Further results on a problem of homogenization with constraints on the gradient
type: Item
instance of: scholarly article
title: Further results on a problem of homogenization with constraints on the gradient
zbMATH Open document ID: 0574.35006
DOI: 10.1007/BF02790187
author: Luciano Carbone
author: S. Salerno
published in: Journal d'Analyse Mathématique
publication date: 1985-01-01T00:00:00Z
review text: Die Verff. untersuchen das von einem reellen Parameter \(h\) abhängige Variationsproblem \[ (1)\quad \min_{u\in S_ h} (\int_{\Omega}f(hx,Du(x))dx+ \int_{\Omega}\alpha(x)u(x)dx). \] Hierbei ist \(\Omega \subset {\mathbb{R}}^ n\) ein Gebiet \(f\), \(\phi\) sind periodische Funktionen, und \[ S_ h:=\{u\in \text{Lip}_{loc}({\mathbb{R}}^ n),\quad | Du(x)| \leq \phi(hx)\text{ fast überall, } u=0 \text{ auf }\partial\Omega\}. \] Die Verff. [J. Math. Anal. Appl. 90, 219- 250 (1982; Zbl 0499.49007)] zeigen unter der Voraussetzung (2) \(0<m<\phi(x)\) \(\forall x\in {\mathbb{R}}^ n\), daß die Lösung \(u_ h\) von (1) für \(h\to \infty\) gegen die Lösung eines Minimierungsproblems konvergiert. In der vorliegenden Arbeit wird dieses Ergebnis erweitert auf den Fall, daß statt (2) die schwächere Bedingung \[ 0<m\leq \phi(x)\quad \forall x\in [0,1]^ n\setminus [-\vartheta,+\vartheta]^ n \] mit einem \(\vartheta\in [0,)\) gilt.
Mathematics Subject Classification ID: 35A15
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 3916807
zbMATH Keywords: homogenization
zbMATH Keywords: constraints on the gradient
zbMATH Keywords: variational problem
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3916807