}\int _\Omega |\chi _j\phi |^2dx\right) . \end{aligned}$$

(2.12)

Let $\alpha _j(x)=(x-a_j)\cdot \big ({\mathbf {A}}(a_j)-{\mathbf {F}}(a_j)\big )$, where ${\mathbf {F}}$ is the vector field in (2.10). Note the useful decomposition

$$\begin{aligned} {\mathbf {A}}(x)-\nabla \alpha _j={\mathbf {F}}(x)+\big ({\mathbf {A}}(x)-{\mathbf {F}}(x)\big ) -\big ({\mathbf {A}}(a_j)-{\mathbf {F}}(a_j)\big ). \end{aligned}$$

By Proposition 2.4, we have in $B(a_j,\ell )\cap \Omega$,

$$\begin{aligned} \begin{aligned} |(\nabla -i\kappa H{\mathbf {A}})(\chi _j\phi ) |^2&=|(\nabla -i\kappa H({\mathbf {A}}-\nabla \alpha _j))(e^{-i\kappa H\alpha _j}\chi _j\phi )|^2\\&\ge (1-\ell ^{\delta })|(\nabla -i\kappa H{\mathbf {F}})e^{-i\kappa H\alpha _j}\chi _j\phi |^2\\&-\ell ^{-\delta }\kappa ^2H^2 \ell ^{2\gamma }\Vert {\mathbf {A}}-{\mathbf {F}}\Vert _{C^{0,\gamma }(\overline{\Omega })}^2|\chi _j\phi |^2\\&\ge (1-\ell ^{\delta })|(\nabla -i\kappa H{\mathbf {F}})(e^{-i\kappa H\alpha _j}\chi _j\phi )|^2 -CH^2\ell ^{(2\gamma -\delta )} |\chi _j\phi |^2. \end{aligned} \end{aligned}$$

(2.13)

Here $\delta >0$ and $\gamma \in (0,1)$ are two parameters to be chosen later.

By Lemma 2.5, we may define a smooth function $\varphi _j$ in $B(a_j,\ell )\cap \Omega$ such that,

$$\begin{aligned} | {\mathbf {F}}(x)-\nabla \varphi _j(x)-|B_0(a_j)|{\mathbf {A}}_0(x-a_j)|\le C \ell ^{1+\alpha }, \end{aligned}$$

where $C>0$ is independent of j.

Consequently, there exists $C>0$ such that, for all j,

$$\begin{aligned} |(\nabla -i\kappa H{\mathbf {F}})(e^{-i\kappa H\alpha _j}\chi _j\phi ) |^2\ge & {} (1-\ell ^{\delta }) |(\nabla -i\kappa H|B_0(a_j)\nonumber \\&\times \,|{\mathbf {A}}_0(x-a_j))e^{-i\kappa H\varphi _j}e^{-i\kappa H\alpha _j}\chi _j\phi |^2\nonumber \\&-\,C\kappa ^2H^2\ell ^{2+2\alpha -\delta }|\chi _j\phi |^2. \end{aligned}$$

(2.14)

Step 2. The case $\mathrm{supp}\phi \subset \{x\in {\overline{\Omega }},~B_0(x)>0\}$ and $\phi =0$ on $\partial \Omega$.

The assumption on the support of $\phi$ yields that $\chi _j\phi \in H^1_0(\Omega )$. Collecting (2.13), (2.14) and the spectral inequality in Lemma 2.1, we get the existence of $C>0$ such that for all j

$$\begin{aligned} \int _\Omega |(\nabla -i\kappa H{\mathbf {A}})(\chi _j\phi )|^2dx\ge & {} (1-2\ell ^{\delta })\kappa H\int _\Omega |B_0(a_j)||\chi _j\phi |^2dx\\&-\,CH^2(\ell ^{2\gamma -\delta } +\kappa ^2\ell ^{2+2\alpha -\delta })\int _\Omega |\chi _j\phi |^2dx. \end{aligned}$$

Since $B_0$ is in $C^{0,\alpha }({\overline{\Omega }})$, we have $B_0(x)=B_0(a_j)+{\mathcal {O}}(\ell ^\alpha )$ in $B(a_j,\ell )$. Thus

$$\begin{aligned} \int _\Omega |(\nabla -i\kappa H{\mathbf {A}})(\chi _j\phi )|^2dx\ge & {} \kappa H\int _\Omega |B_0(x)||\chi _j\phi (x)|^2dx\\&-\,CH^2 (\ell ^\alpha + \ell ^\delta + \ell ^{2\gamma -\delta } +\kappa ^2\ell ^{2+2\alpha -\delta })\int _\Omega |\chi _j\phi (x)|^2dx. \end{aligned}$$

After summation and using that $\sum _j\chi _j^2=1$, we get

$$\begin{aligned}&\int _\Omega |(\nabla -i\kappa H{\mathbf {A}})\phi |^2dx\nonumber \\&\quad \ge \kappa H \left( \int _\Omega |B_0(x)||\phi (x)|^2dx -C(\ell ^\alpha {+} \ell ^\delta + \ell ^{2\gamma -\delta } {+}\kappa ^2\ell ^{2+2\alpha -\delta } {+} \kappa ^{-2} \ell ^{-2}) \int _\Omega |\phi |^2dx\right) . \end{aligned}$$

Hence the goal is to choose, when $\kappa \rightarrow +\infty$ and with $\ell =\kappa ^{-\rho }$, the parameters $\rho , \delta , \gamma$ and $\alpha$ in order to minimize the sum

$$\begin{aligned} \Sigma _0 (\kappa ,\ell ) :=\ell ^\alpha + \ell ^\delta + \ell ^{2\gamma -\delta } +\kappa ^2\ell ^{2+2\alpha -\delta } + \kappa ^{-2} \ell ^{-2}. \end{aligned}$$

(2.15)

If we take $\delta =\gamma$, which corresponds to give the same order for the second and the third terms in $\Sigma _0$, we obtain with $\ell =\kappa ^{-\rho }$

$$\begin{aligned}&\int _\Omega |(\nabla -i\kappa H{\mathbf {A}})\phi |^2dx \ge \kappa H \int _\Omega \Big (|B_0(x)|-C(\kappa ^{-\rho \alpha }+\kappa ^{-\rho \gamma }+\kappa ^{2-(2+2\alpha -\gamma )\rho }\\&\quad +\,\kappa ^{2\rho -2}\Big )|\phi (x)|^2dx. \end{aligned}$$

In the remainder, to minimize the error for the two last terms, we select $\rho$ such that

$$\begin{aligned} 2-(2+2\alpha -\gamma )\rho =2\rho -2, \end{aligned}$$

i.e.

$$\begin{aligned} \rho =4/ (4+2\alpha -\gamma ). \end{aligned}$$

Getting the condition $0<\rho <1$ satisfied leads to the condition $\alpha >\gamma /2$. We select $\gamma =\frac{2}{3}\alpha$. This choice is optimal since

$$\begin{aligned} \sigma (\alpha ):=\max _{0<\gamma <2\alpha }\sigma _0(\alpha ,\gamma )=\sigma _0\left( \alpha ,\frac{2\alpha }{3}\right) =\frac{2\alpha }{3+\alpha }, \end{aligned}$$

where

$$\begin{aligned} \sigma _0(\alpha ,\gamma )=\min \left( \frac{4\alpha }{4+2\alpha -\gamma },\frac{4\gamma }{4+2\alpha -\gamma },\frac{2(2\alpha -\gamma )}{4+2\alpha -\gamma }\right) . \end{aligned}$$

This finishes the proof of Item (2) in Proposition 2.6.

Step 3. The case $\mathrm{supp}\,\phi \subset \{x\in {\overline{\Omega }},~B_0(x)>0\}$.

We continue with the choice $\delta =\gamma =\frac{2}{3}\alpha$ and $\rho =4/(4+2\alpha -\gamma )$. We collect the inequalities in (2.13), (2.14) and Lemma 2.2 and write

$$\begin{aligned} \int _\Omega |(\nabla -i\kappa H{\mathbf {A}})(\chi _j\phi ) |^2dx\ge & {} (1-2\ell ^{2\alpha /3}) \kappa H \int _\Omega \Big (\Theta _0|B_0(a_j)|-C(\kappa H)^{-1/4}\Big )|\chi _j\phi |^2dx\\&-\,CH^2 \kappa ^{-\sigma (\alpha )} \int _\Omega |\chi _j\phi |^2dx. \end{aligned}$$

Since $B_0\in C^{0,\alpha }({\overline{\Omega }})$, we can replace $B_0(a_j)$ by $B_0(x)$ on the support of $\chi _j$ modulo an error ${\mathcal {O}}(\ell ^\alpha )$. We insert the resulting estimate into (2.12) and use that $\sum _j\chi _j^2=1$ to get,

$$\begin{aligned} \int _\Omega |(\nabla -i\kappa H{\mathbf {A}})\phi |^2dx \ge \kappa H \int _\Omega \Big (\Theta _0|B_0(x)|-C(\kappa ^{-\sigma (\alpha )}+\kappa ^{-1/2})\Big )|\phi (x)|^2dx. \end{aligned}$$

Observing that $\sigma (\alpha ) \le \frac{1}{2}$, we have achieved the proof of Item (1) in Proposition 2.6. $\square$

3 Exponential decay

3.1 Main statements

We recall the definition of the de Gennes constant $\Theta _0$ in (1.5), and the two constants $\beta _0,\beta _1$ in (1.4). For all $\lambda \in (0,\beta _0)$, we introduce the two functions on $\omega (\lambda )$:

$$\begin{aligned} t_\lambda (x)=\mathrm{dist}\big (x,\partial {\omega }(\lambda )\big )\quad \mathrm{and}\quad \zeta _\lambda (x)=\mathrm{dist}\big (x,\Omega \cap \partial \omega (\lambda )\big ), \end{aligned}$$

(3.1)

where $\omega (\cdot )$ is the domain introduced in (1.12).

Theorem 3.1

(Exponential decay outside the superconductivity region) Let $c_1$ and $c_2$ be two constants such that $\beta _0^{-1}<c_1<c_2$. Suppose that Assumption 1.2 holds for some $\alpha \in (0,1)$. There exists $\mu _0>0$ and for all $\mu \in (0,\mu _0)$, there exist $\kappa _0>0, C>0$ and ${\hat{\alpha }}>0$ such that, if

$$\begin{aligned} \kappa \ge \kappa _0,\quad c_1\kappa \le H\le c_2\kappa , \end{aligned}$$

and $(\psi ,{\mathbf {A}})_{\kappa ,H}$ is a solution of (1.3), then the following inequalities hold:

• (1) Decay in the interior:

  $$\begin{aligned} \int _{\omega (\lambda )\cap \{t_\lambda (x)\ge \frac{1}{\sqrt{\kappa H}}\}} \Big (|\psi (x)|^2+{\frac{1}{\kappa H}}|(\nabla -i\kappa H{\mathbf {A}})\psi (x)|^2\Big )\exp \Big (2{\hat{\alpha }}\sqrt{\kappa H}t_\lambda (x)\Big )dx \le \frac{C}{\kappa }, \end{aligned}$$

  where $\lambda =\displaystyle \frac{\kappa }{H}+\mu$;

• (2) Decay up to the boundary:

  $$\begin{aligned} \int _{\omega (\beta )\cap \{\zeta _{\beta }(x)\ge \frac{1}{\sqrt{\kappa H}}\}} \Big (|\psi (x)|^2+{\frac{1}{\kappa H}}|(\nabla -i\kappa H{\mathbf {A}})\psi (x) |^2\Big )\exp \Big (2{\hat{\alpha }}\sqrt{\kappa H}\zeta _{\beta }(x)\Big )dx \le \frac{C}{\kappa }, \end{aligned}$$

  where $\beta =\Theta _0^{-1}\left( \displaystyle \frac{\kappa }{H}+\mu \right)$.

Remark 3.2

Theorem 3.1 says that, for $\mu >0$ sufficiently small, bulk superconductivity breaks down in the region $\{x\in \Omega ,~|B_0(x)|\ge \frac{\kappa }{H}+\mu \}$ and that surface superconductivity breaks down in the region $\{x\in \partial \Omega ,~\Theta _0|B_0(x)|\ge \frac{\kappa }{H}+\mu \}$. This is illustrated in Figs. 1 and 2.

Remark 3.3

In the constant magnetic field case, $B_0=1$, Theorem 3.1 is proved by Pan [35], in response to a conjecture by Rubinstein [38, p. 182]. Our proof of Theorem 3.1 is simpler than the one in [35] since we do not use the a priori elliptic $L^\infty$-estimates, whose derivation is not easy (cf. [15, Ch. 11]).

Remark 3.4

On a technical level, one can still avoid to use the $L^\infty$-elliptic estimates in the proof of Theorem 3.1 when the magnetic field is constant, by establishing a weak decay estimate on the order parameter (namely $\Vert \psi \Vert _2={\mathcal {O}}(\kappa ^{-1/4})$). This has been done by Bonnaillie-Noël and Fournais in [10] and then generalized by Fournais–Helffer to non-vanishing continuous magnetic fields in [15, Cor. 12.3.2]. However, in the sign-changing field case and the regime considered in Theorem 3.1, the weak decay estimate as in [10] does not hold.

The substitute of the weak decay estimate in our proof is the use of a (local) gauge transformation. This has been used earlier to estimate the Ginzburg–Landau energy (cf. [9, 29]), and the exponential decay of the order parameter for non-smooth magnetic fields (cf. [6]). We will extend this method for obtaining local estimates in Theorems 4.7 and 4.8.

Remark 3.5

The conclusion in Theorem 1.3 is a simple consequence of Theorem 3.1 and the estimate in Proposition 2.4. Actually, if O is an open set independent of $\kappa$ such that $\overline{O}\subset \omega (\kappa /H)$, then

$$\begin{aligned} O\subset {\omega } \left( \frac{\kappa }{H}+\mu \right) \end{aligned}$$

for $\mu$ sufficiently small, and

$$\begin{aligned} \mathrm{dist}\Big (x,\partial \omega \left( \frac{\kappa }{H}+\mu \right) \Big )\ge c_\mu \quad \mathrm{in}~O, \end{aligned}$$

for a constant $c_\mu >0$.

Similarly, when O is an open set independent of $\kappa$ and

$$\begin{aligned} \overline{O} \subset { \omega }(\kappa /H)\cup \{x\in \partial \Omega ,~\Theta _0|B_0(x)|<\kappa /H\}, \end{aligned}$$

then

$$\begin{aligned} O\subset \omega \left( \Theta _0^{-1}\Big (\frac{\kappa }{H}+\mu \Big )\right) \end{aligned}$$

for $\mu$ sufficiently small, and

$$\begin{aligned} \mathrm{dist}\bigg (x,\partial \omega \left( \Theta _0^{-1}\Big (\frac{\kappa }{H}+\mu \Big )\right) \bigg )\ge {\hat{c}}_\mu \quad \mathrm{in}~O, \end{aligned}$$

for a constant ${\hat{c}}_\mu >0$.

The rest of this section is devoted to the proof of Theorem 3.1, which follows the scheme of the proof of the semi-classical Agmon estimates (cf. [15, Ch. 12] and references therein).

Suppose that the parameters $\kappa$ and H have the same order, i.e.

$$\begin{aligned} \kappa \ge \kappa _0\quad \mathrm{and}\quad c_1\kappa \le H\le c_2\kappa , \end{aligned}$$

where $\kappa _0\ge 1$ is supposed sufficiently large (this condition will appear in the proof below). Suppose also that

$$\begin{aligned} c_2> c_1> \beta _0^{-1}, \end{aligned}$$

where $c_1, c_2$ are fixed constants and $\beta _0$ was introduced in (1.4).

3.2 Useful inequalities

For all $\gamma >0$, we extend to ${\overline{\Omega }}$ the definitions of $t_\gamma$ and $\zeta _\gamma$ given in (3.1) as follows

$$\begin{aligned} t_\gamma (x)= \left\{ \begin{array}{ll} \mathrm{dist}\big (x,\partial \omega (\gamma )\big )&{}\mathrm{if~}x\in \omega (\gamma )\\ 0&{}\mathrm{if~}x\in {\overline{\Omega }}{\setminus }\omega (\gamma )) \end{array}\right. \end{aligned}$$

(3.2)

and

$$\begin{aligned} \zeta _\gamma (x)= \left\{ \begin{array}{ll} \mathrm{dist}\big (x, \Omega \cap \partial \omega (\gamma )\big )&{}\mathrm{if~}x\in \omega (\gamma )\\ 0&{}\mathrm{if~}x\in {\overline{\Omega }}{\setminus }\omega (\gamma ) \end{array}\right. . \end{aligned}$$

(3.3)

In the sequel, we will add conditions on $\gamma$ to ensure that $\omega (\gamma )\not =\emptyset$.

Let ${\tilde{\chi }} \in C^\infty (\mathbb R)$ be a non negative function satisfying

$$\begin{aligned} \tilde{\chi }=0\ \mathrm{on}\ (-\infty ,\frac{1}{2}],\quad \tilde{\chi }=1\ \mathrm{on}\ [1,\infty ). \end{aligned}$$

Define the functions $\chi _\gamma , \eta _\gamma , f_\gamma$ and $g_\gamma$ on $\Omega$ as follows:

$$\begin{aligned} \chi _\gamma (x)= & {} \tilde{\chi }\big (\sqrt{\kappa H} t_\gamma (x)\big ),\quad \eta _\gamma (x)=\tilde{\chi }\big (\sqrt{\kappa H} \zeta _\gamma (x)\big ),\nonumber \\ f_\gamma (x)= & {} \chi _\gamma (x) \exp \big ({\hat{\alpha }} \sqrt{\kappa H}t_\gamma (x)\big )\quad \mathrm{and}\quad g_\gamma (x)=\eta _\gamma (x) \exp \big ({\hat{\alpha }} \sqrt{\kappa H}\zeta _\gamma (x)\big ), \end{aligned}$$

(3.4)

where ${\hat{\alpha }}$ is a positive number whose value will be fixed later.

Let $h\in \{f_\gamma ,g_\gamma \}$. We multiply both sides of the first equation in (1.3) by $h^2{\overline{\psi }}$ and then integrate by parts over $\omega (\gamma )$. We get

$$\begin{aligned} \int _{\omega (\gamma )} \Big ( \big |(\nabla -i\kappa H \mathbf{A})(h\psi ) \big |^2-\kappa ^2h^2|\psi |^2-|\nabla h|^2|\psi |^2 \Big )dx\le 0. \end{aligned}$$

(3.5)

In the computations below, the constant C is independent of ${\hat{\alpha }},\gamma ,\kappa$ and H. We estimate the term involving $\nabla h$ as follows

$$\begin{aligned} \int _{\omega (\gamma )}|\nabla h|^2|\psi |^2dx \le 2 {\hat{\alpha }}^2\kappa H \Vert h \psi \Vert ^2_{L^2(\omega (\lambda ))}+ C \kappa H T(h), \end{aligned}$$

where

$$\begin{aligned} T(h):= {\left\{ \begin{array}{ll} \displaystyle \int _{\omega (\gamma )\cap \{\sqrt{\kappa H} t_\gamma (x)\le 1\}}|\psi (x)|^2dx&{}\mathrm{if}~h=f_\gamma ,\\ &{}\\ \displaystyle \int _{\omega (\gamma )\cap \{\sqrt{\kappa H} \zeta _\gamma (x)\le 1\}}|\psi (x) |^2dx&{}\mathrm{if}~h=g_\gamma . \end{array}\right. } \end{aligned}$$

(3.6)

In this way we infer from (3.5) the following estimate

$$\begin{aligned} \int _{\omega (\gamma )} \Big ( \big |(\nabla -i\kappa H \mathbf{A})(h\psi ) (x)\big |^2-\kappa ^2h(x)^2|\psi (x) |^2- 2 {\hat{\alpha }}^2\kappa Hh(x)^2|\psi (x) |^2\Big )dx\le C \kappa H T(h). \end{aligned}$$

(3.7)

3.3 Decay in the interior

Now we choose

$$\begin{aligned} \gamma =\lambda =\frac{\kappa }{H}+\mu . \end{aligned}$$

Here $0<\mu <\mu _0$ and $\mu _0$ is sufficiently small such that $\mu _0+\frac{1}{c_1}<\beta _0$. This ensures that $\omega (\lambda )\not =\emptyset$.

We choose in (3.7) the function $h=f_\lambda$, where $f_\lambda$ is the function introduced in (3.4). Note that $f_\lambda \psi \in H^1_0(\omega (\lambda ))$. We may apply the result in Proposition 2.6 to $\phi :=f_\lambda \psi$ and infer from (3.7)

$$\begin{aligned} \int _{\omega (\lambda )}\Big (\big (1-C\kappa ^{-\sigma (\alpha )})|B_0(x)|-2 {\hat{\alpha }}^2-\frac{\kappa }{H}\Big )f_\lambda ^2 |\psi |^2dx \le C\int _{\omega (\lambda )\cap \{\sqrt{\kappa H} t_\lambda (x)\le 1\}}|\psi (x)|^2dx. \end{aligned}$$

We then use that $|B_0(x)|\ge \lambda$ in $\omega (\lambda )$ and that $\lambda =\frac{\kappa }{H}+\mu$. Consequently, for $0<\mu<\mu _0, 0<{\hat{\alpha }}<{\hat{\alpha }}_0, \kappa \ge \kappa _0, {\hat{\alpha }}_0$ sufficiently small (for example ${\hat{\alpha }}_0^2 < \mu /4$) and $\kappa _0$ sufficiently large

$$\begin{aligned} \big (1-C\kappa ^{-\sigma (\alpha )})|B_0(x)|- 2{\hat{\alpha }}^2-\frac{\kappa }{H} \ge \frac{\mu }{2}. \end{aligned}$$

Consequently, there exists a constant $C_\mu >0$ such that

$$\begin{aligned} \begin{aligned} \int _{\omega (\lambda )}f_\lambda (x)^2 |\psi (x)|^2dx&\le C_\mu ^{-1}\int _{\omega (\lambda )\cap \{\sqrt{\kappa H} t_\lambda (x)\le 1\}}|\psi (x) |^2dx\\&\le \frac{C}{\sqrt{\kappa H}}\quad \quad \quad \mathrm{by~}(2.8). \end{aligned} \end{aligned}$$

Inserting this into (3.7) [with $h=f_\lambda$ and $T(f_\lambda )$ defined in (3.6)] achieves the proof of Item (1) in Theorem 3.1.

3.4 Decay up to the boundary

Now we prove Item (2) in Theorem 3.1. Here we choose

$$\begin{aligned} \gamma =\beta =\Theta _0^{-1} \left( \frac{\kappa }{H}+\mu \right) . \end{aligned}$$

Note that the estimate in Item (2) of Theorem 3.1 is trivially true if $\omega (\beta )=\emptyset$. So, we assume in the sequel that $\omega (\beta )\not =\emptyset$. This holds if

$$\begin{aligned} H\ge c_1\kappa , \quad c_1> (\Theta _0\beta _1)^{-1}, \end{aligned}$$

and $\mu$ is sufficiently small.

We write (3.7) for $h=g_\beta$, where $g_\beta$ is introduced in (3.4) and $T(g_\beta )$ in (3.6). We apply Proposition 2.6 to $\phi :=g_\beta \psi$ and get

$$\begin{aligned}&\int _{\omega (\beta )}\Big (\big (1-C\kappa ^{-\sigma (\alpha )})\Theta _0|B_0(x)|-C{\hat{\alpha }}^2-\frac{\kappa }{H}\Big )g_\beta (x)^2|\psi (x)|^2dx\nonumber \\&\quad \le C\int _{\omega (\beta )\cap \{\sqrt{\kappa H} \zeta _\beta (x)\le 1\}}|\psi (x)|^2dx. \end{aligned}$$

(3.8)

We decompose the integral over $\omega (\beta )$ as follows

$$\begin{aligned} \int _{\omega (\beta )}=\int _{\omega _\mathrm{int}(\beta )}+\int _{\omega _\mathrm{bnd}(\beta )}, \end{aligned}$$

where

$$\begin{aligned} \omega _\mathrm{int}(\beta ){=}\omega (\beta )\cap \big \{\sqrt{\kappa H}\mathrm{dist}(x,\partial \Omega )\ge 1\big \}\quad \mathrm{and}\quad \omega _\mathrm{bnd}(\beta ){=}\omega (\beta )\cap \big \{\sqrt{\kappa H}\mathrm{dist}(x,\partial \Omega ){<} 1\big \}. \end{aligned}$$

From (3.2), we see that $\zeta _\beta (x)=t_\beta (x)$ and $f_\beta (x)=g_\beta (x)$ in $\omega _\mathrm{int}(\beta )$. Furthermore, from the definition of $\omega (\cdot )$ in (1.12), we see that $\omega (\beta )\subset \omega (\lambda )$ and $t_\beta (x)\le t_\lambda (x)$ on $\omega (\beta )$ if $\beta \ge \lambda$. Hence, by the first item in Theorem 3.1 [which is already proved for all ${\hat{\alpha }}\in (0,{\hat{\alpha }}_0)$],

$$\begin{aligned} \int _{\omega _\mathrm{int}(\beta )}\Big |\big (1-C\kappa ^{-\sigma (\alpha )})\Theta _0|B_0(x)|-2 {\hat{\alpha }}^2-\frac{\kappa }{H}\Big |g_\beta (x)^2 |\psi (x)|^2dx\le \frac{C}{\kappa }. \end{aligned}$$

(3.9)

Thus, we infer from (3.8) (and the bound $|\psi |\le 1$),

$$\begin{aligned} \int _{\omega _\mathrm{bnd}(\beta )}\Big (\big (1-C\kappa ^{-\sigma (\alpha )})\Theta _0|B_0(x)|- 2 {\hat{\alpha }}^2-\frac{\kappa }{H}\Big ) g_\beta (x)^2|\psi (x)|^2dx\le \frac{C}{\kappa }. \end{aligned}$$

But, in $\omega _\mathrm{bnd}(\beta ), \Theta _0|B_0(x)|\ge \frac{\kappa }{H}+\mu$, by definition of $\omega (\beta )$ and $\beta =\Theta _0^{-1}(\frac{\kappa }{H}+\mu )$. Thus, as long as ${\hat{\alpha }}$ is selected sufficiently small, we have

$$\begin{aligned} (1-C\kappa ^{-\sigma (\alpha )}) \Theta _0|B_0(x)|-2 {\hat{\alpha }}^2-\frac{\kappa }{H}\ge \frac{\mu }{2}, \end{aligned}$$

and consequently, for some constant ${\tilde{C}}_\mu >0$,

$$\begin{aligned} \int _{\omega _\mathrm{bnd}(\beta )} g_\beta (x)^2|\psi (x)|^2dx\le \frac{{\tilde{C}}_\mu }{\kappa }. \end{aligned}$$

We insert this estimate and the one in (3.9) into (3.8) to get

$$\begin{aligned} \int _{\omega (\beta )} g_\beta (x)^2|\psi (x)|^2dx\le \frac{{\tilde{C}}_\mu +C}{\kappa }. \end{aligned}$$

Finally, by inserting this estimate into (3.7) [with $h=g_\beta$ and $T(g_\beta )$ defined in (3.6)], we finish the proof of Item (2) in Theorem 3.1.

4 Surface energy

The analysis of surface superconductivity starts with the work of St. James–de Gennes [39], who studied this phenomenon on the ball. In the last two decades, many papers adressed the boundary concentration of the Ginzburg–Landau order parameter for general 2D and 3D samples in the presence of a constant magnetic field. We refer the reader to [3, 12, 14, 16, 18, 19, 32, 35].

In this section, we study surface superconductivity in non-uniform magnetic fields. Our presentation not only generalizes the results known for the constant field case, but also provides local estimates and new proofs, see Theorems 4.7 and 4.8. The most notable novelty in the proofs is that we do not use the $L^\infty$ elliptic estimates.

4.1 The surface energy function

In this subsection, we give the definition of the continuous function $E_\mathrm{surf}:[1,\Theta _0^{-1}]\rightarrow (-\infty ,0]$ introduced by Pan in [35] and which appeared after (1.14) and in Theorem 1.5. $\Theta _0$ is as before the de Gennes constant introduced in (1.5) with property (1.6).

For $b\in [1,\Theta _0^{-1}]$ and $R>0$, we consider the reduced Ginzburg–Landau functional,

$$\begin{aligned} {\mathcal {V}}(U_R)\ni \phi \mapsto {\mathcal {E}}_{b,R}(\phi )=\int _{U_R}\left( b|(\nabla _{(\sigma ,\tau )}+i\tau \mathbf {f})\phi |^2-|\phi |^2+\frac{1}{2}|\phi |^4\right) d\sigma d\tau , \end{aligned}$$

(4.1)

where $\mathbf {f}=(1,0)$ and $U_R$ is the domain,

$$\begin{aligned} U_R=(-R,R)\times (0,+\infty ), \end{aligned}$$

(4.2)

and

$$\begin{aligned} {\mathcal {V}}(U_R)=\left\{ u\in L^2(U_R)~:~(\nabla _{(\sigma ,\tau )}+i\tau \mathbf {f})u\in L^2(U_R)~,~u(\pm R,\cdot )=0\right\} . \end{aligned}$$

(4.3)

We introduce the following ground state energy,

$$\begin{aligned} d(b,R)=\inf \{{\mathcal {E}}_{b,R}(\phi )~:~\phi \in {\mathcal {V}}(U_R)\}. \end{aligned}$$

(4.4)

In [35], it is proved that, for all $b\in [1,\Theta _0^{-1}]$, there exists $E_\mathrm{surf}(b)\in (-\infty ,0]$ such that

$$\begin{aligned} E_\mathrm{surf}(b)=\lim _{R\rightarrow \infty }\frac{d(b,R)}{2R}. \end{aligned}$$

(4.5)

The surface energy function $E_{\mathrm{surf}}(\cdot )$ can be described by a simplified 1D problem as well (cf. [3, 18] and finally [12] for the optimal result). We collect some properties of $E_{\mathrm{surf}}(\cdot )$:

• $E_{\mathrm{surf}}(\cdot )$ is a continuous and increasing function (cf. [19]);

• $E_{\mathrm{surf}}(\Theta _0^{-1})=0$ and $E_{\mathrm{surf}}(b)<0$ for all $b\in [1,\Theta _0^{-1})$ (cf. [14]).

The next theorem gives the existence of some minimizer with good properties (cf. [35, Theorems 4.4 & 5.3]):

Theorem 4.1

There exist positive constants $R_0$ and M such that, for all $b\in [1,\Theta _0^{-1})$ and $R\ge R_0$:

• (1) The functional (4.1) has a minimizer $u_R$ in ${\mathcal {V}}(U_R)$ with the following properties:

  • (a) $u_R\not \equiv 0$;

  • (b) $\Vert u_R\Vert _\infty \le 1$;

  • (c)
    $$\begin{aligned} \frac{1}{R} \int _{U_R\cap \{\tau \ge 3\}}\frac{\tau ^2}{(\ln \tau )^2}\left( |(\nabla _{(\sigma ,\tau )}{+}i\tau \mathbf {f})u_R|^2{+}|u_R(\sigma , \tau )|^2{+}\tau ^2|u_R(\sigma ,\tau )|^4\right) d\sigma d\tau \le M .\end{aligned}$$

• (2) The surface energy function $E_\mathrm{surf}(b)$ satisfies

  $$\begin{aligned} E_\mathrm{surf}(b)\le \frac{d(b,R)}{2R}\le E_\mathrm{surf}(b)+\frac{M}{R}. \end{aligned}$$

The upper bound in Item (2) above results from a property of superadditivity of d(b, R), see [35, Eq. (5.4)]. The lower bound in Item (2) above is not explicitly mentioned in [35], but its derivation is easy [17, Proof of Thm 2.1, Step 2, p. 351] and can be sketched in the following way. Let $R>0$ and $n\in {\mathbb {N}}$. Let $u_R \in H^1_0(U_R)$ be a minimizer of the functional in (4.1). We extend $u_R$ to a function in $H^1_0(U_{(2n+1)R})$ by periodicity as follows

$$\begin{aligned} u_R(x_1+2R,x_2)=u_R(x_1,x_2). \end{aligned}$$

Consequently,

$$\begin{aligned} d(b,(2n+1)R)\le {\mathcal {E}}_{b,(2n+1)R}(u_R)=(2n+1)d(b,R). \end{aligned}$$

Dividing both sides of the preceding inequality by $2(2n+1)R$ and sending n to $+\infty$, we get

$$\begin{aligned} E_{\mathrm{surf}}(b)\le \frac{d(b,R)}{2R}. \end{aligned}$$

4.2 Boundary coordinates

The analysis of the boundary effects is performed in specific coordinates valid in a tubular neighborhood of $\partial \Omega$. We call these coordinates boundary coordinates. For more details on these coordinates, see for instance [15, Appendix F].

For a sufficiently small $t_0>0$, we introduce the open set

$$\begin{aligned} \Omega (t_0)=\{x\in {\mathbb {R}}^2~:~\mathrm{dist}(x,\partial \Omega )<t_0\}. \end{aligned}$$

In the sequel, let $x_0\in \partial \Omega$ be a fixed point. Let $s\mapsto \gamma _{x_0}(s)$ be the parametrization of $\partial \Omega$ by arc-length such that $\gamma _{x_0}(0)=x_0$. Also, let $\nu (s)$ be the unit inward normal of $\partial \Omega$ at $\gamma _{x_0}(s)$. The orientation of $\gamma _{x_0}$ is selected in the counter clock-wise direction, hence

$$\begin{aligned} \mathrm{det}\Big (\gamma _{x_0}'(s),\nu (s)\Big )=1. \end{aligned}$$

Define the transformation

$$\begin{aligned} \Phi _{x_0}: \left[ -\frac{|\partial \Omega |}{2},\frac{|\partial \Omega |}{2}\right) \times (0,t_0)\ni (s,t)\mapsto \gamma _{x_0}(s)+t\nu (s)\in \Omega (t_0). \end{aligned}$$

(4.6)

We may choose $t_0$ sufficiently small (independently from the choice of the point $x_0\in \partial \Omega$) such that the transformation in (4.6) is a diffeomorphism. The Jacobian of this transformation is $|D\Phi _{x_0}|=1-tk (s)$, where k denotes the curvature of $\partial \Omega$. For $x\in \Omega (t_0)$, we put

$$\begin{aligned} \Phi ^{-1}_{x_0}(x)=(s(x),t(x)). \end{aligned}$$

In particular, we get the explicit formulae

$$\begin{aligned} t(x)=\mathrm{dist}(x,\partial \Omega )\quad \mathrm{and}\quad s(x_0)=0. \end{aligned}$$

(4.7)

Using $\Phi _{x_0}$, we may associate to any function $u\in L^2(\Omega )$, a function ${\widetilde{u}}=T_{\Phi _{x_0}} u$ defined in $[-\frac{|\partial \Omega |}{2}, \frac{|\partial \Omega |}{2})\times (0,t_0)$ by,

$$\begin{aligned} {\widetilde{u}}(s,t)=u(\Phi _{x_0}(s,t)). \end{aligned}$$

(4.8)

Also, for every vector field ${\mathbf {A}}\in H^1(\Omega )$, we assign the vector field

$$\begin{aligned} {\tilde{{\mathbf {A}}}}(s,t)=\Big ({\tilde{{\mathbf {A}}}}_1(s,t),{\tilde{{\mathbf {A}}}}_2(s,t)\Big ) \end{aligned}$$

with

$$\begin{aligned} \left\{ \begin{array}{rl} {\tilde{{\mathbf {A}}}}_1(s,t) &{} = a(s,t){\mathbf {A}}\Big ( \Phi _{x_0}(s,t)\Big )\cdot \gamma _{x_0}'(s),\\ {\tilde{{\mathbf {A}}}}_2 (s,t)&{} = {\mathbf {A}}\Big ( \Phi _{x_0}(s,t)\Big )\cdot \nu (s), \end{array} \right. \end{aligned}$$

(4.9)

and

$$\begin{aligned} a(s,t) =1-tk(s). \end{aligned}$$

The following change of variable formulas hold.

Proposition 4.2

For $u\in H^1(\Omega )$ and ${\mathbf {A}}\in H^1(\Omega ;\mathbb R^2)$, we have:

$$\begin{aligned} \int _{\Omega ( t_0)}\left| (\nabla -i{\mathbf {A}})u\right| ^2dx= & {} \int _{0}^{t_0}\int _{-\frac{|\partial \Omega |}{2}}^{\frac{|\partial \Omega |}{2}} \left[ [a(s,t)]^{-2}|(\partial _s-i\tilde{\mathbf {A}}_1){\widetilde{u}}|^2\right. \nonumber \\&\left. +\,|(\partial _t-i{\tilde{{\mathbf {A}}}}_2)\widetilde{u}|^2\right] a(s,t)dsdt, \end{aligned}$$

(4.10)

and

$$\begin{aligned} \int _{\Omega (t_0)} |u(x)|^2dx=\int _{0}^{t_0}\int _{-\frac{|\partial \Omega |}{2}}^{\frac{|\partial \Omega |}{2}} |{\widetilde{u}}(s,t)|^2a(s,t)dsdt. \end{aligned}$$

(4.11)

Recall the vector field ${\mathbf {A}}_0$ introduced in (2.2). Up to a gauge transformation, the vector field ${\mathbf {A}}_0$ admits a useful (local) representation in the coordinate system (s, t).

For $x_0\in \partial \Omega$ and $\ell \in (0,t_0)$, we introduce the set $V_{x_0}(\ell ) \subset \Omega (t_0)$ as follows:

$$\begin{aligned} V_{x_0}(\ell )=\Phi _{x_0}\Big ((-\ell ,\ell )\times (0,\ell )\Big ). \end{aligned}$$

(4.12)

Lemma 4.3

There exists $r_0>0$ such that, for any $x_0$ in $\partial \Omega$, there exists $g_{x_0}$ in $C^\infty ( (-2r_0,\,2r_0)\times (0,r_0))$ such that

$$\begin{aligned}{\tilde{{\mathbf {A}}}}_0(s,t) -\nabla g_{x_0}(s,t) = \left( -t+k(s)\frac{t^2}{2},0\right) \quad \mathrm{in}~(-2r_0,\,2r_0)\times (0,r_0). \end{aligned}$$

Here ${\tilde{{\mathbf {A}}}}_0$ is the vector field associated with ${\mathbf {A}}_0$ by the formulas in (4.9) and one can take $r_0=\min (t_0,\frac{|\partial \Omega |}{4})$.

For the proof of Lemma 4.3, we refer to [15, Proof of Lem. F.1.1]. Note that Lemma F.1.1 in [15] is announced for a more general setting.

We will use Lemma 4.3 to estimate the following Ginzburg–Landau energy of u,

$$\begin{aligned} {\mathcal {G}}_0\big (u,{\mathbf {A}}_0;V_{x_0}(\ell )\big )=\int _{V_{x_0}(\ell )}\Big (|(\nabla -ih_\mathrm{ex}{\mathbf {A}}_0)u|^2-\kappa ^2|u|^2+\frac{\kappa ^2}{2}|u|^4\Big )dx. \end{aligned}$$

(4.13)

Lemma 4.4

There exist constants $C>0, \ell _0>0$ and $\kappa _0>0$ such that, for all $x_0\in \partial \Omega , \ell \in (0,\ell _0), \kappa \ge \kappa _0, \kappa ^2\le h_\mathrm{ex}\le \Theta _0^{-1}\kappa ^2$, and $u\in H^1_0(V_{x_0}(\ell ))\cap L^\infty (V_{x_0}(\ell ))$ satisfying $\Vert u\Vert _\infty \le 1$, the following two inequalities hold:

$$\begin{aligned} {\mathcal {G}}_0\big (u,{\mathbf {A}}_0;V_{x_0}(\ell )\big )\ge {2} \frac{\kappa ^2\ell }{\sqrt{h_\mathrm{ex}}}E_{\mathrm{surf}}\left( \frac{h_\mathrm{ex}}{\kappa ^2}\right) -C\kappa \ell \Big (\ell +\kappa ^3\ell ^4+\kappa \ell ^2\Big ), \end{aligned}$$

(4.14)

and

$$\begin{aligned} {\mathcal {G}}_0\big (u,{\mathbf {A}}_0;V_{x_0}(\ell )\big )\le (1+C\ell ) \frac{\kappa ^2}{h_\mathrm{ex}}{\mathcal {E}}_{h_\mathrm{ex}/\kappa ^2,\sqrt{h_\mathrm{ex}}\ell } ({\widetilde{v}})+C\kappa \ell \Big (\kappa ^3\ell ^4+\kappa \ell ^2\Big ). \end{aligned}$$

(4.15)

where ${\mathcal {E}}_{\cdot ,\cdot }$ is the functional introduced in (4.1) and

$$\begin{aligned} {\widetilde{v}}(\sigma ,\tau )=\exp \left( -ih_\mathrm{ex}g_{x_0}\Big (\frac{\sigma }{\sqrt{h_\mathrm{ex}}},\frac{\tau }{\sqrt{h_\mathrm{ex}}}\Big )\right) {\widetilde{u}}\Big (\frac{\sigma }{\sqrt{h_\mathrm{ex}}},\frac{\tau }{\sqrt{h_\mathrm{ex}}}\Big ). \end{aligned}$$

Here ${\widetilde{u}}$ is the function associated with u by (4.8) and $g_{x_0}$ is introduced in Lemma 4.3.

Proof

Using Proposition 4.2 and the assumptions on u, we may write, for two positive constants $C_0,C$ and for all $0<\ell <\min \big (\frac{1}{2} C_0^{-1},t_0\big )$,

$$\begin{aligned} {\mathcal {G}}_0\big (u,{\mathbf {A}}_0;V_{x_0}(\ell )\big ){\ge } (1-C_0\ell )\int _0^{\ell }\int _{-\ell }^{\ell }\Big (|(\nabla -ih_\mathrm{ex}{\widetilde{{\mathbf {A}}}}_0){\widetilde{u}}|^2{-}\kappa ^2|{\widetilde{u}}|^2+\frac{\kappa ^2}{2}|{\widetilde{u}}|^4\Big )dsdt{-}C\kappa ^2\ell ^3. \end{aligned}$$

Let $g:=g_{x_0}$ be the function defined in Lemma 4.3 and ${\widetilde{w}}(s,t)=e^{-i h_\mathrm{ex}g(s,t)}{\widetilde{u}}(s,t)$. Using the Cauchy-Schwarz inequality, we get the existence of $C>0$ such that

$$\begin{aligned} {\mathcal {G}}_0\big (u,{\mathbf {A}}_0;V_{x_0}(\ell )\big )\ge & {} (1-2C_0\ell )\int _0^{\ell }\int _{-\ell }^{\ell }\Big (|(\nabla +ih_\mathrm{ex} t\mathbf {f}){\widetilde{w}}|^2-\kappa ^2|{\widetilde{w}}|^2+\frac{\kappa ^2}{2}|{\widetilde{w}}|^4\Big )dsdt\\&-C\kappa ^4\ell ^5-C\kappa ^2\ell ^3. \end{aligned}$$

Here $\mathbf {f}=(1,0)$. We apply the change of variables $(\sigma ,\tau )=(\sqrt{h_\mathrm{ex}}s,\sqrt{h_\mathrm{ex}}t)$ and ${\widetilde{v}}(\sigma ,\tau )={\widetilde{w}}(s,t)$ to get

$$\begin{aligned} {\mathcal {G}}_0(u,{\mathbf {A}}_0;V_{x_0}(\ell ))\ge (1-2C_0\ell )\frac{\kappa ^2}{h_\mathrm{ex}}{\mathcal {E}}_{h_\mathrm{ex}/\kappa ^2,R} ({{\widetilde{v}}})-C\kappa ^4\ell ^5-C\kappa ^2\ell ^3, \end{aligned}$$

where $R=h_\mathrm{ex}^\frac{1}{2} \ell$ and ${\mathcal {E}}_{h_\mathrm{ex}/\kappa ^2,R}$ is the functional introduced in (4.1) for $b= {h_\mathrm{ex}}/\kappa ^2$.

Note that we extended ${\widetilde{v}}$ by 0, which is possible because $u\in H^1_0(V_{x_0}(\ell ))$. Using the second Item in Theorem 4.1 and the assumption $C_0 \ell < \frac{1}{2}$, we get

$$\begin{aligned} {\mathcal {G}}_0(u,{\mathbf {A}}_0;V_{x_0}(\ell ))\ge 2 (1-2C_0\ell ) \frac{\kappa ^2}{h_\mathrm{ex}}(h_\mathrm{ex}^\frac{1}{2} \ell )E_{\mathrm{surf}}\left( \frac{h_\mathrm{ex}}{\kappa ^2}\right) -C\kappa ^4\ell ^5-C\kappa ^2\ell ^3. \end{aligned}$$

This proves the lower bound (4.14) in Lemma 4.4.

Similarly, using Lemma 4.3, the Cauchy-Schwarz inequality on the kinetic term and a change of variables, we get the upper bound (4.15) of Lemma 4.4. $\square$

4.3 Existence of surface superconductivity

The proof of Theorem 1.5 follows from the exponential decay stated in Theorem 3.1 and the following result:

Theorem 4.5

Suppose that Assumption 1.2 holds and that $b>\beta _0^{-1}$, where $\beta _0$ is the constant introduced in (1.4). There exists $\rho \in (0,1)$ such that the following is true.

Let $x_0\in \partial \Omega$ such that $\frac{1}{b}< |B_0(x_0)|<\frac{1}{ \Theta _0 b}$. If $(\psi ,{\mathbf {A}})_{\kappa ,H}$ is a minimizer of the functional in (1.1) for $H=b\kappa$, then

$$\begin{aligned} \lim _{\kappa \rightarrow +\infty }\left( 2\kappa ^{1+\rho }\int _{V_{x_0}(\kappa ^{-\rho })} |\psi (x)|^4dx\right) =-2\sqrt{\frac{1}{b|B_0(x_0)|}}E_{\mathrm{surf}}\big (b|B_0(x_0)|\big )>0, \end{aligned}$$

(4.16)

and

$$\begin{aligned} \lim _{\kappa \rightarrow +\infty }\Big (2\kappa ^{\rho -1}{\mathcal {E}}\big (\psi ,{\mathbf {A}};V_{x_0}(\kappa ^{-\rho })\big )\Big )=\sqrt{\frac{1}{b|B_0(x_0)|}}E_{\mathrm{surf}}\big (b|B_0(x_0)|\big )<0. \end{aligned}$$

(4.17)

The proof of Theorem 4.5 will follow from the upper bound in Theorems 4.7 and 4.8 below.

Remark 4.6

Let $\epsilon \in (1,\Theta _0^{-1}-1)$. The convergence in (4.16) and (4.17) is uniform with respect to $x_0\in \{1+\epsilon \le b|B_0|<\Theta _0^{-1}\}\cap \partial \Omega$. This is precisely stated in Theorems 4.7 and 4.8.

4.4 Sharp upper bound on the $L^4$-norm

In this subsection, we will prove:

Theorem 4.7

Suppose that $B_0\in C^{0,\alpha }({\overline{\Omega }})$ for some $\alpha \in (0,1), \rho \in (\frac{3}{3+\alpha },1)$ and

$$\begin{aligned} b\ge \beta _0^{-1},\, \text{ with } \beta _0:=\sup _{x\in {\overline{\Omega }}}|B_0(x)|>0. \end{aligned}$$

There exist $\kappa _0>0$, a function $\mathrm{r}:[\kappa _0,+\infty )\rightarrow \mathbb R_+$ such that $\lim _{\kappa \rightarrow +\infty }\mathrm{r}(\kappa )=0$ and, for all $\kappa \ge \kappa _0$, for all critical point $(\psi ,{\mathbf {A}})_{\kappa ,H}$ of the functional in (1.1) with $H=b\kappa$, and all $x_0\in \partial \Omega$ satisfying

$$\begin{aligned} 1\le b |B_0(x_0)| < \Theta _0^{-1}, \end{aligned}$$

the inequality

$$\begin{aligned} \frac{1}{2\ell }\int _{V_{x_0}(\ell )}|\psi (x)|^4dx\le -2\kappa ^{-1}\sqrt{\frac{1}{b|B_0(x_0)|}}E_{\mathrm{surf}}\Big (b |B_0(x_0)|\Big ) +\kappa ^{-1} \mathrm{r}(\kappa ), \end{aligned}$$

holds with

$$\begin{aligned} \ell = \kappa ^{-\rho } \text { and }\quad V_{x_0}(\ell ) \text { is defined in (4.12)}. \end{aligned}$$

Proof

The proof is reminiscent of the method used by the second author in [26, Sec. 4] (see also [27]). We assume that $B_0(x_0)>0$. The case where $B_0(x_0)<0$ can be treated in the same manner by applying the transformation $u\mapsto \overline{u}$.

Let $\sigma \in (0,1)$ and $\ell =\kappa ^{-\rho }$ as in the statement of Theorem 4.7. Let $f\in C_c^\infty \Big (V_{x_0}\big ((1+\sigma )\ell \big ) \Big )$ be a smooth function satisfying,

$$\begin{aligned} f=1\quad \mathrm{in~} V_{x_0}(\ell ),\quad 0\le f\le 1\hbox { and }|\nabla f|+\sigma \ell |\Delta f|\le \frac{C}{\sigma \ell }\quad \mathrm{in~}V_{x_0}\big ((1+\sigma )\ell \big ). \end{aligned}$$

(4.18)

The function f depends on the parameters $x_0,\ell ,\sigma$ but the constant C is independent of these parameters. We will estimate the following local energy

$$\begin{aligned} {\mathcal {E}}_1(f\psi ,{\mathbf {A}}):={\mathcal {E}}_1\big (f\psi ,{\mathbf {A}}; V_{x_0}((1+\sigma )\ell )\big ), \end{aligned}$$

(4.19)

where, for an open set ${\mathcal {V}}\subset \Omega$,

$$\begin{aligned} \begin{aligned}&{\mathcal {E}}_1(u,{\mathbf {A}}; {\mathcal {V}}):=\int _{{\mathcal {V}}}\left( |(\nabla -i\kappa H{\mathbf {A}})u |^2-\kappa ^2|u|^2+\frac{\kappa ^2}{2}|u|^4\right) dx,\\&{\mathcal {E}}_2(u,{\mathbf {A}}; {\mathcal {V}}):=\int _{{\mathcal {V}}}\left( |(\nabla -i\kappa H{\mathbf {A}})u |^2-\kappa ^2|u|^2+\frac{\kappa ^2}{2}|u|^4\right) dx\\&+(\kappa H)^2\int _\Omega |{{\mathrm{curl}}}{\mathbf {A}}-B_0|^2dx. \end{aligned} \end{aligned}$$

(4.20)

Since $(\psi ,{\mathbf {A}})$ is a solution of (1.3), an integration by parts yields (cf. [16, Eq. (6.2)]),

$$\begin{aligned} {\mathcal {E}}_1(f\psi ,{\mathbf {A}})=\kappa ^2\int _{V_{x_0}((1+\sigma )\ell )}f^2\left( -1+\frac{1}{2}f^2\right) |\psi |^4\,dx+\int _{V_{x_0}((1+\sigma )\ell )}|\nabla f|^2|\psi |^2dx. \end{aligned}$$

(4.21)

Since $f=1$ in $V_{x_0}(\ell )$ and $-1+\frac{1}{2}f^2\le -\frac{1}{2}$ in $V_{x_0}((1+\sigma )\ell )$, we may write

$$\begin{aligned} \int _{V_{x_0}((1+\sigma )\ell )}f^2\left( -1+\frac{1}{2}f^2\right) |\psi |^4dx\le -\frac{1}{2}\int _{V_{x_0}(\ell )}|\psi |^4dx. \end{aligned}$$

We estimate the integral in (4.21) involving $|\nabla f|$ using (4.18) and $\big |\mathrm{supp}\nabla f\big |\le C\sigma \ell ^2$, where $\big |\mathrm{supp}\nabla f\big |$ denotes the area of the support of $\nabla f$. In this way, we infer from (4.21),

$$\begin{aligned} {\mathcal {E}}_1(f\psi ,{\mathbf {A}})\le -\frac{\kappa ^2}{2}\int _{V_{x_0}(\ell )}|\psi |^4dx+C\sigma ^{-1}. \end{aligned}$$

(4.22)

Now we write a lower bound for this energy. We may find a real-valued function $w\in C^{2,\alpha }(\overline{V_{x_0}((1+\sigma )\ell })$ such that

$$\begin{aligned} {\mathcal {E}}_1(f\psi ,{\mathbf {A}})\ge & {} \int _{V_{x_0}((1+\sigma )\ell )}\Big ( (1-C\ell ^\delta )|(\nabla -i\kappa H B_0(x_0){\mathbf {A}}_0)(e^{-i\kappa Hw}f\psi ) |^2\\&-\,\kappa ^2|f\psi |^2+\frac{\kappa ^2}{2}|f\psi |^4\Big )dx\\&-\,C\kappa ^2\Big (\ell ^{2\gamma -\delta }+\kappa ^2\ell ^{2+2 \alpha -\delta }\Big )\int _{V_{x_0}((1+\sigma )\ell )}|f\psi |^2, \end{aligned}$$

where $\gamma \in (0,1)$ is a constant whose choice will be specified later and $\delta >0$.

The details of these computations are given in (2.13) and (2.14).

From now on we choose $\delta =\alpha$, use the lower bound in Lemma 4.4 and the assumption that $H=b\kappa$ to write

$$\begin{aligned} {\mathcal {E}}_1(f\psi ,{\mathbf {A}})\ge & {} 2(1-C\ell ^\alpha )\kappa (1+\sigma )\ell \sqrt{\frac{1}{b|B_0(x_0)|}}E_{\mathrm{surf}}\big (b|B_0(x_0)|\big )\\&-\,C\kappa \ell (\ell +\kappa ^3\ell ^4+\kappa \ell ^2)-C\kappa ^2\Big (\ell ^\alpha +\ell ^{2\gamma -\alpha }+\kappa ^2\ell ^{2+\alpha }\Big )\int _{V_{x_0}((1+\sigma )\ell )}|f\psi |^2. \end{aligned}$$

Using the bound $\Vert f\psi \Vert _\infty \le 1$, we get further

$$\begin{aligned} {\mathcal {E}}_1(f\psi ,{\mathbf {A}})\ge & {} 2(1-C\ell ^\alpha )\kappa (1+\sigma )\ell \sqrt{\frac{1}{b|B_0(x_0)|}} E_{\mathrm{surf}}\big (b|B_0(x_0)|\big )\nonumber \\&-\,C\kappa \ell \Big (\ell +\kappa \ell ^{1+\alpha }+\kappa \ell ^{1+2\gamma -\alpha }+\kappa ^3\ell ^{3+\alpha }\Big ). \end{aligned}$$

(4.23)

To optimize the remainder, we choose $\gamma =\alpha$. Our assumption

$$\begin{aligned} \ell =\kappa ^{-\rho }\quad \mathrm{with}\quad (1+\alpha )^{-1}<3(3+\alpha )^{-1}<\rho <1 \end{aligned}$$

yields that the function

$$\begin{aligned} \Sigma (\kappa ,\ell ) :=\ell ^\alpha +\ell +\kappa \ell ^{1+\alpha }+\kappa ^3\ell ^{3+\alpha } \end{aligned}$$

tends, with $\ell =\kappa ^{-\rho }$, to 0 as $\kappa \rightarrow +\infty$.

Now, coming back to (4.22), we find

$$\begin{aligned} 2\kappa (1+\sigma )\ell \sqrt{\frac{1}{b|B_0(x_0)|}}E_{\mathrm{surf}}\big (b|B_0(x_0)|\big )-C\kappa \ell \Sigma (\kappa ,\ell ) \le -\frac{\kappa ^2}{2} \int _{V_{x_0}(\ell )}|\psi |^4dx+C\sigma ^{-1}. \end{aligned}$$

We rearrange the terms in this inequality, divide by $\kappa ^2 \ell$, and choose $\sigma =\kappa ^{\frac{1}{2}(\rho -1)}$. In this way, we get the upper bound in Theorem 4.7 with, for some constant $C>0$,

$$\begin{aligned} \mathrm{r}(\kappa )=C \Big (\Sigma (\kappa , \kappa ^{-\rho })+ \kappa ^{\frac{1}{2}(\rho -1)} \Big ). \end{aligned}$$

$\square$

4.5 Sharp Lower bound on the $L^4$-norm

In this subsection, we will prove the asymptotic optimality of the upper bound established in Theorem 4.7 by giving a lower bound with the same asymptotics.

We remind the reader of the definition of the domain $V_{x_0}(\ell )$ in (4.12) and the local energy ${\mathcal {E}}_1\big (\psi ,{\mathbf {A}};{\mathcal {V}}\big )$ introduced in (4.20).

Theorem 4.8

Let $1<\epsilon<\Theta _0^{-1}-1, \frac{3}{3+\alpha }<\rho <1$ and $1-\rho<\delta <1$ be constants. Under the assumptions of Theorem 4.7, there exist $\kappa _0>0$, a function $\mathrm{{\hat{r}}}:[\kappa _0,+\infty )\rightarrow \mathbb R_+$ such that $\lim _{\kappa \rightarrow +\infty }\mathrm{{\hat{r}}}(\kappa )=0$ and, for all $\kappa \ge \kappa _0$, for all minimizer $(\psi ,{\mathbf {A}})_{\kappa ,H}$ of the functional in (1.1) with $H=b\kappa$, and all $x_0\in \partial \Omega$ satisfying

$$\begin{aligned} 1+\epsilon \le b|B_0(x_0)| < \Theta _0^{-1}, \end{aligned}$$

the two inequalities

$$\begin{aligned}&\frac{1}{2\ell }\int _{V_{x_0}(\ell )}|\psi (x)|^4dx\ge -2\kappa ^{-1}\sqrt{\frac{1}{b|B_0(x_0)|}}E_{\mathrm{surf}}\Big (b|B_0(x_0)|\Big ) -\kappa ^{-1} \mathrm{{\hat{r}}} (\kappa ),\\&\left| \frac{1}{2\ell }{\mathcal {E}}_1\big (\psi ,{\mathbf {A}};V_{x_0}((1+\sigma )\ell )\big )-\kappa \sqrt{\frac{1}{b|B_0(x_0)|}}E_{\mathrm{surf}}\Big (b|B_0(x_0)|\Big )\right| \le \kappa \mathrm{{\hat{r}}} (\kappa ), \end{aligned}$$

hold, with $\ell =\kappa ^{-\rho }$ and $\sigma =\kappa ^{-\delta }$.

Remark 4.9

Let $c_2>c_1>0$ be fixed constants. The conclusion in Theorem 4.8 remains true if $\ell$ satisfies

$$\begin{aligned} c_1\kappa ^{-\rho }\le \ell \le c_2\kappa ^{-\rho }. \end{aligned}$$

Proof of Theorem 4.8

In the sequel, $\sigma \in (0,1)$ will be selected as a negative power of $\kappa , \sigma =\kappa ^{-\delta }$ for a suitable constant $\delta \in (0,1)$. As the proof of Theorem 4.7, we can assume that $B_0(x_0)>0$. The proof of the lower bound in Theorem 4.8 will be done in four steps.

Step 1: Construction of a trial function.

The construction of the trial function here is reminiscent of that by Sandier–Serfaty in the study of bulk superconductivity (cf. [41]). Define the function

$$\begin{aligned} u(x)= {\mathbf {1}}_{V_{x_0}((1+\sigma )\ell )}(x) \chi \left( \frac{t(x)}{\ell }\right) \exp \Big (i\kappa H w(x)\Big )v_R\circ \Phi _{x_0}^{-1}(x) +\eta _\ell (x)\psi (x)\quad (x\in \Omega ). \end{aligned}$$

(4.24)

Here $V_{x_0}(\cdot )$ is introduced in (4.12), $t(x)=\mathrm{dist}(x,\partial \Omega ), \Phi _{x_0}$ is the coordinate transformation defined in (4.6),

$$\begin{aligned} v_R(s,t)= & {} \exp \Big (i\kappa H g_{x_0}(s,t) \Big ) u_R\big (s\sqrt{B_0(x_0)\kappa H},t\sqrt{B_0(x_0)\kappa H}\big ), \end{aligned}$$

(4.25)

$$\begin{aligned} R= & {} (1+\sigma )\ell \sqrt{{B_0(x_0)}\kappa H}, \end{aligned}$$

(4.26)

and [cf. (4.1)]

$$\begin{aligned} u_R(\cdot ) \text{ is } \text{ a } \text{ minimizer } \text{ of } \text{ the } \text{ reduced } \text{ functional } {\mathcal {E}}_{bB_0(x_0),R}(\cdot ). \end{aligned}$$

The function $g_{x_0}(s,t)$ satisfies the following identity in $\big (-2\ell ,2\ell \big )\times (0,\ell )$ (cf. Lemma 4.3),

$$\begin{aligned} {\tilde{{\mathbf {A}}}}_0(s,t)-\nabla g_{x_0}(s,t)=\Big (-t+\frac{t^2}{2} k(s),0\Big ). \end{aligned}$$

The function $\chi \in C^\infty ([0,\infty ))$ satisfies

$$\begin{aligned} \chi =1~\mathrm{in~}[0,1/2],\quad \chi =0~\mathrm{in~}[1,\infty ), \text{ and } 0\le \chi \le 1. \end{aligned}$$

The function $\eta _\ell$ is a smooth function satisfying

$$\begin{aligned} \eta _\ell (x)=0~\mathrm{in~}V_{x_0}((1+\sigma )\ell ),\quad \eta _\ell (x)=1~\mathrm{in~}\Omega {\setminus }V_{x_0}((1+2\sigma )\ell ),\quad 0\le \eta _\ell (x)\le 1~\mathrm{in~}\Omega , \end{aligned}$$

and

$$\begin{aligned} |\nabla \eta _\ell (x)|+\sigma \ell |\Delta \eta _\ell | \le C\sigma ^{-1}\ell ^{-1}\quad \mathrm{in~}\Omega , \end{aligned}$$

for some constant $C>0$.

Finally, the function w is the sum of two real-valued $C^{2,\alpha }$-functions $w_1$ and $w_2$ in $V_{x_0}((1+\sigma )\ell )$ and satisfying the following estimates

$$\begin{aligned} |{\mathbf {A}}(x)-{\mathbf {F}}(x)-\nabla w_1(x)|\le & {} \frac{C}{\kappa }\ell ^\alpha \quad \mathrm{and}\quad |{\mathbf {F}}(x)-B_0(x_0){\mathbf {A}}_0(x)-\nabla w_2(x)|\le C\ell ^{1+\alpha }\nonumber \\&\mathrm{in~}V_{x_0}((1+\sigma )\ell ). \end{aligned}$$

(4.27)

By Proposition 2.4, we simply define $w_1(x)=(x-x_0)\cdot \big ({\mathbf {A}}(x_0)-{\mathbf {F}}(x_0)\big )$. The fact that the vector field ${\mathbf {A}}_0(x)$ is gauge equivalent to ${\mathbf {A}}_0(x-x_0)$ and Lemma 2.5 ensure the existence of $w_2$.

We decompose the energy ${\mathcal {E}}(u,{\mathbf {A}})$ as follows

$$\begin{aligned} {\mathcal {E}}(u,{\mathbf {A}})={\mathcal {E}}_1(u,{\mathbf {A}})+{\mathcal {E}}_2(u,{\mathbf {A}}), \end{aligned}$$

(4.28)

where

$$\begin{aligned} {\mathcal {E}}_1(u,{\mathbf {A}})={\mathcal {E}}_1\Big (u,{\mathbf {A}};V_{x_0}((1+\sigma )\ell )\Big )\quad \mathrm{and}\quad {\mathcal {E}}_2(u,{\mathbf {A}})={\mathcal {E}}_2\Big (u,{\mathbf {A}};\Omega {\setminus }V_{x_0}((1+\sigma )\ell )\Big ) \end{aligned}$$

(4.29)

are introduced in (4.20).

Step 2: Estimating ${\mathcal {E}}_1(u,{\mathbf {A}})$.

Using the Cauchy-Schwarz inequality and the estimates in (4.27), we get

$$\begin{aligned} {\mathcal {E}}_1(u,{\mathbf {A}})\le {(1+\ell ^\alpha )} {\mathcal {E}}_1\Big ( e^{-i\kappa H w}u,B_0(x_0){\mathbf {A}}_0\Big )+C\Big (\kappa ^2\ell ^{2+\alpha } +\kappa ^4\ell ^{4+\alpha }\Big ). \end{aligned}$$

For estimating the term ${\mathcal {E}}_1\Big ( e^{-i\kappa H w}u,B_0(x_0){\mathbf {A}}_0\Big )$, we write

$$\begin{aligned} {\mathcal {E}}_1\Big ( e^{-i\kappa H w}u,B_0(x_0){\mathbf {A}}_0\Big )={\mathcal {G}}_0\Big ( e^{-i\kappa H w}u,h_\mathrm{ex}{\mathbf {A}}_0;V_{x_0}({\tilde{\ell }})\Big ), \end{aligned}$$

where

$$\begin{aligned} {\tilde{\ell }}=(1+\sigma )\ell ,\quad h_\mathrm{ex}=\kappa H B_0(x_0)\quad \text { and } {\mathcal {G}}_0\text { is introduced in (4.13)}. \end{aligned}$$

We apply Lemma 4.4 and get

$$\begin{aligned} {\mathcal {E}}_1(u,{\mathbf {A}})\le {(1+C\ell ^\alpha )} \frac{1}{b B_0(x_0)}{\mathcal {E}}_{bB_0(x_0),R}\big ({\widetilde{\chi }}_\ell u_R\big ) +C\kappa \ell \Big (\kappa ^3\ell ^{3+\alpha } +\kappa \ell ^{1+\alpha }\Big ), \end{aligned}$$

where

$$\begin{aligned} {\widetilde{\chi }}_\ell (\tau )=\chi \left( \frac{\tau }{\ell \sqrt{\kappa H}}\right) , b=H/\kappa , \text{ and } R=\sqrt{h_\mathrm{ex}}{\tilde{\ell }}, \end{aligned}$$

in conformity with (4.26).

Note that $\mathrm{supp}(1-{\widetilde{\chi }}_\ell ^2)\subset [\ell \sqrt{\kappa H}/2,\,+\infty )$ and $\mathrm{supp}{\widetilde{\chi }}_\ell '\subset [ \ell \sqrt{\kappa H}/2,\ell \sqrt{\kappa H}]$. Using the decay of $u_R$ established in Theorem 4.1, we get

$$\begin{aligned} {\mathcal {E}}_{bB_0(x_0),R}\big ({\widetilde{\chi }}_\ell \,u_R\big )\le {\mathcal {E}}_{bB_0(x_0),R}\big (u_R\big )+C\frac{|\ln (\ell \sqrt{\kappa H})|^2}{\ell \sqrt{\kappa H}}. \end{aligned}$$

Since ${\mathcal {E}}_{bB_0(x_0),R}(u_R)= d (bB_0(x_0),R)$ and $R=(1+\sigma )\ell \sqrt{B_0(x_0)\kappa H}$, Theorem 4.1 yields

$$\begin{aligned} {\mathcal {E}}_1(u,{\mathbf {A}})\le & {} 2\kappa \ell \sqrt{\frac{1}{b|B_0(x_0)|}}\,E_{\mathrm{surf}}\big (bB_0(x_0)\big )\nonumber \\&+\,C\kappa \ell \Big (\ell ^\alpha +\kappa ^3\ell ^{3+\alpha } +\kappa \ell ^{1+\alpha }+\sigma +(\kappa \ell )^{-1}+\frac{|\ln (\ell \sqrt{\kappa H})|^2}{\ell ^2\kappa ^2}\Big ).\qquad \end{aligned}$$

(4.30)

Step 3: Estimating ${\mathcal {E}}_2(u,{\mathbf {A}})$.

Let $V_{x_0}({\tilde{\ell }} )^{\,\complement }:= \Omega {\setminus }V_{x_0}({\tilde{\ell }} )$ and $u=\eta _\ell \psi$. By a straight forward computation, we obtain

$$\begin{aligned}&\int _{V_{x_0}({\tilde{\ell }} )^{\,\complement }}|(\nabla -i\kappa H{\mathbf {A}})\eta _\ell \psi |^2\,dx\\&= \int _{V_{x_0}({\tilde{\ell }} )^{\,\complement }}|\eta _\ell (\nabla -i\kappa H{\mathbf {A}})\psi |^2\,dx-\int _{V_{x_0}({\tilde{\ell }} )^{\,\complement }} |\psi |^2\eta _\ell \Delta \eta _\ell \,dx\\&= \int _{V_{x_0}({\tilde{\ell }} )^{\,\complement }}|(\nabla -i\kappa H{\mathbf {A}})\psi |^2\,dx -\int _{\{t(x)\le \sigma \ell \}\cap V_{x_0}({\tilde{\ell }} )^\complement } |\psi |^2\eta _\ell \Delta \eta _\ell \,dx -\int _{\{t(x)> \sigma \ell \}}|\psi |^2\eta _\ell \Delta \eta _\ell \,dx\\&\le \int _{V_{x_0}({\tilde{\ell }} )^{\,\complement }}|(\nabla -i\kappa H{\mathbf {A}})\psi |^2\,dx\, +\, C. \end{aligned}$$

Here we used the properties of the function $\eta _\ell$, namely that $0\le \eta _\ell \le 1, |\Delta \eta _\ell |={\mathcal {O}}(\sigma ^{-2}\ell ^{-2})$ and $|\{t(x)\le \sigma \ell \}\cap \mathrm{supp}(\nabla \eta _\ell )|={\mathcal {O}}(\sigma ^2\ell ^2)$.

For the integral over $\{t(x)>\sigma \ell \}$, we use that $b|B_0(x_0)|\ge 1+\epsilon$, which in turn allows us to use Theorem 1.3 and prove that the integral of $|\psi |^2$ is exponentially small as $\kappa \rightarrow +\infty$.

Now we use that $\Big |V_{x_0}({\tilde{\ell }} )^{\,\complement }\cap \mathrm{supp}(1-\eta _\ell )\Big |={\mathcal {O}}(\sigma \ell ^2)$ to write

$$\begin{aligned} -\kappa ^2\int _{V_{x_0}({\tilde{\ell }} )^{\,\complement }}|\eta _\ell \psi |^2\,dx&= -\kappa ^2\int _{V_{x_0}({\tilde{\ell }} )^{\,\complement }}|\psi |^2\,dx+\kappa ^2\int _{V_{x_0}({\tilde{\ell }} )^{\,\complement }}(1-\eta _\ell ^2)|\psi |^2\,dx\\&\le -\kappa ^2\int _{V_{x_0}({\tilde{\ell }} )^{\,\complement }}|\psi |^2\,dx+C\kappa ^2\sigma \ell ^2. \end{aligned}$$

This yields

$$\begin{aligned} {\mathcal {E}}_2(u,{\mathbf {A}})\le & {} \int _{\Omega {\setminus }V_{x_0}({\tilde{\ell }} )}\Big (|(\nabla -i\kappa H{\mathbf {A}})\psi |^2-\kappa ^2|\psi |^2+\frac{\kappa ^2}{2}|\psi |^4\Big )\,dx\\&+\,C\Big (1+\kappa ^2\sigma \ell ^2\Big ) +\kappa ^2H^2\int _\Omega |{{\mathrm{curl}}}{\mathbf {A}}-B_0|^2\,dx. \end{aligned}$$

Remembering the definition of ${\mathcal {E}}_2(\psi ,{\mathbf {A}})$ in (4.29), we obtain

$$\begin{aligned} {\mathcal {E}}_2(u,{\mathbf {A}})\le {\mathcal {E}}_2 (\psi ,{\mathbf {A}}) +C\, \Big (1+\kappa ^2\sigma \ell ^2\Big ). \end{aligned}$$

(4.31)

Step 4: Upper bound of the local Ginzburg–Landau energy.

Since $(\psi ,{\mathbf {A}})$ is a minimizer of the functional ${\mathcal {E}}(\cdot ,\cdot )$,

$$\begin{aligned} {\mathcal {E}}(\psi ,{\mathbf {A}})\le {\mathcal {E}}(u,{\mathbf {A}})={\mathcal {E}}_1(u,{\mathbf {A}})+{\mathcal {E}}_2(u,{\mathbf {A}}). \end{aligned}$$

By (4.29), we may write the simple identity ${\mathcal {E}}(\psi ,{\mathbf {A}})={\mathcal {E}}_1(\psi ,{\mathbf {A}})+{\mathcal {E}}_2(\psi ,{\mathbf {A}})$. Using (4.31), we get

$$\begin{aligned} {\mathcal {E}}_1(\psi ,{\mathbf {A}})\le {\mathcal {E}}_1(u,{\mathbf {A}})+C\Big (1+\kappa ^2\sigma \ell ^2\Big ). \end{aligned}$$

Now, we use the estimate in (4.30) to write

$$\begin{aligned}&{\mathcal {E}}_1(\psi ,{\mathbf {A}})\le 2\kappa \ell \sqrt{\frac{1}{b|B_0(x_0)|}}\,E_{\mathrm{surf}}\big (bB_0(x_0)\big ) \nonumber \\&\quad +\,C\kappa \ell \Big (\ell ^\alpha +\kappa ^3\ell ^{3+\alpha } +\kappa \ell ^{1+\alpha }+\sigma +(\kappa \ell )^{-1}+\kappa \sigma \ell +\frac{|\ln (\ell \sqrt{\kappa H})|^2}{\ell ^2\kappa ^2}\Big ).\qquad \end{aligned}$$

(4.32)

Step 5: Lower bound of the $L^4$-norm.

We select

$$\begin{aligned} \ell =\kappa ^{-\rho }\quad \mathrm{and}\quad \sigma =\kappa ^{-\delta }, \end{aligned}$$

with

$$\begin{aligned} \frac{1}{1+\alpha }<\frac{3}{3+\alpha }<\rho<1\quad \mathrm{and}\quad 1-\rho<\delta <1. \end{aligned}$$

In this way, we get that, the restriction ${\bar{\Sigma }} (\kappa ,\kappa ^{-\rho }, \kappa ^{-\delta })$ of

$$\begin{aligned} {\bar{\Sigma }} (\kappa ,\ell ,\sigma ):=\ell ^\alpha +\kappa ^3\ell ^{3+\alpha } +\kappa \ell ^{1+\alpha }+\sigma +(\kappa \ell )^{-1}+\kappa \sigma \ell +\frac{|\ln (\ell \sqrt{\kappa H})|^2}{\ell ^2\kappa ^2}, \end{aligned}$$

(4.33)

tends to 0 as $\kappa \rightarrow +\infty$.

Consequently, we infer from (4.32),

$$\begin{aligned} {\mathcal {E}}_1(\psi ,{\mathbf {A}})\le 2\kappa \, \ell \, \sqrt{\frac{1}{b|B_0(x_0)|}}\,E_{\mathrm{surf}}\big (bB_0(x_0)\big ) +C\, \kappa \ell \, {\bar{\Sigma }}(\kappa ,\kappa ^{-\rho },\kappa ^{-\delta }). \end{aligned}$$

(4.34)

Now, let f be the smooth function satisfying (4.18). Again, using the properties of f and a straightforward computation as in Step 3, we have

$$\begin{aligned} \begin{aligned}&{\mathcal {E}}_1(f\psi ,{\mathbf {A}})\le {\mathcal {E}}_1(\psi ,{\mathbf {A}})+C\Big (1+\kappa ^2\sigma \ell ^2\Big ),\\&\int _{V_{x_0}({\tilde{\ell }} )}f^2\Big (-1+\frac{1}{2}f^2\Big )|\psi |^4\,dx\ge -\frac{1}{2}\int _{V_{x_0}(\ell )}|\psi |^4\,dx -C\sigma \ell ^2. \end{aligned} \end{aligned}$$

(4.35)

Using the lower bound of ${\mathcal {E}}_1(f\psi ;{\mathbf {A}})$ in (4.23), we get from (4.34) and (4.35),

$$\begin{aligned} \left| {\mathcal {E}}_1(\psi ,{\mathbf {A}})-2\kappa \ell \, \sqrt{\frac{1}{b|B_0(x_0)|}}\,E_{\mathrm{surf}}\big (bB_0(x_0)\big )\right| \le C\kappa \ell \,{\bar{\Sigma }} (\kappa ,\kappa ^{-\rho },\kappa ^{-\delta }). \end{aligned}$$

Remembering the definition of ${\mathcal {E}}_1(\psi ,{\mathbf {A}})={\mathcal {E}}_1\big (\psi ,{\mathbf {A}};V_{x_0}((1+\sigma )\ell )\big )$, we get the statement concerning the local energy in Theorem 4.8.

Now we return back to (4.21). Using (4.35), we write

$$\begin{aligned} {\mathcal {E}}_1(\psi ,{\mathbf {A}})+C\Big (1+\kappa ^2\sigma \ell ^2\Big )\ge -\frac{\kappa ^2}{2}\int _{V_{x_0}(\ell )}|\psi |^4\,dx -C\sigma \ell ^2\kappa ^2. \end{aligned}$$

Rearranging the terms, then using (4.34) and (4.33), we arrive at the following upper bound

$$\begin{aligned} \frac{\kappa ^2}{2}\int _{V_{x_0}(\ell )}|\psi (x)|^4\,dx\ge -2\kappa \ell \, \sqrt{\frac{1}{b|B_0(x_0)|}}\,E_{\mathrm{surf}}\big (bB_0(x_0)\big ) +C\,\kappa \,\ell \, {\bar{\Sigma }} (\kappa ,\kappa ^{-\rho },\kappa ^{-\delta }). \end{aligned}$$

Using the remark around (4.33), this finishes the proof of Theorem 4.8. $\square$

5 The superconductivity region: Proof of Theorem 1.7

In this section, we present the proof of Theorem 1.7 devoted to the distribution of the superconductivity in the region

$$\begin{aligned} \{x\in {\overline{\Omega }},~b\,|B_0(x)|<1\}\quad \text {for the applied magnetic field }H=b\kappa . \end{aligned}$$

The proof follows by an analysis similar to the one in Sect. 4, so our presentation will be shorter here.

Remark 5.1

As $\ell \rightarrow 0_+$, the area of ${\mathcal {W}}(x_0,\ell )$ as introduced in (1.18) is

$$\begin{aligned} |{\mathcal {W}}(x_0,\ell )|=4\ell ^2\quad \mathrm{if~}x_0\in \Omega , \end{aligned}$$

and

$$\begin{aligned} |{\mathcal {W}}(x_0,\ell )|=4\ell ^2+o(\ell ^2)\quad \mathrm{if~}x_0\in \partial \Omega . \end{aligned}$$

The proof of Theorem 1.7 is presented in five steps. In the sequel, $\rho \in (\frac{2}{2+\alpha },1)$ and $c_2>c_1>0$ are fixed,

$$\begin{aligned} c_1\kappa ^{-\rho }\le \ell \le c_2\kappa ^{-\rho }\quad \mathrm{and}\quad \sigma =\kappa ^{\frac{\rho -1}{2}}. \end{aligned}$$

(5.1)

We will refer to the condition on $\ell$ by writing $\ell \approx \kappa ^{-\rho }$.

Step 1. Useful estimates.

Let $f\in C_c^\infty \Big ( {\mathcal {W}}\big (x_0,(1+\sigma )\ell \big ) \Big )$ be a smooth function such that

$$\begin{aligned} f=1\quad \mathrm{in~} {\mathcal {W}}(x_0,\ell ),\quad 0\le f\le 1{~\mathrm and~}|\nabla f|\le \frac{C}{\sigma \ell }\quad \mathrm{in~}{\mathcal {W}}\big (x_0,(1+\sigma )\ell \big ). \end{aligned}$$

(5.2)

As in the proof of (4.22), we have

$$\begin{aligned} {\mathcal {E}}_1\Big (f\psi ,{\mathbf {A}};{\mathcal {W}}(x_0,(1+\sigma )\ell )\Big )\le -\frac{\kappa ^2}{2}\int _{{\mathcal {W}}(x_0,\ell )}|\psi (x)|^4\,dx+C\sigma ^{-1}. \end{aligned}$$

(5.3)

Here ${\mathcal {E}}_1$ is introduced in (4.20). Furthermore, by Cauchy’s inequality, we have the following two estimates:

$$\begin{aligned} {\mathcal {E}}_1\Big (f\psi ,{\mathbf {A}};{\mathcal {W}}(x_0,(1+\sigma )\ell )\Big )\le & {} (1+\kappa ^{-\zeta }){\mathcal {E}}_1\Big (\psi ,{\mathbf {A}};{\mathcal {W}}(x_0,(1+\sigma )\ell )\Big )\nonumber \\&+C\kappa ^2\ell ^2\big (\sigma ^{-1}\kappa ^{\zeta }(\kappa \ell )^{-2}+\sigma +\kappa ^{-\zeta }\big ), \end{aligned}$$

(5.4)

and [cf. (4.21)]

$$\begin{aligned} \begin{aligned} {\mathcal {E}}_1\Big (f\psi ,{\mathbf {A}};{\mathcal {W}}(x_0,(1+\sigma )\ell )\Big )&\ge \kappa ^2\int _{{\mathcal {W}}(x_0,(1+\sigma )\ell )}f^2\left( -1+\frac{1}{2}f^2\right) |\psi |^4\,dx\\&\ge -\frac{\kappa ^2}{2}\int _{{\mathcal {W}}(x_0,\ell )}|\psi (x)|^4\,dx-C\sigma \ell ^2\kappa ^2, \end{aligned} \end{aligned}$$

(5.5)

where $\zeta \in (0,1)$ is a constant to be chosen later.

Step 2. The case $B_0(x_0)=0$ .

The upper bound for the integral of $|\psi |^4$ in Theorem 1.7 is trivial since $|\psi |\le 1$ and $g(0)=-\frac{1}{2}$.

We have the obvious inequalities

$$\begin{aligned} {\mathcal {E}}_1\Big (f\psi ,{\mathbf {A}};{\mathcal {W}}(x_0,(1+\sigma )\ell )\Big )\ge & {} \int _{{\mathcal {W}}(x_0,(1+\sigma )\ell )}\Big (-\kappa ^2|f\psi |^2+\frac{\kappa ^2}{2}|f\psi |^4\Big )\,dx\\\ge & {} -\frac{\kappa ^2}{2} \int _{{\mathcal {W}}(x_0,(1+\sigma )\ell )}dx. \end{aligned}$$

Inserting this into (5.4) and selecting $\zeta =\frac{1-\rho }{2}$, we get

$$\begin{aligned} {\mathcal {E}}_1\Big (\psi ,{\mathbf {A}};{\mathcal {W}}(x_0,(1+\sigma )\ell )\Big )\ge -C\kappa ^2\ell ^2\big (\sigma ^{-1}\kappa ^{\zeta }(\kappa \ell )^{-2}+\sigma +\kappa ^{-\zeta }\big )=o(\kappa ^2\ell ^2), \end{aligned}$$

since $\sigma =\kappa ^{\frac{\rho -1}{2}}, \ell \approx \kappa ^{-\rho }$ and $\frac{2}{2+\alpha }<\rho <1$.

Now we prove an upper bound for ${\mathcal {E}}_1\Big (f\psi ,{\mathbf {A}};{\mathcal {W}}(x_0,(1+\sigma )\ell )\Big )$. Let $\eta _\ell$ be a smooth function satisfying

$$\begin{aligned} \eta _\ell (x)=0~\mathrm{in~}{\mathcal {W}}(x_0,(1+\sigma )\ell ),\quad \eta _\ell (x)=1~\mathrm{in~}\Omega {\setminus }{\mathcal {W}}(x_0,(1+2\sigma )\ell ),\quad 0\le \eta _\ell (x)\le 1~\mathrm{in~}\Omega , \end{aligned}$$

(5.6)

and

$$\begin{aligned} |\nabla \eta _\ell (x)|\le C\sigma ^{-1}\ell ^{-1}\quad \mathrm{in~}\Omega , \end{aligned}$$

(5.7)

for some constant $C>0$. We define the function

$$\begin{aligned} u(x)=\exp \big (i\kappa H w(x)\big )f(x) +\eta _\ell (x)\psi (x), \end{aligned}$$

where the function w is the sum of two functions $w_1$ and $w_2$ such that the two inequalities in (4.27) are satisfied in ${\mathcal {W}}(x_0,(1+\sigma )\ell ))$.

We have the obvious decomposition

$$\begin{aligned} {\mathcal {E}}(u,{\mathbf {A}})= & {} {\mathcal {E}}_1\Big (\exp \big (i\kappa H w(x)\big )f(x),{\mathbf {A}};{\mathcal {W}}(x_0,(1+\sigma )\ell )\Big )\\&+ {\mathcal {E}}_2\Big (\eta _\ell (x)\psi (x),{\mathbf {A}};\Omega {\setminus }{\mathcal {W}}(x_0,(1+\sigma )\ell )\Big ), \end{aligned}$$

where ${\mathcal {E}}_1$ and ${\mathcal {E}}_2$ are introduced in (4.20).

We estimate ${\mathcal {E}}_2\Big (\eta _\ell (x)\psi (x),{\mathbf {A}};\Omega {\setminus }{\mathcal {W}}(x_0,(1+\sigma )\ell )\Big )$ as we did in the proof of Theorem 4.8 [cf. Step 3 and (4.31)]. In this way we get

$$\begin{aligned} {\mathcal {E}}_2\Big (\eta _\ell (x)\psi (x),{\mathbf {A}};\Omega {\setminus }{\mathcal {W}}(x_0,(1+\sigma )\ell )\Big )\le & {} {\mathcal {E}}_2\Big (\psi (x),{\mathbf {A}};\Omega {\setminus }{\mathcal {W}}(x_0,(1+\sigma )\ell )\Big )\nonumber \\&+C(\sigma ^{-1}+\sigma \kappa ^2\ell ^2). \end{aligned}$$

(5.8)

For the term ${\mathcal {E}}_1\Big (\exp \big (i\kappa H w(x)\big )f(x),{\mathbf {A}};{\mathcal {W}}(x_0,(1+\sigma )\ell )\Big )$, we argue as in the proof of Theorem 4.8 (Step 2) and write

$$\begin{aligned}&{\mathcal {E}}_1\Big (\exp \big (i\kappa H w(x)\big )f(x),{\mathbf {A}};{\mathcal {W}}(x_0,(1+\sigma )\ell )\Big )\\&\quad \le (1+\ell ^\alpha ){\mathcal {E}}_1\Big (f(x),B_0(x_0){\mathbf {A}}_0;{\mathcal {W}}(x_0,(1+\sigma )\ell )\Big )+C(\kappa ^2\ell ^{2+\alpha }+\kappa ^4\ell ^{4+\alpha }). \end{aligned}$$

Note that

$$\begin{aligned} \begin{aligned} {\mathcal {E}}_1\Big (f(x),B_0(x_0){\mathbf {A}}_0;{\mathcal {W}}(x_0,(1+\sigma )\ell )\Big )&={\mathcal {E}}_1\Big (f(x),0;{\mathcal {W}}(x_0,(1+\sigma )\ell )\Big )\\&\le C\sigma ^{-1}+\kappa ^2\int _{{\mathcal {W}}(x_0,(1+\sigma )\ell )}f^2\Big (-1+\frac{f^2}{2}\Big )\,dx\\&\le C\sigma ^{-1}-\frac{\kappa ^2}{2}|{\mathcal {W}}(x_0,(1+\sigma )\ell )|+C\sigma \kappa ^2\ell ^2. \end{aligned} \end{aligned}$$

Therefore, we get the estimate

$$\begin{aligned} {\mathcal {E}}_1\Big (\exp \big (i\kappa H w(x)\big )f(x),{\mathbf {A}};{\mathcal {W}}(x_0,(1+\sigma )\ell )\Big )\le -(1+\ell ^\alpha )\frac{\kappa ^2}{2}|{\mathcal {W}}(x_0,(1+\sigma )\ell )|\\ +\,C\kappa ^2\ell ^2(\ell ^{\alpha }+\kappa ^2\ell ^{2+\alpha }+\sigma ^{-1}(\kappa \ell )^{-2}+\sigma ), \end{aligned}$$

and consequently

$$\begin{aligned} {\mathcal {E}}(u,{\mathbf {A}})\le & {} -\frac{\kappa ^2}{2}|{\mathcal {W}}(x_0,(1+\sigma )\ell )|+ {\mathcal {E}}_2\Big (\psi (x),{\mathbf {A}};\Omega {\setminus }{\mathcal {W}}(x_0,(1+\sigma )\ell )\Big )\\&+C\kappa ^2\ell ^2(\ell ^{\alpha }+\kappa ^2\ell ^{2+\alpha }+\sigma ^{-1}(\kappa \ell )^{-2}+\sigma ). \end{aligned}$$

Using that ${\mathcal {E}}(\psi ,{\mathbf {A}})\le {\mathcal {E}}(\psi ,{\mathbf {A}})$, we get

$$\begin{aligned} {\mathcal {E}}_1\Big (\psi ,{\mathbf {A}};{\mathcal {W}}(x_0,(1+\sigma )\ell )\Big )\le & {} -\frac{\kappa ^2}{2}|{\mathcal {W}}(x_0,(1+\sigma )\ell )|\nonumber \\&+\,C\kappa ^2\ell ^2(\ell ^{\alpha }+\kappa ^2\ell ^{2+\alpha }+\sigma ^{-1}(\kappa \ell )^{-2}+\sigma ). \end{aligned}$$

(5.9)

We insert this into (5.4), then we substitute the resulting inequality into (5.5). In this way we get

$$\begin{aligned} \int _{{\mathcal {W}}(x_0,\ell )}|\psi |^4\,dx\ge \frac{1}{2}|{\mathcal {W}}(x_0,(1+\sigma )\ell )|-C(\sigma +\sigma ^{-1}(\kappa \ell )^{-2}+\kappa ^2\ell ^{2+\alpha }+\ell ^\alpha ). \end{aligned}$$

The assumption on $\sigma$ and $\ell$ in (5.1) yield that the term on the right hand side above is o(1), hence we get the lower bound for the integral of $|\psi |^4$ in Theorem 1.7. Now, the estimate of the energy follows by collecting the estimates in (5.9) and (5.5).

Step 3. The case $|B_0(x_0)|>0$ : Upper bound.

We use (2.13) and (2.14) with $\gamma =\delta =\alpha$. We obtain, for some $C^{2,\alpha }$ real-valued function w,

$$\begin{aligned} {\mathcal {E}}_1\Big (f\psi ,{\mathbf {A}};{\mathcal {W}}(x_0,(1+\sigma )\ell )\Big )\ge & {} \mathcal (1-\ell ^\alpha ){\mathcal {E}}_1\Big (e^{-i\kappa H w}f\psi ,{\mathbf {A}}_0;{\mathcal {W}}(x_0,(1+\sigma )\ell )\Big )\nonumber \\&-\,C\kappa ^2\ell ^2(\ell ^\alpha +\kappa ^2\ell ^{2+\alpha }). \end{aligned}$$

(5.10)

If $x_0\in \Omega \,$, we get by re-scaling and (2.6) that

$$\begin{aligned} {\mathcal {E}}_1\Big (e^{-i\kappa Hw}f\psi ,{\mathbf {A}}_0;{\mathcal {W}}(x_0,(1+\sigma )\ell )\Big )\ge 4\kappa ^2(1+\sigma )^2\ell ^2 g(b|B_0(x_0)|). \end{aligned}$$

If $x_0\in \partial \Omega \,$, then we may write a lower bound for ${\mathcal {E}}_1\Big (f\psi ,{\mathbf {A}}_0;{\mathcal {W}}(x_0,(1+\sigma )\ell )\Big )$ by converting to boundary coordinates as in Lemma 4.4 and get

$$\begin{aligned}&{\mathcal {E}}_1\Big (e^{-i\kappa Hw}f\psi ,{\mathbf {A}}_0;{\mathcal {W}}(x_0,(1+\sigma )\ell )\Big )\\&\quad \ge \frac{(1-C\ell )}{b|B_0(x_0)|} e^N\Big (b|B_0(x_0)|,2(1+\sigma )\ell \sqrt{|B_0(x_0)|\kappa H}\Big )-C\kappa ^2\ell ^2 \big (\ell +\kappa ^2\ell ^3\big )\\&\quad \ge 4\kappa ^2(1+\sigma )^2\ell ^2 g(b|B_0(x_0)|)-C\kappa ^2\ell ^2\big (\ell +\kappa ^2\ell ^3+(\kappa \ell )^{-1}\big ). \end{aligned}$$

Thus, we infer from (5.10), for $x_0\in {\overline{\Omega }}\,$,

$$\begin{aligned} {\mathcal {E}}_1\Big (f\psi ,{\mathbf {A}};{\mathcal {W}}(x_0,(1+\sigma )\ell )\Big )\ge & {} 4\kappa ^2(1+\sigma )^2\ell ^2 g(b|B_0(x_0)|)\\&-\,C\kappa ^2\ell ^2\big (\ell ^\alpha +\kappa ^2\ell ^{2+\alpha }+(\kappa \ell )^{-1}\big ). \end{aligned}$$

Inserting this into (5.3), we get

$$\begin{aligned} \frac{1}{2}\int _{{\mathcal {W}}(x_0,\ell )}|\psi (x) |^4\,dx\le & {} 4(1+\sigma )^2\ell ^2 g(b|B_0(x_0)|)\\&+\,C\ell ^2\big (\ell ^\alpha +\kappa ^2\ell ^{2+\alpha }+(\kappa \ell )^{-1}+(\kappa \ell )^{-2}\sigma ^{-1}\big ). \end{aligned}$$

Our choice of $\sigma$ and $\ell$ in (5.1) guarantees that the term on the right side above is $o(\ell ^2)\,$. Using Remark 5.1, we get the upper bound in Theorem 1.7.

Remark 5.2

The proof in step 3 is still valid if $|B_0(x_0)|\ge \kappa ^{-2\gamma }, 0<\gamma <1-\rho$ and $Q_{4\kappa ^{-\rho }}(x_0)\subset \Omega .$

Step 4. The case $|B_0(x_0)|>0$ and $x_0\in \partial \Omega \,$ : Lower bound.

For the sake of simplicity, we treat the case $B_0(x_0)>0$. The case $B_0(x_0)<0$ can be treated similarly by taking complex conjugation.

We define the function

$$\begin{aligned} u(x)={\mathbf {1}}_{{\mathcal {W}}(x_0,(1+\sigma )\ell )}(x)\exp \big (i\kappa H w(x)\big )w_R\circ \Phi _{x_0}^{-1}(x) +\eta _\ell (x)\psi (x), \end{aligned}$$

where the function $\eta _\ell$ satisfies (5.6) and (5.7). Similarly as in (4.24), the function w is the sum of two functions $w_1$ and $w_2$, defined in ${\mathcal {W}}(x_0,(1+\sigma )\ell ))$ and satisfying the two inequalities in (4.27). Finally

$$\begin{aligned}w_R(s,t)= \exp \big (i\kappa H g_{x_0}(s,t)\big )\exp \left( \frac{-i\kappa H st}{2}\right) u_R\big (s\sqrt{B_0(x_0)\kappa H},\,t\sqrt{B_0(x_0)\kappa H}\,\big ), \end{aligned}$$

and $g_{x_0}$ is the function satisfying (4.25) in ${\mathcal {W}}(x_0,\ell )$ (by Lemma 4.3). The function $u_R\in H^1_0(Q_{R})$ is a minimizer of the energy $e^D\big (bB_0(x_0),R\big )$ for $R=2(1+\sigma )\sqrt{B_0(x_0)\kappa H}$ (cf. (2.3)). We can estimate ${\mathcal {E}}(u,{\mathbf {A}})$ similarly as we did in the proof of Theorem 4.8 and get

$$\begin{aligned} {\mathcal {E}}(u,{\mathbf {A}})\le & {} 4(1+\sigma )^2\ell ^2\kappa ^2 g\big (bB_0(x_0)\big )+{\mathcal {E}}_2(\psi ,{\mathbf {A}})\\&+C\kappa ^2\ell ^2\big (\ell ^\alpha +\kappa ^2\ell ^{2+\alpha }+\sigma +\sigma ^{-1}(\kappa \ell )^{-2}\big ).\\ {\mathcal {E}}_2(\psi ,{\mathbf {A}})= & {} \int _{\Omega {\setminus }{\mathcal {W}}(x_0,(1+\sigma )\ell )}\Big (|(\nabla -i\kappa H{\mathbf {A}})\psi |^2-\kappa ^2|\psi |^2+\frac{\kappa ^2}{2}|\psi |^4 \Big )\,dx\\&+\,\kappa ^2H^2\int _\Omega |{{\mathrm{curl}}}{\mathbf {A}}-B_0|^2\,dx. \end{aligned}$$

Now we use that ${\mathcal {E}}(\psi ,{\mathbf {A}})\le \min (0,{\mathcal {E}}(u,{\mathbf {A}}))$ to write

$$\begin{aligned} {\mathcal {E}}_1\Big (\psi ,{\mathbf {A}};{\mathcal {W}}(x_0,(1+\sigma )\ell )\Big )\le & {} 4(1+\sigma )^2\ell ^2\kappa ^2 g\big (bB_0(x_0)\big )\nonumber \\&+\,C\kappa ^2\ell ^2\big (\ell ^\alpha +\kappa ^2\ell ^{2+\alpha }+\sigma +\sigma ^{-1}(\kappa \ell )^{-2}\big ). \end{aligned}$$

(5.11)

Now we use (5.4) and (5.5) to obtain

$$\begin{aligned} -\frac{\kappa ^2}{2}\int _{{\mathcal {W}}(x_0,\ell )}|\psi (x)|^4\,dx+C\sigma \ell ^2\kappa ^2\le & {} 4(1+\sigma )^2\ell ^2\kappa ^2 g\big (bB_0(x_0)\big )\nonumber \\&+\,C\kappa ^2\ell ^2\big (\ell ^\alpha +\kappa ^2\ell ^{2+\alpha }+\sigma +\sigma ^{-1}(\kappa \ell )^{-2}\big ). \end{aligned}$$

Remembering that $\sigma =\kappa ^{\frac{\rho -1}{2}}$ and $\ell \approx \kappa ^{-\rho }$ (cf. (5.1)), we get the lower bound for the integral of $|\psi |^4$ as in Theorem 1.7.

For the estimate of the local energy ${\mathcal {E}}_1(\psi ,{\mathbf {A}};{\mathcal {W}}(x_0,(1+\sigma )\ell ))$, we collect the inequalities in (5.11), (5.4), (5.5) and the lower and upper bounds for the integral of $|\psi |^4$.

Remark 5.3

Remark 5.2 holds for Step 4 as well.

Step 5. The case $|B_0(x_0)|>0$ and $x_0\in \Omega \,$ : Lower bound.

In this case ${\mathcal {W}}_{x_0}((1+\sigma )\ell )=Q_{2(1+\sigma )\ell }(x_0)$. We define the following trial state

$$\begin{aligned} u(x)={\mathbf {1}}_{{\mathcal {W}}(x_0,(1+\sigma )\ell )}(x)\exp \big (i\kappa H w(x)\big )w_R(x) +\eta _\ell (x)\psi (x), \end{aligned}$$

where the functions w and $\eta _\ell$ are as in Step 4,

$$\begin{aligned}w_R(s,t)=\left\{ \begin{array}{l} u_R\big (\sqrt{B_0(x_0)\kappa H}\,(x-x_0)\big )~\quad \mathrm{if}~B_0(x_0)>0,\\ ~\\ \overline{u_R\big (\sqrt{B_0(x_0)\kappa H}\,(x-x_0)\big )}\quad ~\mathrm{if}~B_0(x_0)<0, \end{array}\right. \end{aligned}$$

and $u_R\in H^1_0(Q_R)$ is a minimizer of the energy $e^D\big (bB_0(x_0),R\big )$ for $R=2(1+\sigma )\sqrt{B_0(x_0)\kappa H}$ [cf. (2.3)].

We argue as in Step 4 and obtain the lower bound for the integral of $|\psi |^4$ in Theorem 1.7. The details are omitted.