# Distribution of vortices in a type-II superconductor as a free boundary problem: existence and regularity via Nash-Moser theory

### Alexis Bonnet

Université de Cergy-Pontoise, France### Régis Monneau

CERMICS - ENPC, Marne-La-Vallée, France

## Abstract

This paper is concerned with a model describing the distribution of vortices in a Type-II superconductor. These vortices are distributed continuously and occupy an unknown region D with [part]D representing the free boundary. The problem is set as follows: two constants H0 > H1 > 0 are given, to find an open subset D of the smooth bounded open set [ohm] [sub] R2 and a function H defined on [ohm]\D such that {div(F(|[nabla] H|2) [nabla] H) - H=0 in [ohm]\D where the function F is analytic positive increasing H=H0 on [par][ohm] H=H1 on [par]D [par]H[horbar][par]n = [par]D. Here we prove the existence of a solution with a domain D having an analytic boundary. We use the Nash-Moser inverse function theorem applied to a degenerate case.

## Cite this article

Alexis Bonnet, Régis Monneau, Distribution of vortices in a type-II superconductor as a free boundary problem: existence and regularity via Nash-Moser theory. Interfaces Free Bound. 2 (2000), no. 2, pp. 181–200

DOI 10.4171/IFB/17