\(C^{\infty}\)-regularity of the interface of the evolution \(p\)-Laplacian equation
type: Item
instance of: scholarly article
title: \(C^{\infty}\)-regularity of the interface of the evolution \(p\)-Laplacian equation
zbMATH Open document ID: 0929.35071
author: Panagiota Daskalopoulos
author: Richard S. Hamilton
published in: Mathematical Research Letters
publication date: 1999-09-23T00:00:00Z
review text: The authors study the regularity of the interface of the evolution \(p\)-Laplacian equation \[ u_t = \text{div}(| Du| ^{p-2}Du),\qquad (x,t)\in \mathbb R^n\times [0,\infty), \] in the range of exponents \(p>2\) and \(n\geq 2\). The equation becomes degenerate when \(Du=0\), therefore solutions are not always smooth. The \(C^{\infty}\)-regularity of the interface in dimension \(n=1\) is the well-known result. In order to prove regularity properties the authors introduce into consideration a convenient ''pressure'' function \(f\) setting \[ u = (p-2)/(p-1)f^{(p-1)/(p-2)}, \] near the free boundary \(u=0\). The main goal of the article under review is to show that, under certain assumptions on the initial data, the function \(f\), defined as above in terms of \(u\), is smooth up to the interface, for time \(0<t\leq T\), for some \(T>0\). As a consequence, the free boundary is smooth. The proof is based on the idea of introducing an appropriate change of variables which transforms the free boundary problem to an initial value problem with fixed boundary.
Mathematics Subject Classification ID: 35K65
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 35B45
Mathematics Subject Classification ID: 35D10
zbMATH DE Number: 1309087
zbMATH Keywords: change of variables
zbMATH Keywords: \(C^{\infty}\)-regularity
reviewed by: Vladimir N. Grebenev
MaRDI profile type: Publication
full work available at URL: mrl.1998.v5.n5.a11
OpenAlex ID: W2316538340
DOI: 10.4310/MRL.1998.V5.N5.A11
description: scientific article; zbMATH DE number 1309087