On the curvature of the free boundary for the obstacle problem in two dimensions
type: Item
instance of: scholarly article
title: On the curvature of the free boundary for the obstacle problem in two dimensions
zbMATH Open document ID: 1072.35204
author: Björn Gustafsson
author: Makoto Sakai
published in: Monatshefte für Mathematik
publication date: 2004-10-28T00:00:00Z
review text: Let \(B\) be a ball in \(\mathbb{R}^2\) and consider the obstacle problem: \[ u\in C^1(B),\,\,\,u\geq 0,\,\,\,\Delta u=\chi_{\Omega}\,\,\,{\text{in}}\,\,\, B \] where \(\Omega\) is the noncoincidence set and \(\chi_{\Omega}\) is its characteristic function. The condition \(u\in C^1 (B)\) implies \(u=| \nabla u| =0\) on the free boundary \(\partial \Omega \backslash B\). Fixed boundary conditions are prescribed away from \(B\). The aim of the paper is to prove that if \(B\) is centered at the origin and we take the coordinate \(x, y\), the following result holds true: if \(\Omega^+ =\Omega \cap \{y>0\}\) is relatively compact in \(B\), then there exists a function \(r(x)\) such that \[ \Omega^+ =U_{x\in\Omega\cap\mathbb{R}}B(x,r(x))^+, \] where \(B^+(x,r)\) is the half disc with center at the point \((x,0)\) and of radius \(r\), lying in the half plane \(\{y>0\}\).
Mathematics Subject Classification ID: 35R35
Mathematics Subject Classification ID: 35J85
zbMATH DE Number: 2111468
zbMATH Keywords: obstacle problem
zbMATH Keywords: curvature of the free boundary
reviewed by: Antonio Fasano
MaRDI profile type: Publication
full work available at URL: s00605-004-0233-8
OpenAlex ID: W2125351897
DOI: 10.1007/S00605-004-0233-8
description: scientific article; zbMATH DE number 2111468