Partial regularity for minimizers of a class of non autonomous functionals with nonstandard growth
type: Item
instance of: scholarly article
title: Partial regularity for minimizers of a class of non autonomous functionals with nonstandard growth
zbMATH Open document ID: 1383.49047
author: Flavia Giannetti
author: Maria Alessandra Ragusa
author: Atsushi Tachikawa
author: Antonia Passarelli Di Napoli
published in: Calculus of Variations and Partial Differential Equations
publication date: 2018-01-11T00:00:00Z
review text: The paper deals with the regularity of the local minimizers of non-autonomous integral functionals of the form \[ \int_\Omega \Phi^{p(x)}\left(\left(A^{\alpha\, \beta}_{i\,j}(x,u)D_iu^{\alpha}\,D_ju^{\beta}\right)^{\frac{1}{2}}\right) dx, \] where \(\Phi\) is an Orlicz function satisfying both the \(\Delta_2\) and \(\nabla_2\) conditions, \(p(x)\) is continuous and the function \(A(x,s)=\left( A^{\alpha\, \beta}_{i\,j}(x,s)\right) \) is uniformly continuous. Under suitable assumptions on the functions \(\Phi\) and \(p(x)\) the authors prove if \(u \in W^{1,p(x)}( \Omega)\) is a minimizer then \(u \in C^{0,\nu}( \Omega_0)\), for any \(\nu \in (0,1)\) where \( \Omega_0 \subset \Omega\) is an open subset with \(\dim_{\mathcal{H}}(\Omega- \Omega_0)<n-mp\), where \(m= \min_{\Omega}\,p\). Moreover, if the coefficients \(A^{\alpha\, \beta}_{i\,j}\) do not depend on \(u\), then \(\Omega_0=\Omega\).
reviewed by: Elvira Mascolo
Mathematics Subject Classification ID: 49N60
Mathematics Subject Classification ID: 35J50
zbMATH DE Number: 6824798
zbMATH Keywords: no standard growth
zbMATH Keywords: local minimizers
zbMATH Keywords: partial regularity
MaRDI profile type: Publication
full work available at URL: s00526-017-1248-z
OpenAlex ID: W2761221883
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DOI: 10.1007/S00526-017-1248-Z
description: scientific article; zbMATH DE number 6824798