Free-boundary regularity for a problem arising in superconductivity
type: Item
instance of: scholarly article
title: Free-boundary regularity for a problem arising in superconductivity
zbMATH Open document ID: 1072.35203
author: Luis Caffarelli
author: Henrik Shahgholian
author: Jorge A. Salazar
published in: Archive for Rational Mechanics and Analysis
publication date: 2004-09-22T00:00:00Z
review text: For any \(\rho >0\) and \(x_0\) given in \(\mathbb{R}^n\) consider the problem of finding \(u \in C^{1,1}(B_{\rho}(x_0))\) such that \[ \Delta u=\chi_{\{| \nabla u| >0\}} \] and \[ | \nabla u(x_0)| =0,\,\,\,| u(x)-u(x_0)| \leq c(1+| x| ^2) \] with \(c >0\). It is known that for all \((c,\rho, x)\) all solutions \(u\) have uniform \(C^{1,1}\) estimate depending only on \(c\) and \(n\), and that the free boundary \(\partial \{| \nabla u| >0\}\) has locally finite \((n-1)\)-Hausdorff measure. Let \(P(c,\rho,x_0)\) denote the class of all solutions of the problem above and for each of them consider the sets \[ \Omega = \text{int} {\{| \nabla u| >0}\},\,\,\,\,\Omega^c=\mathbb{R}^n\backslash \Omega. \] The density function of \(\Omega^c\) is defined by \[ \delta_{\rho}(u)=\dfrac{MD(\Omega^c\cap B_{\rho})}{\rho} \] where \(MD\) denotes the minimal diameter (i.e. the infimum of the thickness of layers containing the set). The main result of the paper is the following. There exists a modulus of continuity \(\sigma\) such that if \(\delta_{\rho_0}(u)>\sigma(\rho_0)\) for some \(\rho_0<\frac{1}{2}\), then for any \(u\) in the class \(P(c,1,x_0)\) the boundary \(\partial\{| \nabla u| >0\}\) is a \(C^1\) graph in \(B(x_0,c_0\rho^2_0)\), where \(c_0\) depends only on \(c\) and on \(n\). The main tool employed is the blow up technique.
Mathematics Subject Classification ID: 35R35
Mathematics Subject Classification ID: 82D55
Mathematics Subject Classification ID: 35B65
zbMATH DE Number: 2101871
zbMATH Keywords: superconductors
zbMATH Keywords: regularity of the free boundary
reviewed by: Antonio Fasano
MaRDI profile type: Publication
full work available at URL: s00205-003-0287-0
OpenAlex ID: W1976172215
DOI: 10.1007/S00205-003-0287-0
description: scientific article; zbMATH DE number 2101871