Regularity of the stress field for degenerate and/or singular elliptic problems
type: Item
instance of: scholarly article
title: Regularity of the stress field for degenerate and/or singular elliptic problems
author: Q6119171
published in: Annales Mathématiques Blaise Pascal
publication date: 2024-12-12T00:00:00Z
review text: This article investigates locally Lipschitz solutions of the equation\N\[\N\mathrm{div}\, G(\nabla u)=f,\N\]\Nwith \(G\colon \mathbb R^N \to\mathbb R^N\) a monotone field, and gives various sufficient conditions on \(N\), \(G\) and \(f\) which ensure that the stress-field \(G(\nabla u)\) is continuous. These equations are of elliptic type, but for merely monotone fields \(G\), ellipticity might be very degenerate.\N\NMost results presented here assume that \(N=2\) and \(f\) is constant, and establish continuity of the stress field for a broad class of monotone fields \(G\), for instance gradients of convex functions with some regularity or symmetry requirements. For general dimension \(N\geq 2\) and smooth enough right-hand side \(f\), it is assumed that the complement of the set where \(G\) is locally elliptic (in the sense that the symmetric part of \(\nabla G\) is locally bounded and positive definite) is contained in a plane and satisfies some topological constraint.
reviewed by: Xavier Lamy
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 35J62
Mathematics Subject Classification ID: 35J70
Mathematics Subject Classification ID: 35J75
Mathematics Subject Classification ID: 49N99
zbMATH DE Number: 7957319
zbMATH Keywords: regularity
zbMATH Keywords: elliptic PDE
zbMATH Keywords: dimension two
MaRDI profile type: Publication
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DOI: 10.5802/AMBP.427
description: scientific article; zbMATH DE number 7957319