Regularity results up to the boundary for minimizers of \(p(x)\)-energy with \(p(x)>1\)
type: Item
instance of: scholarly article
title: Regularity results up to the boundary for minimizers of \(p(x)\)-energy with \(p(x)>1\)
zbMATH Open document ID: 1359.49013
author: Atsushi Tachikawa
author: Kunihiro Usuba
published in: Manuscripta Mathematica
publication date: 2017-02-02T00:00:00Z
review text: Given a bounded domain \(\Omega\subset\mathbb{R}^m\) with \(C^1\)-smooth boundary, the authors obtain partial regularity in \(\overline{\Omega}\) for the minimizers \(u: \Omega\to \mathbb{R}^n\) of the \(p(x)\)-growth type functional \[ \int_\Omega \left( A^{\alpha\beta}_{ij}(x,u)D_\alpha u^i(x)D_\beta u^j(x)\right)^{p(x)/2}dx \] with a Hölder continuous \(p(x)>1.\) It is assumed that \(A^{\alpha\beta}_{ij}(x,u)\) are bounded Hölder continuous functions on \(\Omega\times \mathbb{R}^n,\) \(A^{\alpha\beta}_{ij}(x,u)=A^{\beta\alpha}_{ji}(x,u)\) and \[ \lambda |\xi|^2\leq A^{\alpha\beta}_{ij}(x,u) \xi^i_\alpha\xi^j_\beta\leq \Lambda |\xi|^2. \] In the particular case when the coefficients satisfy the splitting condition \[ A^{\alpha\beta}_{ij}(x,u)=g^{\alpha\beta}(x) G_{ij}(x,u), \] it is shown that the minimizers have no singular points on \(\partial\Omega\).
reviewed by: Dian K. Palagachev
Mathematics Subject Classification ID: 49N60
Mathematics Subject Classification ID: 35J50
Mathematics Subject Classification ID: 58E20
zbMATH DE Number: 6680077
zbMATH Keywords: \(p(x)\)-growth functional
zbMATH Keywords: minimizer
zbMATH Keywords: partial regularity
MaRDI profile type: Publication
full work available at URL: s00229-016-0855-x
OpenAlex ID: W2469597985
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DOI: 10.1007/S00229-016-0855-X
description: scientific article; zbMATH DE number 6680077