Effects of corners in surface superconductivity

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 $$ {\mathcal {O}}(1) $$ 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.

Almost flat angles in surface superconductivity

“…Hence, before disappearing, superconductivity survives only close to the corner(s). In [CG1,CG2], we proved that, if…”

“…The GL energy functional introduced above was extensively studied for constant or regular magnetic fields in smooth domains (see, e.g., [CDR,CR2,CR3,CR4,FK1] for the 2D case and [AG,FK2,FKP,FMP,P] for the 3D one) and in domains with corners at the boundary [Cor,CG1,CG2,Gia,HK]. More recently, 2D GL models with piecewise-constant magnetic fields were also considered in [Ass1,Ass2,AK,.…”

“…We now focus on the energy asymptotics in the surface superconductivity regime. The most accurate result about is proven in [CG2] (see also the review [Cor]) and reads as follows.…”

“…where α , f is a minimizing pair realizing E 1D , whose existence in discussed in [CG2, section A.1]. Here, we skip most of the details about the one-dimensional models and refer to appendix A or [CR2,CR3,CG1,CG2,Gia] instead (see in particular [CR3, section A]).…”

On the magnetic laplacian with a piecewise constant magnetic field in $\mathbb{R}^3_+$

“…[38,19]). Between the second and third critical fields, i.e., in the regime H c2 ≤ h ex ≤ H c3 , superconductivity survives only close to the boundary of the sample (see e.g., [31,15,16,12,22,13,14]) . Above the third critical field H c3 , the sample goes back to its normal state (see e.g., [28,23,25,17,18,11]).…”