Regularity results for a class of functionals with non-standard growth
type: Item
instance of: scholarly article
title: Regularity results for a class of functionals with non-standard growth
zbMATH Open document ID: 0984.49020
author: Emilio Acerbi
author: Giuseppe Mingione
published in: Archive for Rational Mechanics and Analysis
publication date: 2002-04-23T00:00:00Z
review text: The very interesting paper under review deals with regularity properties of local minimizers for the functional \[ F(u,\Omega)=\int_\Omega f(x,Du) dx \] with \(f(x,z)\) being a Carathéodory integrand which satisfies the non-standard growth condition \[ |z|^{p(x)}\leq f(x,z)\leq L(1+|z|^{p(x)})\qquad \forall z\in{\mathbb R}^n,\quad x\in \Omega, \] where \(p\colon\;\Omega\to(1,+\infty)\) is a continuous function with modulus of continuity \(\omega(R).\) Assuming convexity of \(F\) and \[ \limsup_{R\to 0} \omega(R)\log\left({1}\over{R}\right)=0, \] the authors derive \(C^{0,\alpha}_{\text{ loc}}\)-smoothness \(\forall \alpha\in(0,1),\) of any \(W^{1,1}_{\text{ loc}}\) minimizer of \(F(u,\Omega).\) Further, \(C^2\)-regularity of \(f(x,z)\) with respect to \(z\) and \(p\in C^{0,\alpha},\) \(\alpha\in(0,1],\) imply local Hölder continuity of \(Du.\) The technique used relies on perturbation at the growth exponent \(p(x)\) and precise estimates in the Orlicz space \(L\log L(\Omega)\) due to \textit{T. Iwaniec} and \textit{A. Verde} [J. Funct. Anal. 169, No. 2, 391-420 (1999; Zbl 0961.47020)].
reviewed by: Dian K. Palagachev
Mathematics Subject Classification ID: 49N60
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 49J45
zbMATH DE Number: 1586360
zbMATH Keywords: regularity of minimizers
zbMATH Keywords: non-standard growth conditions
zbMATH Keywords: integral functionals
Wikidata QID: Q58880926
MaRDI profile type: Publication
full work available at URL: s002050100117
OpenAlex ID: W2003751296
DOI: 10.1007/S002050100117
Recommended article: Orlicz spaces, α-decreasing functions, and the Δ2condition
Recommended article: Higher differentiability of minimizers of variational integrals with variable exponents
Recommended article: Q4219752
Recommended article: Q4265706
Recommended article: Regularity of \(\omega \)-minimizers for a class of functionals with non-standard growth
Recommended article: Regularity results for non-autonomous functionals with \(L\log {L}\)-growth and Orlicz Sobolev coefficients
Recommended article: Hölder regularity of quasiminimizers under generalized growth conditions
Recommended article: Q3079716
Recommended article: Q3398919
Recommended article: Partial regularity for minimizers of a class of non autonomous functionals with nonstandard growth
description: scientific article; zbMATH DE number 1586360