Relaxation of singular functionals defined on Sobolev spaces

Hafedh Ben Belgacem

doi:10.1051/cocv:2000102

All issues

Volume 5 (2000)

ESAIM: COCV, 5 (2000) 71-85

Abstract

Free Access

Issue		ESAIM: COCV Volume 5, 2000


Page(s)		71 - 85
DOI		https://doi.org/10.1051/cocv:2000102
Published online		15 August 2002

ESAIM: COCV 5 (2000) 71-85

Relaxation of singular functionals defined on Sobolev spaces

Hafedh Ben Belgacem¹^,2

¹ Département de Mathématiques, Institut Préparatoire aux Études d'Ingénieurs de Sfax, Route Menzel Chaker - Km 0,5, BP. 805, 3000 Sfax, Tunisia; Fax: (00-216) 4. 246. 347.
² Max-Planck Institute for Mathematics in the Sciences, Inselstr. 22-26, 04103 Leipzig, Germany; Hafedh.Belgacem@mis.mpg.de.

Received: 1 December 1998
Revised: 17 November 1999

Abstract

In this paper, we consider a Borel measurable function on the space of $\scriptstyle m\times n$ matrices $\scriptstyle f: M^{m\times n}\rightarrow \bar{\mathbb{R}}$ taking the value $\scriptstyle +\infty$ , such that its rank-one-convex envelope $\scriptstyle Rf$ is finite and satisfies for some fixed $\scriptstyle p>1$ : $\scriptstyle -c_0\leq Rf(F)\leq c(1+\Vert F\Vert^p)\ \hbox{for all}\ F\in M^{m\times n},$ where $\scriptstyle c,c_0>0$ . Let $\scriptstyle\O$ be a given regular bounded open domain of $\scriptstyle \mathbb{R}^n$ . We define on $\scriptstyle W^{1,p}(\O;\mathbb{R}^m)$ the functional $\scriptstyle I(u)=\int_{\O}f(\nabla u(x))\ dx.$ Then, under some technical restrictions on $\scriptstyle f$ , we show that the relaxed functional $\scriptstyle\bar I$ for the weak topology of $\scriptstyle W^{1,p}(\O;\mathbb{R}^m)$ has the integral representation: $\scriptstyle \bar I(u)=\int_{\O}Q[Rf](\nabla u(x))\ dx,$ where for a given function $\scriptstyle g$ , $\scriptstyle Qg$ denotes its quasiconvex envelope.

Résumé

On considère une fonction Borel mesurable $f:M^{m\times n}\rightarrow \overline{\mathbb{R}}$ qui prend la valeur $+\infty$ , dont l'enveloppe rang-1-convexe Rf est finie et satisfait pour un certain p>1, $-c_0 \leq Rf(F)$ $\leq c(1+\Vert F\Vert^p), \forall F\in M^{m\times n},$ avec $c,c_0>0$ fixés. Étant donné un ouvert borné Ω de $\mathbb{R}^n$ , on introduit la fonctionnelle $I(u):=\int_{\Omega}f(\nabla u(x))\ dx,$ pour $u\in W^{1,p}(\Omega;\mathbb{R}^m)$ . On montre alors sous quelques hypothèses supplémentaires concernant f, que la relaxée $\bar I$ de I pour la topolgie faible de $W^{1,p}(\Omega;\mathbb{R}^m)$ admet la représentation suivante : $\bar I(u)=\int_{\Omega}Q{ Rf}(\nabla u(x))\ dx,$ où pour une fonction donnée g, Qg désigne son enveloppe quasi-convexe.

Mathematics Subject Classification: 49-xx

Key words: Rank-one convexity / quasiconvexity / weak lower semicontinuity.

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