Effects of Corners in Surface Superconductivity

TL;DR

This paper analyzes how corners in the boundary of a superconductor influence surface superconductivity, deriving energy asymptotics and highlighting the role of corner angles in the Ginzburg-Landau model.

Contribution

It provides the first detailed asymptotic analysis of the Ginzburg-Landau functional in domains with corners, revealing additional energy contributions from corners.

Findings

Corners add an O(1) energy contribution depending on opening angle.

Derived asymptotics for ground state energy in surface superconductivity regime.

Proposed a model problem in an infinite wedge to analyze corner effects.

Abstract

We study the Ginzburg-Landau functional describing an extreme type-II superconductor wire with cross section with finitely many corners at the boundary. We derive the ground state energy asymptotics up to $ o(1) $ errors in the surface superconductivity regime, i.e., between the second and third critical fields. We show that, compared to the case of smooth domains, each corner provides an additional contribution of order $ \mathcal{O}(1) $ depending on the corner opening angle. The corner energy is in turn obtained from an implicit model problem in an infinite wedge-like domain with fixed magnetic field. We also prove that such an auxiliary problem is well-posed and its ground state energy bounded and, finally, state a conjecture about its explicit dependence on the opening angle of the sector.