The boundary regularity of non-linear parabolic systems. II
type: Item
instance of: scholarly article
title: The boundary regularity of non-linear parabolic systems. II
zbMATH Open document ID: 1194.35085
author: Verena Bögelein
author: Frank Duzaar
author: Giuseppe Mingione
published in: Annales de l'Institut Henri Poincaré. Analyse Non Linéaire
publication date: 2010-02-09T00:00:00Z
review text: This is a sequel of [\textit{V.~Bögelein, F.~Duzaar} and \textit{G.~Mingione}, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 27, No.~1, 201--255 (2010; Zbl 1194.35086)], where the authors have proved a rather general boundary regularity criterion for nonlinear parabolic systems of the form \(u_t-\text{div}\,a(x,t,u,Du)=0\). The boundary data are of Cauchy-Dirichlet type, given by a function \(g\) on the initial time slice and the lateral boundary. The function \(g\) is assumed to be continuous with Hölder continuous spatial derivative and time derivative in some appropriate Morrey space. The growth, regularity and ellipticity assumptions on \(a\) are somewhat stronger than in the first paper, but completely natural. The authors prove existence of regular initial and lateral boundary points for weak solutions, which has been open except for very special cases. More precisely, they estimate the parabolic Hausdorff dimension of the singular set in terms of the Hölder exponents from the assumptions, finding that, once these exponents are not too small, the singular sets have lower dimension than the boundary components they are subsets of. Which means that almost every boundary point must be regular. The proofs involve technically involved continuations of methods many of which have been developped by the authors in earlier work, including fractional difference quotients and the relations between fractional Sobolev and Nikolskii spaces.
reviewed by: Andreas Gastel
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 35K51
Mathematics Subject Classification ID: 35K59
Mathematics Subject Classification ID: 35K65
zbMATH DE Number: 5667950
zbMATH Keywords: parabolic system
zbMATH Keywords: Cauchy-Dirichlet problem
zbMATH Keywords: boundary regularity
zbMATH Keywords: Morrey space
zbMATH Keywords: parabolic Hausdorff dimension of the singular set
zbMATH Keywords: fractional difference quotients
zbMATH Keywords: fractional Sobolev and Nikolskii spaces
Wikidata QID: Q58880852
MaRDI profile type: Publication
full work available at URL: j.anihpc.2009.09.002
OpenAlex ID: W2071617189
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DOI: 10.1016/J.ANIHPC.2009.09.002
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description: scientific article; zbMATH DE number 5667950