Continuity properties of minimizers of integral functionals in a limit case
type: Item
instance of: scholarly article
title: Continuity properties of minimizers of integral functionals in a limit case
zbMATH Open document ID: 0865.49008
published in: Journal of Mathematical Analysis and Applications
publication date: 1997-06-16T00:00:00Z
review text: The authors prove the continuity of the minimizers \(u\in w^{1,p}(\Omega)\) for an integral functional of the form \({\mathcal F}(u)=\int_\Omega f(x,u(x),Du(x))dx\), where \(\Omega\) is an open subset of \(\mathbb{R}^n\). The function \(f\) is assumed to be a Carathéodory function of the form \(f_0+f_1\), where \(f_0\) satisfies a polynomial growth condition in the second and the third variable, and \(|f_1(x,\eta,\xi)|\leq g(x)|\xi|\), where \(g\) is a positive function in the Lorentz space \(L^{{n\over p-1},{1\over p-1}}(\Omega)\). The results obtained are sharpened versions of known continuity results by \textit{E. De Giorgi} [Mem. Accad. Sci. Torino, Cl. Sci. Fis. Mat. Nat., III. Ser. 3, 25-43 (1957)] and the first author [Boll. Unione Mat. Ital., VII. Ser. B8, No. 2, 257-270 (1994; Zbl 0799.35056)] concerning weak solutions of non-homogeneous elliptic partial differential equations. The motivation for the paper is the study of the regularity properties of elliptic equations which are the Euler-Lagrange equations of integral functionals. As an application of the techniques developed, the authors prove the continuity of solutions of the nonlinear equation \(\text{div }A(x,u,Du)+H(x,u)=\text{div }h\), under the hypotheses that \(h\in L^{{n\over p-1},{1\over p-1}}(\Omega)\), \(p>1\), \(A\) and \(H\) satisfy suitable monotonicity and polynomial growth assumptions.
reviewed by: Paolo Piccione
Mathematics Subject Classification ID: 49J40
Mathematics Subject Classification ID: 35D10
zbMATH DE Number: 930238
zbMATH Keywords: weak solutions of nonlinear elliptic equations
zbMATH Keywords: Lorentz spaces
zbMATH Keywords: \(A\)-minimizers
zbMATH Keywords: continuity of the minimizers
zbMATH Keywords: Euler-Lagrange equations
author: Vincenzo Ferone
author: N. Fusco
MaRDI profile type: Publication
full work available at URL: jmaa.1996.0301
OpenAlex ID: W2024927164
DOI: 10.1006/JMAA.1996.0301
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description: scientific article; zbMATH DE number 930238