Partial regularity for minimizers of convex integrals with \(L\log L\)-growth
type: Item
instance of: scholarly article
title: Partial regularity for minimizers of convex integrals with \(L\log L\)-growth
zbMATH Open document ID: 0954.49026
DOI: 10.1007/PL00001420
published in: NoDEA. Nonlinear Differential Equations and Applications
publication date: 2001-02-08T00:00:00Z
review text: For a domain \(\Omega\subset \mathbb{R}^n\) the authors discuss the regularity properties of local minimizers \(u\in W^1_{\text{loc}}(\Omega, \mathbb{R}^n)\) of the variational integral \[ {\mathcal F}(u)= \int_\Omega \ln(1+|\nabla u|)|\nabla u|dx\tag{1} \] which is of nearly linear growth. In [\textit{M. Fuchs} and \textit{G. Seregin}, Calc. Var. Partial Differ. Equ. 6, No. 2, 171-187 (1998; Zbl 0929.49022)] partial regularity was established up to dimension \(n=4\). In the present paper the blow-up procedure is refined by a very careful choice of the quantity defining the excess which leads to partial \(C^1\)-regularity without any restriction on the dimension of the domain \(\Omega\). It is worth remarking that in [\textit{G. Mingione} and \textit{F. Siepe}, Z. Anal. Anwend. 18, No. 4, 1083-1100 (1999; preceding review)] even full \(C^1\)-regularity for local minimizers of (1) was proved by using the special structure of (1). Further extensions concerning functionals of the type \[ {\mathcal G}(u)= \int_\Omega g(|\nabla u|) dx\tag{2} \] with \(g\) being of nearly linear growth were presented in [\textit{M. Fuchs} and \textit{G. Mingione}, Manuscr. Math. 102, No. 2, 227-250 (2000)], for example, full \(C^1\)-regularity is true for \[ g(t)= t\ln(1+ \ln..(1+ \ln(1+ t))), \] i.e., for any iterated logarithm.
Mathematics Subject Classification ID: 49N60
Mathematics Subject Classification ID: 35J50
Mathematics Subject Classification ID: 46E30
zbMATH DE Number: 1439285
zbMATH Keywords: convex integrals
zbMATH Keywords: \(L\log L\)-growth
zbMATH Keywords: integral functional
zbMATH Keywords: local minimizers
zbMATH Keywords: variational integral
zbMATH Keywords: partial regularity
zbMATH Keywords: iterated logarithm
Wikidata QID: Q58880928
author: Luca Esposito
author: Giuseppe Mingione
reviewed by: M. Fuchs
MaRDI profile type: Publication
full work available at URL: pl00001420
OpenAlex ID: W2011679247
description: scientific article; zbMATH DE number 1439285