Magnetic Ginzburg-Landau energy with a periodic rapidly oscillating and diluted pinning term

TL;DR

This paper analyzes the Ginzburg-Landau energy in 2D for heterogeneous type II superconductors with a rapidly oscillating, diluted pinning term, calculating critical fields and vorticity defect locations across different scales.

Contribution

It provides a detailed multi-scale analysis of vorticity defect locations and critical fields in a heterogeneous superconductor with a complex pinning landscape using a perturbative approach.

Findings

Calculated the first critical magnetic field linking vorticity defects to field strength.

Proved the quantized vorticity defects depend standardly on the applied magnetic field.

Determined the macroscopic, mesoscopic, and microscopic locations of vorticity defects in the model.

Abstract

We study the 2D full Ginzburg-Landau energy with a periodic rapidly oscillating, discontinuous and [strongly] diluted pinning term using a perturbative argument. This energy models the state of an heterogeneous type II supercon-ductor submitted to a magnetic field. We calculate the value of the first critical field which links the presence of vorticity defects with the intensity of the applied magnetic field. Then we prove a standard dependance of the quantized vorticity defects with the intensity of the applied field. Our study includes the case of a London solution having several minima. The macroscopic location of the vor-ticity defects is understood with the famous Bethuel-Brezis-H{\'e}lein renormalized energy. The mesoscopic location, i.e., the arrangement of the vorticity defects around the minima of the London solution, is the same than in the homogenous case. The microscopic location is exactly the same than in the heterogeneous case without magnetic field. We also compute the value of secondary critical fields that increment the quantized vorticity. 1. Introdution This article studies the pinning phenomenon in type-II superconducting composites. Superconductivity is a property that appears in certain materials cooled below a critical temperature. These materials are called superconductors. Superconductivity is characterized by a total absence of electrical resistance and a perfect diamagnetism. Unfortunately, when the imposed conditions are too intense, superconductivity is destroyed in certain areas of the material called vorticity defects. We are interested in type II superconductors which are characterized by the fact that the vorticity defects first appear in small areas. Their number increases with the intensity of the conditions imposed until filling the material. For example, when the intensity h ex of an applied magnetic field exceeds a first threshold, the first vorticity defects appear: the magnetic field begins to penetrate the superconductor. The penetration is done along thin wires and may move resulting an energy dissipation. These motions may be limited by trapping the vorticity defects in small areas. The behavior of a superconductor is modeled by minimizers of a Ginzburg-Landau type energy. In order to study the presence of traps for the vorticity defects we consider an energy including a pinning term that models impurities in the supercon-ductor. These impurities would play the role of traps for the vorticity defects. We are thus lead to the subject of this article: the type-II superconducting composites with impurities.

Equations

$$\begin{array}[]{cccc}\mathcal{E}_{\varepsilon,h_{\rm ex}}:&\mathscr{H}&\to&\mathbb{R}^{+}\\ &(u,A)&\mapsto&\displaystyle\dfrac{1}{2}\int_{\Omega}|\nabla u-\imath Au|^{2}+\dfrac{1}{2\varepsilon^{2}}(a_{\varepsilon}^{2}-|u|^{2})^{2}+|{\rm curl}(A)-h_{\rm ex}|^{2}\end{array},$$

$$\{\begin{array}{ll}-{\Delta }^2{\xi }_0+\Delta {\xi }_0=0& \text{ in }\Omega \\ \Delta {\xi }_0=1& \text{ on }\partial \Omega \\ {\xi }_0=0& \text{ on }\partial \Omega \end{array}.$$

$${\lambda }^{1/4}|\ln \epsilon |\to 0\text{ and }|\ln (\lambda \delta )|=\mathcal{O}(\ln |\ln \epsilon |),$$

$$K\ge 1$$

$$\frac{\ln (\delta \sqrt{h_{\mathrm{ex}}})}{\ln (\ln h_{\mathrm{ex}})}\to -\infty .$$

$${\xi }_0$$

$$(u_{\epsilon },A_{\epsilon })$$

$$\mathbf{D}\in {\Lambda }_d:=\left\{{\mathbf{D}}^{\prime }\in {\left\{\lceil \frac{d}{N_0}\rceil ;\lfloor \frac{d}{N_0}\rfloor \right\}}^{N_0}\right|\sum _{k=1}^{N_0}{D}_k^{\prime }=d\}.$$

$$W_{p,D}^{\mathrm{meso}}:\{(a_1,...,a_D)\in ({\mathbb{R}}^2{)}^D|a_i\ne a_j\text{ for }i\ne j\}\to \mathbb{R}$$

$${\left[h_{\mathrm{ex}}-{\mathtt{K}}_k^{(\mathtt{I})}\right]}^{-}\to 0\text{ and }{\left[h_{\mathrm{ex}}-{\mathtt{K}}_{k+1}^{(\mathtt{I})}\right]}^{+}\to 0.$$

$${\left[h_{\mathrm{ex}}-{\mathtt{K}}_L^{(\mathtt{I})}\right]}^{-}\to 0\text{ and }{\left[h_{\mathrm{ex}}-{\mathtt{K}}_1^{(\mathtt{II})}\right]}^{+}\to 0.$$

$${\left[h_{\mathrm{ex}}-{\mathtt{K}}_l^{(\mathtt{II})}\right]}^{-}\to 0\text{ and }{\left[h_{\mathrm{ex}}-{\mathtt{K}}_{l+1}^{(\mathtt{II})}\right]}^{+}\to 0.$$

$${\mathrm{deg}}_{\Gamma }(g):=\frac{1}{2\pi }{\int }_{\Gamma }g\wedge {\partial }_{\tau }g\in \mathbb{Z}.$$

$$\begin{array}[]{cccc}a_{\varepsilon}:&\mathbb{R}^{2}&\to&\{b,1\}\\ &x&\mapsto&\begin{cases}b&\text{if }x\in\omega_{\varepsilon}\\ 1&\text{otherwise}\end{cases}\end{array}.$$

$$(u,A)\stackrel{\mathrm{gauge}}{\sim }(u^{\prime },A^{\prime })⟺\{\begin{array}{l}\exists \phi \in H^2(\Omega ,\mathbb{R})\text{ s.t.}\\ u^{\prime }=u{\text{e}}^{ı\phi }\text{ and }{A}^{\prime }=A+\nabla \phi \end{array}.$$

$$\{\begin{array}{ll}\mathrm{div}(A)=0& \text{in }\Omega \\ A\cdot \nu =0& \text{on }\partial \Omega \end{array}.$$

$$\Vert A{\Vert }_{H^1(\Omega ,{\mathbb{R}}^2)}\le C\Vert \mathrm{curl}(A){\Vert }_{L^2(\Omega )}$$

$$\Vert A{\Vert }_{H^2(\Omega ,{\mathbb{R}}^2)}\le C\Vert \mathrm{curl}(A){\Vert }_{H^1(\Omega )}.$$

$$\text{A∈H1(Ω,R2) is a solution of eqrefJaugeCoulomb}⟺\exists \xi \in H_0^1\cap H^2(\Omega ,\mathbb{R})\text{ s.t. }A={\nabla }^{\perp }\xi .$$

$$\{\begin{array}{ll}-(\nabla -ıA_{\epsilon }{)}^2{u}_{\epsilon }=\frac{u_{\epsilon }}{{\epsilon }^2}(a_{\epsilon }^2-|u_{\epsilon }{|}^2{)}^2& \text{in }\Omega \\ (\nabla -ıA)u_{\epsilon }\cdot \nu =0& \text{on }\Omega \\ -{\nabla }^{\perp }{h}_{\epsilon }=u_{\epsilon }\wedge (\nabla -ıA_{\epsilon })u_{\epsilon }& \text{in }\Omega \\ h_{\epsilon }=h_{\mathrm{ex}}& \text{on }\partial \Omega \end{array}.$$

$$\{\begin{array}{ll}-\Delta u_{\epsilon }=\frac{u_{\epsilon }}{{\epsilon }^2}(a_{\epsilon }^2-|u_{\epsilon }{|}^2{)}^2-2ı(A_{\epsilon }{u}_{\epsilon }\cdot \nabla u_{\epsilon })-|A_{\epsilon }{|}^2{u}_{\epsilon }& \text{in }\Omega \\ {\partial }_{\nu }{u}_{\epsilon }=0& \text{on }\Omega \end{array}.$$

$$\Vert \nabla u{\Vert }_{L^{\infty }(\Omega )}^2\le C_{\Omega }\left(\Vert \Delta u{\Vert }_{L^{\infty }(\Omega )}+\Vert u{\Vert }_{L^{\infty }(\Omega )}\right)\Vert u{\Vert }_{L^{\infty }(\Omega )}.$$

$$\Vert \nabla u_{\epsilon }{\Vert }_{L^{\infty }(\Omega )}\le \frac{C_{\Omega }}{\epsilon }.$$

$${\mathcal{E}}_{\epsilon ,h_{\mathrm{ex}}}(u,A)=E_{\epsilon }(u)+\frac{1}{2}{\int }_{\Omega }-2(u\wedge \nabla u)\cdot A+|u{|}^2|A{|}^2+|\mathrm{curl}(A)-h_{\mathrm{ex}}{|}^2$$

$$E_{\epsilon }(u)=\frac{1}{2}{\int }_{\Omega }|\nabla u{|}^2+\frac{1}{2{\epsilon }^2}(a_{\epsilon }^2-|u{|}^2{)}^2.$$

$$\{\begin{array}{ll}-\Delta U=\frac{U}{{\epsilon }^2}(a_{\epsilon }^2-|U{|}^2)& \text{in }\Omega \\ {\partial }_{\nu }U=0& \text{on }\partial \Omega \end{array}.$$

$$E_{\epsilon }(U_{\epsilon }v)=E_{\epsilon }(U_{\epsilon })+F_{\epsilon }(v)$$

$$F_{\epsilon }(v):=\frac{1}{2}{\int }_{\Omega }{U}_{\epsilon }^2|\nabla v{|}^2+\frac{U_{\epsilon }^4}{2{\epsilon }^2}(1-|v{|}^2{)}^2.$$

$${\mathcal{F}}_{\epsilon ,h_{\mathrm{ex}}}(v,A)$$

$$\{\begin{array}{l}|v_{\epsilon }|\in W^{2,1}(\Omega ,\mathbb{C})\\ \Vert \nabla |v_{\epsilon }|{\Vert }_{L^{\infty }(\Omega )}=\mathcal{O}({\epsilon }^{-1})\end{array}.$$

Full text

The author would like to thank Lia Bronsard, Vincent Millot, Petru Mironescu and Etienne Sandier for fruitful discussions.

1. Introdution

This article studies the pinning phenomenon in type-II superconducting composites.

Superconductivity is a property that appears in certain materials cooled below a critical temperature. These materials are called superconductors. Superconductivity is characterized by a total absence of electrical resistance and a perfect diamagnetism. Unfortunately, when the imposed conditions are too intense, superconductivity is destroyed in certain areas of the material called vorticity defects.

We are interested in type II superconductors which are characterized by the fact that the vorticity defects first appear in small areas. Their number increases with the intensity of the conditions imposed until filling the material. For example, when the intensity $h_{\rm ex}$ of an applied magnetic field exceeds a first threshold, the first vorticity defects appear: the magnetic field begins to penetrate the superconductor. The penetration is done along thin wires and may move resulting an energy dissipation. These motions may be limited by trapping the vorticity defects in small areas.

The behavior of a superconductor is modeled by minimizers of a Ginzburg-Landau type energy. In order to study the presence of traps for the vorticity defects we consider an energy including a pinning term that models impurities in the superconductor. These impurities would play the role of traps for the vorticity defects. We are thus lead to the subject of this article: the type-II superconducting composites with impurities.

The case of an infinite long homogenous type II superconducting cylinder was intensively studied in mathematics by various authors since the 90’s [see [16] for a guide to the litterature]. Namely, the present work deals with a cylindrical superconductor $\mathcal{S}=\Omega\times\mathbb{R}$ [whose section is $\Omega\subset\mathbb{R}^{2}$] submitted to a vertical magnetic field $(0,0,h_{\rm ex})$. Under these considerations, the vorticity defects are thin vertical cylinder. Thus their study may be done via a 2D problem formulated on $\Omega\subset\mathbb{R}^{2}$. Following the works of various authors [see [14], [1], [11]], for a small parameter $\varepsilon>0$ [$\varepsilon\to 0$ in this article] and $h_{\rm ex}=h_{\rm ex}(\varepsilon)\geq 0$, we are interested in the description of the [global] minimizers of the functional

[TABLE]

where [see Section 2 for more detailed notation]

• •

$\Omega\subset\mathbb{R}^{2}$ is a smooth bounded simply connected open set,

• •

$\mathscr{H}:=H^{1}(\Omega,\mathbb{C})\times H^{1}(\Omega,\mathbb{R}^{2})$,

• •

$a_{\varepsilon}:\Omega\to\{1,b\}$ [$b\in(0,1)$ is independent of $\varepsilon$] is a periodic diluted pinning term [see Figure 1 and Section 2.3 for a construction of $a_{\varepsilon}$]. The impurities are the connected components of $\omega_{\varepsilon}:=a_{\varepsilon}^{-1}(\{b\})$. In the definition of $a_{\varepsilon}$, $\delta=\delta(\varepsilon)\underset{\varepsilon\to 0}{\to}0$ is the parameter of period, $\lambda=\lambda(\varepsilon)\underset{\varepsilon\to 0}{\to}0$ is the parameter of dilution and $0\in\omega\subset\mathbb{R}^{2}$ is a smooth bounded simply connected open set which gives the form of the impurities.

We focus on a strongly diluted case [$\lambda^{1/4}|\ln\varepsilon|\to 0$] with not too small connected components of $\omega_{\varepsilon}$ [$|\ln(\lambda\delta)|=\mathcal{O}(\ln|\ln\varepsilon|)$] but with a sufficiently small parameter of the period [see (4)].

Under these considerations, if $(u_{\varepsilon},A_{\varepsilon})$ minimizes $\mathcal{E}_{\varepsilon,h_{\rm ex}}$, then the vorticity defects may be interpreted as the set $\{|u_{\varepsilon}|<b/2\}$.

As said above, our study takes place in the extrem type II case $\varepsilon\to 0$ and we also assume a divergent upper bound for $h_{\rm ex}$. Vorticity defects appear for minimizers above a critical valued $H_{c_{1}}=[{b^{2}|\ln\varepsilon|+(1-b^{2})|\ln(\lambda\delta)|}]/({2\|\xi_{0}\|_{L^{\infty}(\Omega)}})+\mathcal{O}(1)$ [see Corollary 64 and (75)]. Here $\xi_{0}\in H^{1}_{0}\cap H^{2}$ is called the London solution and is the unique solution of the London equation

[TABLE]

The value $H_{c_{1}}$ is calculated by a standard balance of the energetic costs of a configuration without vorticity defects [$|u|\geq b/2$] with well prepared competitors having an arbitrary number of quantized vorticity defects. Here quantization as to be interpreted by the degree of $u$ around a vorticity defect. It is an observable quantity related with the circulation of the superconducting current.

In order to lead the study, the set $\Lambda:=\{z\in\Omega\,|\,\xi_{0}(z)=\min\xi_{0}\}\subset\Omega$ is of major interest [it is standard to prove that, in $\Omega$, $-1<\xi_{0}<0$]. From Lemma 4.4 in [17] and Lemma 4 in [15] we have the following :

Lemma 1**.**

The set $\Lambda$ is finite. Moreover there exist $\eta>0$ and $M\geq 1$ s.t. for $a\in\Omega$ we have $\xi_{0}(a)\geq\min\xi_{0}+\eta{\rm dist}(a,\Lambda)^{M}$ 111In Lemma 4 in [15], $M$ is just a positive number, but $\xi\in C^{0}(\overline{\Omega})$, and then, up to consider $\eta>0$ sufficiently small, we may assume $M\geq 1$..

We write $N_{0}:={\rm Card}(\Lambda)$ and $\Lambda=\{p_{1},...,p_{N_{0}}\}$.

We may give a simple picture of the emergence of the vorticity defects. The first vorticity defects appear close to $H_{c_{1}}$. If $N_{0}=1$ then there is first a unique vorticity defect and it is close to $\Lambda$. If $N_{0}\geq 2$ the situation is less clear: we first have $d_{1}^{\star}\in\{1,...,N_{0}\}$ vorticity defect and each of them is located close to $d_{1}^{\star}$ elements of $\Lambda$. By increasing the intensity of the applied field $h_{\rm ex}$ by a bounded quantity we increment the number of vorticity defects until filling $\Lambda$.

Once each elements of $\Lambda$ is close to a vorticity defect, then by increasing $h_{\rm ex}$ of a $\mathcal{O}(\ln|\ln\varepsilon|)$, additional defects appear one by one.

We may now state the main theorems of the present work. For simplicity of the presentation the theorems are not stated on their most general form [see Theorem 4].

These main results are obtained assuming that $\lambda,\delta$ and $h_{\rm ex}$ satisfy

[TABLE]

[TABLE]

and when $h_{\rm ex}\to\infty$ we need

[TABLE]

Namely, in order to meet Hypothesis (2), (3) and (4), we may think $\lambda\simeq|\ln\varepsilon|^{-s},\delta\simeq|\ln\varepsilon|^{-t}$ with $s>4$ and $t>1/2$.

We need also assume that

[TABLE]

in the sense that for $p\in\Lambda$, letting ${\rm Hess}_{\xi_{0}}(p)$ be the Hessian matrix of $\xi_{0}$ at $p$, the quadratic form $Q_{p}(z)=z\cdot{\rm Hess}_{\xi_{0}}(p)z$ is a definite positive quadratic form. Note that if (5) holds then we may take $M=2$ in Lemma 1.

The strategy of this work is based on a perturbative argument. This argument applies for families of quasi-minimizers of the energy with some regularity assumptions [see Theorem 4]. In particular, we cannot have a sharp profil near a zero of a quasi-minimizer since such profil does not make any sense for quasi-minimizer. Therefore we cannot speak about an ad-hoc notion of vortices s.t. "isolated zeros". However with a natural $L^{\infty}$-bound on the gradient of quasi-minimizers, the notion of vorticity defects is sufficiently robust to give them a nice description.

For simplicity of the presentation we first state the main results for a family $\{(u_{\varepsilon},A_{\varepsilon})\,|\,0<\varepsilon<1\}\subset\mathscr{H}$ s.t.

[TABLE]

Theorem 1**.**

Assume that (5) holds and $\lambda,\delta,h_{\rm ex},K$ satisfy (2), (3) and (4). There exists $\mathcal{D}_{K,b}>1$ s.t. for $\{(u_{\varepsilon},A_{\varepsilon})\,|\,0<\varepsilon<1\}\subset\mathscr{H}$ satisfying (6), for sufficiently small $\varepsilon$, there exits $d_{\varepsilon}\in\mathbb{N}$ s.t. if $d_{\varepsilon}=0$ then $|u_{\varepsilon}|\geq b/2$ in $\Omega$, and if $d_{\varepsilon}\in\mathbb{N}^{*}$ then there exists a set of $d_{\varepsilon}$ points, $\mathcal{Z}_{\varepsilon}=\{z^{\varepsilon}_{1},...,z^{\varepsilon}_{d_{\varepsilon}}\}\subset\Omega$, s.t. for $\mu>0$ sufficiently small and independent of $\varepsilon$ we have:

1. (1)

$d_{\varepsilon}\leq\mathcal{D}_{K,b}$** 2. (2)

$\{|u_{\varepsilon}|<b/2\}\subset\cup B(z^{\varepsilon}_{i},\varepsilon^{\mu})\subset\Omega$, 3. (3)

$|z^{\varepsilon}_{i}-z^{\varepsilon}_{j}|\geq h_{\rm ex}^{-1}\ln h_{\rm ex}$* for $i\neq j$,* 4. (4)

${\rm dist}(z^{\varepsilon}_{i},\Lambda)\leq h_{\rm ex}^{-1/2}{\ln h_{\rm ex}}{}$* for all $i$,* 5. (5)

${\rm deg}_{\partial B(z^{\varepsilon}_{i},\varepsilon^{\mu})}(u_{\varepsilon})=1$* for all $i$.*

Moreover:

1. (1)

There is $\eta_{\omega,b}>0$ depending only on $\omega$ and $b$ s.t. for all $i$ we have $B(z^{\varepsilon}_{i},\eta_{\omega,b}\lambda\delta)\subset\omega_{\varepsilon}$. 2. (2)

If for a sequence $\varepsilon=\varepsilon_{n}\downarrow 0$ we have $h_{\rm ex}=\mathcal{O}(1)$ then $d_{\varepsilon}=0$ for small $\varepsilon$.

From Theorem 1 we know that, for small $\varepsilon$, if $\{|u_{\varepsilon}|<b/2\}\neq\emptyset$, then the vorticity defects are contained in small disks which are well separated, trapped by the impurities and located near $\Lambda$. The second theorem gives sharper informations related with the location of these disks. We divide the second theorem in three parts:

• •

Macroscopic location: We know that the disks are near $\Lambda$, for some $p\in\Lambda$, how many disks are near $p$ ?

• •

Mesoscopic location: For $p\in\Lambda$, how the disks near $p$ are they organized ? What is their inter-distance ?

• •

Microscopic location: We know that the disks are trapped by the inclusion $\omega_{\varepsilon}$, what is their location inside $\omega_{\varepsilon}$.

These questions are related with the crucial notion of renormalized energy [see Section 6].

Theorem 2**.**

**[Direct part]

**Assume that (5) holds and $\lambda,\delta,h_{\rm ex},K$ satisfy (2), (3) and (4). Assume also $h_{\rm ex}\to\infty$.

Let $\{(u_{\varepsilon},A_{\varepsilon})\,|\,0<\varepsilon<1\}\subset\mathscr{H}$ satisfying (6) and let $\varepsilon=\varepsilon_{n}\downarrow 0$ be a sequence. Since $d=d_{\varepsilon}\leq\mathcal{D}_{K,b}$, up to pass to a subsequence, we may assume that $d$ is independent of $\varepsilon$. Assume $d>0$.

Macroscopic location.* Recall that $\Lambda=\{p_{1},...,p_{N_{0}}\}$ and for $k\in\{1,...,N_{0}\}$ we let $D_{k}:={\rm deg}_{\partial B(p_{k},2\ln(h_{\rm ex})/\sqrt{h_{\rm ex}})}(u_{\varepsilon})$. Write ${\bf D}=(D_{1},...,D_{N_{0}})$. Up to pass to a subsequence we may assume that ${\bf D}$ is independent of $\varepsilon$. We then have:*

• •

The distribution of the disks $B(z^{\varepsilon}_{i},\varepsilon^{\mu})$ around the elements of $\Lambda$ is the most homogenous possible :

[TABLE]

Here, for $x\in\mathbb{R}$, we wrote $\lceil x\rceil$ for the ceiling of $x$ and $\lfloor x\rfloor$ for the floor of $x$.

• •

There exists a renormalized energy $\mathcal{W}_{d}:\Lambda_{d}\to\mathbb{R}$ [see (106)] s.t. ${\bf D}$ minimizes $\mathcal{W}_{d}$.

Mesoscopic location.* The mesoscopic location is the same than in the homogenous case. Namely, for $p\in\Lambda$ s.t. ${\rm deg}_{\partial B(p,2\ln(h_{\rm ex})/\sqrt{h_{\rm ex}})}(u_{\varepsilon})=D>0$, there exists a renormalized energy [see Section 6.2]*

[TABLE]

s.t., denoting $\ell:=\sqrt{\dfrac{D}{h_{\rm ex}}}$ and for $z_{i}^{\varepsilon}\in B(p,2\ln(h_{\rm ex})/\sqrt{h_{\rm ex}})$ letting $\breve{z}_{i}^{\varepsilon}:=\dfrac{z_{i}^{\varepsilon}-p}{\ell}$, we have $\breve{\bf z}^{\varepsilon}=(\breve{z}_{1}^{\varepsilon},...,\breve{z}_{D}^{\varepsilon})$ [assuming $z_{i}^{\varepsilon}\in B(p,2\ln(h_{\rm ex})/\sqrt{h_{\rm ex}})\Leftrightarrow i\in\{1,...,D\}$] which converges to a minimizer of $W^{\rm meso}_{p,D}$. In particular $\ell$ is the typical interdistance between two close $z^{\varepsilon}_{i},z^{\varepsilon}_{j}$.

Microscopic location.* We know that, for $i\in\{1,...,d\}$, $B(z^{\varepsilon}_{i},\eta_{\omega,b}\lambda\delta)\subset\omega_{\varepsilon}$. Moreover for $i\neq j$ we have $|z^{\varepsilon}_{i}-z^{\varepsilon}_{j}|\geq\ln(h_{\rm ex})h_{\rm ex}^{-1}\gg\lambda\delta$. Then each connected component of $\omega_{\varepsilon}$ contains at most one disk $B(z^{\varepsilon}_{i},\varepsilon^{\mu})$.*

There exists a renormalized energy $W^{\rm micro}:\omega\to\mathbb{R}$ [see Section 6.3] s.t. for $i\in\{1,...,d\}$, letting $y^{\varepsilon}_{i}\in\delta\cdot\mathbb{Z}^{2}$ be s.t. $B(z^{\varepsilon}_{i},\eta_{\omega,b}\lambda\delta)\subset y^{\varepsilon}_{i}+\lambda\delta\omega$ and $\hat{z}^{\varepsilon}_{i}:=\dfrac{z^{\varepsilon}_{i}-y^{\varepsilon}_{i}}{\lambda\delta}\in\omega$ we have

• •

$W^{\rm micro}(\hat{z}^{\varepsilon}_{i})\to\displaystyle\min_{\omega}W^{\rm micro}$,

• •

Up to pass to a subsequence, there is $a_{i}\in\omega$ s.t. $\hat{z}^{\varepsilon}_{i}\to a_{i}$ and $a_{i}$ minimizes $W^{\rm micro}$.222For example if $\omega$ is a disk then $a_{i}$ is the center of the disk [7] .

**[Optimality of the renormalized energies]

**Consider a sequence $\varepsilon=\varepsilon_{n}\downarrow 0$ previously fixed [in order to have ${\bf D}$ independent of $\varepsilon$] and assume $d\neq 0$. We let

• •

${\bf D}^{\prime}\in\Lambda_{d}$* be a minimizer of $\mathcal{W}_{d}$,*

• •

for $k\in\{1,...,N_{0}\}$ s.t. $D_{k}^{\prime}\geq 1$, ${\bf a}_{k}^{\prime}$ be a minimizer of $W^{\rm meso}_{p_{k},D_{k}^{\prime}}$,

• •

$a_{0}$* be a minimizer of $W^{\rm micro}$.*

Then, for $\varepsilon=\varepsilon_{n}$, there exist $(u_{\varepsilon}^{\prime},A_{\varepsilon}^{\prime})\in\mathscr{H}$ and $d$ distinct points of $\Omega$, $\{z_{1}^{\prime},...,z_{d}^{\prime}\}=\{{z^{\varepsilon}_{1}}^{\prime},...,{z^{\varepsilon}_{d}}^{\prime}\}\subset\omega_{\varepsilon}$, s.t.

• •

$\mathcal{E}_{\varepsilon,h_{\rm ex}}(u_{\varepsilon}^{\prime},A_{\varepsilon}^{\prime})\leq\inf_{\mathscr{H}}\mathcal{E}_{\varepsilon,h_{\rm ex}}+o(1)$,

• •

$\{|u_{\varepsilon}^{\prime}|\leq b/2\}\subset\cup B(z_{i}^{\prime},\sqrt{\varepsilon})\subset\cup_{p\in\Lambda}B(p,{\ln(h_{\rm ex})}/\sqrt{h_{\rm ex}})$,

• •

for $k\in\{1,...,N_{0}\}$, $D_{k}^{\prime}={\rm deg}_{\partial B(p_{k},2\ln(h_{\rm ex})/\sqrt{h_{\rm ex}})}(u_{\varepsilon}^{\prime})$,

• •

${\rm deg}_{\partial B(z_{i}^{\prime},\sqrt{\varepsilon})}(u_{\varepsilon}^{\prime})=1$* for all $i$,*

• •

writing for $p_{k}\in\Lambda$ [s.t. $D^{\prime}_{k}\geq 1$] and $z^{\prime}_{i}\in B(p_{k},\ln(h_{\rm ex})/\sqrt{h_{\rm ex}})$, $\breve{z}_{i}^{\prime}:=(z_{i}-p_{k})/\sqrt{D_{k}/h_{\rm ex}}$ and $\breve{\bf z}_{p_{k}}^{\prime}:=\{\breve{z}^{\prime}_{i}\,|\,z_{i}^{\prime}\to p_{k}\}$333We used a little abuse of notation for the simplicity of the presentation., we have $\breve{\bf z}_{p_{k}}^{\prime}\to{\bf a}_{k}^{\prime}$,

• •

For $i\in\{1,...,d\}$, letting $y^{\varepsilon}_{i}\in\delta\cdot\mathbb{Z}^{2}$ be s.t. $z^{\prime}_{i}\in y^{\varepsilon}_{i}+\lambda\delta\cdot\omega$ and $\hat{z}^{\prime}_{i}:=\dfrac{z^{\prime}_{i}-y^{\varepsilon}_{i}}{\lambda\delta}\in\omega$ we have $\hat{z}_{i}^{\prime}\to a_{0}$.

The third theorem underline the link between the number $d$ and $h_{\rm ex}$. In this theorem we write, for $x\in\mathbb{R}$, $[x]^{+}=\max(x,0)$ and $[x]^{-}=\min(x,0)$.

Theorem 3**.**

Assume that $\Omega$ satisfies (5), $\lambda,\delta,h_{\rm ex},K$ satisfy (2), (3) and (4).

There are integers $L\in\{1,...,N_{0}\}$, $0=d_{0}^{\star}<d_{1}^{\star}<\cdots<d_{L}^{\star}=N_{0}$ [$d_{k}^{\star}\in\mathbb{N}$ is independent of $\varepsilon$] and critical fields [depending on $\varepsilon$] ${\tt K}^{\tt(I)}_{1}<\cdots<{\tt K}^{\tt(I)}_{L}<{\tt K}^{\tt(II)}_{1}<{\tt K}^{\tt(II)}_{2}<\cdots$ [see (126) and (127) for the expressions of ${\tt K}^{\tt(I)}_{k}$ and ${\tt K}^{\tt(II)}_{k}$] s.t. for $\{(u_{\varepsilon},A_{\varepsilon})\,|\,0<\varepsilon<1\}\subset\mathscr{H}$ a family satisfying (6) and for a sequence $\varepsilon=\varepsilon_{n}\downarrow 0$:

• •

If $d_{\varepsilon}=0$ for small $\varepsilon$, then $[h_{\rm ex}-{\tt K}^{\tt(I)}_{1}]^{+}\to 0$.

• •

If $d_{\varepsilon}>0$ for small $\varepsilon$, then $[h_{\rm ex}-{\tt K}^{\tt(I)}_{1}]^{-}\to 0$.

• •

Assume $L\geq 2$. For $k\in\{1,...,L-1\}$, if for small $\varepsilon$ we have $d_{k-1}^{\star}<d_{\varepsilon}\leq d_{k}^{\star}$, then

[TABLE]

• •

For $L\geq 1$, if for small $\varepsilon$ we have $d_{L-1}^{\star}<d_{\varepsilon}\leq d_{L}^{\star}=N_{0}$, then

[TABLE]

• •

Let $l\in\mathbb{N}^{*}$. If for small $\varepsilon$ we have $d_{\varepsilon}=N_{0}+l$, then

[TABLE]

Remark 2*.*

A more complete statement for $d_{\varepsilon}\in\{0,...,N_{0}\}$ may be found in Proposition 68.

2. Notation

2.1. Sets, vectors and numbers

• •

We identify the real plan $\mathbb{R}^{2}$ with $\mathbb{C}$ and we denote by $\mathbb{S}^{1}$ the unit circle in $\mathbb{C}$.

• •

For $\mathscr{U}\subset\mathbb{R}^{2}$, $N\in\mathbb{N}\setminus\{0;1\}$, $(\mathscr{U}^{N})^{*}:=\{(z_{1},...,z_{N})\in\mathscr{U}^{N}\,|\,z_{i}\neq z_{j}\text{ for }i\neq j\}$.

• •

For $k\in\{1;2\}$, $\mathcal{H}^{k}$ is the $k$-dimensional Hausdorff measure.

• •

If $(a_{1},a_{2}),(b_{1},b_{2})\in\mathbb{R}^{2}$, then $|(a_{1},a_{2})|=\sqrt{a_{1}^{2}+a_{2}^{2}}$, $(a_{1},a_{2})^{\bot}=(-a_{2},a_{1})$, $(a_{1},a_{2})\cdot(b_{1},b_{2})=a_{1}b_{1}+a_{2}b_{2}$ and $(a_{1},a_{2})\wedge(b_{1},b_{2})=a_{1}b_{2}-a_{2}b_{1}$.

• •

For $\mathscr{U}\subset\mathbb{R}^{2}$, $\overline{\mathscr{U}}$ is the closure of $\mathscr{U}$ w.r.t. $|\cdot|$

• •

For $\emptyset\neq\mathscr{U},\mathscr{V}\subset\mathbb{R}^{2}$ and $x_{0}\in\mathbb{R}^{2}$ we write ${\rm dist}(\mathscr{U},\mathscr{V}):=\inf\{|x-y|\,|\,x\in\mathscr{U},\,y\in\mathscr{V}\}$ and ${\rm dist}(x_{0},\mathscr{V}):={\rm dist}(\{x_{0}\},\mathscr{V})$.

• •

For $\Gamma\subset\mathbb{R}^{2}$ a Jordan curve we let:

• –

${\rm int}(\Gamma)$, the interior of $\Gamma$, be the bounded open set $\mathscr{U}\subset\mathbb{R}^{2}$ s.t. $\Gamma=\partial\mathscr{U}$ where $\partial\mathscr{U}$ is the boundary of $\mathscr{U}$.

• –

$\nu$ be the outward normal unit vector of ${\rm int}(\Gamma)$

• –

$\tau$ be the direct unit tangent vector of $\Gamma$ ($\tau=\nu^{\bot}$)

• •

If $S$ is a finite set then ${\rm Card}(S)$ is the cardinal of $S$.

• •

If $x\in\mathbb{R}$, then we write $\lceil x\rceil:=\min\{m\in\mathbb{Z}\,|\,m\geq x\}$, the ceiling of $x$, and $\lfloor x\rfloor:=\max\{m\in\mathbb{Z}\,|\,m\leq x\}$, the floor of $x$.

• •

If $x\in\mathbb{R}$, then we write $[x]^{+}=\max(x,0)$ and $[x]^{-}=\min(x,0)$.

2.2. Functions

• •

For $\mathscr{U}\subset\mathbb{R}^{2}$ a smooth open set and $K\subset\mathbb{C}$, $H^{1}(\mathscr{U},K)=\{u\in H^{1}(\mathscr{U},\mathbb{C})\,|\,u(x)\in K\text{ for a.e. }x\in\mathscr{U}\}$ where $H^{1}(\mathscr{U},\mathbb{C})$ is the Classical Sobolev space of the first order modeled on the Lebesgue space $L^{2}$.

For $k\in\mathbb{N}^{*}$ and $p\in[1;\infty]$ we use the standard notation for the higher order Sobolev space $H^{k}(\mathscr{U},K)$ modeled on $L^{2}$ and $W^{k,p}(\mathscr{U},K)$ for the Sobolev space of order $k$ modeled on $L^{p}$.

• •

We use the standard notation for the differential operators: ”$\nabla$” for the gradient, ”${\rm curl}$” for the curl, ”${\rm div}$” for the divergence, "$\partial_{\tau}=\tau\cdot\nabla$" for the tangential derivative, "$\partial_{\nu}=\nu\cdot\nabla$" for the normal derivative…

• •

For $\mathscr{U}\subset\mathbb{R}^{2}$ a smooth bounded open set we let ${\rm tr}_{\partial\mathscr{U}}:H^{1}(\mathscr{U},\mathbb{C})\to H^{1/2}(\partial\mathscr{U},\mathbb{C})$ be the [surjective] trace operator. For $\Gamma$ a connected component of $\partial\mathscr{U}$ and $u\in H^{1}(\mathscr{U},\mathbb{C})$, we let ${\rm tr}_{\Gamma}(u)$ be the restriction of ${\rm tr}_{\partial\mathscr{U}}(u)$ to $\Gamma$.

We write $H_{0}^{1}(\mathscr{U},\mathbb{C}):=\{u\in H^{1}(\mathscr{U},\mathbb{C})\,|\,{\rm tr}_{\partial\mathscr{U}}(u)=0\}$.

• •

For $u:\Omega\to\mathbb{C}$ a function we let $\underline{u}:=\begin{cases}u&\text{if }|u|\leq 1\\ u/|u|&\text{if }|u|>1\end{cases}$.

• •

For $\Gamma\subset\mathbb{R}^{2}$ a Jordan curve and $g\in H^{1/2}(\Gamma,\mathbb{S}^{1})$, the degree of $g$ is defined as

[TABLE]

For a smooth and bounded open set $\mathscr{U}\subset\mathbb{R}^{2}$, $\Gamma$ a connected component of $\partial\mathscr{U}$ and $u\in H^{1}(\mathscr{U},\mathbb{C})$, if there exists $\eta>0$ s.t. $g:={\rm tr}_{\Gamma}(u)$ satisfies $|g|\geq\eta$ , then $g/|g|\in H^{1/2}(\Gamma,\mathbb{S}^{1})$ and we write ${\rm deg}_{\Gamma}(u):={\rm deg}_{\Gamma}(g/|g|)$.

When $\mathscr{U},\mathscr{V}\subset\mathbb{R}^{2}$ are smooth bounded simply connected open sets s.t. $\overline{\mathscr{V}}\subset\mathscr{U}$ and $u\in H^{1}(\mathscr{U}\setminus\overline{\mathscr{V}},\mathbb{S}^{1})$, then we write [without ambiguity] ${\rm deg}(u)$ instead of ${\rm deg}_{\Gamma}(u)$ for any Jordan curve $\Gamma\subset\overline{\mathscr{U}}\setminus{\mathscr{V}}$ s.t. $\mathscr{V}\subset{\rm int}(\Gamma)$.

2.3. Construction of the pinning term

Let

1. $\bullet$

$\delta=\delta(\varepsilon)\in(0,1),\,\lambda=\lambda(\varepsilon)\in(0,1)$; 2. $\bullet$

$\omega\subset\mathbb{R}^{2}$ be a smooth bounded and simply connected open set s.t. $(0,0)\in\omega$ and $\overline{\omega}\subset Y:=(-1/2,1/2)^{2}$.

For $m\in\mathbb{Z}^{2}$ we denote $Y_{m}^{\delta}:=\delta m+\delta\cdot Y$ and $\displaystyle\omega_{\varepsilon}=\bigcup_{\begin{subarray}{c}m\in\mathbb{Z}^{2}\text{ s.t.}\\ Y_{m}^{\delta}\subset\Omega\end{subarray}}[\delta m+\lambda\delta\cdot\omega]$. For $b\in(0,1)$ we define

[TABLE]

2.4. Asymptotic

• $\bullet$

In this article $\varepsilon\in(0;1)$ is a small number. We are essentially interested in the asymptotic $\varepsilon\to 0$.

• $\bullet$

The notation $o(1)$ means a quantity depending on $\varepsilon$ which tends to [math] when $\varepsilon\to 0$.

• $\bullet$

The notation $o[f(\varepsilon)]$ means a quantity $g(\varepsilon)$ s.t. $\dfrac{g(\varepsilon)}{f(\varepsilon)}=o(1)$.

• $\bullet$

The notation $\mathcal{O}[f(\varepsilon)]$ means a quantity $g(\varepsilon)$ s.t. $\dfrac{g(\varepsilon)}{f(\varepsilon)}$ is bounded for small $\varepsilon$.

3. Classical facts and the strongest theorem

Gauge invariance and Coulomb Gauge

It is standard to quote the gauge invariance of the energy $\mathcal{E}_{\varepsilon,h_{\rm ex}}$. Namely, two configurations $(u,A),(u^{\prime},A^{\prime})\in\mathscr{H}$ are gauge equivalent, denoted by $(u,A)\stackrel{{\scriptstyle{\rm gauge}}}{{\sim}}(u^{\prime},A^{\prime})$, if there exists a gauge transformation from $(u,A)$ to $(u^{\prime},A^{\prime})$:

[TABLE]

Two gauge equivalent configurations describe the same physical state. Then, physical quantities are those which are gauge invariant. For example, if $(u,A)\in\mathscr{H}$, then $|u|$, $|\nabla u-\imath Au|$, ${\rm curl}(A)$ and then $\mathcal{E}_{\varepsilon,h_{\rm ex}}(u,A)$, $\{|u|<b/2\}$ also are gauge invariants.

In the context the Ginzburg-Landau energy, a classical choice of gauge is the Coulomb gauge. We say that $(u,A)$ is in the Coulomb gauge if

[TABLE]

One may prove [see Proposition 3.2 in [16]] that, for $(u,A)\in\mathscr{H}$, there exists $\varphi\in H^{2}(\Omega,\mathbb{R})$ s.t. $A^{\prime}:=A+\nabla\varphi$ satisfies (7). Then, letting $u^{\prime}=u\e^{\imath\varphi}$, we have $(u^{\prime},A^{\prime})$ which is in the Coulomb gauge and $(u,A)\stackrel{{\scriptstyle{\rm gauge}}}{{\sim}}(u^{\prime},A^{\prime})$.

One of the main motivations in using the Coulomb gauge comes from the fact that $\|{\rm curl}(A)\|_{L^{2}}$ controls $\|A\|_{H^{1}}$. Namely there exists $C\geq 1$ [which depends only on $\Omega$] s.t. if $A$ satisfies (7) then [see Proposition 3.3 in [16]]

[TABLE]

and

[TABLE]

Moreover we have an easy representation of $A\in H^{1}(\Omega,\mathbb{R}^{2})$ satisfying (7)

[TABLE]

Basic description of a minimizer

We first note that, by direct minimization, for all $a_{\varepsilon}\in L^{\infty}(\Omega,[b;1])$, $\varepsilon,h_{\rm ex}>0$, the minimization problem of $\mathcal{E}_{\varepsilon,h_{\rm ex}}$ in $\mathscr{H}$ admits [at least] a solution $(u_{\varepsilon},A_{\varepsilon})\in\mathscr{H}$.

Writing $h_{\varepsilon}:={\rm curl}(A_{\varepsilon})$, it is standard to check that a such minimizer solves:

[TABLE]

Using a maximum principle, we may get the following proposition:

Proposition 3**.**

Let $\varepsilon,h_{\rm ex}>0$ and $a\in L^{\infty}(\Omega,[b,1])$. If $(u_{\varepsilon},A_{\varepsilon})$ is a minimizer of $\mathcal{E}(u,A)=\displaystyle\dfrac{1}{2}\int_{\Omega}|\nabla u-\imath Au|^{2}+\dfrac{1}{2\varepsilon^{2}}(a^{2}-|u|^{2})^{2}+|{\rm curl}(A)-h_{\rm ex}|^{2}$ in $\mathscr{H}$ then $|u_{\varepsilon}|\leq 1$ in $\Omega$.

On the other hand, if $(u_{\varepsilon},A_{\varepsilon})$ is a minimizer of $\mathcal{E}_{\varepsilon,h_{\rm ex}}$ in the Coulomb gauge, then it solves

[TABLE]

A fundamental bound in the study concerns $\|\nabla u_{\varepsilon}\|_{L^{\infty}(\Omega)}$. We have the following lemma which is a Gagliardo-Nirenberg type inequality with homogenous Neumann boundary condition.

Lemma 4**.**

444The proof of Lemma 4 is done by first using $\Phi:\mathbb{D}\to\Omega$, a conformal representation of $\Omega$ on the unit disk $\mathbb{D}$. Then we extend $\tilde{u}:=u\circ\Phi$ in the disk $B(0,2)$ by letting $u^{\prime}(x)=\tilde{u}(x/|x|)$ for $x\in B(0,2)\setminus\mathbb{D}$. By using the boundary condition we have $u^{\prime}\in H^{2}(B(0,2),\mathbb{C})$. And finally one may conclude by using an interior version of Lemma 4 [Lemma A.1 in [3]].

Let $\Omega\subset\mathbb{R}^{2}$ be a smooth bounded simply connected open set. There exists $C_{\Omega}\geq 1$ s.t. if $u\in H^{2}(\Omega)$ is s.t. $\partial_{\nu}u=0$ on $\partial\Omega$ then

[TABLE]

Consequently, with Lemma 4 [up to change the value of $C_{\Omega}$], for $\varepsilon,h_{\rm ex}>0$ and $a_{\varepsilon}\in L^{\infty}(\Omega,[b^{2},1])$, if $(u_{\varepsilon},A_{\varepsilon})\in\mathscr{H}$ minimizes $\mathcal{E}_{\varepsilon,h_{\rm ex}}$ is in the Coulomb gauge and is s.t. $\|A_{\varepsilon}\|_{L^{\infty}(\Omega)}\leq 1/\varepsilon$ [which is the case in the present work] then

[TABLE]

In the homogenous case as well as in the case without magnetic field, Estimate (13) is crucial to describe vorticity defects. It is the same in the present work. More precisely, the main result [Theorem 4] states that the three above theorems are true replacing $(u_{\varepsilon},A_{\varepsilon})$ that minimizes $\mathcal{E}_{\varepsilon,h_{\rm ex}}$ in $\mathscr{H}$ by any configuration $(\tilde{u}_{\varepsilon},\tilde{A}_{\varepsilon})$ s.t. $E_{\varepsilon}(\tilde{u}_{\varepsilon},\tilde{A}_{\varepsilon})=\inf_{\mathscr{H}}\mathcal{E}_{\varepsilon,h_{\rm ex}}+o(1)$ with two extra hypotheses on $|\tilde{u}_{\varepsilon}|$ : $\|\nabla|\tilde{u}_{\varepsilon}|\|_{L^{\infty}(\Omega)}=\mathcal{O}(\varepsilon^{-1})$ and $|\tilde{u}_{\varepsilon}|\in W^{2,1}(\Omega)$ [see (17)]

Lassoued-Mironescu decoupling

In order to study pinned Ginzburg-Landau type energies, a nice trick was initiated by Lassoued and Mironescu in [12]. Before explaining this trick we have to do a direct calculation for $(u,A)\in\mathscr{H}$:

[TABLE]

with

[TABLE]

The Lassoued-Mironescu decoupling is obtained by first minimizing $E_{\varepsilon}$ in $H^{1}(\Omega,\mathbb{C})$. It is clear that $E_{\varepsilon}$ admits minimizers and if $U$ minimizes $E_{\varepsilon}$ then it satisfies

[TABLE]

By an energetic argument it is easy to prove that, if $U$ minimizes $E_{\varepsilon}$ in $H^{1}(\Omega,\mathbb{C})$, then $b\leq|U|\leq 1$. Moreover from (15), $U\wedge\nabla U=0$, i.e. $U=|U|\e^{\imath\theta}$ with $\theta\in\mathbb{R}$.

Then one may consider a scalar minimizer $U_{\varepsilon}:\Omega\to[b,1]$. This scalar minimizer may be seen as a regularization of $a_{\varepsilon}$ [see Proposition 7].

Using this scalar minimizer one may get the well known Lassoued-Mironescu decoupling: for $v\in H^{1}(\Omega,\mathbb{R})$ we have

[TABLE]

with

[TABLE]

Using this decoupling, one may prove that, for $\varepsilon>0$, there exists a unique positive minimizer $U_{\varepsilon}:\Omega\to[b,1]$ of $E_{\varepsilon}$ in $H^{1}(\Omega,\mathbb{R})$.

On the other hand, from (14) and (16), for $(u,A)\in\mathscr{H}$ and $v=u/U_{\varepsilon}$ we have:

[TABLE]

It is easy to check that $\mathcal{F}_{\varepsilon,h_{\rm ex}}(v,A)$ is gauge invariant. This functional is of major interest in the study since $(v,A)$ minimizes $\mathcal{F}_{\varepsilon,h_{\rm ex}}$ in $\mathscr{H}$ if and only if $(U_{\varepsilon}v,A)$ minimizes $\mathcal{E}_{\varepsilon,h_{\rm ex}}$ in $\mathscr{H}$.

An easy comparaison argument implies that if $(v_{\varepsilon},A_{\varepsilon})$ minimizes $\mathcal{F}_{\varepsilon,h_{\rm ex}}$ then $\|v_{\varepsilon}\|_{L^{\infty}(\Omega)}\leq 1$.

From now on we focus on the study of the minimizer of $\mathcal{F}_{\varepsilon,h_{\rm ex}}$. Namely we have the following theorem.

Theorem 4**.**

Assume that (5) holds and $\lambda,\delta,h_{\rm ex},K$ satisfy (2), (3) and (4).

Let $\{(v_{\varepsilon},A_{\varepsilon})\,|\,0<\varepsilon<1\}\subset\mathscr{H}$ be s.t. $\mathcal{F}(v_{\varepsilon},A_{\varepsilon})\leq\inf_{\mathscr{H}}\mathcal{F}+o(1)$. Assume also that

[TABLE]

Then Theorems 1, 2 and 3 hold for $u_{\varepsilon}=U_{\varepsilon}v_{\varepsilon}$.

Remark 5*.*

Theorem 4 may be rephrased in term of $U_{\varepsilon}$. Let $(h_{\rm ex})_{0<\varepsilon<1}\subset(0,\infty)$, $\{(u_{\varepsilon},A_{\varepsilon})\,|\,0<\varepsilon<1\}\subset\mathscr{H}$ and let $v_{\varepsilon}:=u_{\varepsilon}/U_{\varepsilon}\in H^{1}(\Omega,\mathbb{C})$. On the one hand, from the decoupling (16), we have $\{(u_{\varepsilon},A_{\varepsilon})\,|\,0<\varepsilon<1\}\subset\mathscr{H}$ is s.t. $\mathcal{E}_{\varepsilon,h_{\rm ex}}(u_{\varepsilon},A_{\varepsilon})\leq\inf_{\mathscr{H}}\mathcal{E}_{\varepsilon,h_{\rm ex}}+o(1)$ if and only $\{(v_{\varepsilon},A_{\varepsilon})\,|\,0<\varepsilon<1\}$ is s.t. $\mathcal{F}_{\varepsilon,h_{\rm ex}}(v_{\varepsilon},A_{\varepsilon})\leq\inf_{\mathscr{H}}\mathcal{F}_{\varepsilon,h_{\rm ex}}+o(1)$. On the other hand $v_{\varepsilon}$ satisfies (17) if and only if we have $|u_{\varepsilon}|\in W^{2,1}(\Omega,\mathbb{C})$ and $\|\nabla|u_{\varepsilon}|\|_{L^{\infty}(\Omega)}=\mathcal{O}(\varepsilon^{-1})$.

4. Plan of the article and proof of Theorem 4

The proof of Theorem 4 is done in several steps. It is based on a perturbative argument by replacing the energy $\mathcal{F}_{\varepsilon,h_{\rm ex}}$ with an energy $\tilde{\mathcal{F}}_{\varepsilon,h_{\rm ex}}$. This step is called the energetic cleaning [Section 5.1]. The functional $\tilde{\mathcal{F}}_{\varepsilon,h_{\rm ex}}$ is a perturbation of $\mathcal{F}_{\varepsilon,h_{\rm ex}}$: for $(v_{\varepsilon},A_{\varepsilon})\in\mathscr{H}$ which is in the Coulomb gauge and s.t. $\mathcal{F}_{\varepsilon,h_{\rm ex}}(v_{\varepsilon},A_{\varepsilon})=\mathcal{O}(h_{\rm ex}^{2})$ we have $\tilde{\mathcal{F}}_{\varepsilon,h_{\rm ex}}(v_{\varepsilon},A_{\varepsilon})-\mathcal{F}_{\varepsilon,h_{\rm ex}}(v_{\varepsilon},A_{\varepsilon})=o(1)$ [see Proposition 8]. In particular we have $\mathcal{F}_{\varepsilon,h_{\rm ex}}(v_{\varepsilon},A_{\varepsilon})\leq\inf_{\mathscr{H}}\mathcal{F}_{\varepsilon,h_{\rm ex}}+o(1)$ if and only if $\tilde{\mathcal{F}}_{\varepsilon,h_{\rm ex}}(v_{\varepsilon},A_{\varepsilon})\leq\inf_{\mathscr{H}}\tilde{\mathcal{F}}_{\varepsilon,h_{\rm ex}}+o(1)$.

In section 5.2 we apply a vortex ball construction of Sandier-Serfaty [Proposition 10] and we follow the strategy of Sandier-Serfaty developed in [15] to prove that the vorticity of a reasonable configuration is bounded [see Theorem 5].

Once the bound on the vorticity yields, we adapt a result of Serfaty [17] which gives a decomposition of $\tilde{\mathcal{F}}_{\varepsilon,h_{\rm ex}}(v_{\varepsilon},A_{\varepsilon})$ in term of $F_{\varepsilon}(v_{\varepsilon})$ and the location of the vorticity defects [Proposition 11].

The decomposition obtained in Proposition 11 allows to focus the study on the energy $F_{\varepsilon}$ which ignores the magnetic field. From this point, the study of a configuration $(v_{\varepsilon},A_{\varepsilon})$ is done for a major part via classical results based on the case without magnetic field [as in [4]]. To this end we adapt to our case some standard estimates ignoring the magnetic field, in particular the crucial notion of Renormalized energies is presented Section 6.

With these preliminary results, in Section 7, for $d\in\mathbb{N}^{*}$, we construct competitors $(v_{\varepsilon},A_{\varepsilon})\in\mathscr{H}$ with $d$ quantized vorticity defects and then we get a sharp upper bound [see Proposition 39]:

[TABLE]

Here ${\bf J_{0}}\&M_{\Omega}$ are independent of $\varepsilon$ and $d$, $\mathscr{L}_{1}(d)\&\mathscr{L}_{2}(d)$ are independent of $\varepsilon$ and $H^{0}_{c_{1}}$ is the leading term in the expression of the first critical field.

With the above upper bound for the minimal energy, the heart of the work consists in getting lower bounds for quasi-minimizers. Before getting such lowers bounds we adapt to our case some tools in Section 8: an $\eta$-ellipticity result is proved [Proposition 40], a construction of ad-hoc bad-discs is done [Proposition 42] and the strong effect of the dilution is expressed by various result in Section 8.3.

In Section 9 we begin the proof of the theorems. The part of Theorem 4 related with Theorem 1 is a direct consequence of Propositions 52, 53, 55 and 56 [and also Corollary 65].

The part of Theorem 4 related with Theorem 2 is given by Corollary 62 and Proposition 39.

The part of Theorem 4 related with Theorem 3 is a direct consequence of Corollary 65 and Propositions 68$\&$69.

5. Some preliminaries

5.1. Energetic cleaning

In order to do the cleaning step, we have to get some estimates. Our goal is to study quasi-minimizer of $\mathcal{F}_{\varepsilon,h_{\rm ex}}$. To keep a simple presentation, we write $\mathcal{F}$ instead of $\mathcal{F}_{\varepsilon,h_{\rm ex}}$ and $F$ instead of $F_{\varepsilon}$ when there is no ambiguity.

From (8), (9) and classical elliptic regularity arguments we have the following proposition.

Proposition 6**.**

Let $\{(v_{\varepsilon},A_{\varepsilon})\,|\,0<\varepsilon<1\}\subset\mathscr{H}$ be a family of configuration in the Coulomb gauge. Then there is $\xi_{\varepsilon}\in H^{1}_{0}\cap H^{2}(\Omega,\mathbb{R})$ s.t. $A_{\varepsilon}=\nabla^{\bot}\xi_{\varepsilon}$. Moreover, if for some $h_{\rm ex}=h_{\rm ex}(\varepsilon)$ we have

[TABLE]

*then there exists ${C}$ [independent of $\varepsilon$] s.t. *

[TABLE]

Consequently, for $p\in[1,\infty)$, there exists $C_{p}>1$ [independent of $\varepsilon$] s.t.

[TABLE]

Moreover, up to increase the value of $C>1$ [independently of $\varepsilon$], we have

[TABLE]

And if ${\rm curl}(A_{\varepsilon})\in H^{1}(\Omega)$ then

[TABLE]

In particular, for further use, note that if ${\rm curl}(A_{\varepsilon})\in H^{1}(\Omega)$ then $\xi_{\varepsilon}\in H^{1}_{0}\cap H^{2}\cap W^{1,\infty}(\Omega)$ and

[TABLE]

In order to do the cleaning step we need to underline the fact that $U_{\varepsilon}$ may be seen as a regularization of $a_{\varepsilon}$ in $W^{1,\infty}$ with estimates that become bad when approaching $\partial\omega_{\varepsilon}$.

Proposition 7**.**

There exist $C_{b},s_{b}>0$ depending only on $b$ and $\Omega$ s.t. for $\varepsilon,r>0$ we have:

[TABLE]

[TABLE]

[TABLE]

Proof.

Estimate (24) is a consequence of Lemma 4. The proof of (25) is the same than Proposition 2 in [9]. Estimate (26) is proved in Appendix A. ∎

Since the 2-dimensional Hausdorff measure of $\omega_{\varepsilon}$ satisfies $\mathcal{H}^{2}(\omega_{\varepsilon})=\mathcal{O}(\lambda^{2})$, from (25), for $p\in[1,\infty[$, we have the following crucial estimate

[TABLE]

We are now in position to do the cleaning step. We assume that $\{(v_{\varepsilon},A_{\varepsilon})\,|\,0<\varepsilon<1\}\subset\mathscr{H}$ is a family of configuration in the Coulomb gauge which satisfies (18). We denote $\alpha_{\varepsilon}=U_{\varepsilon}^{2}$ and $\rho_{\varepsilon}=|v_{\varepsilon}|$. From direct computations, by splitting the integrals with the identity $\alpha_{\varepsilon}=(\alpha_{\varepsilon}-1)+1$ and using $(1-\rho_{\varepsilon})^{4}\leq(1-\rho_{\varepsilon}^{2})^{2}$, we have the existence of $C\geq 1$ [independent of $\varepsilon$] s.t.

[TABLE]

and

[TABLE]

By combining (28) and (29) we immediately get the following proposition.

Proposition 8**.**

If $(v_{\varepsilon},A_{\varepsilon})$ is in the Coulomb gauge and satisfies (18) then

[TABLE]

with $C$ which is independent of $\varepsilon$ and

[TABLE]

Remark 9*.*

1. (1)

One may claim that $\tilde{\mathcal{F}}$ is not gauge invariant if $\alpha_{\varepsilon}\not\equiv 1$. 2. (2)

Note that if $\lambda^{1/4}|\ln\varepsilon|\to 0$ and if $h_{\rm ex}=\mathcal{O}(|\ln\varepsilon|)$ then for $(v_{\varepsilon},A_{\varepsilon})\in\mathscr{H}$ which is in the Coulomb gauge and satisfies (18) we have $\tilde{\mathcal{F}}(v_{\varepsilon},A_{\varepsilon})-\mathcal{F}(v_{\varepsilon},A_{\varepsilon})=o(1)$ without hypothesis on $\delta\in(0;1)$.

5.2. Bound on the vorticity and energetic decomposition

By applying Proposition 1 in [15] with $U_{\varepsilon}\geq b$ we immediately get the following proposition which does not need any assumption for $\lambda,\delta\in(0;1)$.

Proposition 10**.**

Assume $h_{\rm ex}\leq C_{0}|\ln\varepsilon|$ with $C_{0}\geq 1$ which is independent of $\varepsilon$. Let $\{(v_{\varepsilon},A_{\varepsilon})\,|\,0<\varepsilon<1\}$ be a family s.t. $\mathcal{F}(v_{\varepsilon},A_{\varepsilon})\leq C_{0}|\ln\varepsilon|^{2}$.

Then there exist $C,\varepsilon_{0}>0$ [depending only on $\Omega$, $b$ and $C_{0}$] s.t. for $\varepsilon<\varepsilon_{0}$ we have either $|v_{\varepsilon}|\geq 1-|\ln\varepsilon|^{-2}$ in $\Omega$ or there exists a finite family of disjoint disks $\{B_{i}\,|\,i\in\mathcal{J}\}$ with $\mathcal{J}\subset\mathbb{N}^{*}$ [$\mathcal{J}$ depends on $\varepsilon$] and $B_{i}:=B(a_{i},r_{i})$ satisfying :

1. (1)

$\{|v_{\varepsilon}|<1-|\ln\varepsilon|^{-2}\}\subset\cup B_{i}$** 2. (2)

$\sum r_{i}<|\ln\varepsilon|^{-10}$, 3. (3)

writing $h_{\varepsilon}={\rm curl}(A_{\varepsilon})$, $\rho_{\varepsilon}=|v_{\varepsilon}|$ and $v_{\varepsilon}=\rho_{\varepsilon}\e^{\imath\varphi_{\varepsilon}}$ [$\varphi_{\varepsilon}$* is locally defined] we have*

[TABLE]

with $d_{i}={\rm deg}_{\partial B_{i}}(v)$ if $B_{i}\subset\Omega$ and [math] otherwise.

By following the argument of Sandier and Serfaty [15], we get the main result of this section.

Theorem 5**.**

Assume that $\lambda,\delta$ satisfy (2) and $\delta^{2}|\ln\varepsilon|\leq 1$. Assume also Hypothesis (3) holds for $h_{\rm ex}$ with some $K\geq 1$.

Then there exist $\varepsilon_{K}>0$ and $\mathcal{M}_{K}\geq 1$ [independent of $\varepsilon$] s.t. if $\{(v_{\varepsilon},A_{\varepsilon})\,|\,0<\varepsilon<1\}\subset\mathscr{H}$ is a family in the Coulomb gauge satisfying $\mathcal{F}(v_{\varepsilon},A_{\varepsilon})\leq\inf_{\mathscr{H}}\mathcal{F}+K\ln|\ln\varepsilon|$ then for $0<\varepsilon<\varepsilon_{K}$ we have

[TABLE]

Moreover, if $|v_{\varepsilon}|\not>1-|\ln\varepsilon|^{-2}$ in $\Omega$, then letting $\{B_{i}\,|\,i\in\mathcal{J}\}$ be a family of disks given by Proposition 10, for $0<\varepsilon<\varepsilon_{K}$, we have $d_{i}\geq 0$ for all $i\in\mathcal{J}$ and there is $s_{0}>0$ [depending only on $\Omega$] s.t. if $i\in\mathcal{J}$ is s.t. $d_{i}\neq 0$ then ${\rm dist}(B_{i},\Lambda)\leq\mathcal{M}_{K}|\ln\varepsilon|^{-s_{0}}$.

The proof of this theorem is postponed in Appendix B.

We let

[TABLE]

Note that if $\{(v_{\varepsilon},A_{\varepsilon})\,|\,0<\varepsilon<1\}$ is a family of quasi-minimizers then

[TABLE]

The discs given by Proposition 10 are "too large" for our strategy. Indeed one of the main argument is a construction of bad discs in the spirit of [4] which links $x_{\varepsilon}\in\{|v_{\varepsilon}|\leq 1/2\}$ with the energetic cost in a ball $B(x_{\varepsilon},\varepsilon^{\mu})$ with small $\mu>0$. Namely if $x_{\varepsilon}\in\{|v_{\varepsilon}|<1-|\ln\varepsilon|^{-2}\}\subset\cup B_{i}$ then the energetic cost in a ball $B(x_{\varepsilon},\varepsilon^{\mu})$ is not sufficiently large comparing to our error term.

In the next proposition we present the good framework of vortex balls required in the study. The first step in the study is an energetic decomposition valid under some assumptions [no assumption on $\delta\in(0;1)$ is required].

Proposition 11**.**

Let $C_{0}>1$, $(v_{\varepsilon})_{0<\varepsilon<1}\subset H^{1}(\Omega,\mathbb{C})$ and $h_{\rm ex}>0$ be s.t.

[TABLE]

Assume furthermore that $\lambda^{1/4}|\ln\varepsilon|\to 0$ and, for $\varepsilon\in(0;1)$, either $|v_{\varepsilon}|>1/2$ in $\Omega$ or $v_{\varepsilon}$ admits a family of valued disks $\{(B(a_{i},r_{i}),d_{i})\,|\,i\in\mathcal{J}\}$ [$\mathcal{J}$ is finite] s.t. :

• $\bullet$

the disks $B_{i}=B(a_{i},r_{i})$ are pairwise disjoint

• $\bullet$

$\{|v_{\varepsilon}|\leq 1/2\}\subset\cup_{i\in\mathcal{J}}B_{i}$**

• $\bullet$

$\sum_{i\in\mathcal{J}}r_{i}<|\ln\varepsilon|^{-10}$**

• $\bullet$

For $i\in\mathcal{J}$, letting $d_{i}=\begin{cases}{\rm deg}_{\partial B_{i}}(v)&\text{if }B_{i}\subset\Omega\\ 0&\text{otherwise}\end{cases}$, we assume $\sum_{i\in\mathcal{J}}|d_{i}|\leq C_{0}$.

Then, if $(\xi_{\varepsilon})_{\varepsilon}\subset H^{1}_{0}\cap H^{2}\cap W^{1,\infty}(\Omega,\mathbb{R})$ is s.t.

[TABLE]

writing $\zeta_{\varepsilon}:=\xi_{\varepsilon}-h_{\rm ex}\xi_{0}$ we have in the case $|v_{\varepsilon}|\not>1/2$ in $\Omega$:

[TABLE]

where for $\zeta\in H^{1}_{0}\cap H^{2}(\Omega)$ we denoted

[TABLE]

And if $|v|>1/2$ in $\Omega$ then

[TABLE]

The proof of Proposition 11 is an adaptation of an argument of Serfaty [17] [section 4]. The proof is presented Appendix C

Before going further, we state a result which will be useful in this article and whose proof is left to the reader.

Lemma 12**.**

For $v\in H^{1}(\Omega,\mathbb{C})$, $0<\varepsilon<1$ and $h_{\rm ex}>0$, there exists a unique potential $A_{v,\varepsilon,h_{\rm ex}}=A_{v}\in H^{1}(\Omega,\mathbb{R}^{2})$ s.t. $(v,A_{v})$ is in the Coulomb gauge and satisfies

[TABLE]

Moreover $A_{v}$ is the unique solution of the minimization problem

[TABLE]

and from (9) and (10) we have $A_{v}=\nabla^{\bot}\xi_{v}$ with $\xi_{v}\in H^{1}_{0}\cap H^{2}\cap W^{1,\infty}(\Omega,\mathbb{R})$.

Remark 13*.*

Assume $\lambda,\delta$ satisfy (2), $\delta^{2}|\ln\varepsilon|\leq 1$ and Hypothesis (3) holds. Consider $\{(v_{\varepsilon},A_{\varepsilon})\,|\,0<\varepsilon<1\}\subset\mathscr{H}$ a family in the Coulomb gauge satisfying $\mathcal{F}(v_{\varepsilon},A_{\varepsilon})\leq\inf_{\mathscr{H}}\mathcal{F}+\mathcal{O}(\ln|\ln\varepsilon|)$.

• •

From Theorem 5, either $|v_{\varepsilon}|>1-|\ln\varepsilon|^{-2}$ in $\Omega$ or the family of disjoint disks given by Proposition 10 satisfies the properties of the family of discs used in Proposition 11.

• •

Let $A_{v_{\varepsilon}}=\nabla^{\bot}\xi_{v_{\varepsilon}}\in H^{1}(\Omega,\mathbb{R}^{2})$ be given by Lemma 12. Then with (9)$\&$(39) we have $A_{v_{\varepsilon}}\in L^{\infty}(\Omega)$ and $\|A_{v_{\varepsilon}}\|_{L^{\infty}(\Omega)}\leq C|\ln\varepsilon|$ where $C$ depends only on $\Omega$.

As noted by Serfaty [17], with the help of the decomposition given by Proposition 11, we may prove that $h_{\rm ex}^{2}{\bf J_{0}}$ is almost the minimal energy of a vortex less configuration.

Corollary 14**.**

Let $\mathscr{H}^{0}:=\left\{(\rho\e^{\imath\varphi},A)\,|\,\rho\in H^{1}(\Omega,[0,\infty)),\,\varphi\in H^{1}(\Omega,\mathbb{R})\text{ and }A\in H^{1}(\Omega,\mathbb{R}^{2})\right\}$. Note that $\mathscr{H}^{0}$ is gauge invariant. Assume $\lambda^{1/4}|\ln\varepsilon|\to 0$.

1. (1)

Let $\varepsilon=\varepsilon_{n}\downarrow 0$. Assume $h_{\rm ex}=\mathcal{O}(|\ln\varepsilon|)$ and for each $\varepsilon$ let $(v_{\varepsilon},\nabla^{\bot}\xi_{\varepsilon})\in\mathscr{H}^{0}$ be s.t. $\xi_{\varepsilon}\in H^{1}_{0}\cap H^{2}\cap W^{1,\infty}(\Omega,\mathbb{R})$ with $\|\nabla\xi_{\varepsilon}\|_{L^{\infty}(\Omega)}=\mathcal{O}(|\ln\varepsilon|)$. Writing $\zeta_{\varepsilon}:=\xi_{\varepsilon}-h_{\rm ex}\xi_{0}$ we have:

[TABLE]

Thus, if $\mathcal{F}(v_{\varepsilon},\nabla^{\bot}\xi_{\varepsilon})\leq h_{\rm ex}^{2}{\bf J_{0}}+o(1)$ then $\zeta_{\varepsilon}\to 0$ in $H^{2}(\Omega)$, $|v_{\varepsilon}|\to 1$ in $H^{1}(\Omega)$ and, up to pass to a subsequence, there exists ${\tt v}\in\mathbb{S}^{1}$ s.t. $v_{\varepsilon}\to{\tt v}$ in $H^{1}(\Omega)$. 2. (2)

We have $\inf_{\mathscr{H}^{0}}\mathcal{F}=h_{\rm ex}^{2}{\bf J_{0}}+o(1)$.

Proof.

We prove the first assertion. Estimate (41) is a direct consequence of Proposition 11.

For sake of simplicity of the presentation we drop the subscript $\varepsilon$. If $\mathcal{F}(v,\nabla^{\bot}\xi)\leq h_{\rm ex}^{2}{\bf J_{0}}+o(1)$, then $F(v)+\|\zeta\|_{H^{2}(\Omega)}=o(1)$ and then $\zeta\to 0$ in $H^{2}(\Omega)$, $|v|\to 1$ in $H^{1}(\Omega)$. Moreover $\|\nabla v\|_{L^{2}(\Omega)}=o(1)$ and $\|v\|_{L^{2}(\Omega)}=\mathcal{O}(1)$. This clearly implies the remaining part of the assertion.

We prove the second assertion. We first claim, by the definition of ${\bf J_{0}}$, that using the configuration $(1,h_{\rm ex}\nabla^{\bot}\xi_{0})\in\mathscr{H}^{0}$ we have $\inf_{\mathscr{H}^{0}}\mathcal{F}\leq h_{\rm ex}^{2}{\bf J_{0}}+o(1)$.

By the gauge invariance of $\mathscr{H}^{0}$ we may consider a family of quasi-minimizer $\{(v_{\varepsilon},A_{\varepsilon})\,|\,0<\varepsilon<1\}\subset\mathscr{H}^{0}$ which is in the Coulomb gauge. We write $(v_{\varepsilon},A_{\varepsilon})=(v,A)$. Let $(\tilde{v},\tilde{A})\in\mathscr{H}^{0}$ be defined by $\tilde{v}=\underline{v}$ and $\tilde{A}$ is the unique solution of (40) associated to $\tilde{v}$.

By direct calculations we have: $\mathcal{F}(\tilde{v},\tilde{A})\leq\mathcal{F}(\tilde{v},A)\leq\mathcal{F}(v,A)\leq h_{\rm ex}^{2}{\bf J_{0}}+o(1)$.

Moreover, by denoting $h:={\rm curl}(\tilde{A})$, we have $\nabla h=\alpha\tilde{v}\wedge(\nabla^{\bot}\tilde{v}-\tilde{A}^{\bot}\tilde{v})$ in $\Omega$ and $h=h_{\rm ex}$ on $\partial\Omega$. Then $\|h\|_{H^{1}(\Omega)}=\mathcal{O}(|\ln\varepsilon|)$ and using (22) we get $\|\tilde{A}\|_{H^{2}(\Omega)}=\mathcal{O}(|\ln\varepsilon|)$.

We are then able to apply the first assertion to get $\mathcal{F}(\tilde{v},\tilde{A})\geq h_{\rm ex}^{2}{\bf J_{0}}+o(1)$.

∎

5.3. Pseudo vortex structure

We assume $\lambda^{1/4}|\ln\varepsilon|\to 0$. Let $\{(v_{\varepsilon},A_{\varepsilon})\,|\,0<\varepsilon<1\}\subset\mathscr{H}$ be a family of configurations in the Coulomb gauge satisfying (34). We assume that $|v_{\varepsilon}|\not>1/2$ in $\Omega$ and that there exists $\{(B(a_{i},r_{i}),d_{i})\,|\,i\in\mathcal{J}\}$ as in Proposition 11. Then Proposition 11 gives a decomposition of $\mathcal{F}(v,A)$. Except in the crucial hypothesis $\sum r_{i}<|\ln\varepsilon|^{-10}$, the radii $r_{i}$ do not play any role as well as the disks "$B(a_{i},r_{i})$" associated to a zero degree. We thus introduce an ad-hoc notion of pseudo vortex.

Definition 15*.*

We assume that we have either $\varepsilon=\varepsilon_{n}\downarrow 0$ or $0<\varepsilon<1$. We consider $(v_{\varepsilon})_{\varepsilon}\subset H^{1}(\Omega,\mathbb{C})$, $(h_{\rm ex})_{\varepsilon}\subset(1,\infty)$ satisfying (34).

Let $\{B_{i}=B(a_{i},r_{i})\,|\,i\in\mathcal{J}\}$ be a family of disks as in Proposition 11 and let $d_{i}=d^{(\varepsilon)}_{i}\in\mathbb{Z}$ be the associated "degrees" defined in Proposition 11. We denote $\mathcal{J}^{\prime}=\mathcal{J}^{\prime}_{\varepsilon}:=\{i\in\mathcal{J}\,|\,d_{i}\neq 0\}$ [note that we have ${\rm Card}(\mathcal{J}^{\prime}_{\varepsilon})\leq\sum|d_{i}|=\mathcal{O}(1)$].

If $\mathcal{J}^{\prime}\neq\emptyset$, then we say that $\{{({\bf a},{\bf d})}\}=\{(a_{i},d_{i})\,|\,i\in\mathcal{J}^{\prime}\}$ is a set of pseudo vortices of $v_{\varepsilon}$.

For a fixed configuration ${({\bf a},{\bf d})}$ of pseudo vortices, Serfaty studied in [17] the minimization problem of $\tilde{V}_{({\bf a},{\bf d})}$ [defined in (37)]. We have the following result [Proposition 4.2 in [17]].

Proposition 16**.**

Let ${({\bf a},{\bf d})}=\{(a_{i},d_{i})\,|\,i\in\mathcal{J}^{\prime}\}\subset\Omega\times\mathbb{Z}^{*}$ be a configuration s.t. $1\leq{\rm Card}(\mathcal{J}^{\prime})<\infty$ and $a_{i}\neq a_{j}$ for $i\neq j$. Then $\tilde{V}_{({\bf a},{\bf d})}(\zeta)$ is minimal for $\zeta=\zeta_{({\bf a},{\bf d})}$ which satisfies

[TABLE]

[Here $\delta_{a}$ is the Dirac mass at $a\in\mathbb{R}^{2}$]

And we have $\tilde{V}[\zeta_{({\bf a},{\bf d})}]=\pi\sum_{i\in\mathcal{J}^{\prime}}d_{i}\zeta_{({\bf a},{\bf d})}(a_{i})$.

In order to prove the above proposition, Serfaty introduced for $a\in\Omega$ the function $\zeta^{a}\in H^{1}_{0}\cap H^{2}(\Omega)$ which is the unique solution of

[TABLE]

In particular we have $\zeta^{a}\leq 0$ in $\Omega$. It is easy to see that $\zeta_{({\bf a},{\bf d})}=\sum_{i\in\mathcal{J}^{\prime}}d_{i}\zeta^{a_{i}}$ is the unique solution of (42).

Lemma 4.6 in [17] gives important properties related with $\zeta^{a}$ and $\zeta_{({\bf a},{\bf d})}$:

Proposition 17**.**

For $s\in(0,1)$, there exists $C_{s}>0$ s.t. for $a,b\in\Omega$

[TABLE]

and

[TABLE]

Consequently there exists $C>0$ depending only on $\Omega$ s.t., if $\zeta_{({\bf a},{\bf d})}$ is the unique solution of (42), then

[TABLE]

For a further use we need the following lemma.

Lemma 18**.**

Let ${({\bf a},{\bf d})}$ as in Proposition 16 then $\zeta_{({\bf a},{\bf d})}\in H^{1}_{0}\cap H^{2}\cap W^{1,\infty}(\Omega,\mathbb{R})$ and there is $C\geq 1$ depending only on $\Omega$ s.t.

[TABLE]

Proof.

Let ${({\bf a},{\bf d})}$ be as in Proposition 16, with Proposition 17 we have $\zeta_{({\bf a},{\bf d})}=\sum d_{i}\zeta^{a_{i}}\in H^{1}_{0}\cap H^{2}$ and $\|\zeta_{({\bf a},{\bf d})}\|_{H^{2}(\Omega)}\leq C\sum_{i}|d_{i}|$ where $C$ depends only on $\Omega$.

Moreover, for $a\in\Omega$, from (42), we have $\Delta\zeta_{({\bf a},{\bf d})}=\zeta_{({\bf a},{\bf d})}-\sum d_{i}\ln|x-a_{i}|-R_{({\bf a},{\bf d})}$ where $R_{({\bf a},{\bf d})}$ is the harmonic extension of ${\rm tr}_{\partial\Omega}(-\sum d_{i}\ln|x-a_{i}|)$ in $\Omega$.

Consequently there exists $C\geq 1$ depending only on $\Omega$ s.t.

[TABLE]

and therefore by elliptic regularity and a Sobolev embedding we get the result. ∎

Until now, the only way to get a nice magnetic potential associated to a function $v$ was to consider $A_{v}=A_{v,\varepsilon,\alpha}\in H^{2}(\Omega,\mathbb{R}^{2})$, the unique solution of (40). The previous results give that, after the cleaning step, we can do asymptotically as well by using a magnetic potential depending on a pseudo vortices structure of $v$ instead of $v$ itself [see Remark 20].

Definition 19*.*

Let $N\geq 1$ and ${({\bf a},{\bf d})}\in(\Omega^{N})^{*}\times(\mathbb{Z}^{*})^{N}$, $h_{\rm ex}>0$. Then we define $A_{({\bf a},{\bf d})}:=h_{\rm ex}\nabla^{\bot}\xi_{0}+\nabla^{\bot}\zeta_{({\bf a},{\bf d})}$ where $\zeta_{({\bf a},{\bf d})}$ is the unique solution of $\eqref{LondonEqModifie}$, the potential associated to ${({\bf a},{\bf d})}$.

Remark 20*.*

Let $C_{0}>1$ and $(v_{\varepsilon})_{0<\varepsilon<1}\subset H^{1}(\Omega,\mathbb{C})$, $h_{\rm ex}>0$ satisfying (34) be s.t. $(v_{\varepsilon})_{0<\varepsilon<1}$ admits a set of pseudo vortices $({({\bf a},{\bf d})}_{\varepsilon})_{0<\varepsilon<1}$ with $\sum|d_{i}|\leq C_{0}$. We write $v\&{({\bf a},{\bf d})}$ instead of $v_{\varepsilon}\&{({\bf a},{\bf d})}_{\varepsilon}$.

Assume $\min{\rm dist}(a_{i},\partial\Omega)>|\ln\varepsilon|^{-1}$ in order to have $\|\nabla\zeta_{{({\bf a},{\bf d})}}\|_{L^{\infty}(\Omega)}=\mathcal{O}(|\ln\varepsilon|)$ [with Lemma 18] and $\lambda^{1/4}|\ln\varepsilon|\to 0$.

For $0<\varepsilon<1$, let $A_{v}\in H^{1}(\Omega,\mathbb{R}^{2})$ be the unique solution of (40) and $A_{{({\bf a},{\bf d})}}$ be defined in Definition 19. Then we have $A_{{({\bf a},{\bf d})}}=\nabla^{\bot}\xi_{{({\bf a},{\bf d})}}$ and $A_{v}=\nabla^{\bot}\xi_{v}$ where $\xi_{{({\bf a},{\bf d})}},\xi_{v}\in H^{1}_{0}\cap H^{2}\cap W^{1,\infty}(\Omega,\mathbb{R})$ satisfy the hypotheses of Proposition 11 [here we used (9)$\&$(39)]. Therefore we have the following inequalities

[TABLE]

[TABLE]

In particular we have $\mathcal{F}(v,A_{v})=\mathcal{O}(|\ln\varepsilon|^{2})$ and $\mathcal{F}(v,A_{{({\bf a},{\bf d})}})=\mathcal{O}(|\ln\varepsilon|^{2})$.

5.4. Cluster of pseudo vortices

From a standard result for the homogenous case, it is expected that, for a reasonable magnetic field, the asymptotic location of pseudo vortices of a studied configuration is a subset of $\Lambda$. This problem is related to the macroscopic location of the pseudo vortices. To treat this problem we use an ad-hoc notion of cluster of pseudo vortices.

Definition 21*.*

Let $N,\tilde{N}_{0}\in\mathbb{N}^{*}$, $\tilde{N}_{0}\leq N$, ${({\bf p},{\bf D})}\in(\overline{\Omega}^{\tilde{N}_{0}})^{*}\times\mathbb{Z}^{\tilde{N}_{0}}$, $\varepsilon=\varepsilon_{n}\downarrow 0$ and ${({\bf a},{\bf d})}_{\varepsilon}\in(\Omega^{N})^{*}\times\mathbb{Z}^{N}$ s.t. ${\bf d}$ is independent of $\varepsilon$. We say that $({({\bf a},{\bf d})}_{\varepsilon})_{\varepsilon}$ admits a cluster structure on ${({\bf p},{\bf D})}$ if

• •

for $i\in\{1,...,N\}$, $\lim a_{i}$ exists, $\lim a_{i}\in\{p_{1},...,p_{\tilde{N}_{0}}\}$ and we write for $k\in\{1,...,\tilde{N}_{0}\}$, $S_{k}:=\{i\in\{1,...,N\}\,|\,a_{i}\to p_{k}\}$

• •

for $k\in\{1,...,\tilde{N}_{0}\}$ $S_{k}\neq\emptyset$,

• •

for $k\in\{1,...,\tilde{N}_{0}\}$, $D_{k}=\sum_{i\in S_{k}}d_{i}$.

Remark 22*.*

In this article we will use the notion of cluster structure with ${({\bf a},{\bf d})}$ as in Proposition 11 and ${\bf p}\subset\Lambda$.

Proposition 23**.**

Let $N\geq 1$, $\varepsilon=\varepsilon_{n}\downarrow 0$, ${({\bf a},{\bf d})}_{\varepsilon}\in(\Omega^{N})^{*}\times\mathbb{Z}^{N}$ s.t. $\sum|d_{i}|$ is bounded independently of $\varepsilon$.

1. (1)

If $({({\bf a},{\bf d})}_{\varepsilon})_{\varepsilon}$ admits a cluster structure on ${({\bf p},{\bf D})}$ [and then ${\bf d}$ is independent of $\varepsilon$] then ${({\bf p},{\bf D})}$ is unique [up to change the order]. We say that ${({\bf p},{\bf D})}$ is the cluster of $({({\bf a},{\bf d})}_{\varepsilon})_{\varepsilon}$. 2. (2)

Up to pass to a subsequence, there exist $1\leq\tilde{N}_{0}\leq N$ and ${({\bf p},{\bf D})}\in(\overline{\Omega}^{\tilde{N}_{0}})^{*}\times\mathbb{Z}^{\tilde{N}_{0}}$ s.t. ${({\bf p},{\bf D})}$ is the cluster of $({({\bf a},{\bf d})}_{\varepsilon})_{\varepsilon}$. 3. (3)

If ${({\bf p},{\bf D})}$ is the cluster of $({({\bf a},{\bf d})}_{\varepsilon})_{\varepsilon}$ then, denoting $\chi:=\max_{k}\max_{i\in S_{k}}|a^{\varepsilon}_{i}-p_{k}|$, we have

[TABLE]

and

[TABLE]

where $C$ depends only on $N$, $\sum|d_{i}|$ and $\Omega$.

Proof.

The two first assertions are obvious. Estimate (43) is direct by noting that $\xi_{0}$ a Lipschitzian function in $\Omega$. Estimate (44) is a direct consequence of Proposition 17. ∎

We then have:

Corollary 24**.**

Assume that $\lambda,\delta,h_{\rm ex}$ satisfy (2) and (3) for some $K\geq 0$ independent of $\varepsilon$. Assume also $\delta^{2}|\ln\varepsilon|\leq 1$.

Let $\{(v_{\varepsilon},A_{\varepsilon})\,|\,0<\varepsilon<1\}\subset\mathscr{H}$ be a family s.t. $\mathcal{F}(v_{\varepsilon},A_{\varepsilon})\leq\inf_{\mathscr{H}}\mathcal{F}+K\ln|\ln\varepsilon|$ which is in the Coulomb gauge and let $\{({\bf a}_{\varepsilon},{\bf d}_{\varepsilon})={({\bf a},{\bf d})}\,|\,0<\varepsilon<1\}$ be a family of pseudo vortices associated to $\{(v_{\varepsilon},A_{\varepsilon})\,|\,0<\varepsilon<1\}$ [indexed on $\mathcal{J}=\mathcal{J}_{\varepsilon}$ possibly empty].

1. (1)

Letting $A_{v_{\varepsilon}}\in H^{1}(\Omega,\mathbb{R}^{2})$ be defined by Lemma 12 we have

[TABLE]

And then

[TABLE] 2. (2)

Assume furthermore that ${({\bf a},{\bf d})}$ admits a cluster structure on ${({\bf p},{\bf D})}$. Then we have

[TABLE]

Proof.

The lower bounds (45) and (46) are direct consequences of Theorem 5, Lemma 12, Remark 13 and Propositions 6$\&$11$\&$16.

Estimate (47) is a direct consequence of Proposition 23 and (45).∎

We then have the following corollary.

Corollary 25**.**

Assume that $\lambda,\delta,h_{\rm ex}$ satisfy (2) and (3). Assume also $\delta^{2}|\ln\varepsilon|\leq 1$.

Let $(v_{\varepsilon})_{0<\varepsilon<1}\subset H^{1}(\Omega,\mathbb{C})$ be s.t. $|v_{\varepsilon}|\not>1/2$ in $\Omega$ and assume the existence of $(B_{\varepsilon})_{0<\varepsilon<1}\subset H^{1}(\Omega,\mathbb{R}^{2})$ s.t. $(v_{\varepsilon},B_{\varepsilon})$ is in the Coulomb gauge and $\mathcal{F}(v_{\varepsilon},B_{\varepsilon})\leq\inf_{\mathscr{H}}\mathcal{F}+\mathcal{O}(\ln|\ln\varepsilon|)$. Assume also that $({\bf a}_{\varepsilon},{\bf d}_{\varepsilon})=({\bf a},{\bf d})$ are pseudo-vortices as in Definition 15 for $v_{\varepsilon}$ [note that we thus have $\sum|d_{i}|=\mathcal{O}(1)$], then

[TABLE]

where $A_{({\bf a},{\bf d})}:=h_{\rm ex}\nabla^{\bot}\xi_{0}+\nabla^{\bot}\zeta_{({\bf a},{\bf d})}$.

Consequently we get

[TABLE]

Proof.

Corollary 25 is a direct consequence of $\inf_{\mathscr{H}}\mathcal{F}\leq h_{\rm ex}^{2}{\bf J_{0}}$, Corollary 24 and Propositions 11$\&$17. ∎

Remark 26*.*

We may state an analog of Corollary 25 if ${({\bf a},{\bf d})}$ admits a structure of cluster.

6. Renormalized energies

6.1. Macroscopic renormalized energy [at scale $1$]

We consider in this section:

• $\bullet$

$N\in\mathbb{N}^{*}$, ${\bf z}={\bf z}^{(n)}\in(\Omega^{N})^{*}:=\{(z_{1},...,z_{N})\subset\Omega\,|\,z_{i}\neq z_{j}\text{ pour }i\neq j\}$,

• $\bullet$

${\bf d}=(d_{1},...,d_{N})\in\mathbb{Z}^{N}$.

• $\bullet$

$\hbar=\hbar({\bf z}):=\min_{i}{\rm dist}(z_{i},\partial\Omega)$

We are going to deal with functions defined in the set $\Omega$ perforated by disks with radius ${\tilde{r}}={\tilde{r}}_{n}\downarrow 0$:

[TABLE]

We assume

[TABLE]

For a radius ${\tilde{r}}>0$ s.t. (50) is satisfied, we consider the set of functions

[TABLE]

and

[TABLE]

In this section we are interested in the minimization of the Dirichlet functional in $\mathcal{I}^{\rm deg}_{\tilde{r}}$ and $\mathcal{I}^{\rm Dir}_{\tilde{r}}$.

Before beginning we state an easy result proved by direct minimization [the proof is left to the reader, see [4]].

Proposition 27**.**

For $N\geq 1$, ${\bf(z,d)}\in(\Omega^{N})^{*}\times\mathbb{Z}^{N}$ and ${\tilde{r}}>0$ s.t. (50) is satisfied, the following minimization problems admit solutions:

[TABLE]

and

[TABLE]

Moreover, these solutions are unique up to the multiplication by an $\mathbb{S}^{1}$ constant.

6.1.1. Study of $I^{\rm deg}_{\tilde{r}}$ and $I^{\rm Dir}_{\tilde{r}}$

Following [4], it is standard to define the canonical harmonic map associated to ${\bf(z,d)}$.

Definition 28*.*

Let $N\in\mathbb{N}^{*}$ and ${\bf(z,d)}\in(\Omega^{N})^{*}\times\mathbb{Z}^{N}$. A function $w^{\bf(z,d)}_{\star}\in\cap_{0<p<2}W^{1,p}(\Omega,\mathbb{S}^{1})\cap C^{\infty}(\Omega\setminus\{z_{1},...,z_{N}\},\mathbb{S}^{1})$ is the canonical harmonic map associated to the singularities ${\bf(z,d)}$ if

[TABLE]

Remark 29*.*

In this framework, it is classic to define $\Phi^{\bf(z,d)}_{\star}$ [with the notation of Definition 28], the unique solution of

[TABLE]

This function satisfies $\nabla^{\bot}\Phi^{\bf(z,d)}_{\star}=w^{\bf(z,d)}_{\star}\wedge\nabla w^{\bf(z,d)}_{\star}$. Moreover, by denoting $R_{\bf(z,d)}$ the unique solution of

[TABLE]

we have $\Phi^{\bf(z,d)}_{\star}(z)=\sum_{i}d_{i}\ln|z-z_{i}|+R_{\bf(z,d)}(z)$.

We first study the asymptotic behavior of minimizers of $I^{\rm deg}_{\tilde{r}}{\bf(z,d)}$ when ${\tilde{r}}\to 0$.

Proposition 30**.**

Let $N\in\mathbb{N}^{*}$, ${\bf(z,d)}={\bf(z,d)}^{(n)}\subset(\Omega^{N})^{*}\times\mathbb{Z}^{N}$ and $\hbar:=\min_{i}{\rm dist}(z_{i},\partial\Omega)$. We assume that $\sum_{i}|d_{i}|=\mathcal{O}(1)$.

For ${\tilde{r}}>0$ s.t. (50) is satisfied, we may consider $w^{\bf(z,d)}_{\tilde{r}}$, the unique solution of the problem

[TABLE]

of the form

[TABLE]

We thus have the existence of $C>0$ [depending only on $\Omega,N$ and the bound of $\sum_{i}|d_{i}|$] s.t.

[TABLE]

We denote

[TABLE]

and we have

[TABLE]

[TABLE]

Moreover, if there exists $\eta>0$ [independent of $n$] s.t. $\hbar>\eta$ then (56) may be refined into

[TABLE]

The proof of Proposition 30 is in Appendix D.1.

By adapting the proof of Proposition 5.1 in [17] we have

Proposition 31**.**

For $N\geq 1$, there exists an application $W^{\tt macro}_{N}=W^{\tt macro}:(\Omega^{N})^{*}\times\mathbb{Z}^{N}\to\mathbb{R}$ s.t. for sequences ${\bf(z,d)}={\bf(z,d)}^{(n)}\in(\Omega^{N})^{*}\times\mathbb{Z}^{N}$ and ${\tilde{r}}={\tilde{r}}_{n}\to 0$ satisfying (50) and s.t. ${\bf d}$ is independent of $n$, there exists $C\geq 1$ [depending only on $N$, $\sum|d_{i}|$ and $\Omega$] s.t.

[TABLE]

with

[TABLE]

[TABLE]

Proposition 31 is proved in D.2. We immediately obtain from Proposition 31 the following corollary.

Corollary 32**.**

Under the hypotheses of Proposition 31 and assuming that there exists $C_{1}>0$ [independent of $r$] s.t. $\dfrac{{\tilde{r}}(1+|\ln\hbar|)}{\hbar}\leq C_{1}$, there is $C>1$ [depending only on $\Omega$, $N$, $\sum_{i}|d_{i}|$ and $C_{1}$] s.t. $\displaystyle\int_{\Omega_{\tilde{r}}}|\nabla w^{\bf(z,d)}_{\star}|^{2}\leq C|\ln{\tilde{r}}|$.

We end this section by linking $I^{\rm deg}_{\tilde{r}}$ and $I^{\rm Dir}_{\tilde{r}}$.

Proposition 33**.**

Let $N\geq 1$, ${\bf z}\in(\Omega^{N})^{*}$ and ${\tilde{r}}={\tilde{r}}_{n}\downarrow 0$ satisfying (50). Assume $\dfrac{{\tilde{r}}}{\hbar}\to 0$ and if $N\geq 2$, we also assume $\dfrac{{\tilde{r}}}{\min_{i\neq j}|z_{i}-z_{j}|}\to 0$.

Let

[TABLE]

Assume furthermore

[TABLE]

Then for ${\bf d}\in\mathbb{Z}^{N}$ [independent of $n$], there exists $C>1$ [depending only on $\Omega$, $N$ and $\sum|d_{i}|$] s.t.

[TABLE]

Proposition 33 is proved Appendix D.3.

6.1.2. Macroscopic renormalized energy and cluster of vortices

We first state an easy lemma.

Lemma 34**.**

1. (1)

Let $N\in\mathbb{N}^{*}$ and ${\bf d}\in\mathbb{Z}^{N}$. Let $\chi>0$ and ${\bf z},{\bf z}^{\prime}\in(\Omega^{N})^{*}$ be s.t. for $i\in\{1,...,N\}$ we have $|z_{i}-z_{i}^{\prime}|\leq\chi$. Then we have

[TABLE] 2. (2)

Let $1\leq\tilde{N}_{0}\leq N$, ${\bf p}\in(\Omega^{\tilde{N}_{0}})^{*}$, ${\bf(z,d)}={\bf(z,d)}^{(n)}\in(\Omega^{N})^{*}\times\mathbb{Z}^{N}$ be s.t. ${\bf d}$ is independent of $n$ and for $i\in\{1,...,N\}$ there exists $k\in\{1,...,\tilde{N}_{0}\}$ s.t. $z_{i}\to p_{k}$. We let $\chi:=\max_{i}{\rm dist}(z_{i},\{p_{1},...,p_{\tilde{N}_{0}}\})$.

For $k\in\{1,...,\tilde{N}_{0}\}$ we let $D_{k}:=\displaystyle\sum_{z_{i}\to p_{k}}d_{i}$ and ${\bf D}=(D_{1},...,D_{\tilde{N}_{0}})$. Then we have

[TABLE]

Proof.

The first assertion is obtained with the help of the maximum principle and the bound $|R_{\bf(z,d)}-R_{({\bf z}^{\prime},{\bf d})}|\leq\sum_{i}|d_{i}|\dfrac{\chi}{\max\{\hbar({\bf z}),\hbar({\bf z}^{\prime})\}}$ on $\partial\Omega$. The second assertion assertion follows by the same way. ∎

With Lemma 34 we may exploit a structure of cluster for $W^{\rm macro}$.

Proposition 35**.**

Let $1\leq\tilde{N}_{0}\leq N$, ${\bf p}\in(\Omega^{\tilde{N}_{0}})^{*}$ [independent of $n$] and write

[TABLE]

Let ${\bf(z,d)}={\bf(z,d)}^{(n)}\in(\Omega^{N})^{*}\times\mathbb{Z}^{N}$ be s.t. ${\bf d}$ is independent of $n$ and for $i\in\{1,...,N\}$ there exists $k\in\{1,...,\tilde{N}_{0}\}$ s.t. $z_{i}\to p_{k}$. We denote $\chi:=\max_{i}{\rm dist}(z_{i},\{p_{1},...,p_{\tilde{N}_{0}}\})$.

For $k\in\{1,...,\tilde{N}_{0}\}$ we denote $D_{k}:=\displaystyle\sum_{z_{i}\to p_{k}}d_{i}$ and ${\bf D}=(D_{1},...,D_{\tilde{N}_{0}})$. Then there exists $C\geq 1$ [depending only on $\Omega,N$ and $\sum|d_{i}|$] s.t.

[TABLE]

Proof.

We have

[TABLE]

It is easy to check that

[TABLE]

with $H\leq 4\left(\sum_{i}|d_{i}|\right)^{2}\dfrac{\chi}{\gamma_{\bf p}}$ for sufficiently large $n$.

On the other hand, from Lemma 34 [second assertion], we have $\|R_{\bf(z,d)}-R_{{({\bf p},{\bf D})}}\|_{L^{\infty}(\Omega)}\leq\sum_{i}|d_{i}|\dfrac{\chi}{\max\{\hbar({\bf z}),\hbar({\bf p})\}}$. From standard pointwise estimates for the gradient of harmonic functions [see (166)] there exists $C\geq 1$ depending only on $\Omega$, $\sum|D_{k}|$ and $N$ [here we used $1\leq\tilde{N}_{0}\leq N$] s.t. for $z_{i}\to p_{k}$ we have $\left|R_{{({\bf p},{\bf D})}}(z_{i})-R_{{({\bf p},{\bf D})}}(p_{k})\right|\leq C\chi\dfrac{1+|\ln[\hbar({\bf p})]|}{\hbar({\bf p})}$.

Then, up to change the value of $C$, we have

[TABLE]

By combining (61) and (62) we get the result. ∎

6.2. Mesoscopic renormalized energy [at scale $h_{\rm ex}^{-1/2}$]

From the work of Sandier and Serfaty we may obtain mesoscopic informations. To this end we need to assume a non degeneracy assumption for minimal points of $\xi_{0}$. So we assume in this section that Hypothesis (5) holds.

Let

[TABLE]

For $p\in\Lambda$, by applying Lemma 11.1 in [16] in the disk $B(p,\eta_{\Omega})$, we get the following proposition.

Proposition 36**.**

Assume that Hypothesis (5) holds. Let $D\in\mathbb{N}^{*}$ and $h_{\rm ex}\uparrow\infty$ when $\varepsilon\to 0$. Then for $p\in\Lambda$ and $R=R(\varepsilon)\to 0$ s.t. $R\sqrt{h_{\rm ex}}\to\infty$ we have

[TABLE]

with

[TABLE]

and

[TABLE]

where $Q_{p}(x):=x\cdot{\rm Hess}_{\xi_{0}}(p)x$, ${\rm Hess}_{\xi_{0}}(p)$ is the Hessian matrix of $\xi_{0}$ at $p$.

Moreover the infimum in (64) is reached and if ${\bf z}^{\varepsilon}\in[B(p,R)^{D}]^{*}$ is s.t.

[TABLE]

then for all sequence $\varepsilon=\varepsilon_{n}\downarrow 0$, up to pass to a subsequence, denoting $\ell=\sqrt{\dfrac{D}{h_{\rm ex}}}$ and $\breve{z}_{i}^{\varepsilon}=\dfrac{z_{i}^{\varepsilon}-p}{\ell}$, we have $\breve{\bf z}^{\varepsilon}=(\breve{z}_{1}^{\varepsilon},...,\breve{z}_{D}^{\varepsilon})$ which converges to a minimizer of $W^{\rm meso}_{p,D}$. In particular $|\breve{z}_{i}^{\varepsilon}|\leq C_{\Omega,D}$ with $C_{\Omega,D}>0$ which depends only on $\Omega$ and $D$.

6.3. Microscopic renormalized energy [at scale $\lambda\delta$]

The location of the vorticity defects at scale $\lambda\delta$ [inside a connected component of $\omega_{\varepsilon}$] is given by the microscopic renormalized energy exactly as in the case without magnetic field. In order to define the microscopic renormalized energy we need some notation. Recall that the pinning term $a_{\varepsilon}:\Omega\to\{b,1\}$ is obtained [see Section 2.3] from a smooth bounded simply connected set $\omega$ s.t. $0\in\omega\subset\overline{\omega}\subset Y:=(-1/2,1/2)^{2}$. The construction of the pinning term uses two parameters $\delta=\delta(\varepsilon)$ [the parameter of period] and $\lambda=\lambda(\varepsilon)$ [the parameter of dilution]. For $x_{0}\in\omega$ and a sequence $\varepsilon=\varepsilon_{n}\downarrow 0$, we consider $\hat{x}_{\varepsilon}\in\omega$ s.t. $\hat{x}_{\varepsilon}\to x_{0}\in\omega$.

Let $m_{\varepsilon}\in\mathbb{Z}^{2}$ be s.t. the cell $Y_{\varepsilon}=\delta(m_{\varepsilon}+Y)$ satisfies $\overline{Y}_{\varepsilon}\subset\Omega$. We then denote $z_{\varepsilon}=\delta[m_{\varepsilon}+\lambda\hat{x}_{\varepsilon}]$. It is proved in [7] [see Estimates (9) and (10)] that for $R=R_{\varepsilon}\gg\lambda\delta$ and $r=r_{\varepsilon}\ll\lambda\delta$, denoting $\hat{R}=R/(\lambda\delta)$, $\hat{r}=r/(\lambda\delta)$, $\mathcal{D}_{\varepsilon}=B(\delta m_{\varepsilon},R)\setminus\overline{B(z_{\varepsilon},r)}$, $\hat{\mathcal{D}}_{\varepsilon}=B(0,\hat{R})\setminus\overline{B(\hat{x}_{\varepsilon},\hat{r})}$ and $\hat{\mathcal{D}}=B(0,\hat{R})\setminus\overline{B(x_{0},\hat{r})}$:

[TABLE]

Moreover from the main result in [8], we have the existence of an application $\tilde{W}^{\rm micro}:\omega\to\mathbb{R}$ [depending only on $\omega$ and $b$] s.t.

[TABLE]

where $\displaystyle f_{\omega}(\hat{R}):=\inf_{\begin{subarray}{c}w\in H^{1}[B(0,{\hat{R}})\setminus\overline{\omega},\mathbb{S}^{1}]\\ {\rm deg}(w)=1\end{subarray}}\frac{1}{2}\int_{B(0,\hat{R})\setminus\overline{\omega}}|\nabla w|^{2}$.

It is clear that there exists $C_{\omega}\in\mathbb{R}$ [depending only on $\omega$] s.t. when $\hat{R}\to\infty$ we have $f_{\omega}(\hat{R})=\pi\ln(\hat{R})+C_{\omega}$.

Then, by denoting $W^{\rm micro}(x_{0}):=\tilde{W}^{\rm micro}(x_{0})+C_{\omega}$, we get from (69) :

[TABLE]

Moreover, from [9] we know that $W^{\rm micro}$ admits minimizers in $\omega$.

7. Sharp upper bound: construction of a test function

From now on we assume that Hypothesis (5) holds. We thus may use for $p\in\Lambda$ and $D\in\mathbb{N}^{*}$ the constant $C_{p,D}$ defined in (65). We denote also $C_{p,0}:=0$.

We let for $d\in\mathbb{N}^{*}$ :

[TABLE]

[TABLE]

where, for $x\in\mathbb{R}$, $\lceil x\rceil$ is the ceiling of $x$, $\lfloor x\rfloor$ is the floor of $x$, $W^{\rm macro}(\cdot)$ is defined in Proposition 31 and $\tilde{V}[\zeta_{({\bf p},{\bf D})}]$ is defined in Proposition 17.

We now state an easy lemma whose proof is left to the reader.

Lemma 37**.**

Let $d\in\mathbb{N}^{*}$ and ${\bf D}\in\Lambda_{d}$. Then the following quantities are independent of ${\bf D}$:

[TABLE]

[TABLE]

Moreover: $d\leq N_{0}\Longleftrightarrow\mathscr{L}_{1}(d)=0\Longleftrightarrow\mathscr{L}_{2}(d)=\overline{\mathcal{W}}_{d}$.

Notation 38*.*

We let $\mathscr{L}_{1}(0)=\mathscr{L}_{2}(0)=0$.

The main result of this section is the following proposition.

Proposition 39**.**

Assume that $h_{\rm ex}=\mathcal{O}(|\ln\varepsilon|)$, $h_{\rm ex}\to+\infty$,

[TABLE]

and assume that Hypothesis (5) holds.

Let $d\in\mathbb{N}^{*}$ and let ${\bf D}\in\Lambda_{d}$ be a minimizer of the minimizing problem (72).

For $0<\varepsilon<1$, there exists $(v_{\varepsilon},A_{\varepsilon})\in\mathscr{H}$ which is in the Coulomb gauge with $d$ vortices of degree $1$ s.t.

[TABLE]

with $M_{\Omega}:=2\pi\|\xi_{0}\|_{L^{\infty}(\Omega)}$ and

[TABLE]

where

[TABLE]

$\gamma$* is a universal constant defined in Lemma IX.1 [4] and $W^{\rm micro}$ is defined in Section 6.3. *

Proposition 39 is proved in Appendix E.

8. Tool box

The proof of the main theorems of this article is done in a classic way: by matching upper and lower bounds. A [sharp] upper bound is obtained by Proposition 39. Getting a sharp lower bound is the most challenging part of the proof. It needs the proof of several facts related with the vorticity defects of a family of quasi-minimizers [quantization, localization, size …].

In this section we present some technical and quite classical results adapted to our situation.

8.1. An $\eta$-ellipticity property

In this section we focus on quasi-minimizers. We let $h_{\rm ex}=\mathcal{O}(|\ln\varepsilon|)$ and we consider $\{(v_{\varepsilon},A_{\varepsilon})\,|\,0<\varepsilon<1\}$ be a family of quasi-minimizers for $\mathcal{F}$, i.e.,

[TABLE]

We assume that for all $\varepsilon\in(0;1)$, $(v_{\varepsilon},A_{\varepsilon})$ is in the Coulomb gauge and that $v_{\varepsilon}\in H^{1}(\Omega,\mathbb{C})$ is s.t.

[TABLE]

The major result of this section is a key tool in this article: an $\eta$ ellipticity property.

Proposition 40**.**

Let $h_{\rm ex}=\mathcal{O}(|\ln\varepsilon|)$ and let $\{(v_{\varepsilon},A_{\varepsilon})\,|\,0<\varepsilon<1\}\subset\mathscr{H}$ be a family in the Coulomb gauge satisfying (77) and (78).

For $\eta\in(0,1)$ there exist $\varepsilon_{\eta}>0$ and $C_{\eta}>0$ [depending on the bound of $\varepsilon\|\nabla|v_{\varepsilon}|\|_{L^{\infty}(\Omega)}$] s.t. for $0<\varepsilon<\varepsilon_{\eta}$, if $z\in\Omega$ is s.t.

[TABLE]

then $|v_{\varepsilon}(z)|>\eta$.

Proposition 40 is proved in Appendix F.

By combining Proposition 40 with Theorem 5 we get immediately a first step in the [macroscopic] localization of the vorticity defects. In order to apply Theorem 5 we need assume

[TABLE]

Corollary 41**.**

Assume that $\lambda,\delta$ and $h_{\rm ex}$ satisfy (79) and let $\{(v_{\varepsilon},A_{\varepsilon})\,|\,0<\varepsilon<1\}\subset\mathscr{H}$ be s.t. (77) and (78) hold. There exist $0<\varepsilon_{0}\leq\varepsilon_{K}$ and $M\geq 1$ s.t. for $0<\varepsilon<\varepsilon_{0}$, letting $\tilde{\Lambda}_{\varepsilon}:=\Lambda\cap\cup_{d_{i}\neq 0}B(a_{i},2\mathcal{M}_{K}|\ln\varepsilon|^{-s_{0}})$ where the $(a_{i},d_{i})$’s [depend on $\varepsilon$] are given by Proposition 10 and $\varepsilon_{K}\&\mathcal{M}_{K}\&s_{0}$ are given by Theorem 5, we have

[TABLE]

Proof.

We argue by contradiction and we assume that there exist $\varepsilon=\varepsilon_{n}\downarrow 0$ and a sequence $((v_{\varepsilon},A_{\varepsilon}))_{\varepsilon}\subset\mathscr{H}$ s.t. (77) and (78) hold and s.t. for all $n$ there exists

[TABLE]

Since (77) and (78) are gauge invariant we may assume that, for all $\varepsilon$, $(v_{\varepsilon},A_{\varepsilon})$ is in the Coulomb gauge.

Let $\mathcal{B}:=\{(B(a_{i},r_{i}),d_{i})\,|\,i\in\mathcal{J}\}$ be given by Proposition 10. Write $B_{i}:=B(a_{i},r_{i})$ for $i\in\mathcal{J}$. Note that by Theorem 5, from the quasi-minimality of $(v_{\varepsilon},A_{\varepsilon})$, for $\varepsilon$ sufficiently small, we have $d_{i}\geq 0$ for all $i$ and $d:=\sum|d_{i}|=\sum d_{i}=\mathcal{O}(1)$. Up to pass to a subsequence, we may thus assume that $d$ is independent of $\varepsilon$.

From the definition of $\tilde{\Lambda}_{\varepsilon}$, we have

[TABLE]

Note that from Theorem 5 we have $\mathcal{F}(v_{\varepsilon},0)=\mathcal{O}(|\ln\varepsilon|^{2})$. Then we may use Proposition 10 for the configuration $(v_{\varepsilon},0)\in\mathscr{H}$ to get a covering $\cup_{i\in\tilde{\mathcal{J}}}\tilde{B}_{i}$ of $\{|v_{\varepsilon}|<1-|\ln\varepsilon|^{-2}\}$ with disjoint disks $\tilde{B}_{i}=B(\tilde{a}_{i},\tilde{r}_{i})$, $\sum\tilde{r}_{i}<|\ln\varepsilon|^{-10}$.

Therefore there is $\rho\in[2\mathcal{M}_{K}|\ln\varepsilon|^{-\tilde{s}_{0}};(2\mathcal{M}_{K}+6)|\ln\varepsilon|^{-\tilde{s}_{0}}]$ s.t.

[TABLE]

In particular $|v_{\varepsilon}|\geq 1-|\ln\varepsilon|^{-2}$ on $\bigcup_{p\in\tilde{\Lambda}_{\varepsilon}}\partial B(p,\rho)$. Thus, writing $\tilde{d}_{i}:={\rm deg}_{\partial\tilde{B}_{i}}(v_{\varepsilon})$ when $\tilde{B}_{i}\subset\Omega$, we get for $p\in\tilde{\Lambda}_{\varepsilon}$

[TABLE]

Note that for sufficiently large $n$ we have $B(z_{0},\sqrt{\varepsilon})\cap\bigcup_{p\in\tilde{\Lambda}_{\varepsilon}}B(p,\rho)=\emptyset$.

On the other hand, since $\sum\tilde{r}_{i}<|\ln\varepsilon|^{-10}$, we have for $\tilde{B}_{i}\subset\Omega$

[TABLE]

Using Proposition 40 we obtain

[TABLE]

where $C_{1/2}>0$ is given by Proposition 40 with $\eta=1/2$. Estimate (80) is in contradiction with (49).

∎

8.2. Construction of the $\varepsilon^{s}$-bad discs

As in the previous section we assume that $\lambda,\delta$ and $h_{\rm ex}$ satisfy (79). In this section we establish the existence of $\varepsilon^{s}$-bad discs associated to a quasi-minimizing sequence. The construction of the bad discs requires the hypotheses: $|v_{\varepsilon}|\in W^{2,1}(\Omega)$.

An $\varepsilon^{s}$-bad discs family associated to a familly $\{(v_{\varepsilon},A_{\varepsilon})\,|\,0<\varepsilon<1\}\subset\mathscr{H}$ consists in sets of discs that have small diameters [a roots of $\varepsilon$] s.t. for fix $\varepsilon$ the discs are "well separated", the union of the discs is a covering of $\{|v|\leq 1/2\}$ and each "heart" of a disc intersects $\{|v|\leq 1/2\}$. Such sets of discs give thus a nice visualization of $\{|v|\leq 1/2\}$.

In the next section [Section 9], adding an extra hypothesis on $\lambda,\delta$ and $h_{\rm ex}$ we get some informations in terms of location and quantification of the $\varepsilon^{s}$-bad discs.

Proposition 42**.**

Assume that $\lambda,\delta$ and $h_{\rm ex}$ satisfy (79). There exists $M_{0}\in\mathbb{N}^{*}$ s.t. for $\mu\in(0,1/2)$, if $\{(v_{\varepsilon},A_{\varepsilon})\,|\,0<\varepsilon<1\}$ is in the Coulomb gauge and agrees (17)$\&$(77), then there exist $\varepsilon_{\mu}>0$ and $C_{\mu}\geq 1$ [independent of $\varepsilon$] s.t. for $0<\varepsilon<\varepsilon_{\mu}$, there is $J_{\mu}=J_{\mu,\varepsilon}\subset\{1,...,M_{0}\}$ [possibly empty] s.t. if $J_{\mu}=\emptyset$ then $|v|>1/2$ in $\Omega$ and if $J_{\mu}\neq\emptyset$ then there are $\{z_{i}\,|\,i\in J_{\mu}\}\subset\Omega$, a set of mutually distinct points, and $r\in[\varepsilon^{\mu},\varepsilon^{\mu_{*}}]$ with $\mu_{*}:=2^{-L_{0}^{2}}\mu$ verifying:

1. (1)

$|z_{i}-z_{j}|\geq r^{3/4}$* for $i,j\in J_{\mu}$, $i\neq j$,* 2. (2)

$\{|v_{\varepsilon}|\leq 1/2\}\subset\cup_{J_{\mu}}B(z_{i},r)\subset\Omega$* and, for $i\in J_{\mu}$, $B(z_{i},r/4)\cap\{|v_{\varepsilon}|\leq 1/2\}\neq\emptyset$,* 3. (3)

For $i\in J_{\mu}$ we have $\displaystyle r\int_{\partial B(z_{i},r)}|\nabla v_{\varepsilon}|^{2}+\dfrac{1}{2\varepsilon^{2}}(1-|v_{\varepsilon}|^{2})^{2}\leq C_{\mu}$ and $|v|\geq 1-|\ln\varepsilon|^{-2}$ on $\partial B(z_{i},r)$.

Proposition 42 is proved in Appendix G. We have the following standard estimate.

Proposition 43**.**

Assume (79) and let $\{(v_{\varepsilon},A_{\varepsilon})\,|\,0<\varepsilon<1\}$ be as in Proposition 42. Fix $\mu\in(0,1/2)$ and let $\varepsilon_{\mu}$, $C_{\mu}$ be given by Proposition 42. For $0<\varepsilon<\varepsilon_{\mu}$ we consider $J_{\mu}$, $\{z_{i}\,|\,i\in J_{\mu}\}\subset\Omega$ and $r$ obtained in Proposition 42. We denote $d_{i}:={\rm deg}_{\partial B(z_{i},r)}(v_{\varepsilon})$.

There exists $c_{\mu,b}\geq 1$ independent of $\varepsilon$ s.t. for $\varepsilon<\varepsilon_{\mu}$ we have

[TABLE]

[TABLE]

and then

[TABLE]

Moreover there is $0<\tilde{\varepsilon}_{\mu}\leq\varepsilon_{\mu}$ s.t. for $0<\varepsilon<\tilde{\varepsilon}_{\mu}$ we have

[TABLE]

and

[TABLE]

Proof.

It is classical to get (81) from Proposition 42.3 and the Cauchy Schwartz inequality. Estimate (82) follows from Proposition 42 $\&$ Lemma VI.1 in [2] and (83) is a consequence of (82).

The proof of (84) is done arguing by contradiction with the construction of a comparaison function $\tilde{v}:=\begin{cases}v&\text{in }\Omega\setminus B(z_{i_{0}},r)\\ \tilde{\rho}\e^{\imath\tilde{\phi}}&\text{in }B(z_{i_{0}},r)\end{cases}$ s.t. $\tilde{v}\in H^{1}(\Omega,\mathbb{C})$ and $F(\tilde{v},B(z_{i_{0}},r))=\mathcal{O}(1)$ where we assumed $d_{i_{0}}=0$.

Since $(v,A)$ is a quasi-minimizer of $\mathcal{F}$ we have $\mathcal{F}(v,A)\leq\mathcal{F}(\tilde{v},A)+o(1)$.

On the other hand, by direct calculations $\mathcal{F}(v,A)-\mathcal{F}(\tilde{v},A)=F(v,B(z_{i_{0}},r))-F(\tilde{v},B(z_{i_{0}},r))+o(1)$. Consequently $F(v,B(z_{i_{0}},r))=\mathcal{O}(1)$ which is in contradiction with $F(v,B(z_{i_{0}},r))\geq C_{1/2}|\ln\varepsilon|$ [given by Proposition 40] for small $\varepsilon$.

We now prove (85). From (83) we have $\sum_{J_{\mu}}|d_{i}|\left[\pi(1-\mu)|\ln\varepsilon|-c_{\mu,b}\right]\leq\dfrac{\mathcal{M}_{K}|\ln\varepsilon|}{b^{2}}$. Since $\mu\in(0,1/2)$, the last estimate gives the result for $\varepsilon>0$ sufficiently small.∎

8.3. Lower bounds in perforated disks

The goal of this section is to get lower bounds for $\frac{1}{2}\int_{\mathcal{D}}\alpha|\nabla v|^{2}$ where $\mathcal{D}$ is a perforated disk s.t. $\mathcal{D}\subset\Omega$ and $|v|\geq 1/2$ in $\mathcal{D}$.

The starting point of the argument is an estimate on circles. Let $\tilde{b}\in(0,1)$, $\beta\in L^{\infty}((0,2\pi),[\tilde{b},1])$. With Lemma D.7 in [6], for $\varphi\in H^{1}((0,2\pi),\mathbb{R})$ s.t. $\varphi(2\pi)-\varphi(0)=2\pi$, we have the following lower bound:

[TABLE]

In order to use (86) we need to do a preliminary analysis.

For $\alpha=U_{\varepsilon}^{2}\in L^{\infty}(\Omega,[b^{2},1])$, using Lemma E.1 in [6], we have the existence of $C\geq 1$ [independent of $\varepsilon$] s.t.

[TABLE]

From now on, in all this section, we consider a sequence $\varepsilon=\varepsilon_{n}\downarrow 0$, $\lambda,\delta,h_{\rm ex}$ and $((v_{\varepsilon},A_{\varepsilon}))_{\varepsilon}\subset\mathscr{H}$ satisfying the hypotheses of Proposition 42 [namely (17), (77) and (79)]. We drop the subscript $\varepsilon$ writing $(v,A)$ instead of $(v_{\varepsilon},A_{\varepsilon})$

Recall that $\eta_{\Omega}$ is defined in (63) and consider

[TABLE]

We then denote $\mathscr{R}:=B(x_{\varepsilon},R)\setminus\overline{B(x_{\varepsilon},r)}\subset\Omega$.

Assume $|v|\geq 1/2$ in $\mathscr{R}$ and let $d:={\rm deg}_{\mathscr{R}}(v)$. From the proof of Proposition 42 [see (189) in Appendix G], there exists $1/2<t_{\varepsilon}<1$, $t_{\varepsilon}=1+o(1)$ s.t. $t_{\varepsilon}\in{\rm Im}(|v|)\cap[1-2/|\ln\varepsilon|;1-1/|\ln\varepsilon|]$ and

[TABLE]

and since $\mathcal{H}^{2}(\{|v|\leq t_{\varepsilon}\})=o(1)$ we then have

[TABLE]

Remark 44*.*

Since $\mathcal{H}^{1}[V(t_{\varepsilon})]=o(1)$, for sufficiently small $\varepsilon$, if $\Gamma$ [resp. $U$] is a connected component of $V(t_{\varepsilon})$ [resp. $\{|v|\leq t_{\varepsilon}\}$] which intersects $\mathscr{R}$ then $\Gamma$ is a Jordan curve [resp. $\partial U$ is a union of connected components of $V(t_{\varepsilon})$].

We have the following lemma:

Lemma 45**.**

Assume $x_{\varepsilon},r,R$ satisfy (88) and we assume $|v|\geq 1/2$ in $\mathscr{R}$. Then, for $s\in(r,R)$, letting

[TABLE]

we have

[TABLE]

Proof.

Let $s\in(r,R)$ be s.t. $\mathcal{H}^{1}(K_{s})>0$ and denote $\wideparen{\mathcal{K}}_{s}:=\{x_{\varepsilon}+s\e^{\imath\theta}\,|\,\theta\in K_{s}\}\subset\partial B(x_{\varepsilon},s)$. Then $\mathcal{H}^{1}(\wideparen{\mathcal{K}}_{s})=s\mathcal{H}^{1}(K_{s})$.

On the one hand, letting $\mathcal{V}_{\mathscr{R}}(t_{\varepsilon})$ be the union of the connected components of $\{|v|\leq t_{\varepsilon}\}$ which intersect $\mathscr{R}$, we have $\wideparen{\mathcal{K}}_{s}=\mathcal{V}_{\mathscr{R}}(t_{\varepsilon})\cap\partial B(x_{\varepsilon},s)$.

On the other hand, by Remark 44, $\partial\mathcal{V}_{\mathscr{R}}(t_{\varepsilon})$ is a union of connected components of $V(t_{\varepsilon})$ which are Jordan curves. Among these Jordan curves, we may select the maximal curves w.r.t. the inclusion of their interior. We denote these maximal curves by $\Gamma_{1},...,\Gamma_{N}$ and we let for $i\in\{1,...,N\}$, $\mathcal{V}_{i}:=\overline{{\rm int}(\Gamma_{i})}$. We then obtain $\mathcal{V}_{\mathscr{R}}(t_{\varepsilon})\subset\cup_{i=1}^{N}\mathcal{V}_{i}$ and thus $\wideparen{\mathcal{K}}_{s}\subset\cup_{i=1}^{N}[\partial B(x_{\varepsilon},s)\cap\mathcal{V}_{i}]$.

For $i\in\{1,...,N\}$, we fix $x_{i}\in\mathcal{V}_{i}$ and we define the disk $B_{i}:=\overline{B(x_{i},{\rm diam}(\mathcal{V}_{i}))}$. It is clear that $\mathcal{V}_{i}\subset B_{i}$ . Consequently

[TABLE]

We claim that $2{\rm diam}(\mathcal{V}_{i})\leq\mathcal{H}^{1}(\Gamma_{i})$. Since the curves $\Gamma_{i}$ are pairwise disjoint, we have $\sum_{i=1}^{N}\mathcal{H}^{1}(\Gamma_{i})\leq\mathcal{H}^{1}[V(t_{\varepsilon})]$.

We may now conclude:

[TABLE]

∎

The next proposition is one of the major use of the dilution [$\lambda\to 0$].

Proposition 46**.**

Let $x_{\varepsilon},r,R$ satisfying (88) and assume $|v|\geq 1/2$ in $\mathscr{R}$. We write $d:={\rm deg}_{\mathscr{R}}(v)$ and, in $\mathscr{R}$, we let $w:=v/|v|\,\&\,\rho:=|v|$.

1. (1)

If $r\geq\delta/3$ and if $\mathcal{H}^{1}[V(t_{\varepsilon})]/r+(1-t^{2}_{\varepsilon})+\lambda=o[\ln(R/r)]$ then

[TABLE] 2. (2)

If $r=o(1)$ and if $\mathcal{H}^{1}[V(t_{\varepsilon})]/r+(1-t^{2}_{\varepsilon})=o[\ln(R/r)]$ then

[TABLE]

Proof.

We prove the first assertion. We claim that, up to replace $v$ with $\underline{v}$, we may assume $|v|\leq 1$ in $\Omega$. Moreover, if $d=0$ then there is nothing to prove. We then assume $d\neq 0$.

We write $v=\rho\e^{\imath d\varphi}$ where $\varphi$ is locally defined and its gradient is globally defined. Letting $x_{\varepsilon}+\mathbb{R}^{+}:=\{x_{\varepsilon}+s\,|\,s\geq 0\}$, we may assume $\varphi\in H^{1}(\mathscr{R}\setminus(x_{\varepsilon}+\mathbb{R}^{+}),\mathbb{R})$. For $s\in(r,R)$, we let $\varphi_{s}(\theta)=\varphi(x_{\varepsilon}+s\e^{\imath\theta})$, $\rho_{s}(\theta)=|v(x_{\varepsilon}+s\e^{\imath\theta})|$ and $\alpha_{s}(\theta)=\alpha(x_{\varepsilon}+s\e^{\imath\theta})$. Then $\varphi_{s}\in H^{1}((0,2\pi),\mathbb{R})$ is s.t. $\varphi_{s}(2\pi)-\varphi_{s}(0)=2\pi$ and we immediately get

[TABLE]

From (86) with $\beta:=\alpha_{s}\rho_{s}^{2}$ we get

[TABLE]

Since $b^{2}/4\leq\alpha_{s}\rho_{s}^{2}\leq 1$ we have

[TABLE]

On the one hand, from Lemma 45 we have

[TABLE]

On the other hand, using (87), there is $C\geq 1$ [independent of $\varepsilon$] s.t.$\displaystyle\int_{0}^{2\pi}{1-\alpha_{s}}\leq C\lambda$. Then

[TABLE]

We thus get

[TABLE]

The second assertion is obtain exactly in the same way than the first one. Indeed, since $\alpha$ plays no role in the statement, we may use the same argumentation with $\lambda=0$ and $\delta>0$ an arbitrary small number. ∎

We now state the reformulation of Proposition 46 by replacing the annular $\mathscr{R}$ with a perforated disk.

Corollary 47**.**

Let $D_{0}\in\mathbb{N}^{*}$ be independent of $\varepsilon$, $0<r=r_{\varepsilon}<R=R_{\varepsilon}$ be s.t. $r=o(R)$, $N=N_{\varepsilon}\in\mathbb{N}^{*}$ be s.t. $N\leq D_{0}$ and $z_{1}=z_{1}^{\varepsilon},...,z_{N}=z_{N}^{\varepsilon}$ be s.t. $|z_{i}-z_{j}|\geq 8r$ for $i\neq j$.

Let $y=y_{\varepsilon}\in\Omega$ and assume $z_{1},...,z_{N}\in B(y,R)\subset B(y,4R)\subset B(y,\eta_{\Omega})\subset\Omega$. We let $\mathcal{D}:=B(y,2R)\setminus\cup_{i=1}^{N}\overline{B(z_{i},r)}$.

Assume $\rho=|v|\geq 1/2$ in $\mathcal{D}$. For $i\in\{1,...,N\}$, we let $d_{i}:={\rm deg}_{\partial B(z_{i},r)}(v)$. We also assume $d_{i}>0$ for all $i\in\{1,...,N\}$ and $\sum_{i=1}^{N}d_{i}\leq D_{0}$. Write $v=\rho w$ in $\mathcal{D}$.

Then there exists $C_{0}>0$ depending only on $D_{0}$ s.t. :

1. (1)

If $r\geq\delta/3$ and $\mathcal{H}^{1}[V(t_{\varepsilon})]/r+(1-t^{2}_{\varepsilon})+\lambda=o[\ln(R/r)]$ then, for sufficiently small $\varepsilon$, we have

[TABLE] 2. (2)

If $\mathcal{H}^{1}[V(t_{\varepsilon})]/r+(1-t^{2}_{\varepsilon})=o[\ln(R/r)]$ then, for sufficiently small $\varepsilon$, we have

[TABLE]

Proof.

We claim that, up to replace $v$ with $\underline{v}$, we may assume $|v|\leq 1$ in $\Omega$.

We first proceed to a scaling with the conformal mapping:

[TABLE]

We then let $\hat{z}_{i}:=\Phi(z_{i})$, $\hat{r}:=r/R$, $\hat{\mathcal{D}}:=\Phi[\mathcal{D}]=B(0,2)\setminus\cup_{i=1}^{N}\overline{B(\hat{z}_{i},\hat{r})}$, $\hat{\alpha}:=\alpha\circ\Phi^{-1}$ and $\hat{v}:=v\circ\Phi^{-1}$.

If $N=1$ or $N\geq 2$ and $|\hat{z}_{i}-\hat{z}_{j}|\geq 4\times 10^{-2D_{0}}$ for $i\neq j$ then, letting $\tilde{\Omega}:=B(0,4)$, $\eta_{\tilde{\Omega}}=10^{-1}$, we may apply Proposition 46.1

[TABLE]

This estimate is the desired result with $C_{0}=\pi D_{0}^{2}|\ln(2\times 10^{-2D_{0}})|+1$.

If we are not in the previous case, i.e. $N\geq 2$ and there exists $i\neq j$ s.t. $|\hat{z}_{i}-\hat{z}_{j}|<4\times 10^{-2D_{0}}$, then we apply the separation process presented Appendix C [Section C.3.1] in [6] to the domain $\hat{\mathcal{D}}$ with $\eta_{\rm stop}:=10^{-2D_{0}}$.

The key ingredient in the separation process is a variant of Theorem IV.1 in [4] [stated with $P=9$, the general case $P\in\mathbb{N}\setminus\{0,1\}$ is left to the reader]:

Lemma 48**.**

Let $N\geq 2$, $P\in\mathbb{N}\setminus\{0,1\}$, $x_{1},...,x_{N}\in\mathbb{R}^{2}$ and $\eta>0$. There are $\kappa\in\{P^{0},...,P^{N-1}\}$ and $\emptyset\neq J\subset\{1,...,N\}$ s.t.

[TABLE]

The separation process is an iterative selection of points in $\{\hat{z}_{1},...,\hat{z}_{N}\}$ associated to the construction of a good radius.

We initialize the process by letting $\eta_{0}:=\hat{r}$, $M_{0}:=N$ and $J_{0}=\{1,...,M_{0}\}$.

For $k\geq 1$ [where $k$ is the index in the iterative process] we construct a set $\emptyset\neq J_{k}\subsetneq J_{k-1}$, $M_{k}:={\rm Card}(J_{k})$ and 3 numbers

$\kappa_{k}\in\{9^{1},...,9^{M_{k-1}-1}\}$, $\eta_{k}^{\prime}:=\dfrac{1}{4}\displaystyle\min_{\begin{subarray}{c}i,j\in J_{k-1}\\ i\neq j\end{subarray}}|\hat{z}_{i}-\hat{z}_{j}|$ and $\eta_{k}:=2\kappa_{k}\eta_{k}^{\prime}$.

These objects are obtained with Lemma 48 with $P=9$, $N=M_{k-1}={\rm Card}(J_{k-1})$, $\{x_{1},...,x_{N}\}=\{z_{i}\,|\,i\in J_{k-1}\}$, $J=J_{k}$, $\eta=\eta_{k}$, $\kappa=\kappa_{k}$ The process stops at the end of Step $K_{0}\geq 1$ if $M_{K_{0}}=1$ or $M_{K_{0}}\geq 2$ and $\displaystyle\min_{\begin{subarray}{c}i,j\in J_{K_{0}}\\ i\neq j\end{subarray}}|\hat{z}_{i}-\hat{z}_{j}|>4\eta_{\rm stop}$.

By construction, we have for $1\leq k\leq K_{0}$, $\emptyset\neq J_{k}\subsetneq J_{k-1}$ and $\eta_{k-1}\leq\eta_{k}^{\prime}<\eta_{k}$. In particular, since ${\rm Card}(J_{0})\leq D_{0}$, we get $K_{0}\leq D_{0}-1$.

By definition, for $k\in\{1,...,K_{0}\}$ we have $2\cdot 9\eta_{k}^{\prime}\leq\eta_{k}\leq 9^{D_{0}}\eta_{k}^{\prime}$. We let

[TABLE]

and then $\eta_{0}\geq\eta_{\rm stop}=10^{-2D_{0}}$. For $k\in\{0,...,{K_{0}}-1\}$ and $i\in J_{k}$ we denote $\mathscr{R}_{i,k}:=B(\hat{z}_{i},\eta_{k+1}^{\prime})\setminus\overline{B(\hat{z}_{i},\eta_{k})}$, and, for $i\in J_{K_{0}}$, $\mathscr{R}_{i}:=B(\hat{z}_{i},\eta_{0})\setminus\overline{B(\hat{z}_{i},\eta_{K_{0}})}$. By construction, the previous rings are pairwise disjoint. From Proposition 46.1 we have for $k\in\{0,...,{K_{0}}-1\}$ and $i\in J_{k}$ :

[TABLE]

And for $i\in J_{K_{0}}$:

[TABLE]

By summing the previous lower bound we get the result. As for Proposition 46, the second assertion is obtained in a similar way than the first assertion. ∎

8.4. Lower bounds in a perforated domain

In this section we state a lower bound for a weighted Dirichlet energy in the domain $\Omega$ perforated by small [but not too small] disks. The philosophy of this lower bound is that in the case which interest us we may ignore the weight if the perforations are not too small ; it is an effect of the dilution $\lambda\to 0$.

Proposition 49**.**

Let $\beta\in(0,1)$, $(\tilde{\alpha}_{n})_{n}\subset L^{\infty}(\Omega,[\beta^{2},1])$ be s.t.

[TABLE]

Let $N\in\mathbb{N}^{*}$ and ${\bf(z,d)}={\bf(z,d)}^{(n)}\subset(\Omega^{N})^{*}\times\mathbb{Z}^{N}$ be s.t. ${\bf d}$ is independant of $n$. We denote $\hbar:=\min_{i}{\rm dist}(z_{i},\partial\Omega)$.

Assume the existence of ${\tilde{r}}>0$ s.t. ${\tilde{r}}=o(1)$, (50) holds and s.t. there is $C_{1}>0$ [independent of $n$] satisfying $\dfrac{{\tilde{r}}|\ln{\tilde{r}}|}{\hbar}\leq C_{1}$. Write $\Omega_{{\tilde{r}}}:=\Omega\setminus\cup\overline{B(z_{i},{\tilde{r}})}$.

Let $(u_{n})_{n}\subset H^{1}(\Omega,\mathbb{C})$ satisfying $|u_{n}|\geq\dfrac{1}{2}$ in $\Omega_{\tilde{r}}$ and ${\rm deg}_{\partial B(z_{i},{\tilde{r}})}(u_{n})=d_{i}$ for all $i$.

Assume also

[TABLE]

Then

[TABLE]

with $\Phi^{\bf(z,d)}_{\star}$ is defined in Remark 29 and $X$ is defined in (57).

Proposition 49 is proved Appendix H.

9. Study of the $\varepsilon^{s}$-bad discs

In this section, in addition to the assumption (79) on $\lambda,\delta$ and $h_{\rm ex}$, we assume that (4) holds. This [technical] hypothesis (4) is a little bit more restrictive than (73) [$\delta\sqrt{h_{\rm ex}}\to 0$] used to get a nice upper bound.

Let $\varepsilon=\varepsilon_{n}\downarrow 0$ and let $((v,A))_{\varepsilon}=((v_{\varepsilon},A_{\varepsilon}))_{\varepsilon}$ be a sequence that agrees (17) and (77). Let also $\mu\in(0,1/2)$.

Since (17) and (77) are gauge invariant we may assume that $(v,A)$ is in the Coulomb gauge.

The goal of this section is to prove that, for sufficiently small $\varepsilon\&\mu$, if $J_{\mu}\neq\emptyset$ then $d_{i}=1$ $\&$ ${\rm dist}(z_{i},\Lambda)\leq\ln(h_{\rm ex})/\sqrt{h_{\rm ex}}$ $\&$ $z_{i}\in\omega_{\varepsilon}$ for all $i\in J_{\mu}$ and for $i\neq j$, $|z_{i}-z_{j}|\geq\ln(h_{\rm ex})/h_{\rm ex}$ with a "uniform" distribution of the $z_{i}$’s around $\Lambda$.

With the notation of Proposition 42 we let $\Omega_{r}:=\Omega\setminus\cup_{i\in J_{\mu}}\overline{B(z_{i},r)}$ and $d:=\sum_{i\in J_{\mu}}|d_{i}|$.

In view of the goal of this section we may argue on subsequences. First note that from (84) we have $d_{i}\neq 0$ for all $i$. Up to pass to a subsequence, from (85), we may assume that $J_{\mu}\neq\emptyset$ and independent of $\varepsilon$ as well as the $d_{i}$’s.

Since we are interested here only on informations related with $|v|$ and the $d_{i}$’s, we may consider that $(v,A)$ is in the Coulomb gauge and we may also change the potential vector. Namely, we may assume that $A=\nabla^{\bot}\xi$ with $\xi=\xi_{\varepsilon}\in H^{1}_{0}\cap H^{2}(\Omega,\mathbb{R})$ is the unique solution of (40). Note that (77) still holds.

Consequently, ${\rm curl}(A)\in H^{1}$ and then with (11)$\&$(22): $\|\xi\|_{H^{3}(\Omega)}\leq C\|{\rm curl}(A_{\varepsilon})\|_{H^{1}(\Omega)}\leq C|\ln\varepsilon|$.

From Proposition 11 and letting $\zeta=\zeta_{\varepsilon}:=\xi-h_{\rm ex}\xi_{0}$

[TABLE]

Proposition 17 infers $\tilde{V}_{\bf(z,d)}(\zeta)=\mathcal{O}(1)$. Consequently

[TABLE]

In particular we have $\mathcal{F}(v,\nabla^{\bot}\xi)\leq h_{\rm ex}^{2}{\bf J_{0}}+o(1)$, thus with (91) we get

[TABLE]

From Corollary 41 and Propositions 42$\&$43 we deduce $-\sum d_{i}\xi_{0}(z_{i})=\|\xi_{0}\|_{L^{\infty}(\Omega)}\sum d_{i}+o(1)$ and we immediately obtained

[TABLE]

On the other hand, from Proposition 39, we have

[TABLE]

By combining (91) and (94) we get

[TABLE]

In conclusion, from (82) in conjunction with (95) we obtain

[TABLE]

We first have the following proposition.

Proposition 50**.**

Assume

[TABLE]

where $\mathcal{D}_{K,b}=\dfrac{3\mathcal{M}_{K}}{b^{2}}$ and $\mathcal{M}_{K}$ is as in Theorem 5.

Then there exists $\tilde{\varepsilon}_{\mu}^{\prime}>0$ s.t. for $0<\varepsilon<\tilde{\varepsilon}_{\mu}^{\prime}$ if $J_{\mu}\neq\emptyset$ then

1. (1)

$d_{i}>0$* for all $i$,* 2. (2)

${\rm dist}(z_{i},\omega_{\varepsilon})<\sqrt{\varepsilon}$.

Proof.

**Step 1. We prove that $d_{i}>0$ for all $i$

**

We argue by contradiction and we assume the existence of an extraction still denoted by $\varepsilon=\varepsilon_{n}\downarrow 0$ s.t. $J_{-}:=\{i\in J_{\mu}\,|\,d_{i}<0\}\neq\emptyset$ [from (84), for $0<\varepsilon<\tilde{\varepsilon}_{\mu}$, we have $d_{i}\neq 0$ for all $i\in J_{\mu}$].

From (93) we thus obtain: $\sum_{i\in J_{\mu}\setminus J_{-}}d_{i}\geq d+1$. Then, with the help of (83), we obtain

[TABLE]

Consequently (95) implies $d(1+o(1))\geq(d+2)(1-\mu)-o(1)$. This inequality gives $\mu\geq\dfrac{2}{d+2}-o(1)$ which is in contradiction with $0<\mu<(\mathcal{D}_{K,b}+1)^{-1}$ for sufficiently small $\varepsilon>0$ [here we used $\mathcal{D}_{K,b}\geq\mathcal{M}_{K}\geq d$].

**Step 2. We prove that ${\rm dist}(z_{i},\omega_{\varepsilon})<\sqrt{\varepsilon}$ for all $i$

**

We argue by contradiction and we assume the existence of a subsequence still denoted by $\varepsilon=\varepsilon_{n}\downarrow 0$ and $i_{0}\in J_{\mu}$ s.t. ${\rm dist}(z_{i_{0}},\omega_{\varepsilon})\geq\sqrt{\varepsilon}$. From (25) we have $\inf_{B(z_{i_{0}},r)}\alpha\geq 1-o(|\ln\varepsilon|^{-2})$. Consequently using (83) we get $F(v,B(z_{i_{0}},r))\geq d_{i_{0}}\pi(1-\mu)|\ln\varepsilon|-\mathcal{O}(1)$. Then $F(v)\geq\pi b^{2}(1-\mu)d|\ln\varepsilon|+\pi(1-b^{2})(1-\mu)d_{i_{0}}|\ln\varepsilon|-\mathcal{O}(1)$.

From (95) we obtain

[TABLE]

The last estimate implies $\mu\geq\dfrac{1-b^{2}}{b^{2}d+1-b^{2}}+o(1)$ which is in contradiction with $\mu\leq\dfrac{1-b^{2}}{2(\mathcal{D}_{K,b}+1)}$ for $\varepsilon>0$ sufficiently small. ∎

Definition 51*.*

• •

For $i\in J_{\mu}$we let $y_{i}\in\delta\cdot\mathbb{Z}^{2}$ be the unique point s.t. $z_{i}\in B(y_{i},\delta/2)$. Since ${\rm dist}(z_{i},\omega_{\varepsilon})<\sqrt{\varepsilon}$ for all $i$, $y_{i}$ is well defined.

• •

We denote also $\tilde{J}\subseteq J_{\mu}$ a set of indices s.t. $\cup_{i\in J_{\mu}}B(z_{i},r)\subset\cup_{k\in\tilde{J}}B(y_{k},2\lambda\delta)$ and for $k,l\in\tilde{J}$ s.t. $k\neq l$ we have $y_{k}\neq y_{l}$. We then let for $k\in\tilde{J}$, $\tilde{J}_{k}:=\{i\in J_{\mu}\,|\,z_{i}\in B(y_{k},2\lambda\delta)\}$.

• •

We may also select "good indices" in order to get well separated centers $y_{k}$’s. Using Lemma 48 with $P=17,\eta=\delta$, there exists a set $\emptyset\neq J^{(y)}\subset J_{\mu}$ and a number $\kappa\in\{1,17,...,17^{{\rm Card}(J_{\mu})-1}\}$ [dependent on $\varepsilon$] s.t.

[TABLE]

We denote, for $k\in J^{(y)}$, $\tilde{d}_{k}:={\rm deg}_{\partial B(y_{k},\kappa\delta)}(v)$.

• •

There exists also $\{J_{k}\,|\,k\in J^{(y)}\}$, a partition of $J_{\mu}$ with non empty sets [dependent on $\varepsilon$], s.t.

[TABLE]

We are going to prove that $\tilde{J}=J_{\mu}$ and for all $k\in J^{(y)}$ we have ${J}_{k}=\tilde{J}_{k}$.

Proposition 52**.**

Assume (97), for $\varepsilon>0$ sufficiently small, if $J_{\mu}\neq\emptyset$ then $d_{i}=1$ for all $i\in J_{\mu}$.

Proof.

We argue by contradiction and we assume the existence of a subsequence [still denoted by $\varepsilon=\varepsilon_{n}\downarrow 0$] s.t. for all $\varepsilon$ there exits $i_{0}\in J_{\mu}$ s.t. $d_{i_{0}}\geq 2$.

From Corollary 47.2 applied in $B(y_{k},2\lambda\delta)\setminus\cup_{i\in\tilde{J}_{k}}\overline{B(z_{i},r)}$ :

[TABLE]

We then get $F(v)\geq\pi b^{2}(d|\ln\varepsilon|+|\ln r|)+\mathcal{O}(|\ln(\lambda\delta)|)$. Since $|\ln\varepsilon|=\mathcal{O}(|\ln r|)$ and $|\ln(\lambda\delta)|+\ln h_{\rm ex}=o(|\ln\varepsilon|)$, this estimate is in contradiction with (95) for sufficiently small $\varepsilon$. ∎

Proposition 53**.**

Assume $\mu$ satisfies (97) and $J_{\mu}\neq\emptyset$. Then for sufficiently small $\varepsilon>0$ we have ${\rm dist}({\bf z},\Lambda)\leq\dfrac{\ln h_{\rm ex}}{\sqrt{h_{\rm ex}}}$.

The proof of the proposition uses the following obvious lemma whose proof is left to the reader.

Lemma 54**.**

1. (1)

Let $N\in\mathbb{N}^{*}$, ${\bf D}\in\mathbb{N}^{N}$ and for $k\in\{1,...,N\}$ let $N_{k}\in\mathbb{N}^{*}$ and ${\bf d}^{(k)}\in\mathbb{N}^{N_{k}}$ be s.t. $D_{k}=\sum_{i}d_{i}^{(k)}$. Then we have

[TABLE]

Moreover the equality holds if and only if for all $k\in\{1,...,N\}$ and for all $i\in\{1,...,N_{k}\}$ we have $d_{i}^{(k)}\in\{0,D_{k}\}$. 2. (2)

Let $N,d\in\mathbb{N}^{*}$ and denote $\displaystyle E_{d}:=\min_{{{\bf D}\in\mathbb{N}^{N},\,\sum D_{k}=d}}\sum_{k=1}^{N}D_{k}^{2}$. Then we have for ${\bf D}\in\mathbb{N}^{N}$ s.t. $\sum D_{k}=d$:

[TABLE]

Proof of Proposition 53.

We argue by contradiction and we assume the existence of a subsequence [still denoted by $\varepsilon=\varepsilon_{n}\downarrow 0$] and $i_{0}\in J_{\mu}$ s.t. ${\rm dist}(z_{i_{0}},\Lambda)>\dfrac{\ln h_{\rm ex}}{\sqrt{h_{\rm ex}}}$.

Then there exists $\eta>0$ [independent of $\varepsilon$] s.t. $h_{\rm ex}\xi_{0}(z_{i_{0}})\geq-h_{\rm ex}\|\xi_{0}\|_{L^{\infty}(\Omega)}+4\eta(\ln h_{\rm ex})^{2}$. Consequently: $-2\pi h_{\rm ex}\sum\xi_{0}(z_{i})\leq 2\pi dh_{\rm ex}\|\xi_{0}\|_{L^{\infty}(\Omega)}-4\eta(\ln h_{\rm ex})^{2}.$

From (92) we get [for small $\varepsilon$]

[TABLE]

Using (82) we get

[TABLE]

We let $\chi:=10\max_{k\in\tilde{J}}{\rm dist}(y_{k},\Lambda)$ and for $p\in\Lambda$, $D_{p}:={\rm deg}_{\partial B(p,\chi)}(v)$, $J_{p}:=\{k\in J^{(y)}\,|\,y_{k}\in B(p,\chi)\}$. For a latter use we claim that $\chi\geq\ln(h_{\rm ex})/\sqrt{h_{\rm ex}}$ and then

[TABLE]

We have [see Definition 51 for notation]

[TABLE]

It is clear that, for $k\in\tilde{J}$, we may use Corollary 47.2 in $B(y_{k},2\lambda\delta)\setminus\cup_{i\in\tilde{J}_{k}}\overline{B(z_{i},r)}$ in order to get

[TABLE]

Let $k\in\tilde{J}$, from (25) and Proposition 46.2 we obtain

[TABLE]

Let $p\in\Lambda$ be s.t. $D_{p}\neq 0$, Corollary 47.1 gives

[TABLE]

From Propositions 30$\&$31$\&$49 [with (99)] we deduce

[TABLE]

From Lemma 54.1 we have $d\leq\sum_{k\in\tilde{J}}{\rm deg}_{\partial B(y_{k},2\lambda\delta)}(v)^{2}\leq\sum_{p\in\Lambda}\sum_{k\in J_{p}}\tilde{d}_{k}^{2}\leq\sum_{p\in\Lambda}D_{p}^{2}$. Then we get

[TABLE]

This estimate is in contradiction with (98) for sufficiently small $\varepsilon$. ∎

Proposition 55**.**

Assume $\mu$ satisfies (97) and let $\varepsilon=\varepsilon_{n}\downarrow 0$ be a sequence.

1. (1)

If ${\rm Card}(J_{\mu})\geq 2$ then for $\varepsilon>0$ sufficiently small and for $i\neq j$, $|z_{i}-z_{j}|\geq h_{\rm ex}^{-1}\ln h_{\rm ex}$. 2. (2)

For $\varepsilon>0$ sufficiently small we have for $p\in\Lambda$, ${\rm deg}_{\partial B(p,h_{\rm ex}^{-1/2}\ln h_{\rm ex})}(v)\in\{\lfloor d/N_{0}\rfloor;\lceil d/N_{0}\rceil\}$.

The proof of Proposition 55 is postponed to Appendix I.

Since $\lambda\delta h_{\rm ex}\to 0$, Proposition 55 implies that each cell of period contains at most a disc $B(z_{i},r)$ with $i\in J_{\mu}$.

Following the argument in [6] [proof of the third part in Proposition 3.6, see Appendix D-Section 4.5], we may refined Proposition 50.2.

Proposition 56**.**

Assume $\mu$ satisfies (97), then there is $\eta_{\omega,b}>0$ depending only on $\omega$ and $b$ s.t. for $i\in J_{\mu}$ we have $B(z_{i},2\eta_{\omega,b}\lambda\delta)\subset\omega_{\varepsilon}$.

Corollary 57**.**

Assume $\mu$ satisfies (97). Then we have

[TABLE]

Moreover

[TABLE]

Proof.

We have

[TABLE]

For $i\in J_{\mu}$, from Corollary 46.2 :

[TABLE]

Since, by Proposition 55, the discs $B(z_{i},\lambda^{2}\delta^{2})$ are pairwise disjoint, we obtain with (95):

[TABLE]

This estimate is equivalent to (103).

We are going to prove (104). We argue by contradiction and we assume the existence of an extraction still denoted $\varepsilon=\varepsilon_{n}\downarrow 0$, $t\in(0,1)$ and $(x_{n})_{n}\subset\Omega\setminus\cup_{i\in J_{\mu}}B(z_{i},2\lambda^{2}\delta^{2})$ s.t. $|v_{\varepsilon_{n}}(x_{n})|<t$.

By Proposition 40, there exists $C_{t}>0$ s.t. for sufficiently large $n$:

[TABLE]

Moreover, for $n$ sufficiently large to get $\sqrt{\varepsilon}_{n}<\lambda^{2}\delta^{2}$, we have $[B(x_{n},\sqrt{\varepsilon}_{n})\cap\Omega]\subset\Omega\setminus\cup_{i\in J_{\mu}}B(z_{i},\lambda^{2}\delta^{2})$. This inclusion is in contradiction with (103) and (105). ∎

From Proposition 56, for $i\in J_{\mu}$, we have $\hat{z}_{i}:=\dfrac{z_{i}-y_{i}}{\lambda\delta}\in\omega$ where $y_{i}\in\delta\mathbb{Z}^{2}$ is s.t. $z_{i}\in B(y_{i},\lambda\delta)$. Moreover, up to consider an extraction, we may assume that, for $i\in J_{\mu}$, there exits $\hat{z}^{0}_{i}\in\omega$ s.t. $\hat{z}_{i}\to\hat{z}_{i}^{0}$.

We start with the following proposition.

Proposition 58**.**

We have the following sharp lower bound:

[TABLE]

where $\overline{\mathcal{W}}_{d}=\min_{\Lambda_{d}}\mathcal{W}_{d}$ is defined in (72) and

[TABLE]

where for $p\in\Lambda$, $D\in\mathbb{N}^{*}$, $C_{p,D}$ is defined in (65), $C_{p,0}:=0$ and $\tilde{V}[\zeta_{({\bf p},{\bf D})}]$ is defined in Proposition 17.

We split the proof of Proposition 58 in several lemmas.

The first step is the following lemma consisting in a "macroscopic/mesoscopic" version of Proposition 58.

Lemma 59**.**

Let $\rho=|v|$ and $w=v/\rho$ in $\Omega\setminus\cup_{i\in J_{\mu}}\overline{B(y_{i},\delta/3)}$ . We then have

[TABLE]

Proof.

On the one hand, from Proposition 53 and letting $\chi:=h_{\rm ex}^{-1/4}$ we have $|v|\geq 1/2$ in $\Omega\setminus\cup_{p\in\Lambda}\overline{B(p,\chi)}$. Then, from Proposition 49, we have

[TABLE]

On the other hand, from Proposition 55, if ${\rm Card}(J_{\mu})\geq 2$ then, for $i,j\in J_{\mu}$ with $i\neq j$, we have $|y_{i}-y_{j}|\geq h_{\rm ex}^{-1}\ln(h_{\rm ex})-2\lambda\delta$.

Consequently, if $D_{p}={\rm deg}_{\partial B(p,\eta_{\Omega})}(v)\neq 0$ [$\eta_{\Omega}$ is defined in (63)], letting $J_{p}:=\{i\in J_{\mu}\,|\,z_{i}\in B(p,\eta_{\Omega})\}$, $\mathcal{D}_{p}:=B(p,\chi)\setminus\cup_{i\in J_{p}}\overline{B(y_{i},h_{\rm ex}^{-1})}$,

[TABLE]

$\hat{v}=v\circ\Phi^{-1}$, $\hat{\alpha}=\alpha\circ\Phi^{-1}$, $\hat{\mathcal{D}}_{p}:=\Phi(\mathcal{D}_{p})$ and $\hat{y}_{i}:=\Phi(y_{i})$ for $y_{i}\in B(p,\chi)$, then we may apply Proposition 49. Writing $(\hat{\bf y}_{p},{\bf 1}):=\{(\hat{y}_{i},1)\,|\,i\in J_{p}\}$, Proposition 49 gives:

[TABLE]

where $W^{\rm macro}_{D_{p},\mathbb{D}}$ is the macroscopic renormalized energy in the unit disc $\mathbb{D}$ with $D_{p}$ points.

From Proposition 1 in [13] we have

[TABLE]

Using Proposition 53, we get for $i\in J_{p}$, $|\hat{y}_{i}|\leq\dfrac{h_{\rm ex}^{-1/2}\ln h_{\rm ex}}{\chi}=o(1)$ and then

[TABLE]

For $i\in J_{\mu}$, we let $\mathscr{R}_{i}:=B(y_{i},h_{\rm ex}^{-1})\setminus\overline{B(y_{i},\delta/3)}$. With Proposition 46.1 we obtain

[TABLE]

By combining (107), (108), (109) and (110) the result is proved. ∎

The second step is a "microscopic" version of Proposition 58.

Lemma 60**.**

If $r\leq{\tilde{r}}\leq\lambda^{2}\delta^{2}$, then :

[TABLE]

where, for $i\in J_{\mu}$, $\mathscr{R}_{i}:=B(y_{i},\delta/3)\setminus\overline{B(z_{i},{\tilde{r}})}$.

Proof.

We first note that in order to prove Lemma 60 [up to replace $v$ by $\underline{v}$] we may assume $\rho=|v|\leq 1$. We may also assume

[TABLE]

since in the contrary case there is nothing to prove.

Fix $i\in J_{\mu}$ and let $v_{\star}$ be a minimizer of $\displaystyle F_{\varepsilon}(\cdot,\mathscr{R}_{i})$ in $H^{1}(\mathscr{R}_{i},\mathbb{C})$ with the Dirichlet boundary condition ${\rm tr}_{\partial\mathscr{R}_{i}}(\cdot)={\rm tr}_{\partial\mathscr{R}_{i}}(v)$. Note that such minimizers exist and we have $F_{\varepsilon}(v_{\star},\mathscr{R}_{i})\leq F_{\varepsilon}(v,\mathscr{R}_{i})=\mathcal{O}(|\ln(\lambda\delta)|)$.

The key ingredient consists in noting that since $v_{\star}$ is a minimizer of a weighted Ginzburg-Landau type energy we may thus use a sharp interior $\eta$-ellipticity result. Namely, following the strategy of [9] to prove Lemma 1 [see Appendix C in [9]], by using the first part of the proof [the interior argument which does not required any information on ${\rm tr}_{\partial\mathscr{R}_{i}}(v_{\star})$], we get

[TABLE]

Write in $\tilde{\mathscr{R}}_{i}$: $v_{\star}=\rho_{\star}w_{\star}$ where $w_{\star}\in H^{1}(\tilde{\mathscr{R}},\mathbb{S}^{1})$.

Note that by (2) [namely $|\ln(\lambda\delta)|=\mathcal{O}(\ln|\ln\varepsilon|)$] we have $|\ln(\lambda\delta)|^{3}/|\ln\varepsilon|=o(1)$ and then from (111) $\&$ (112) [and aslo $\rho_{\star}\leq 1$] we have

[TABLE]

We then immediately get:

[TABLE]

It suffices now to claim that from (70) we have

[TABLE]

in order to get $F(v,\mathscr{R}_{i})\geq\pi\left(|\ln(3\lambda)|+b^{2}|\ln(\lambda\delta/{\tilde{r}})|\right)+W^{\rm micro}(\hat{z}_{i}^{0})+o(1)$. By summing these lower bounds we get the result. ∎

Lemma 61**.**

There exits $r\leq{\tilde{r}}=o(\lambda^{2}\delta^{2})$ s.t. for $i\in J_{\mu}$ we have

[TABLE]

Proof.

We first note that we have

[TABLE]

The above estimate is proved by contradiction and assuming the existences of an extraction [still denoted by $\varepsilon=\varepsilon_{n}\downarrow 0$] and of a sequence $R_{n}\uparrow\infty$ s.t.

[TABLE]

From (83) we get

[TABLE]

Using Lemmas 59 and 60 we get an estimate which contradicts (95).

By a classical argument, for sufficiently small $\varepsilon$, there exists $\sqrt{r}\leq{\tilde{r}}\leq r^{1/4}$ s.t. for $i\in J_{\mu}$

[TABLE]

Arguing as in the proof of Proposition 42 [Step 3 in Appendix G] it is clear that we may assume $|v|\geq 1-|\ln\varepsilon|^{-2}$ on $\partial B(z_{i},{\tilde{r}})$ for $i\in J_{\mu}$.

We now define for $i\in J_{\mu}$, $\rho_{i}:={\rm tr}_{\partial B(z_{i},{\tilde{r}})}(|v|)$, $w_{i}:={\rm tr}_{\partial B(z_{i},{\tilde{r}})}(v/|v|)$. We immediately get

[TABLE]

On the other hand, since ${\rm deg}(w_{i})=1$, there exists $\phi_{i}=\phi_{i,\varepsilon}\in H^{1}((0,2\pi),\mathbb{R})$ s.t. $\phi_{i}(0)=\phi_{i}(2\pi)\in[0,2\pi)$ and $w_{i}\left(z_{i}+{\tilde{r}}\e^{\imath\theta}\right)=\e^{-\imath(\theta+\phi_{i}(\theta))}$. A direct calculation gives:

[TABLE]

The last equalities imply $\phi_{i}^{\prime}\to 0$ in $L^{2}(0,2\pi)$ and then $\phi_{i}-\phi_{i}(0)\to 0$ in $L^{2}(0,2\pi)$. Hence, up to pass to a subsequence, we get the existence of $\theta_{i}\in[0,2\pi]$ s.t. $\phi_{i}\to\theta_{i}$ in $H^{1}(0,2\pi)$.

We now define $\tilde{w}_{i}\in H^{1}(B(z_{i},2{\tilde{r}})\setminus\overline{B(z_{i},{\tilde{r}})},\mathbb{S}^{1})$ by

[TABLE]

A direct calculation gives $\int_{B(z_{i},2{\tilde{r}})\setminus\overline{B(z_{i},{\tilde{r}})}}|\nabla\tilde{\phi}_{i}|^{2}=o(1)$ and then

[TABLE]

Let $\tilde{\rho}_{i}\in H^{1}[B(z_{i},2{\tilde{r}})\setminus\overline{B(z_{i},{\tilde{r}})},\mathbb{R}^{+}]$ be s.t. $\tilde{\rho}_{i}(z_{i}+s\e^{\imath\theta}):=\tilde{\rho}_{i}(z_{i}+{\tilde{r}}\e^{\imath\theta})\dfrac{2{\tilde{r}}-s}{{\tilde{r}}}+\dfrac{s-{\tilde{r}}}{{\tilde{r}}}$.

We then have $F[\tilde{\rho}_{i},B(z_{i},2{\tilde{r}})\setminus\overline{B(z_{i},{\tilde{r}})}]=o(1)$. Consequently, letting $v_{i}:=\tilde{\rho}_{i}\tilde{w}_{i}\in H^{1}[B(z_{i},2{\tilde{r}})\setminus\overline{B(z_{i},{\tilde{r}})},\mathbb{C}]$ we have

[TABLE]

In order to conclude we let $u_{i}:=\begin{cases}v_{i}&\text{in }B(z_{i},2{\tilde{r}})\setminus\overline{B(z_{i},{\tilde{r}})}\\ v&\text{in }B(z_{i},{\tilde{r}})\end{cases}$.

It is clear that $u_{i}(z_{i}+2{\tilde{r}}\e^{\imath\theta})=\e^{\imath\theta_{i}}\e^{\imath\theta}$ and then, using Lemma IX.1 in [4], we get

[TABLE]

The last estimate ends the proof of the lemma. ∎

Proof of Proposition 58.

From the three previous lemmas we have

[TABLE]

On the other hand, with Corollary 24 [estimate (47)] we get

[TABLE]

where $\zeta_{({\bf p},{\bf D})}$ is defined in Proposition 16.

From Proposition 36 [estimate (64)], for $p\in\Lambda$ s.t. $D_{p}\geq 2$, we have:

[TABLE]

By combining (114), (115) and (116) [and also $\xi_{0}\leq 0$] we obtain

[TABLE]

It suffices to see that, since ${\bf D}\in\Lambda_{d}$, from the definition of $\mathscr{L}_{1}(d)$ we have

[TABLE]

in order to deduce from (117) that

[TABLE]

where $\mathcal{W}_{d}({\bf D})$ is defined in (106). This estimate with the definition of $H^{0}_{c_{1}}$ and $\overline{\mathcal{W}}_{d}$ [see (72)$\&$(75)$\&$(76)] ends the proof of the proposition. ∎

10. The first critical field and the location of the vorticity defects

We assume that $\lambda,\delta,h_{\rm ex}$ satisfy (2) and (3) for some $K\geq 0$ independent of $\varepsilon$. We assume also (4). We consider a sequence $\varepsilon=\varepsilon_{n}\downarrow 0$.

As in the previous section we focus on sequences of quasi-minimizers of $\mathcal{F}$. For simplicity we write $(v,A)$ instead of $(v_{\varepsilon},A_{\varepsilon})$. We assume that (17)$\&$(77) holds and since (17)$\&$(77) are gauge invariant we may also assume that $(v,A)$ is in the Coulomb gauge.

From above results, for a fixed $\mu>0$ sufficiently small [satisfying (97)] and for $\varepsilon>0$ sufficiently small, there exists a [finite] set $\mathcal{Z}\subset\Omega$, depending on $\varepsilon$ and possibly empty s.t. letting $d:={\rm Card}(\mathcal{Z})$ [we write $\mathcal{Z}=\{z_{1},...,z_{2}\}$]:

• •

If $d=0$, then $|v|\geq 1/2$ in $\Omega$.

• •

If $d>0$, then $|z_{i}-z_{j}|\gtrsim h_{\rm ex}^{-1}\ln h_{\rm ex}$ if $i\neq j$, $|v|\geq 1/2$ in $\Omega\setminus\cup_{i=1}^{d}\overline{B(z_{i},\varepsilon^{\mu})}$ and ${\rm deg}_{\partial B(z,\varepsilon^{\mu})}(v)=1$ for $z\in\mathcal{Z}$.

Moreover $d=\mathcal{O}(1)$. Then if needed, up to pass to a subsequence, we may assume that $d$ is independent of $\varepsilon$.

By combining Corollary 14, Propositions 36, 39, 55 and 58 we get the following corollary.

Corollary 62**.**

Assume $\lambda,\delta,h_{\rm ex}$ satisfy (2) and (3) for some $K\geq 0$ independent of $\varepsilon$. Let $\varepsilon=\varepsilon_{n}\downarrow 0$ and and let $((v_{\varepsilon},A_{\varepsilon}))_{\varepsilon}\subset\mathscr{H}$ be a sequence satisfying (17)$\&$(77). Assume that $d$ is independent of $\varepsilon$. Without loss of generality we may assume that $(v_{\varepsilon},A_{\varepsilon})$ is in the Coulomb gauge. We have

[TABLE]

Moreover, if $d\neq 0$ then:

• •

We have ${\bf D}\in\Lambda_{d}$ [see (71)] and ${\bf D}$ minimises $\mathcal{W}_{d}$ in $\Lambda_{d}$ where $\mathcal{W}_{d}$ is defined in (106).

• •

For $p\in\Lambda$ s.t. $D_{p}>0$ and $i\in J_{p}$, we denote $\breve{z}_{i}:=(z_{i}-p)\sqrt{D_{p}/h_{\rm ex}}$ and $\breve{\bf z}_{p}:=\{\breve{z}_{i}\,|\,i\in J_{p}\}$. Then, up to pass to a subsequence, $\breve{\bf z}_{p}$ converges to a minimizer of $W^{\rm meso}_{p,D_{p}}$ defined in (66).

• •

For $i\in\{1,...,d\}$, we write $\hat{z}_{i}:=(z_{i}-y_{i})/(\lambda\delta)\in\omega$ where $y_{i}\in\delta\mathbb{Z}^{2}$ is s.t. $z_{i}\in B(y_{i},\lambda\delta)$. Then, up to pass to a subsequence, $\hat{z}_{i}$ converges to a minimizer of $W^{\rm micro}$.

For a further use, we claim that for $d_{0}\geq 0$, from Proposition 39, there exits a configuration $(v^{0},A^{0})\in\mathscr{H}$ which is in the Coulomb gauge s.t.

[TABLE]

Recall that, from Lemma 37, for $d\neq 0$, we have $d\in\{1,...,N_{0}\}$ if and only if $\mathscr{L}_{1}(d)=0$ and $\mathscr{L}_{2}(d)=\overline{\mathcal{W}}_{d}$. For further use we state another lemma whose proof is left to the reader:

Lemma 63**.**

For $0\leq d<d^{\prime}$ we let :

1. (1)

$\displaystyle\Delta^{(1)}_{d}:=\dfrac{\mathscr{L}_{1}(d+1)-\mathscr{L}_{1}(d)}{M_{\Omega}}=\dfrac{\pi}{M_{\Omega}}\left\lfloor\dfrac{d}{N_{0}}\right\rfloor$. 2. (2)

$\displaystyle\Delta^{(1)}_{d^{\prime},d}:=\dfrac{\mathscr{L}_{1}(d^{\prime})-\mathscr{L}_{1}(d)}{M_{\Omega}(d^{\prime}-d)}=\dfrac{\pi}{M_{\Omega}(d^{\prime}-d)}\sum_{k=d}^{d^{\prime}-1}\left\lfloor\dfrac{k}{N_{0}}\right\rfloor$. 3. (3)

$\displaystyle\Delta^{(2)}_{d}:=\dfrac{\mathscr{L}_{2}(d+1)-\mathscr{L}_{2}(d)}{M_{\Omega}}$* and $\displaystyle\Delta^{(2)}_{d}-\dfrac{\overline{\mathcal{W}}_{d+1}-\overline{\mathcal{W}}_{d}}{M_{\Omega}}=$*

[TABLE] 4. (4)

$\displaystyle\Delta^{(2)}_{d^{\prime},d}:=\dfrac{\mathscr{L}_{2}(d^{\prime})-\mathscr{L}_{2}(d)}{M_{\Omega}(d^{\prime}-d)}$* thus, if $d^{\prime}\leq N_{0}$, then $\Delta^{(2)}_{d^{\prime},d}=\dfrac{\overline{\mathcal{W}}_{d^{\prime}}-\overline{\mathcal{W}}_{d}}{M_{\Omega}(d^{\prime}-d)}$.*

By using (118) and (119) we easily get the following corollary.

Corollary 64**.**

Let $\varepsilon=\varepsilon_{n}\downarrow 0$, $\lambda$, $\delta$, $h_{\rm ex}$ and $((v_{\varepsilon},A_{\varepsilon}))_{\varepsilon}\subset\mathscr{H}$ be as in Corollary 62.

Assume that $d$ is independent of $\varepsilon$. Then we have for $d^{\prime}>d$

[TABLE]

*Then, letting $\chi$ be s.t. $h_{\rm ex}=H^{0}_{c_{1}}(1+\chi)$ [$\chi=o(1)$ from (3)], we have thus *

[TABLE]

If $d>d^{\prime}\geq 0$ then

[TABLE]

We are now in position to give an asymptotic value for the first critical field. Indeed with Corollary 64 [(120) with $d=0\&d^{\prime}\in\{1,...,N_{0}\}$ and (121) with $d\geq 1\&d^{\prime}=0$].

Recall that we write, for $x\in\mathbb{R}$, $[x]^{+}=\max(x,0)$ and $[x]^{-}=\min(x,0)$

Corollary 65**.**

Denote $H_{c_{1}}:=H^{0}_{c_{1}}+\min_{d\in\{1,...,N_{0}\}}{\dfrac{\overline{\mathcal{W}}_{d}}{dM_{\Omega}}}$. Let $\{(v_{\varepsilon},A_{\varepsilon})\,|\,0<\varepsilon<1\}\subset\mathscr{H}$ be a family of quasi-minimizers satisfying (17).

1. (1)

If for sufficiently small $\varepsilon$ we have $d=0$ then $[h_{\rm ex}-H_{c_{1}}]^{+}\to 0$. 2. (2)

If for sufficiently small $\varepsilon$ we have $d>0$ then $[h_{\rm ex}-H_{c_{1}}]^{-}\to 0$.

Proof.

The corollary is a direct consequence of Corollary 64 taking $d^{\prime}\in\{1,...,N_{0}\}$ which minimizes $\Delta^{(2)}_{d^{\prime},0}=W_{d^{\prime}}/(M_{\Omega}d^{\prime})$ in (120) for the first assertion and $d^{\prime}=0$ in (121) for the second. ∎

10.1. Secondary critical fields for $d\in\{1,...,N_{0}\}$

If $N_{0}=1$, if $h_{\rm ex}$ is near $H_{c_{1}}$ and if $d>0$, then it is standard to prove that $d=1$. If $N_{0}\geq 2$ and $d\in\{1,...,N_{0}\}$, then the situation is more involved: we have no a priori sharp informations about the number of vorticity defects and their [macroscopic] location. The goal of this section is to get such informations.

10.1.1. Preliminaries

Note that for $0\leq d<d^{\prime}\leq N_{0}$ we have $\Delta^{(1)}_{d^{\prime},d}=0$ and $\Delta^{(2)}_{d^{\prime},d}=\dfrac{\overline{\mathcal{W}}_{d^{\prime}}-\overline{\mathcal{W}}_{d}}{M_{\Omega}(d^{\prime}-d)}$.

Rephrasing Corollary 64 for $d,d^{\prime}\in\{0,...,N_{0}\}$ we have the following key lemma.

Lemma 66**.**

Let $\varepsilon=\varepsilon_{n}\downarrow 0$, $\lambda$, $\delta$, $h_{\rm ex}$ and $((v_{\varepsilon},A_{\varepsilon}))_{\varepsilon}\subset\mathscr{H}$ be as in Corollary 62.

Assume ${\rm Card}(\mathcal{Z})=d$ is independent of $\varepsilon$ then the following properties hold:

1. (1)

If $0\leq d^{\prime}<d$ then, letting $\overline{\mathcal{W}}_{0}:=0$, we have $h_{\rm ex}\geq H^{0}_{c_{1}}+\dfrac{\overline{\mathcal{W}}_{d}-\overline{\mathcal{W}}_{d^{\prime}}}{M_{\Omega}(d-d^{\prime})}+o(1)$.

In particular taking $d^{\prime}=0$ we get $h_{\rm ex}\geq H^{0}_{c_{1}}+\dfrac{\overline{\mathcal{W}}_{d}}{M_{\Omega}d}+o(1)$. 2. (2)

If $d<N_{0}$ and $d<d^{\prime}\leq N_{0}$ then $h_{\rm ex}\leq H^{0}_{c_{1}}+\dfrac{\overline{\mathcal{W}}_{d^{\prime}}-\overline{\mathcal{W}}_{d}}{M_{\Omega}(d^{\prime}-d)}+o(1)$. 3. (3)

If $N_{0}\geq 2$, $N_{0}\geq d^{\prime}>d\geq 1$ then

[TABLE] 4. (4)

If $N_{0}\geq 2$ and $N_{0}\geq d^{\prime}>d\geq 1$ then

[TABLE] 5. (5)

If $N_{0}\geq 2$ and $0\leq d<d^{\prime}<d^{\prime\prime}\leq N_{0}$ then we have the following convex combination

[TABLE]

Consequenlty $\dfrac{\overline{\mathcal{W}}_{d^{\prime\prime}}-\overline{\mathcal{W}}_{d}}{d^{\prime\prime}-d}$ is between $\dfrac{\overline{\mathcal{W}}_{d^{\prime\prime}}-\overline{\mathcal{W}}_{d^{\prime}}}{d^{\prime\prime}-d^{\prime}}$ and $\dfrac{\overline{\mathcal{W}}_{d^{\prime}}-\overline{\mathcal{W}}_{d}}{d^{\prime}-d}$.

Proof.

The two first assertions are obtained with Corollary 64. The remaining part of the lemma consists in basic calculations. ∎

10.1.2. First step in the definition of the critical fields

Assume $N_{0}\geq 2$. We are going to define some energetic levels [in term of $\overline{\mathcal{W}}_{d}$] related with the number of vorticity defects and their [macroscopic] location.

We denote $d^{\star}_{0}:=0$, $\mathscr{S}_{1}:=\{1,...,N_{0}\}$, $\mathscr{K}_{1}^{\star}:=\min_{d\in\mathscr{S}_{1}}\dfrac{\overline{\mathcal{W}}_{d}}{d}=\min_{d\in\mathscr{S}_{1}}\dfrac{\overline{\mathcal{W}}_{d}-\overline{\mathcal{W}}_{d_{0}^{\star}}}{d-d_{0}^{\star}}$, $\mathscr{S}^{\star}_{1}:=\{d\in\mathscr{S}_{1}\,|\,{\overline{\mathcal{W}}_{d}}/{d}=\mathscr{K}_{1}^{\star}\}$ and $\mathscr{D}_{1}:=\{\,{\bf D}\in\Lambda_{d}\,|\,d\in\mathscr{S}^{\star}_{1}\text{ and }{\bf D}\text{ minimizes }\mathcal{W}_{d}\}$. We let also $d_{1}^{\star}:=\max\mathscr{S}^{\star}_{1}$ and ${\mathscr{D}}_{1}^{\star}:={\mathscr{D}}_{1}\cap\tilde{\Lambda}_{d_{1}^{\star}}$.

If $d_{1}^{\star}=N_{0}$ we are going to prove that for $h_{\rm ex}\geq H_{c_{1}}+o(1)$ [but $h_{\rm ex}$ not too large], then there is exactly one vorticity defect close to each point of $\Lambda$. In the contrary case [$1\leq d_{1}^{\star}<N_{0}$], then there are other critical fields which govern the number of vorticity defects.

If $d_{1}^{\star}<N_{0}$, then $\mathscr{S}_{2}:=\{d_{1}^{\star}+1,...,N_{0}\}\neq\emptyset$. For $d\in\mathscr{S}_{2}$ we let $\mathscr{K}_{2}(d):=\dfrac{\overline{\mathcal{W}}_{d}-\overline{\mathcal{W}}_{d^{\star}_{1}}}{d-d^{\star}_{1}}$, $\mathscr{S}_{2}^{\star}:=\left\{d\in\mathscr{S}_{2}\,|\,\mathscr{K}_{2}(d)=\min_{\mathscr{S}_{2}}\mathscr{K}_{2}\right\}$, $d_{2}^{\star}:=\max\mathscr{S}_{2}^{\star}$ and $\mathscr{K}_{2}^{\star}:=\mathscr{K}_{2}(d_{2}^{\star})$.

We denote $\mathscr{D}_{2}:=\{{\bf D}\in\Lambda_{d}\,|\,d\in\mathscr{S}^{\star}_{2}\text{ and }{\bf D}\text{ minimizes }\mathcal{W}_{d}\}$ and ${\mathscr{D}}_{2}^{\star}:=\mathscr{D}_{2}\cap\Lambda_{d_{2}^{\star}}$.

We claim that for $d\in\mathscr{S}_{2}$ we have $\overline{\mathcal{W}}_{d}/d>\overline{\mathcal{W}}_{d_{1}^{\star}}/d_{1}^{\star}$. Then, with Lemma 66.3, we get $\mathscr{K}_{2}(d)>\overline{\mathcal{W}}_{d_{1}^{\star}}/d_{1}^{\star}$. In particular

[TABLE]

If $d_{2}^{\star}=N_{0}$ then we stop the construction. In the contrary case, for $d\in\mathscr{S}_{3}:=\{d_{2}^{\star}+1,...,N_{0}\}\neq\emptyset$ we have $\mathscr{K}_{2}(d)>\mathscr{K}_{2}(d_{2}^{\star})$.

We continue the iterative construction. For $k\geq 2$, assume that we have $1<d_{k-1}^{\star}<d_{k}^{\star}<N_{0}$, we let $\mathscr{S}_{k+1}:=\{d_{k}^{\star}+1,...,N_{0}\}\neq\emptyset$ and we assume that for $d\in\mathscr{S}_{k+1}$:

[TABLE]

For $d\in\mathscr{S}_{k+1}$ we let $\mathscr{K}_{k+1}(d):=\dfrac{\overline{\mathcal{W}}_{d}-\overline{\mathcal{W}}_{d^{\star}_{k}}}{d-d^{\star}_{k}}$,

[TABLE]

$d_{k+1}^{\star}:=\max\mathscr{S}_{k+1}^{\star}$ and $\mathscr{K}_{k+1}^{\star}:=\mathscr{K}_{k+1}(d_{k+1}^{\star})$.

We define also

[TABLE]

From (123) we have

[TABLE]

Then, from Lemma 66.5 with $d=d_{k-1}^{\star}$, $d^{\prime}=d_{k}^{\star}$ and $d^{\prime\prime}=d_{k+1}^{\star}$, we get that $\mathscr{K}_{k}(d_{k+1}^{\star})$ is between $\mathscr{K}_{k}^{\star}$ and $\mathscr{K}_{k+1}^{\star}$. Consequently, with (124) we get

[TABLE]

We stop the construction at Step $L$ s.t. $d_{L}^{\star}=N_{0}$. Since $1\leq d_{k}^{\star}<d_{k+1}^{\star}\leq N_{0}$, it is clear that a such $L$ exists and $1\leq L\leq N_{0}$.

We then have two possibilities: $L=1$ or $L\in\{2,...,N_{0}\}$. If $L\geq 2$ then, for $k\in\{1,...,L-1\}$, (125) holds. We also claim that $(1,...,1)\in\mathscr{D}_{L}$.

Lemma 67**.**

Let $k\in\{1,...,L\}$, assume that $d_{k}^{\star}-d_{k-1}^{\star}\geq 2$ and fix $d_{k-1}^{\star}<d<d_{k}^{\star}$. We have

[TABLE]

Moreover, if $d\notin\mathscr{S}_{k}^{\star}$, then

[TABLE]

Proof.

From Lemma 66.5, $\mathscr{K}_{k}^{\star}$ is between $\dfrac{\overline{\mathcal{W}}_{d}-\overline{\mathcal{W}}_{d^{\star}_{k-1}}}{d-d^{\star}_{k-1}}$ and $\dfrac{\overline{\mathcal{W}}_{d_{k}^{\star}}-\overline{\mathcal{W}}_{d}}{d_{k}^{\star}-d}$. On the other hand, from the definition of $d_{k}^{\star}$, $\mathscr{K}_{k}^{\star}\leq\dfrac{\overline{\mathcal{W}}_{d}-\overline{\mathcal{W}}_{d^{\star}_{k-1}}}{d-d^{\star}_{k-1}}$. Clearly the first part of the lemma holds. If $d\notin\mathscr{S}_{k}^{\star}$ then, by definition, $\mathscr{K}_{k}^{\star}<\dfrac{\overline{\mathcal{W}}_{d}-\overline{\mathcal{W}}_{d^{\star}_{k-1}}}{d-d^{\star}_{k-1}}$. ∎

10.1.3. Main result

For $k\in\{1,...,L\}$ we let

[TABLE]

and we let also

[TABLE]

Recall that the $\mathscr{K}_{k}^{\star}$’s are defined in Section 10.1.2 and $\Delta^{(1)}_{N_{0}}\&\Delta^{(2)}_{N_{0}}$ in Lemma 63. Note that $H_{c_{1}}={\tt K}^{\tt(I)}_{1}$.

Proposition 68**.**

Assume that (5) holds and $\lambda,\delta,h_{\rm ex},K$ satisfy (2), (3) and (4).

Let $\{(v_{\varepsilon},A_{\varepsilon})\,|\,0<\varepsilon<1\}\subset\mathscr{H}$ be a family satisfying (17)$\&$(77) which is in the Coulomb gauge. Assume $d_{\varepsilon}={\rm Card}(\mathcal{Z}_{\varepsilon})\in\{1,...,N_{0}\}$.

We denote ${\bf D}=(D_{1},...,D_{N_{0}})$ with $D_{l}={\rm deg}_{\partial B(p_{l},\eta_{\Omega})}(v)$ [$\eta_{\Omega}$ is defined in (63)].

1. (1)

Assume $L=1$. For sufficiently small $\varepsilon>0$ we have ${\bf D}\in\mathscr{D}_{1}$.

Moreover, if $\varepsilon=\varepsilon_{n}\downarrow 0$ is a sequence s.t. $d_{\varepsilon}$ is independent of $\varepsilon$ and $d_{\varepsilon}\neq N_{0}$ [i.e. ${\bf D}\neq(1,...,1)$] then $\left[h_{\rm ex}-{\tt K}^{\tt(I)}_{1}\right]^{+}\to 0$. 2. (2)

Assume $L\geq 2$. For $k\in\{1,...,L-1\}$, if $d^{\star}_{k-1}<d_{\varepsilon}\leq d_{k}^{\star}$ for small $\varepsilon$ or for a sequence indexed by $\varepsilon=\varepsilon_{n}\downarrow 0$, then

[TABLE]

Moreover, for sufficiently small $\varepsilon$, ${\bf D}\in\mathscr{D}_{k}$. And if ${\bf D}\in\mathscr{D}_{k}\setminus\mathscr{D}^{\star}_{k}$ [i.e. $d^{\star}_{k-1}<d_{\varepsilon}<d_{k}^{\star}$] then

[TABLE] 3. (3)

If $d^{\star}_{L-1}<d_{\varepsilon}\leq d_{L}^{\star}=N_{0}$ for small $\varepsilon$ or for a sequence indexed by $\varepsilon=\varepsilon_{n}\downarrow 0$, then

[TABLE]

Moreover, for sufficiently small $\varepsilon$, ${\bf D}\in\mathscr{D}_{L}$. And if $d_{\varepsilon}<N_{0}$ [i.e ${\bf D}\neq(1,...,1)$] then

[TABLE]

In particular, for sufficiently small $\varepsilon$, we have ${\bf D}\in\cup_{l=1}^{L}\mathscr{D}_{l}$.

Proof.

We prove the first item arguing by contradiction. First note that if $N_{0}=1$ then there is nothing to prove. Assume thus $N_{0}\geq 2\,\&\,L=1$ and let $\{(v_{\varepsilon},A_{\varepsilon})\,|\,0<\varepsilon<1\}$ be as in the proposition. Assume there exists $\varepsilon=\varepsilon_{n}\downarrow 0$ s.t. ${\bf D}\notin\mathscr{D}_{1}$. Up to pass to a subsequence we may assume that ${\bf D}$ is independent of $\varepsilon$.

From Corollary 62, for sufficiently small $\varepsilon$, ${\bf D}$ minimizes $\mathcal{W}_{d}$ and then, from the definition of $\mathscr{D}_{1}$, we get $d\notin\mathscr{S}^{\star}_{1}$. Consequently $\overline{\mathcal{W}}_{N_{0}}/N_{0}<\overline{\mathcal{W}}_{d}/d$ and thus, from Lemma 66.2$\&$66.3 [with $d^{\prime}=N_{0}$], we get the existence of $t>0$ s.t. $h_{\rm ex}\leq H_{c_{1}}-t$. This last estimate is in contradiction with Corollary 65.2. Thus ${\bf D}\in\mathscr{D}_{1}$ for sufficiently small $\varepsilon$. The rest of the first assertion is a direct consequence of $d\in\mathscr{S}^{\star}_{1}\setminus\{N_{0}\}$ and Lemma 66.2$\&$66.4 [with $d^{\prime}=N_{0}$].

We now prove the second assertion. Assume $L\geq 2$. For $k\in\{1,...,L-1\}$, if $d^{\star}_{k-1}<d\leq d_{k}^{\star}$, then, from Lemma 66.1 [with $d^{\prime}=d_{k-1}^{\star}$] and Lemma 66.2 [with $d^{\prime}=d_{k+1}^{\star}$], we get

[TABLE]

From the definition of $d_{k}^{\star}$ we have ${\mathscr{K}_{k}^{\star}}{}\leq\dfrac{\overline{\mathcal{W}}_{d}-\overline{\mathcal{W}}_{d_{k-1}^{\star}}}{d-d_{k-1}^{\star}}$ and then the lower bound in (132) gives the first convergence in (128).

On the other hand, if $d=d_{k}^{\star}$ then, from the definition of $\mathscr{K}_{k+1}^{\star}$, the upper bound in (132) gives the second convergence in (128).

If $d\neq d^{\star}_{k}$, using Lemma 66.5 [with $d<d_{k}^{\star}<d_{k+1}^{\star}$] we obtain that $\dfrac{\overline{\mathcal{W}}_{d_{k+1}^{\star}}-\overline{\mathcal{W}}_{d}}{d_{k+1}^{\star}-d}$ is between $\dfrac{\overline{\mathcal{W}}_{d_{k}^{\star}}-\overline{\mathcal{W}}_{d}}{d_{k}^{\star}-d}$ and $\mathscr{K}_{k+1}^{\star}$. But, from Lemma 67, we get $\dfrac{\overline{\mathcal{W}}_{d_{k}^{\star}}-\overline{\mathcal{W}}_{d}}{d_{k}^{\star}-d}\leq\mathscr{K}_{k}^{\star}$. Since from (125) we have $\mathscr{K}_{k+1}^{\star}>\mathscr{K}_{k}^{\star}$, we obtain $\dfrac{\overline{\mathcal{W}}_{d_{k+1}^{\star}}-\overline{\mathcal{W}}_{d}}{d_{k+1}^{\star}-d}\leq\mathscr{K}^{\star}_{k+1}$. Therefore the upper bound of (132) gives the second convergence in (128).

We now demonstrate that, for sufficiently small $\varepsilon$, ${\bf D}\in\mathscr{D}_{k}$ arguing by contradiction. We assume the existence of sequence $\varepsilon=\varepsilon_{n}\downarrow 0$ s.t. $d_{k-1}^{\star}<d\leq d_{k}^{\star}$ with $k\in\{1,...,L-1\}$, ${\bf D}$ is independent of $\varepsilon$ and ${\bf D}\notin\mathscr{D}_{k}$. From Corollary 62, ${\bf D}$ minimizes $\mathcal{W}_{d}$ and then, from the definition of $\mathscr{D}_{k}$, we get $d\notin\mathscr{S}^{\star}_{k}$ [then $d<d_{k}^{\star}$].

On the one hand, with Lemma 66.1 [with $d^{\prime}=d_{k-1}^{\star}$] and Lemma 66.2 [with $d^{\prime}=d_{k}^{\star}$] we have

[TABLE]

On the other hand, with Lemma 67, we have $\dfrac{\overline{\mathcal{W}}_{d}-\overline{\mathcal{W}}_{d^{\star}_{k}}}{d-d^{\star}_{k}}<\dfrac{\overline{\mathcal{W}}_{d}-\overline{\mathcal{W}}_{d_{k-1}^{\star}}}{d-d_{k-1}^{\star}}$. This inequality gives a contradiction.

Lemma 66.2 [with $d^{\prime}=d_{k}^{\star}$] and Lemma 67 give immediately (129).

We now treat the last item of the proposition and we assume $d_{L-1}^{\star}<d\leq d_{L}^{\star}=N_{0}$. From (121) [with $d^{\prime}=d_{L-1}^{\star}$] we get $h_{\rm ex}-H_{c_{1}}^{0}\geq\Delta^{(2)}_{d,d_{L-1}^{\star}}+o(1)$. On the other hand, from the definition of $\mathscr{K}_{L}^{\star}$, we get

[TABLE]

Before ending the proof of (130) we prove that (131) holds and, for sufficiently small $\varepsilon$, ${\bf D}\in\mathscr{D}_{L}$. Assume that there exists $\varepsilon=\varepsilon_{n}\downarrow 0$ s.t. ${\bf D}$ is independent of $\varepsilon$ and $d_{L-1}^{\star}<d<N_{0}$.

From Lemma 66.2 [with $d^{\prime}=N_{0}$] we have

[TABLE]

Using (133) with (134) we get ${\mathscr{K}_{L}^{\star}}{}\leq{(\overline{\mathcal{W}}_{N_{0}}-\overline{\mathcal{W}}_{d})}/(N_{0}-d)$. Lemma 67 [with $d_{L-1}^{\star}<d<N_{0}$] gives $(\overline{\mathcal{W}}_{N_{0}}-\overline{\mathcal{W}}_{d})/(N_{0}-d)\leq\mathscr{K}_{L}^{\star}$. Therefore, $(\overline{\mathcal{W}}_{N_{0}}-\overline{\mathcal{W}}_{d})/(N_{0}-d)=\mathscr{K}_{L}^{\star}$ and then by combining (133) and (134) we deduce that, if for some sequence $\varepsilon=\varepsilon_{n}\downarrow 0$ we have $d_{L-1}^{\star}<d<N_{0}$, then (131) holds.

Arguing as above, [using (119) with $d_{0}=N_{0}$], one may prove that for sufficiently small $\varepsilon$ we have $d\in\mathscr{S}_{L}^{\star}$ and thus ${\bf D}\in\mathscr{D}_{L}$.

We complete the proof of (130). Assume that $h_{\rm ex}$ is sufficiently large in order to have $d=N_{0}$ [here we used (131)]. It suffices to use (120) [with $d=N_{0}$ and $d^{\prime}=N_{0}+1$] in order to get the remaining part of (130).

∎

10.2. Secondary critical fields for $d\geq N_{0}+1$

The case $d\geq N_{0}+1$ is easier to handle than the case $1\leq d\leq N_{0}$.

For $k\in\mathbb{N}^{*}$, we let

[TABLE]

where $\Delta^{(1)}_{N_{0}+k}\&\Delta^{(2)}_{N_{0}+k}$ are defined in Lemma 63. We have the following proposition.

Proposition 69**.**

Assume that (5) holds and $\lambda,\delta,h_{\rm ex},K$ satisfy (2), (3) and (4).

Let $\{(v_{\varepsilon},A_{\varepsilon})\,|\,0<\varepsilon<1\}\subset\mathscr{H}$ be a family satisfying (17)$\&$(77) which is in the Coulomb gauge.

Let $k\in\mathbb{N}^{*}$. If for a sequence $\varepsilon=\varepsilon_{n}\downarrow 0$ we have $d_{\varepsilon}=N_{0}+k$ then

[TABLE]

Proof.

The proposition is a direct consequence of (120) [with $d=N_{0}+k$ and $d^{\prime}=N_{0}+k+1$] and (121) [with $d=N_{0}+k$ and $d^{\prime}=N_{0}+k-1$].

∎

Appendix A Proof of Estimate (26)

Consider a conformal mapping $\Phi:\mathbb{D}\to\Omega$. From a result of Painlevé [see Footnote 4 page 4], the maps $\Phi$ and $\Phi^{-1}$ may be extended in $\overline{\Omega}$ and $\overline{\mathbb{D}}$ by smooth maps. Then there exists $C_{\star}\geq 1$ s.t.

[TABLE]

Write $\tilde{a}_{\varepsilon}:=a_{\varepsilon}\circ\Phi$ and $\tilde{U}_{\varepsilon}:=U_{\varepsilon}\circ\Phi$. Since the function $\tilde{U}_{\varepsilon}$ is a minimizers of $\tilde{E}_{\varepsilon}$, the analog of $E_{\varepsilon}$ in $\mathbb{D}$, $\tilde{U}_{\varepsilon}$ is a solution of

[TABLE]

with $w={\rm Jac}\,\Phi$ is the Jacobian of $\Phi$.

Define $V_{\varepsilon}:B(0,2)\to[b^{2},1]$ by

[TABLE]

Then $-\Delta V_{\varepsilon}=-\Delta\tilde{U}_{\varepsilon}$ in $\mathbb{D}$ and $-\Delta V_{\varepsilon}(x)=-|x|^{-4}\Delta\tilde{U}_{\varepsilon}({x}/{|x|^{2}})$ in $B(0,2)\setminus\mathbb{D}$. Thus $V_{\varepsilon}\in H^{2}(B(0,2),\mathbb{C})$.

First note that if $r\leq\varepsilon$, then (26) is given by (24).

Let $r>\varepsilon$ and $x_{0}\in\Omega$ be s.t. ${\rm dist}(x_{0},\partial\omega_{\varepsilon})>r$. Let $\eta:=a_{\varepsilon}(x_{0})-V_{\varepsilon}$ in $B(x_{0},r/2)$. From Lemma A.1 in [3] and (25) we get for $x\in B(x_{0},r/4)$ :

[TABLE]

In the previous estimate the constants are independent of $\varepsilon,r$ and $x_{0}$. From (135) we then get (26).

Appendix B Proof of Theorem 5

Assume that (5) holds and $\lambda,\delta,h_{\rm ex},K$ satisfy (2), (3) and $\delta^{2}|\ln\varepsilon|\leq 1$.

Consider a family of configurations $\{(v_{\varepsilon},A_{\varepsilon})\,|\,0<\varepsilon<1\}\subset\mathscr{H}$ which is in the Coulomb gauge and s.t.

[TABLE]

We drop the subscript $\varepsilon$. From Lemma 12, we may consider $A_{v}\in H^{1}(\Omega,\mathbb{R}^{2})$ s.t. $(v,A_{v})$ is in the Coulomb gauge and (39) holds.

We then have

[TABLE]

Proposition 10 gives the existence of $C,\varepsilon_{0}>0$ [independent of $\varepsilon$] s.t., for $\varepsilon<\varepsilon_{0}$, there exists a family of disjoint disks $\{B_{i}\,|\,i\in\mathcal{J}\}$ with $B_{i}=B(a_{i},r_{i})$ satisfying :

1. (1)

$\{|v|<1-|\ln\varepsilon|^{-2}\}\subset\cup B_{i}$ 2. (2)

$\sum r_{i}<|\ln\varepsilon|^{-10}$, 3. (3)

writing $\rho=|v|$ and $v=\rho\e^{\imath\varphi}$ we have

[TABLE]

where $d_{i}={\rm deg}_{\partial B_{i}}(v)$ if $B_{i}\subset\Omega$ and [math] otherwise.

From now on, the notation $C$ stands for a positive constant independent of $\varepsilon$ whose value may change from one line to another.

B.1. A substitution lemma

As in [15], we first state a substitution lemma.

Lemma 70**.**

There exists $(\tilde{v},\tilde{A})\in\mathscr{H}$ which is in the Coulomb gauge and s.t., writing, $\rho=|v|,\,v=\rho\e^{\imath\varphi}$ and $\tilde{\rho}=|\tilde{v}|,\,\tilde{v}=\tilde{\rho}\e^{\imath\tilde{\varphi}}$ we have

1. (1)

$(\tilde{v},\tilde{A})$* satisfies (39) and $\tilde{\rho}\leq 1$,* 2. (2)

$\tilde{\rho}=1$* and $\varphi=\tilde{\varphi}$ in $\Omega\setminus\cup B_{i}$,* 3. (3)

$\|\rho(\nabla\varphi-A_{v})-\tilde{\rho}(\nabla\tilde{\varphi}-\tilde{A})\|_{L^{2}(\Omega)}^{2}=o(1)$, 4. (4)

$\|{\rm curl}(A_{v})-{\rm curl}(\tilde{A})\|_{L^{2}(\Omega)}^{2}\leq C|\ln\varepsilon|^{-2}$, 5. (5)

$\mathcal{F}(\tilde{v},\tilde{A})\leq\mathcal{F}(v,A_{v})+o(1)$.

Lemma 70 is proved in [15] [Lemma 1] for $\alpha\equiv 1$. The adaptation to our case is presented below.

Proof of Lemma 70.

The proof of the lemma follows the same lines than in [15].

We define a continuous function $\chi_{\varepsilon}=\chi:[0,1]\to[0,1]$ by letting

[TABLE]

We then let $\tilde{v}:=\dfrac{\chi(\rho)}{\rho}v\in H^{1}(\Omega,\mathbb{C})$ and we let $\tilde{A}=A_{\tilde{v}}$ given by Lemma 12. Letting $\tilde{h}={\rm curl}(\tilde{A})$ we then get

[TABLE]

Exactly as in [15] we have

[TABLE]

As in [15], from (7), (39) and (137) we obtain PDE of the second order satisfied by $A$ and $\tilde{A}$.

By considering the difference of these PDE we get

[TABLE]

From (20), (136) and (138), the RHS of (139) is bounded in $L^{2}(\Omega)$ by $\dfrac{C}{|\ln\varepsilon|}$.

Since $(\tilde{A}-A)\cdot\nu=0$ on $\partial\Omega$, by elliptic regularity, we deduce Assertions 3$\&$4 of the lemma.

The end of the proof is exactly as in [15] ∎

From now on we replace $(v,A_{v})$ with $(\tilde{v},\tilde{A})$ and we claim that the valued disks given by Proposition 10 is valid for $(v,A_{v})$ and $(\tilde{v},\tilde{A})$ and getting the conclusions of Theorem 5 for $(\tilde{v},\tilde{A})$ implies the same for $(v,A)$.

In order to simplify the presentation we write $(v,A)$ instead of $(\tilde{v},\tilde{A})$.

B.2. Energetic Decomposition

We have the following lower bound:

Proposition 71**.**

Let $h:={\rm curl}(A)$, $h_{0}:=\Delta\xi_{0}=1+\xi_{0}$, $f:=h-h_{\rm ex}h_{0}$ and let $\{B_{i}=B(a_{i},r_{i})\,|\,i\in\mathcal{J}\}$ be the disks given by Proposition 10. We have:

[TABLE]

where

[TABLE]

This estimate is the starting point of the main argument of [15].

Proof of Proposition 71.

Let $\tilde{\Omega}:=\Omega\setminus\cup\overline{B_{i}}$. With (141) we get

[TABLE]

On the other hand, letting $f:=h-h_{\rm ex}h_{0}$ and since $\alpha|\nabla v-\imath Av|^{2}\geq|\nabla h|^{2}$, we get

[TABLE]

Before refining the above lower bound we make some preliminary claims. We first note that from (137) we have $\|h-h_{\rm ex}\|_{H^{1}(\Omega)}^{2}\leq C\|\nabla v-\imath Av\|_{L^{2}(\Omega)}^{2}=\mathcal{O}(h_{\rm ex}^{2})$. Then $\|f\|_{H^{1}(\Omega)}^{2}=\mathcal{O}(h_{\rm ex}^{2})$. Consequently for $g\in\{f,h\}$ we have

[TABLE]

We also observe that

[TABLE]

With (23) we get $\|A\|_{L^{\infty}(\Omega)}\leq Ch_{\rm ex}$ and then [with (137)]

[TABLE]

If $B_{i}\subset\Omega$ we have

[TABLE]

And then

[TABLE]

If $B_{i}\not\subset\Omega$, then $\|h_{0}-1\|_{L^{\infty}(B_{i}\cap\Omega)}\leq Cr_{i}$ and

[TABLE]

By combining (144) with (145) we deduce:

[TABLE]

We used that if $B_{i}\not\subset\Omega$ then $d_{i}=0$.

We end the preliminary claims by noting that

[TABLE]

On the one hand, since $-\Delta f+f=-\Delta h+h$, we have with (142), (143), (146), (147) and integrations by parts:

[TABLE]

On the other hand, since $\|f\|_{L^{4}(\Omega)}\leq Ch_{\rm ex}$, we get $\displaystyle\int_{\cup B_{i}}f^{2}=o(h_{\rm ex}^{-4})$, and this estimate ends the proof.

∎

B.3. Estimate related with the signs of the $d_{i}$’s

By Proposition 71 we have:

[TABLE]

Denote $I_{+}:=\{i\in\mathcal{J}\,|\,d_{i}>0\}$, $I_{-}:=\{i\in\mathcal{J}\,|\,d_{i}<0\}$, $D:=\sum_{\mathcal{J}}|d_{i}|$, $D_{+}:=\sum_{i\in I_{+}}d_{i}$ and $D_{-}:=\sum_{i\in I_{-}}|d_{i}|$.

With (148) we obtain $2h_{\rm ex}D_{+}\|\xi_{0}\|_{L^{\infty}(\Omega)}\geq b^{2}D|\ln\varepsilon|\left(1-\frac{C\ln|\ln\varepsilon|}{|\ln\varepsilon|}\right)+o(1)$ and then:

[TABLE]

B.4. Estimate related with ${\rm dist}(a_{i},\Lambda)$

From Lemma 1, there exist $\eta>0$ and $M\geq 1$ s.t., for $a\in\Omega$, $\xi_{0}(a)\geq\min\xi_{0}+\eta{\rm dist}(a,\Lambda)^{M}$.

We let $I_{0}:=\{i\in I\,|\,{\rm dist}(a_{i},\Lambda)<|\ln\varepsilon|^{-\frac{1}{2M}}\}$ and $D_{0}:=\sum_{i\in I_{0}}|d_{i}|$.

If $i\notin I_{0}$, then $|\xi_{0}(a_{i})|\leq\|\xi_{0}\|_{L^{\infty}(\Omega)}-\dfrac{\eta}{\sqrt{|\ln\varepsilon|}}$. We thus have

[TABLE]

From (148) we may deduce

[TABLE]

and consequently

[TABLE]

B.5. Estimate of the two last terms in (148)

We let $t\geq|\ln\varepsilon|^{-\frac{1}{2M}}\geq{|\ln\varepsilon|}^{-1/2}$ and then $t\geq\delta$ since $\delta|\ln\varepsilon|^{1/2}\leq 1$.

On the one hand, from Lemma E.1 in [6], by denoting $\mathscr{C}_{t}$ a circle with radius $t$ we get:

[TABLE]

We assume now that the center of $\mathscr{C}_{t}$ is in $\Lambda$ and $t$ is s.t. $\mathscr{C}_{t}\subset\tilde{\Omega}=\Omega\setminus\cup\overline{B_{i}}$. We denote also $B_{t}$ the disk bounded by $\mathscr{C}_{t}$. On $\mathscr{C}_{t}$ we have $|v|=1$ and then $v=\e^{\imath\varphi}$ with $\varphi$ locally defined.

By direct calculations, we have [with $f=h-h_{\rm ex}h_{0}$, $\nu$ the outward normal unit vector to $\mathscr{C}_{t}$ and $\tau=\nu^{\bot}$ ]:

[TABLE]

On the other hand $\displaystyle\int_{\mathscr{C}_{t}}\alpha^{-1}\partial_{\nu}h_{0}=\int_{B_{t}}h_{0}+\int_{\mathscr{C}_{t}}(\alpha^{-1}-1)\partial_{\nu}h_{0}$. Note that

[TABLE]

Then for $\varepsilon>0$ sufficiently small: $\displaystyle-\int_{\mathscr{C}_{t}}\alpha^{-1}\partial_{\nu}f+\int_{B_{t}}f\geq 2\pi d_{t}-C\lambda h_{\rm ex}t$. Consequently we obtain

[TABLE]

and thus, by denoting $m_{t}:=\displaystyle\int_{\mathscr{C}_{t}}\alpha^{-2}$, we get

[TABLE]

Since $2\pi t\leq m_{t}\leq b^{-4}2\pi t$, for sufficiently $\varepsilon>0$ small we obtain

[TABLE]

Following exactly the argument in [15] we get

[TABLE]

With (148) and $\xi_{0}(a_{i})\leq-\|\xi_{0}\|_{L^{\infty}(\Omega)}$ there are $C_{1},C_{2}>0$ [ independent of $\varepsilon$] s.t.

[TABLE]

This estimate implies $D\leq\dfrac{C_{1}}{C_{2}}$. Therefore with (149) and (150) we get the three first assertion of the theorem.

It remains to get (32) whose proof follows the same lines as in [15] [Section 4].

Appendix C Proof of Proposition 11

Let $C_{0}>1$, $(v_{\varepsilon})_{0<\varepsilon<1}\subset H^{1}(\Omega,\mathbb{C})$, $(h_{\rm ex})_{0<\varepsilon<1}\subset(0,\infty)$ and $(\xi_{\varepsilon})_{0<\varepsilon<1}\subset H^{1}_{0}\cap H^{2}\cap W^{1,\infty}(\Omega,\mathbb{R})$ be s.t. (34) and (35) hold. For simplicity of the presentation we omit the index $\varepsilon$.

Let $\{(B(a_{i},r_{i}),d_{i})\,|\,i\in\mathcal{J}\}$ be as in the proposition and write $B_{i}:=B(a_{i},r_{i})$.

In this proof the letter "$C$" stands for a quantity bounded by a power of $C_{0}$ whose value may differ from one line to another.

We let $A=\nabla^{\bot}\xi$ and $\tilde{\Omega}:=\begin{cases}\Omega\setminus\cup\overline{B_{i}}&\text{if }|v|\not>1/2\text{ in }\Omega\\ \Omega&\text{if }|v|>1/2\text{ in }\Omega\end{cases}$. The heart of the proof consists in estimating the quantity $\int_{\Omega}(v\wedge\nabla v)\cdot A$ in (30).

We first get with the help of (34) and (35) that if $|v|\not>1/2$ in $\Omega$ then $\int_{\cup B_{i}}v\wedge\nabla v\cdot A=o(1)$.

We also claim that, letting $w:=v/|v|$ in $\tilde{\Omega}$: $\int_{\tilde{\Omega}}(v\wedge\nabla v-w\wedge\nabla w)\cdot A=o(1)$.

In particular, if $|v|>1/2$ in $\Omega$ then we have $\int_{\Omega}(v\wedge\nabla v)\cdot A=o(1)$. We thus assume that $|v|\not>1/2$ in $\Omega$.

Then, with an integration by part we get

[TABLE]

For $B_{i}\subset\Omega$ we immediately have :

[TABLE]

We now define $u:=\begin{cases}v&\text{in }\tilde{\Omega}\\ u_{i}&\text{in }B_{i}\cap\Omega\end{cases}$ where $u_{i}$ is the harmonic extension of ${\rm tr}_{\partial(B_{i}\cap\Omega)}(v)$ in $B_{i}\cap\Omega$. By the Dirichlet principle we have for all $i$:

[TABLE]

It is easy to check that $(w\wedge\nabla^{\bot}w)\cdot\nu=|u|^{-2}(u\wedge\nabla^{\bot}u)\cdot\nu$ on $\cup_{i}\partial B_{i}$. For $i\in\mathcal{J}$, we let

[TABLE]

From (35) we get

[TABLE]

Our goal is now to estimate $\int_{\partial(B_{i}\cap\Omega)}f_{i}(w\wedge\nabla^{\bot}w)\cdot\nu$. We first consider the case where $i\in\mathcal{J}$ is s.t. $|u|\geq 1/2$ in $B_{i}\cap\Omega$. In this case we may write in $B_{i}$: $u=|u|\e^{\imath\phi}$ with $\phi\in H^{1}(B_{i},\mathbb{R})$, $\|\phi\|_{H^{1}(B_{i})}\leq C|\ln\varepsilon|$. We then have with (156) and an integration by parts

[TABLE]

We now assume $i\in\mathcal{J}$ is s.t. $|u|\not\geq 1/2$ in $B_{i}\cap\Omega$. By smoothness of $|u_{i}|^{2}\in C^{\infty}(B_{i}\cap\Omega,\mathbb{R})$, there exists $t_{i}\in]1/5,1/4[$, a regular value of $|u_{i}|^{2}$, s.t. $\omega_{i}:=\{|u_{i}|^{2}<t_{i}\}\neq\emptyset$. We denote $D_{i}:=\Omega\cap[B_{i}\setminus\overline{\omega}_{i}$]. Since $|u|^{2}\geq 1/4$ on $\partial B_{i}\cap\Omega$ we have $\partial D_{i}=(\partial B_{i}\cap\Omega)\cup\partial\omega_{i}\cup(\partial\Omega\cap\overline{D_{i}})$.

Letting $W:=\dfrac{u}{|u|}\wedge\nabla^{\bot}\left(\dfrac{u}{|u|}\right)$ we then get

[TABLE]

It is standard to check that ${\rm div}\left(W\right)=0$ in $D_{i}$. Moreover:

[TABLE]

Consequently using (158) we may deduce

[TABLE]

On the other hand, from (156), $\xi=0$ on $\partial\Omega$ and ${\rm div}\left(u\wedge\nabla^{\bot}u\right)=-2\partial_{1}u\wedge\partial_{2}u$ in $B_{i}\cap\Omega$, we get

[TABLE]

We may conclude by using (153), (154), (157), (159) and (160):

[TABLE]

The rest of the proof is exactly the same than in [17].

Appendix D Proof of some results of Section 6.1.1

D.1. Proof of Proposition 30

We use the same notation than in Proposition 30. In this proof, the letter $C$ is a quantity which depends only on $\Omega$, $N$ and $\sum_{i}|d_{i}|$, its value may change from one line to another.

We argue as in [13]. We let $\Phi^{\bf(z,d)}_{\star}\in\cap_{0<p<2}W^{1,p}(\Omega,\mathbb{R})\cap H_{\rm loc}^{1}(\Omega\setminus\{z_{1},...,z_{N}\},\mathbb{R})$ be the unique solution of

[TABLE]

and let $\Phi_{\tilde{r}}\in H^{1}(\Omega_{\tilde{r}},\mathbb{R})$ be the unique solution of

[TABLE]

We then have $\nabla^{\bot}\Phi^{\bf(z,d)}_{\star}=w^{\bf(z,d)}_{\star}\wedge\nabla w^{\bf(z,d)}_{\star}$ and $\nabla^{\bot}\Phi^{\bf(z,d)}_{\tilde{r}}=w^{\bf(z,d)}_{\tilde{r}}\wedge\nabla w^{\bf(z,d)}_{\tilde{r}}$. It is important to note that if $w\in H^{1}(\Omega_{\tilde{r}},\mathbb{S}^{1})$, then $|\nabla w|=|w\wedge\nabla w|$.

We may decompose $\Phi^{\bf(z,d)}_{\star}$ as $\Phi^{\bf(z,d)}_{\star}=\sum_{i}d_{i}\Phi_{z_{i}}$ where, for $z\in\Omega$, $\Phi_{z}$ is the unique solution of

[TABLE]

With a standard pointwise bound for the gradient of an harmonic function [see (2.31) in [10]] we have $\|\nabla\Phi_{z_{i}}\|_{L^{\infty}(\Omega\setminus\overline{B(z_{i},{\tilde{r}})})}\leq C\dfrac{\|\Phi_{z_{i}}\|_{L^{\infty}(\Omega\setminus\overline{B(z_{i},{\tilde{r}}/4)})}}{{\tilde{r}}}$. Thus

[TABLE]

Moreover, it is easy to check that $\Phi_{z_{i}}=\ln|x-z_{i}|+R_{z_{i}}$ where $R_{z_{i}}$ is the harmonic extension of $-\ln|x-z_{i}|_{|\partial\Omega}$. From (162) and by the maximum principle we get for ${\tilde{r}}<\min\left\{[{\rm diam}(\Omega)]^{-1};1/4\right\}$

[TABLE]

which proves (56).

If there is $\eta>0$ s.t. $\hbar>\eta$, then $\|R_{z_{i}}\|_{C^{1}(\Omega)}\leq C_{\eta}$ where $C_{\eta}$ which depends only on $\eta$ and $\Omega$. We thus get $\|\nabla\Phi^{\bf(z,d)}_{\star}\|_{L^{\infty}(\Omega_{\tilde{r}})}\leq\dfrac{\tilde{C}_{\eta}}{{\tilde{r}}}$ [where $\tilde{C}_{\eta}$ depends only on $\eta$, $N$, $\sum|d_{i}|$ and $\Omega$] and this estimates implies (60).

We now define $R_{\bf(z,d)}:=\sum_{i}d_{i}R_{z_{i}}$ in order to have $\Phi^{\bf(z,d)}_{\star}=\sum_{i}d_{i}\ln|x-z_{i}|+R_{\bf(z,d)}$.

From Lemma I.4 in [4] we have

[TABLE]

If $N=1$, then the first term of the RHS in (164) is [math]. Otherwise, as in [17] [Proposition 5.1], we have

[TABLE]

And for $i\in\{1,...,N\}$, by harmonicity of $R_{\bf(z,d)}$, for $0<\rho<\dfrac{\hbar}{2}$ we get

[TABLE]

Then

[TABLE]

We let

[TABLE]

By combining (164), (165) and (167) we get

[TABLE]

From (163) and (169) we immediately get

[TABLE]

On the other hand, for $i\in\{1,...,N\}$, we have [with (166)]

[TABLE]

Using $X$ defined in (57), from (170) and (171), we get

[TABLE]

From (172) we deduce (59) and since $\int_{\partial\Omega}(\varphi_{\star}-\varphi_{\tilde{r}})=0$, with a Poincaré inequality we obtain (58).

D.2. Proof of Proposition 31

Let ${\bf(z,d)}={\bf(z,d)}^{(n)}\in(\Omega^{N})^{*}\times\mathbb{Z}^{N}$ and denote $\hbar:=\min_{i}{\rm dist}(z_{i},\partial\Omega)>0$. Assume that $d_{1},...,d_{N}$ are independent of $n$. Let ${\tilde{r}}={\tilde{r}}_{n}\to 0$ be s.t (50) holds.

In this proof the letter $C$ stands for a quantity which depends only on $\Omega$, $N$, $C_{1}$ and $\sum_{i}|d_{i}|$, its value may change from one line to another.

By Remark 29 and an integration by parts we have

[TABLE]

For $i_{0}\in\{1,...,N\}$, we fix $x_{i_{0}}\in\partial B(z_{i_{0}},{\tilde{r}})$. Then [with $\nabla^{\bot}\Phi^{\bf(z,d)}_{\star}=w^{\bf(z,d)}_{\star}\wedge\nabla w^{\bf(z,d)}_{\star}$]

[TABLE]

On the one hand, arguing as in the proof of (169), we get for $z\in\partial B(z_{i_{0}},{\tilde{r}})$ :

[TABLE]

Then, using (171), we obtain

[TABLE]

On the other hand, for $i_{0}\in\{1,...,N\}$

[TABLE]

and with (166) we get $\displaystyle\left|R_{\bf(z,d)}(x_{i_{0}})-R_{\bf(z,d)}(z_{i_{0}})\right|\leq\dfrac{C(1+|\ln\hbar|){\tilde{r}}}{\hbar}$. We then immediately get:

[TABLE]

With (174), (175) and (176) we may prove that (173) may be rewritten into

[TABLE]

where "$\mathcal{O}(X)$" is quantity bounded by $CX$ with $C$ depending only on $N,\Omega$ and $\sum|d_{i}|$.

D.3. Proof of Proposition 33

Let ${\bf(z,d)}={\bf(z,d)}^{(n)}\in(\Omega^{N})^{*}\times\mathbb{Z}^{N}$, ${\tilde{r}}\downarrow 0$ and $\eta>0$ be as in the proposition.

In this proof the letter $C$ stands for a quantity which depends only on $\Omega$, $N$ and $\sum_{i}|d_{i}|$, its value may change from one line to another.

We first claim that, for $i\neq j$, $B(z_{i},\eta)\cap B(z_{j},\eta)\neq\emptyset$, $B(z_{i},\eta)\subset\Omega$ and $\eta=\chi{\tilde{r}}$ with $\chi\to\infty$. In particular we assume $n$ sufficiently large to have $\eta>{\tilde{r}}$.

Since $\nabla^{\bot}\Phi^{\bf(z,d)}_{\star}=w^{\bf(z,d)}_{\star}\wedge\nabla w^{\bf(z,d)}_{\star}$, for $i_{0}\in\{1,...,N\}$ and $z\in\Omega\setminus\{z_{1},...,z_{N}\}$, we have

[TABLE]

For $j\in\{1,...,N\}$, let $\theta_{j}$ be the main determination of the argument of $\dfrac{z-z_{j}}{|z-z_{j}|}$ and let $\mathcal{R}$ be an harmonic conjugate of $R_{\bf(z,d)}$. In $\Omega\setminus\{z_{1},...,z_{N}\}$ we have

[TABLE]

Then for $z\in B(z_{i_{0}},\eta)\setminus\{z_{i_{0}}\}$ we have $w^{\bf(z,d)}_{\star}(z)=\left(\dfrac{z-z_{i_{0}}}{|z-z_{i_{0}}|}\right)^{d_{i_{0}}}\e^{\imath\varphi_{i_{0}}(z)}$ with $\varphi_{i_{0}}=\sum_{j\neq i_{0}}d_{j}\tilde{\theta}_{j}+\mathcal{R}+{\rm Cte}_{i_{0}}$ where, for $j\neq i_{0}$, $\tilde{\theta}_{j}$ is a determination of the argument of $\dfrac{z-z_{i}}{|z-z_{i}|}$ which is globally defined in $B(z_{i_{0}},\eta)$. Note that $\varphi_{i_{0}}\in H^{1}(B(z_{i_{0}},\eta),\mathbb{R})$.

On the other hand, by direct calculations, we have $\left\|\sum_{j\neq i_{0}}d_{j}\nabla\tilde{\theta}_{j}\right\|_{L^{\infty}(B(z_{i_{0}},\eta))}\leq\dfrac{C}{\eta}$ and, since $R_{\bf(z,d)}$ is harmonic, we also have from the definition of $\mathcal{R}$

[TABLE]

We thus deduce

[TABLE]

We switch to polar coordinates by letting for $i\in\{1,...,N\}$ and $\rho\in]{\tilde{r}},\eta[$, $\tilde{\varphi}_{i}(\rho,\theta):=\varphi_{i}(z_{i}+\rho\e^{\imath\theta})$. We then get, by (177) and a mean value argument, the existence of $\rho_{n}\in]\sqrt{\chi}{\tilde{r}},\eta[$ s.t.

[TABLE]

We let $Z:=\dfrac{1}{\ln\chi}\left[\dfrac{\eta(|\ln(\hbar)|+1)}{\hbar}+1\right]^{2}$ and by assumption we have $Z\to 0$.

We denote, for $i\in\{1,...,N\}$, $m_{i}=\displaystyle\dfrac{1}{2\pi}\int_{0}^{2\pi}\tilde{\varphi}_{i}(\rho_{n},\theta)\,{\rm d}\theta$ in order to have

[TABLE]

We then define $\phi_{i}\in H^{1}(B(z_{i},\rho_{n})\setminus\overline{B(z_{i},{\tilde{r}})},\mathbb{R})$ using polar coordinates:

[TABLE]

For $z_{i}+s\e^{\imath\theta}\in B(z_{i},\rho_{n})\setminus\overline{B(z_{i},{\tilde{r}})}$, we let $\phi_{i}(z_{i}+s\e^{\imath\theta}):=\tilde{\phi}_{i}(s,\theta)$. By standard calculations we get $\displaystyle\int_{B(z_{i},\rho_{n})\setminus\overline{B(z_{i},{\tilde{r}})}}|\nabla\phi_{i}|^{2}\leq CZ$.

We conclude by defining $v=\begin{cases}w^{\bf(z,d)}_{\star}&\text{in }\Omega\setminus\cup\overline{B(z_{i},\rho_{n})}\\ u_{i}\e^{\imath\phi_{i}}&\text{in }B(z_{i},\rho_{n})\setminus\overline{B(z_{i},{\tilde{r}})}\end{cases}$ with $u_{i}(z)=\left(\dfrac{z-z_{i}}{|z-z_{i}|}\right)^{d_{i}}$. It is clear that $v\in H^{1}(\Omega_{\tilde{r}},\mathbb{S}^{1})$ and that for $i\in\{1,...,N\}$ we have $v(z_{i}+{\tilde{r}}\e^{\imath\theta})={\rm Cte}_{i}u_{i}$ [with ${\rm Cte}_{i}=\e^{\imath m_{i}}$]. Note that since ${\rm deg}_{\partial B(z_{i},{\tilde{r}})}(w^{\bf(z,d)}_{\star})=d_{i}$ we have

[TABLE]

and

[TABLE]

Consequently using (177) and $\rho_{n}<\eta$ we obtain

[TABLE]

Thus $\displaystyle\dfrac{1}{2}\int_{\Omega_{\tilde{r}}}|\nabla v|^{2}\leq\dfrac{1}{2}\int_{\Omega_{\tilde{r}}}|\nabla w^{\bf(z,d)}_{\star}|^{2}+CZ$. The last estimate and (59) end the proof.

Appendix E Proof of Proposition 39

Proof.

Step 1. Selection of "good" points

Let $d\in\mathbb{N}^{*}$ and consider ${\bf D}\in\Lambda_{d}$ which minimizes (72).

For $k\in\{1,...,N_{0}\}$, if $D_{k}\geq 1$ we let $(\tilde{z}^{(k)}_{1},...,\tilde{z}^{(k)}_{D_{k}})\in[B(p_{k},h_{\rm ex}^{-1/4})^{D_{k}}]^{*}$ which minimizes the infimum in the left hand side of (64) with $R=h_{\rm ex}^{-1/4}$, $p=p_{k}$ and $D=D_{k}$.

We then have the existence of $C$ [depending only on $\Omega$ and $d$] s.t. $|p_{k}-\tilde{z}^{(k)}_{i}|\leq Ch_{\rm ex}^{-1/2}$ and if $D_{k}\geq 2$ then $|\tilde{z}^{(k)}_{i}-\tilde{z}^{(k)}_{j}|\geq h_{\rm ex}^{-1/2}/C$ for $i\neq j$.

We may choose [in an arbitrary way] $z_{i}^{(k)}\in B(\tilde{z}_{i}^{(k)},\delta)\cap[\delta(\mathbb{Z}\times\mathbb{Z})]$. Since $\delta\sqrt{h}_{\rm ex}\to 0$, we still have [up to change the value $C$] $|p_{k}-z^{(k)}_{i}|\leq Ch_{\rm ex}^{-1/2}$ and if $D_{k}\geq 2$ then $|z^{(k)}_{i}-\tilde{z}^{(k)}_{j}|\geq h_{\rm ex}^{-1/2}/C$ for $i\neq j$.

For $i\in\{1,...,D_{k}\}$ we let $x_{i}^{(k)}:=z_{i}^{(k)}+\lambda\delta x_{0}$ where $x_{0}\in\omega$ is an arbitrary point of minimum of $W^{\rm micro}$ [defined in (70)].

Step 2. Construction of the test function

We construct test functions in subdomains of $\Omega$ and then we glue the test functions.

• •

We let $w^{\rm macro}_{h_{\rm ex}}\in H^{1}(\Omega_{h_{\rm ex}^{-1}}({\bf z}),\mathbb{S}^{1})$ be a minimizer of $I_{h_{\rm ex}^{-1}}^{\rm Dir}{\bf(z,d)}$ [defined in (52)] with ${\bf d}=(1,...,1)\in\mathbb{Z}^{d}$ and ${\bf z}\in(\Omega^{d})^{*}$ is a $d$-tuple s.t. $\{z_{1},...,z_{d}\}=\{z_{i}^{(k)}\,|\,k\in\{1,...,N_{0}\}\text{ s.t. }D_{k}\geq 1\text{ and }i\in\{1,...,D_{k}\}\}$.

• •

For $k\in\{1,...,N_{0}\}$ s.t. $D_{k}\geq 1$ and $i\in\{1,...,D_{k}\}$, we let $w_{k,i}^{\rm micro}\in H^{1}[B(z_{i}^{(k)},h_{\rm ex}^{-1})\setminus\overline{B(x_{i}^{(k)},\lambda\delta^{2})},\mathbb{S}^{1}]$ be a minimizer of the right hand side of (67) with $z_{\varepsilon}=z_{i}^{(k)}$, $x_{\varepsilon}=x_{i}^{(k)}$, $R=h_{\rm ex}^{-1}$ and $r=\lambda\delta^{2}$ [from (73) we have $R/r\to\infty$].

We let also $u_{k,i}\in H^{1}[B(x_{i}^{(k)},\lambda\delta^{2}),\mathbb{C}]$ be a minimizer of

[TABLE]

with the Dirichlet boundary condition $u(x_{i}^{(k)}+\lambda\delta^{2}\e^{\imath\theta})=\e^{\imath\theta}$.

By considering well chosen constants ${\rm Cte}_{k,i}^{(1)}$, ${\rm Cte}_{k,i}^{(2)}$ and ${\rm Cte}_{k}$, we may glue the above test functions and we define $v\in H^{1}(\Omega,\mathbb{C})$ :

[TABLE]

Step 3. The energy of the test function

We first note that the configuration ${\bf(z,d)}$ is s.t. $\hbar({\bf z})>\dfrac{1}{2}{\rm dist}(\Lambda,\partial\Omega)$ and for $i\neq j$ we have $\dfrac{h_{\rm ex}^{-1}}{|z_{i}-z_{j}|}\to 0$, then we may apply Propositions 30, 31 and 33. We may also use Proposition 35. From these propositions we get

[TABLE]

For $k\in\{1,...,N_{0}\}$ s.t. $D_{k}\geq 1$ and $i\in\{1,...,D_{k}\}$ with (67), (68) and (69) we get:

[TABLE]

From Lemma IX.1 in [4] and (25) [with $|\nabla v|\leq C\varepsilon^{-1}$], for $k\in\{1,...,N_{0}\}$ s.t. $D_{k}\geq 1$ we have

[TABLE]

where $\gamma\in\mathbb{R}$ is a universal constant.

In conclusion, by combining (178), (179) and (180) [note $\lambda\delta h_{\rm ex}\to 0$]:

[TABLE]

Step 4. Definition of the magnetic potential and conclusion

Let $A_{({\bf z},{\bf 1})}$ be given by Definition 19 with ${({\bf a},{\bf d})}=({\bf z},{\bf 1})$. It is clear that we have

[TABLE]

where $C$ depends only on $d$ and $\Omega$.

Consequently, for $\varepsilon>0$ sufficiently small and $C_{0}>\pi d$ we have $F(v)\leq C_{0}|\ln\varepsilon|$. Therefore, with Remark 20, the configuration $(v,A_{({\bf z},{\bf 1})})\in\mathscr{H}$ is s.t. $\mathcal{F}(v,A_{({\bf z},{\bf 1})})\leq\mathcal{F}(v,0)+o(1)\leq C_{0}|\ln\varepsilon|^{2}+\mathcal{H}^{2}(\Omega)h_{\rm ex}^{2}$.

Using Proposition11 and Lemma 18 we get

[TABLE]

where $\zeta_{({\bf z},{\bf 1})}$ is the unique solution of (42) with ${({\bf a},{\bf d})}={({\bf z},{\bf 1})}$.

We now use Assertion 3 of Proposition 23 in order to get $\tilde{V}[\zeta_{({\bf z},{\bf 1})}]=\tilde{V}[\zeta_{({\bf p},{\bf D})}]+o(1)$ and then

[TABLE]

We claim that, from the choice of the points $z_{i}^{(k)},\tilde{z}_{i}^{(k)}$ we have $\xi_{0}(z_{i}^{(k)})-\xi_{0}(\tilde{z}_{i}^{(k)})=\mathcal{O}(\delta/\sqrt{h_{\rm ex}})$. Thus with Proposition 36 we have

[TABLE]

We may now conclude:

[TABLE]

This estimate ends the proof of the proposition.∎

Appendix F Proof of Proposition 40

Let $h_{\rm ex}$ and $(v_{\varepsilon},A_{\varepsilon})$ be as in Proposition 40. Note that we may assume that $A_{\varepsilon}=A_{v_{\varepsilon}}$ given by Lemma 12 and then $\|A_{\varepsilon}\|_{L^{\infty}(\Omega)}=\mathcal{O}(h_{\rm ex})$. We drop the subscript $\varepsilon$. We first note that, by smoothness of $\Omega$, there is $t_{0}>0$, s.t. letting $\Omega_{t_{0}}:=\{x\in\mathbb{R}^{2}\,|\,{\rm dist}(x,\Omega)<t_{0}\}$, we may extend by reflexion $v\in H^{1}(\Omega,\mathbb{C})$ into $u\in H^{1}(\Omega_{t_{0}},\mathbb{C})$ letting $u=v$ in $\Omega$ and $u=v\circ\mathcal{S}_{\Omega}$ in $\Omega_{t_{0}}\setminus\overline{\Omega}$ where

[TABLE]

Here $\Pi:\Omega_{t_{0}}\setminus\overline{\Omega}\to\partial\Omega$ is the orthogonal projection on $\partial\Omega$ and, for $\sigma\in\partial\Omega$, $\nu_{\sigma}$ is the normal outward at $\sigma$.

Lemma 72**.**

Let $C_{0}\geq 1$ and let $\{(v_{\varepsilon},A_{\varepsilon})\,|\,0<\varepsilon<1\}$ be a family in the Coulomb gauge of quasi-minimizers of $\mathcal{F}$ in $\mathscr{H}$ for an intensity of the applied field $h_{\rm ex}=h_{\rm ex}(\varepsilon)\geq 0$ s.t. $\|\nabla|v|\|_{L^{\infty}(\Omega)}\leq C_{0}\varepsilon^{-1}$.

Under these hypotheses, for $\eta\in(0,1)$ there exists $\varepsilon_{\eta},C_{\eta}>0$ [depending on $C_{0}$] s.t. for $0<\varepsilon<\varepsilon_{\eta}$, if $z\in\Omega$ is s.t.

[TABLE]

with $u=\begin{cases}v&\text{in }\Omega\\ v\circ\mathcal{S}_{\Omega}&\text{in }\Omega_{t_{0}}\setminus\overline{\Omega}\end{cases}$, then $|v(z)|>\eta$.

In order to prove Proposition 40 we need the following lemma.

Lemma 73**.**

There exists $\varepsilon_{\Omega}>0$ depending only on $\Omega$ s.t. for $0<\varepsilon<\varepsilon_{\Omega}$, $z\in\Omega$ and $v\in H^{1}(\Omega,\mathbb{C})$, by defining $u$ as in Lemma 72, the following inequality holds:

[TABLE]

Proof of Lemma 73.

In order to prove the lemma it suffices to check that by smoothness of $\Omega$ we have $\|\nabla(\mathcal{S}_{\Omega}^{-1})\|_{L^{\infty}(\Omega)},\|{\rm jac}\,(\mathcal{S}_{\Omega}^{-1})\|_{L^{\infty}(\Omega)}=1+o(1)$. We then immediately obtain

[TABLE]

On the other hand, if $x\in B(z,\sqrt{\varepsilon}/2)\setminus\Omega$ then $|\mathcal{S}_{\Omega}(x)-z|\leq[1+o(1)]\sqrt{\varepsilon}/2\leq\sqrt{\varepsilon}$ for sufficiently small $\varepsilon>0$ [depending only on $\Omega$]. Then $\mathcal{S}_{\Omega}[B(z,\sqrt{\varepsilon}/2)\setminus\Omega]\subset B(z,\sqrt{\varepsilon})\cap\Omega$. The lemma follows from the monotonicity of the integral. ∎

By combining both lemmas we get Proposition 40.

Proof of Lemma 72.

We argue by contradiction and we assume the existence of $\eta\in(0,1)$, $\varepsilon=\varepsilon_{n}\downarrow 0$ s.t. for all $n\geq 1$ there are $(v,A)=(v_{n},A_{n})\in\mathscr{H}$, $z=z_{n}\in\Omega$ and $h_{\rm ex}=h_{\rm ex}^{(n)}\geq 0$ s.t. $(v,A)$ is a quasi-minimizers of $\mathcal{F}$ in $\mathscr{H}$ satisfying:

[TABLE]

with $u=u_{n}=\begin{cases}v&\text{in }\Omega\\ v\circ\mathcal{S}_{\Omega}&\text{in }\Omega_{t_{0}}\setminus\overline{\Omega}\end{cases}$ and $|v(z)|\leq\eta$. Up to replace $v$ by $\underline{v}$ we may assume $|v|\leq 1$ in $\Omega$.

We are going to prove that (183) implies

[TABLE]

On the other hand, $\|\nabla|v|\|_{L^{\infty}(\Omega)}=\mathcal{O}(\varepsilon^{-1})$ and then, from an argument in [4] [Theorem III.3], we will get, for sufficiently large $n$, $|v(z)|>\eta$. Clearly this contradiction will end the proof.

Since for $n\geq 1$ we have $\displaystyle\int_{\varepsilon^{3/4}/2}^{\sqrt{\varepsilon}/2}\dfrac{{\rm d}\rho}{\rho}\,\rho\int_{\partial B(z,\rho)}|\nabla u|^{2}+\dfrac{b^{2}}{\varepsilon^{2}}(1-|u|^{2})^{2}\leq\dfrac{|\ln\varepsilon|}{n}$, there exists $\displaystyle\rho_{n}\in(\varepsilon^{3/4},\sqrt{\varepsilon}/2)$ s.t. $\displaystyle\rho_{n}\int_{\partial B(z,\rho_{n})}|\nabla u|^{2}+\dfrac{b^{2}}{\varepsilon^{2}}(1-|u|^{2})^{2}\leq\dfrac{4}{n}$. Then we get :

[TABLE]

We switch in polar coordinate and we denote $\tilde{u}(\theta):=u(z+\rho_{n}\e^{\imath\theta})$. Estimate (185) becomes

[TABLE]

On the one hand, $|\partial_{\theta}|\tilde{u}||^{2}\leq|\partial_{\theta}\tilde{u}|^{2}$ and then $\displaystyle\int_{0}^{2\pi}|\partial_{\theta}|\tilde{u}||\leq\dfrac{2\sqrt{2\pi}}{\sqrt{n}}$. Consequently in $[0,2\pi]$ we get $(1-|\tilde{u}|^{2})^{2}\geq\max_{[0,2\pi]}(1-|\tilde{u}|^{2})^{2}-\dfrac{2\sqrt{2\pi}}{\sqrt{n}}$. From (186) we deduce

[TABLE]

and thus for sufficiently large $n$ we get $0\leq\max_{[0,2\pi]}(1-|\tilde{u}|^{2})^{2}\leq\dfrac{100}{\sqrt{n}}$.

For a further use we define

[TABLE]

By direct calculations we have

[TABLE]

On the other hand, for $n$ sufficiently large, $|{u}|^{2}\geq\dfrac{1}{2}$ in $\partial B(z,\rho_{n})$. We thus may compute the degree of $u$ on $\partial B(z,\rho_{n})$ and we find $\left|{\rm deg}_{\partial B(z,\rho_{n})}(u)\right|=\mathcal{O}\left(\dfrac{1}{n}\right)$ which implies, for sufficiently large $n$, ${\rm deg}_{\partial B(z,\rho_{n})}(u)=0$. Consequently, we may write $u=|u|\e^{\imath\varphi}$ with $\varphi=\varphi_{n}\in H^{1}(\partial B(z,\rho_{n}),\mathbb{R})$. Moreover, up to multiply $u$ by a constant in $\mathbb{S}^{1}$, we may assume $\int_{\partial B(z,\rho_{n})}\varphi=0$.

We then consider $\tilde{\varphi}:[0,2\pi]\to\mathbb{R}$ defined by $\tilde{\varphi}(\theta)=\varphi(z+\rho_{n}\e^{\imath\theta})$, and thus

[TABLE]

Since $\displaystyle\int_{0}^{2\pi}\tilde{\varphi}=0$, this estimate implies $\displaystyle\int_{0}^{2\pi}\tilde{\varphi}^{2}=\mathcal{O}\left(\dfrac{1}{n}\right)$.

Letting $\psi=\psi_{n}:B(z,{\rho_{n}})\to\mathbb{R}$, $z+\rho\e^{\imath\theta}\mapsto\dfrac{\rho}{\rho_{n}}\tilde{\varphi}(\theta)$, it is direct to check $\displaystyle\int_{B(z,\rho_{n})}|\nabla\psi|^{2}=\mathcal{O}\left(\dfrac{1}{n}\right)$.

We are now in position to end the proof by considering $V=V_{n}={\chi}_{n}\e^{\imath\psi}\in H^{1}(B(z,\rho_{n}),\mathbb{C})$ in order to have $V=v$ on $\partial B(z,\rho_{n})\cap\Omega$,

[TABLE]

and [with $\|A\|_{L^{\infty}(\Omega)}=\mathcal{O}(h_{\rm ex})$]

[TABLE]

Since $V=v$ on $\partial B(z,\rho_{n})\cap\Omega$ we have $w:=\begin{cases}v&\text{in }\Omega\setminus B(z,\rho_{n})\\ V&\text{in }B(z,\rho_{n})\cap\Omega\end{cases}\in H^{1}(\Omega,\mathbb{C})$. Considering the comparison configuration $(\tilde{w},A)$, from the quasi-minimality of $(v,A)$ and the above estimates we get

[TABLE]

Since $\rho_{n}>\varepsilon^{3/4}$ we get (184) and thus this estimate ends the proof. ∎

Appendix G Proof of Proposition 42

The proof of the proposition is an adaptation of the arguments presented in [2] [Section V] and also used in [17] [Proposition 3.2]. It is also inspired of the bad disk construction in [4]. Let $\mu$, $\lambda$, $\delta$, $(v,A)$ and $h_{\rm ex}$ be as in the proposition.

**Step 1. A first covering of $\{|v|\leq 1/2\}$

**

For $0<\varepsilon<\varepsilon_{1/2}$ [$\varepsilon_{1/2}>0$ is given by Proposition 40 with $\eta=1/2$] we consider a covering of $\Omega$ by disks $\{B(x^{\varepsilon}_{1},4\sqrt{\varepsilon}),...,B(x^{\varepsilon}_{N_{\varepsilon}},4\sqrt{\varepsilon})\}$ s.t., for $i\neq j$, $B(x^{\varepsilon}_{i},\sqrt{\varepsilon})\cap B(x^{\varepsilon}_{j},\sqrt{\varepsilon})=\emptyset$ and $x^{\varepsilon}_{i}\in\Omega$.

For the simplicity of the presentation we omit the dependance in $\varepsilon$.

We say that $B(x_{i},4\sqrt{\varepsilon})$ is a bad disk if $\tilde{E}_{\varepsilon}[v,B(x_{i},8\sqrt{\varepsilon})\cap\Omega]>C_{1/2}|\ln\varepsilon|$ where for a disk $B$ we denoted

[TABLE]

and $C_{1/2}>0$ is given by Proposition 40 with $\eta=1/2$. Let

[TABLE]

We make two fundamental claims:

1. (1)

There exists $M_{0}\geq 1$ [independent of $\varepsilon$] s.t. ${\rm Card}(J^{\prime})\leq M_{0}$. 2. (2)

If $B(x_{i},4\sqrt{\varepsilon})$ is not a bad disk then $|v|\geq 1/2$ in $B(x_{i},4\sqrt{\varepsilon})$.

The first claim is a direct consequence of (32) and $B(x^{\varepsilon}_{i},\sqrt{\varepsilon})\cap B(x^{\varepsilon}_{j},\sqrt{\varepsilon})=\emptyset$ for $i\neq j$.

The second claim is given by Proposition 40. Then $\cup_{i\in J^{\prime}}B(x_{i},4\sqrt{\varepsilon})$ is covering of $\{|v|\leq 1/2\}$ and ${\rm Card}(J^{\prime})\leq M_{0}$.

Up to drop some disks, we may always assume that for $i\in J^{\prime}$ we have $B(x_{i},4\sqrt{\varepsilon})\cap\{|v|\leq 1/2\}\neq\emptyset$. Consequently using Corollary 41, for $i\in J^{\prime}$ and $0<\varepsilon<\min\{\varepsilon_{0},\varepsilon_{1/2}\}$ [$\varepsilon_{0}$ given by Corollary 41] we have ${\rm dist}(x_{i},\Lambda)=\mathcal{O}(|\ln\varepsilon|^{-s_{0}})$.

If $|v|>1/2$ in $\Omega$ then there is nothing to prove. We then assume $J^{\prime}\neq\emptyset$.

**Step 2. Separation process

**

We replace the above bad disks with disks having same centers and with a radius $\varepsilon^{\mu}$. Let $\varepsilon_{\mu}^{(1)}>0$ be s.t. $\min\{\varepsilon_{0},\varepsilon_{1/2}\}\geq\varepsilon_{\mu}^{(1)}$, for $0<\varepsilon<\varepsilon_{\mu}^{(1)}$ we have $4\sqrt{\varepsilon}<\varepsilon^{\mu}$ and

[TABLE]

In particular $\cup_{i\in J^{\prime}}B(x_{i},\varepsilon^{\mu})$ is a covering of $\{|v|\leq 1/2\}$.

The goal of this step is to get a covering of $\{|v|\leq 1/2\}$ with disks $B(x_{i},\varepsilon^{s})$ where $i\in\tilde{J}=\tilde{J}_{\varepsilon}\subset J^{\prime}$, $s=s_{\varepsilon}=2^{-K}\mu$, $K=K_{\varepsilon}\in\{0,...,M_{0}-1\}$ and s.t. for $i,j\in\tilde{J}$, $i\neq j$, we have

[TABLE]

If ${\rm Card}(J^{\prime})=1$ or (188) is satisfied with $s=\mu$ [i.e. $K=0$] then we let $\tilde{J}=J^{\prime}$ and we obtained the desired result of this step. Otherwise, there are $i_{0},j_{0}\in J^{\prime}$ [with $i_{0}<j_{0}$] s.t. $|x_{i_{0}}-x_{j_{0}}|<\varepsilon^{\mu/2}$. In this case we let $J^{(1)}:=J^{\prime}\setminus\{i_{0}\}$ and we claim that ${\rm Card}(J^{(1)})={\rm Card}(J^{\prime})-1$.

If ${\rm Card}(J^{(1)})=1$ or ${\rm Card}(J^{(1)})>1$ with (188) holds with $s=2^{-1}\mu$ [i.e. $K=1$] for all $i,j\in J^{(1)}$ [$i\neq j$] then the goal of this step is done with $\tilde{J}=J^{(1)}$ and $s=2^{-1}\mu$.

Otherwise, there exits $i_{0},j_{0}\in J^{(1)}$ [with $i_{0}<j_{0}$] s.t. $|x_{i_{0}}-x_{j_{0}}|<\varepsilon^{s/2}$. We then let $J^{(2)}:=J^{(1)}\setminus\{i_{0}\}$ and thus ${\rm Card}(J^{(2)})={\rm Card}(J^{(1)})-1$.

By noting that ${\rm Card}(J^{\prime})\leq M_{0}$, the above process stops after at most $M_{0}-1$ iteration. We thus get the existence of $K=K_{\varepsilon}\in\{0,...,M_{0}-1\}$ and $\emptyset\neq J^{(K)}=J^{(K)}_{\varepsilon}\subset J^{\prime}$ s.t. ${\rm Card}(J^{(K)})=1$ or (188) is satisfied with $s=s_{\varepsilon}=2^{-K}\mu$ and $i,j\in J^{(K)}$ [$i\neq j$].

We then denote $\tilde{J}:=J^{(K)}$, $s=2^{-K}\mu$ and we fix $0<\varepsilon_{\mu}^{(2)}\leq\varepsilon_{\mu}^{(1)}$ s.t. for $0<\varepsilon<\varepsilon_{\mu}^{(2)}$ we have

[TABLE]

In particular $B(x_{i},\varepsilon^{s/4})\subset\Omega$ for $i\in\tilde{J}$.

Step 3. Definition of $r$

With Corollary 5.2 in [5], for a.e. $t\in{\rm Image}(|v|)$ the set $V(t):=\{x\in\Omega\,|\,|v(x)|=t\}$ is a finite union of curve. Moreover if a such curve is included in $\Omega$ then it is a Jordan curve.

Following the same strategy as in [2] [Lemma V.1], we have the existence of $t_{\varepsilon}\in[1-2|\ln\varepsilon|^{-2},1-|\ln\varepsilon|^{-2}]$ s.t. $V(t_{\varepsilon})$ is a finite union of Jordan curves s.t.

[TABLE]

We fix $0<\varepsilon_{\mu}^{(3)}\leq\varepsilon_{\mu}^{(2)}$ s.t. for $0<\varepsilon<\varepsilon_{\mu}^{(3)}$ we have $C\varepsilon|\ln\varepsilon|^{5}\leq 10^{-2}\varepsilon^{s}$.

We denote for $i\in\tilde{J}$

[TABLE]

From the continuity of $|v|$, it is clear that $[\varepsilon^{s},\varepsilon^{2s/3}]=\mathcal{A}_{i}\cup\mathcal{B}_{i}\cup\mathcal{C}_{i}$ where

[TABLE]

and

[TABLE]

We first claim that, since the function $\rho\mapsto\rho$ is increasing, we have

[TABLE]

Then $\mathcal{H}^{1}(\mathcal{C}_{i})=\mathcal{O}(\varepsilon|\ln\varepsilon|^{5/2})$.

On the other hand one may prove that if $I$ is a connected components of $\mathcal{B}_{i}$, then there is $\rho_{1},\rho_{2}$ s.t. $I=[\rho_{1},\rho_{2}]$. Since straight lines are geodesics, we obviously get

[TABLE]

Moreover one may prove that if $[\rho_{1},\rho_{2}]$ and $[\rho_{1}^{\prime},\rho_{2}^{\prime}]$ are distinct connected component of $\mathcal{B}_{i}$ and if $\Gamma$ is a connected component of $V(t_{\varepsilon})$ s.t. $\Gamma\cap\overline{B(x_{i},\rho_{2})}\setminus B(x_{i},\rho_{1})\neq\emptyset$ then $\Gamma\cap\overline{B(x_{i},\rho_{2}^{\prime})}\setminus B(x_{i},\rho_{1}^{\prime})=\emptyset$ [here we used (189)]. One may conclude: $\mathcal{H}^{1}(\mathcal{B}_{i})\leq\mathcal{H}^{1}(V(t_{\varepsilon}))\leq C\varepsilon|\ln\varepsilon|^{5}$.

Consequently

[TABLE]

Fix $0<\varepsilon_{\mu}^{(4)}\leq\varepsilon_{\mu}^{(3)}$ s.t. for $0<\varepsilon<\varepsilon_{\mu}^{(4)}$ we have $\mathcal{H}^{1}(\mathcal{A}_{i})\geq\varepsilon^{2s/3}-\varepsilon^{s}-\sqrt{\varepsilon}$.

Define

[TABLE]

It is clear that $\mathcal{H}^{1}(\mathcal{A})\geq\varepsilon^{2s/3}-\varepsilon^{s}-M_{0}\sqrt{\varepsilon}$

Since $\rho\mapsto 1/\rho$ is decreasing we have

[TABLE]

Consequently, there exist $r=r_{\mu,\varepsilon}\in\mathcal{A}$, $C_{\mu}\geq 1$ [$C_{\mu}$ is independent of $\varepsilon$] and $0<\varepsilon_{\mu}^{(5)}\leq\varepsilon_{\mu}^{(4)}$ s.t. for $0<\varepsilon<\varepsilon_{\mu}^{(5)}$ we have

[TABLE]

We finally let $J_{\mu}:=\tilde{J}$, with (188) and (192) the result is proved.

Appendix H Proof of Proposition 49

The proof is an adaptation of the proof of (VI.21) in [2].

Let $\tilde{\alpha}=\tilde{\alpha}_{n}\in L^{\infty}(\Omega,[\beta^{2};1])$, ${\bf(z,d)}={\bf(z,d)}^{(n)}\in(\Omega^{N})^{*}\times\mathbb{Z}^{N}$ and $u=u_{n}\in H^{1}(\Omega,\mathbb{C})$ be as in the proposition.

We first claim that up to consider $\underline{u}$ instead of $u$ we may assume $|u|\leq 1$ in $\Omega$. Note also that if $\int_{\Omega_{\tilde{r}}}|\nabla u|^{2}\geq\beta^{-2}\int_{\Omega_{\tilde{r}}}|\nabla w^{\bf(z,d)}_{\star}|^{2}$, then there is nothing to prove. We thus may assume

[TABLE]

Let $w:={u}/{|u|}\in H^{1}(\Omega_{\tilde{r}},\mathbb{S}^{1})$. From Lemma I.1 in [4] we have $w\wedge\nabla w=\nabla^{\bot}\Phi^{\bf(z,d)}_{\star}+\nabla H$ with $H=H_{\varepsilon}\in H^{1}(\Omega_{\tilde{r}},\mathbb{R})$ and

[TABLE]

Let $\Phi_{\tilde{r}}$ be the unique solution of (161). We have $\displaystyle\int_{\Omega_{\tilde{r}}}\nabla H\cdot\nabla^{\bot}\Phi_{\tilde{r}}=0$. Then letting $\rho=|u|$:

[TABLE]

But, from (172), there exists $C\geq 1$ s.t. $\displaystyle\left|\int_{\Omega_{\tilde{r}}}\nabla H\cdot(\nabla^{\bot}\Phi^{\bf(z,d)}_{\star}-\nabla^{\bot}\Phi_{\tilde{r}})\right|\leq C\|\nabla H\|_{L^{2}(\Omega_{\tilde{r}})}\sqrt{X}$ where $X$ is defined in (57).

Consequently, letting $\tilde{C}:=4C^{2}/\beta^{2}$ we get

[TABLE]

Therefore

[TABLE]

On the other hand, using (56) and Corollary 32, we get

[TABLE]

and with (193):

[TABLE]

The proposition is thus proved.

Appendix I Proof of Proposition 55

We prove the first assertion and we assume ${\rm Card}(J_{\mu})\geq 2$. We let $\chi_{1}:=2h_{\rm ex}^{-1}\ln h_{\rm ex}$, $\chi_{2}:=2h_{\rm ex}^{-1/2}\ln h_{\rm ex}$ and $\Omega_{\chi_{2}}=\Omega\setminus\cup_{p\in\Lambda}\overline{B(p,\chi_{2})}$.

In order to get sufficiently sharp estimates to prove the proposition, we decompose $\Omega_{r}$ in several subdomains. To this aim, we distinguish two cases for $p\in\Lambda$ : either ${\rm Card}(J_{p}^{(y)})\geq 2$ or ${\rm Card}(J_{p}^{(y)})\in\{0,1\}$ where $J_{p}^{(y)}:=\{k\in J^{(y)}\,|\,y_{k}\in B(p,\chi_{2})\}$ [the $y_{k}$’s are introduced in Definition 51].

If $p\in\Lambda$ is s.t. ${\rm Card}(J_{p}^{(y)})\geq 2$, then with Lemma 48 [with $P=17$ and $\eta=\chi_{1}/2$], there are $\kappa_{p}=\kappa_{p,\varepsilon}\in\{17^{0},...,17^{N_{0}-1}\}$ and $\tilde{J}_{p}^{(y)}\subset J_{p}^{(y)}$ s.t.

[TABLE]

We then let $\mathcal{D}_{p}:=B(p,\chi_{2})\setminus\cup_{k\in\tilde{J}_{p}^{(y)}}\overline{B(y_{k},\kappa_{p}\chi_{1})}$ and, for $k\in\tilde{J}_{p}^{(y)}$, we write $\underline{d}_{k}:={\rm deg}_{\partial B(y_{k},\kappa_{p}\chi_{1})}(v)$. We denote also $D_{p}:=\sum_{k\in\tilde{J}_{p}^{(y)}}\underline{d}_{k}$

If $p\in\Lambda$ is s.t. $J_{p}^{(y)}=\{k\}$, then we let $\mathcal{D}_{p}=B(p,\chi_{2})\setminus\overline{B(y_{k},\kappa\delta)}$ with $\kappa$ given by Definition 51. We let also $D_{p}:=\underline{d}_{k}:={\rm deg}_{\partial B(y_{k},\kappa\delta)}(v)$.

Recall that we denoted (see Definition 51), for $k\in J^{(y)}$, $\tilde{d}_{k}:={\rm deg}_{\partial B(y_{k},\kappa\delta)}(v)$. Consequently, if $J_{p}^{(y)}=\{k\}$, then $D_{p}=\underline{d}_{k}=\tilde{d}_{k}$.

If $J_{p}^{(y)}=\emptyset$ then we denote $D_{p}=0$ and $\mathcal{D}_{p}=B(p,\chi_{2})$.

The heart of the proof consists in proving that $\underline{d}_{k}=1$ for all $k$. Indeed, we know that if $i\in J_{\mu}$ then ${\rm deg}_{\partial B(z_{i},r)}(v)=1$. Consequently $\underline{d}_{k}$ is the number of points $z_{i}$ contained in a disk of radius at least $\chi_{1}$.

We let:

• •

$\mathcal{R}:=\bigcup_{k\in J^{(y)}}B(y_{k},\kappa\delta)\setminus\bigcup_{i\in J_{\mu}}\overline{B(z_{i},r)}$, $\kappa$ given in Definition 51.

• •

For $p\in\Lambda$ s.t. ${\rm Card}(J_{p}^{(y)})\geq 2$ and for $k\in\tilde{J}_{p}^{(y)}$ we denote

[TABLE]

Moreover, by construction, we have [for sufficiently small $\varepsilon$]

[TABLE]

Thus

[TABLE]

From (101) and (102) we have

[TABLE]

If $J_{p}^{(y)}=\{k\}$, then with Corollary 47.1 we get

[TABLE]

And if ${\rm Card}(J_{p}^{(y)})\geq 2$, still with Corollary 47.1:

[TABLE]

We continue by dealing with the case ${\rm Card}(J_{p}^{(y)})\geq 2$. From Corollary 47.1 applied in $\mathcal{Q}_{k,p}$ for $k\in\tilde{J}_{p}^{(y)}$ [with (194)] we get

[TABLE]

In order to end the proof, using Propositions 30 $\&$ 31 $\&$ 49, we get

[TABLE]

We let

[TABLE]

From (195), (196), (197), (198), (199) and (200) we get

[TABLE]

Since $\underline{d}_{k},\tilde{d}_{l}\geq 1$ for all $k,l$, from Lemma 54.1 we have $\sum_{p\in\Lambda}D_{p}^{2}\geq\Delta\geq\tilde{\Delta}\geq d$ and moreover

[TABLE]

and

[TABLE]

On the other hand since for $p\in\Lambda$ s.t. $J_{p}^{(y)}=\{k\}$ we have $\underline{d}_{k}=\tilde{d}_{k}$, we get

[TABLE]

Then (96) gives

[TABLE]

Since $|\ln\chi_{1}|=\ln({h_{\rm ex}})+\mathcal{O}[\ln(\ln h_{\rm ex})]$ and $|\ln\chi_{2}|=\ln\sqrt{h_{\rm ex}}+\mathcal{O}[\ln(\ln h_{\rm ex})]$ we obtain

[TABLE]

From Lemma 54.2 and the definition of $\mathscr{L}_{1}(d)$ [see Lemma 37], we have

[TABLE]

Using (202) in (201), (4) and $\tilde{\Delta}-d\geq 0\&\Delta-\tilde{\Delta}\geq 0$ we get $\tilde{\Delta}-d=\Delta-\tilde{\Delta}=0$ and then $\Delta=d$, i.e. $\underline{d}_{k}=1$ for all $k$.

On the other hand, with the help of (201) we may write

[TABLE]

We may thus deduce $\dfrac{\mathscr{L}_{1}(d)}{\pi}+\dfrac{d-\sum_{p\in\Lambda}D_{p}^{2}}{2}=0$ and then, with Lemma 54.2, for $p\in\Lambda$ we have $D_{p}\in\{\lfloor d/N_{0}\rfloor;\lceil d/N_{0}\rceil\}$.