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On the Ginzburg-Landau Energy of Corners

Michele Correggi, Emanuela L. Giacomelli, Ayman Kachmar

TL;DR

This work analyzes how boundary corners affect surface superconductivity within the Ginzburg-Landau framework by connecting two asymptotic effective models: a corner-energy model and a sector-model energy. It proves that, at the critical threshold $ mu= Theta_0$, the corner and sector energies converge to a common finite limit $E_{ ext{sector}}( Theta_0)$, and establishes the existence of a minimizer at the threshold while showing $E_{ ext{sector}}( Theta_0)<0$. The results imply continuity of the full GL energy across the threshold for domains with a single corner and refute a proposed linear dependence of the corner energy on the opening angle near threshold. These findings bridge linear and nonlinear regimes near onset, clarifying how corner geometry governs the onset and localization of superconductivity in high-field regimes.

Abstract

It is a well known fact that the geometry of a superconducting sample influences the distribution of the surface superconductivity for strong applied magnetic fields. For instance, the presence of corners induces geometric terms described through effective models in sector-like regions. We study the connection between two effective models for the offset of superconductivity and for surface superconductivity introduced in \cite{BNF} and \cite{CG2}, respectively. We prove that the transition between the two models is continuous with respect to the magnetic field strength, and, as a byproduct, we deduce the existence of a minimizer at the threshold for both effective problems. Furthermore, as a consequence, we disprove a conjecture stated in \cite{CG2} concerning the dependence of the corner energy on the angle close to the threshold.

On the Ginzburg-Landau Energy of Corners

TL;DR

This work analyzes how boundary corners affect surface superconductivity within the Ginzburg-Landau framework by connecting two asymptotic effective models: a corner-energy model and a sector-model energy. It proves that, at the critical threshold , the corner and sector energies converge to a common finite limit , and establishes the existence of a minimizer at the threshold while showing . The results imply continuity of the full GL energy across the threshold for domains with a single corner and refute a proposed linear dependence of the corner energy on the opening angle near threshold. These findings bridge linear and nonlinear regimes near onset, clarifying how corner geometry governs the onset and localization of superconductivity in high-field regimes.

Abstract

It is a well known fact that the geometry of a superconducting sample influences the distribution of the surface superconductivity for strong applied magnetic fields. For instance, the presence of corners induces geometric terms described through effective models in sector-like regions. We study the connection between two effective models for the offset of superconductivity and for surface superconductivity introduced in \cite{BNF} and \cite{CG2}, respectively. We prove that the transition between the two models is continuous with respect to the magnetic field strength, and, as a byproduct, we deduce the existence of a minimizer at the threshold for both effective problems. Furthermore, as a consequence, we disprove a conjecture stated in \cite{CG2} concerning the dependence of the corner energy on the angle close to the threshold.
Paper Structure (18 sections, 15 theorems, 183 equations, 2 figures)

This paper contains 18 sections, 15 theorems, 183 equations, 2 figures.

Table of Contents

  1. Introduction and Main Results
  2. Effective models
  3. Main result
  4. Organization of the paper
  5. Preliminaries
  6. One-dimensional energies
  7. Magnetic Laplacian in a sector
  8. Another corner effective energy at the threshold $\mu = \Theta_0$
  9. Corner effective model
  10. Corner energy at the threshold $\mu = \Theta_0$
  11. An effective model on the wedge
  12. Sector effective model
  13. Basic properties
  14. Proof of \ref{['theo: main']}
  15. Lower bound
  16. ...and 3 more sections

Key Result

Theorem 1.1

For any angle $\beta \in (0, \pi)$ such that eq: condition holds, we have that

Figures (2)

  • Figure 1: The infinite sector $\mathcal{S}_\beta$ with opening angle $\beta$ and the corresponding coordinate frame.
  • Figure 2: The corner region $\Gamma_{\beta}(L,\ell)$ and the associated coordinate system. The opening angle $\widehat{AVB}$ is equal to $\beta$ and the side lengths are $|\overline{AV}| = |\overline{VB}| = L$ and $|\overline{AC}| = |\overline{EB}| = \ell$.

Theorems & Definitions (32)

  • Theorem 1.1: Effective energies
  • Corollary 1.2
  • Remark 1.3: Non-triviality of $E_{\mathrm{corner}}$
  • Remark 1.4: Continuity of $E^{\mathrm{GL}}$
  • Remark 1.5: Conjecture on $E_{\mathrm{corner}}$
  • Remark 1.6: Magnetic steps
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • ...and 22 more