Boundary regularity for elliptic systems under a natural growth condition
type: Item
instance of: scholarly article
title: Boundary regularity for elliptic systems under a natural growth condition
zbMATH Open document ID: 1248.35073
author: Lisa Beck
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 2012-05-08T00:00:00Z
full work available at URL: https://arxiv.org/abs/1002.1935
review text: This paper is concerned with the existence of regular boundary points for the gradient of bounded vector-valued weak solutions of the Dirichlet problem to the second order elliptic system, \(-\)div \( a(\cdot, u, D u)=b(\cdot,u,D u)\), in a bounded domain \(\Omega\subset\mathbb R^n\), \((n=2,3,4)\), of class \(C^{1,\alpha}\), where \(1/2\leq \alpha<1\), with boundary value \(u_0\in C^{1,\alpha}(\bar\Omega;\mathbb{R}^N)\). The coefficients \(a\) are assumed to be Hölder continuous with exponent \(\alpha\) with respect to the first two variables, and of class \(C^1\) in the last variable, and to satisfy quadratic growth and ellipticity conditions. The existence of regular boundary points for \(Du\) is proved if \(b\) is a Carathéodory map that satisfies \(|b(x,v,z)|\leq L+L_2|z|^2\), for all \((x,v,z)\in \Omega\times \mathbb{R}^N\times\mathbb{R}^{nN}\) such that \(|v|\leq M\), where \(L\) is the upper bound of \(D_za\), \(M\) is the \(L^\infty\)-bound of the weak solution \(u\), \(L_2\) is such that \(2L_2M<\nu\), with \(\nu\) being the ellipticity constant. The strategy of the proof relies on the accomplishment of fractional Sobolev estimates for tangential and full derivatives of \(u\), and an interpolation technique.
reviewed by: Luisa Consiglieri
Mathematics Subject Classification ID: 35J47
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 35J57
Mathematics Subject Classification ID: 35D30
zbMATH DE Number: 6031767
zbMATH Keywords: regularity theory for elliptic systems
zbMATH Keywords: dimension reduction
zbMATH Keywords: regular boundary points
MaRDI profile type: Publication
OpenAlex ID: W2136148434
arXiv ID: 1002.1935
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DOI: 10.1007/S10231-010-0163-0
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description: scientific article; zbMATH DE number 6031767