A $Γ$-convergence result for 2D type-I superconductors

TL;DR

The paper establishes a $\Gamma$-convergence result for a 2D non-standard Modica-Mortola type energy derived from the Ginzburg-Landau model of type-I superconductors in the regime of non-vanishing $\kappa$. It shows that, as $\varepsilon\to0$, the energy concentrates on the interface between superconducting and normal regions and converges to a perimeter functional with a cell-constant density $\sigma_{\kappa}$, i.e. $\mathcal{E}_{\varepsilon} \Gamma$-converges to $\sigma_{\kappa}P(\{\rho=0\},\mathbb{T}^2)$. The analysis combines Modica-Mortola-type compactness, a blow-up-based $\Gamma$-liminf, and a carefully constructed $\Gamma$-limsup via cell problems and gauge-fixing, including vertical Dirichlet reductions and horizontal periodic boundary considerations. This work provides a foundational step toward extending previous small-$\kappa$ results to the non-vanishing-$\kappa$ regime and clarifies how the Ginzburg-Landau parameters influence the interfacial energetics in type-I superconductors.

Abstract

We consider a 2D non-standard Modica-Mortola type functional. This functional arises from the Ginzburg-Landau theory of type-I superconductors in the case of an infinitely long sample and in the regime of comparable penetration and coherence lengthes. We prove that the functional $Γ$-converges to the perimeter functional. This result is a first step in understanding how to extend the results of Conti, Goldman, Otto, Serfaty (2018) to the regime of non vanishing Ginzburg-Landau parameter $κ$.

Figures (4)

• Figure 1: A typical competitor $(u_n,A_n)_n$ for $\sigma_{\kappa}$. Competitors enjoy fixed boundary conditions at distance $\delta$ from the transition (the zone in blue). Competitors for $\sigma_{\kappa,\delta}^{\text{per }}$ moreover are such that $u_n$ coincides with a given real profile $U_n$ near the top and bottom boundaries of $Q_1$.
• Figure 2: Details of the sets $R_n$ and $Q_{\delta/2,N}$ inside $Q_{1,N}$.
• Figure 3: Sketch of the construction near a corner of $\partial E$. For simplicity in this picture $\zeta=0$.
• Figure 4: Example of a polyhedral set $E$ (in blue) such that both $E$ and $E^c$ have multiple connected components, $E_1$, $E_2$, $E_3$, and $F_1$, $F_2$, $F_3$ respectively. The set $T$ is represented by the double line around $\partial E$. The phase $\theta_n$ is defined locally on each $F_k,\, k=1\dots 3$. On any $F_k, \, k=1\dots 3$, to ensure that the phase is defined up to a multiple of $2\pi$, each connected component of $E$ which is enclosed by $F_k$ is subject to the flux quantization condition \ref{['e:localquantization']}. In particular, in this example $F_3$ forces a quantization condition on $E_1, E_2$ and $E_3$, while $F_2$ forces a quantization condition on $E_2$. Since $F_1$ is simply connected, its phase is already quantized and no quantization condition is enforced.

Theorems & Definitions (31)

• Theorem 1.1
• Remark 1.2
• Remark 1.3
• Remark 1.4
• Theorem 3.1
• Proposition 3.2
• proof
• Lemma 3.3
• proof
• proof : Proof of Theorem \ref{['th:compactess']}