Relaxation of singular functionals defined on Sobolev spaces

Free access article

Issue		ESAIM: COCV Volume 5, 2000


Page(s)		71 - 85
DOI		10.1051/cocv:2000102

DOI: 10.1051/cocv:2000102

ESAIM: COCV, January 2000, Vol. 5, p. 71-85

Relaxation of singular functionals defined
on Sobolev spaces

Relaxation de fonctionnelles singulières sur des espaces de Sobolev

Hafedh Ben Belgacem
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.
Present address: Max-Planck Institute for Mathematics in the Sciences, Inselstr. 22-26, 04103 Leipzig, Germany (Hafedh.Belgacem@mis.mpg.de)

Received December 1, 1998. Revised November 17, 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 $\scriptstyle p>1$ :

$\begin{displaymath}\scriptstyle -c_0\leq Rf(F)\leq c(1+\Vert F\Vert^p)\ \hbox{for all}\ F\in M^{m\times n},\end{displaymath}$

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

$\begin{displaymath}\scriptstyle I(u)=\int_{\O}f(\nabla u(x))\ dx.\end{displaymath}$

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:

$\begin{displaymath}\scriptstyle \bar I(u)=\int_{\O}Q[Rf](\nabla u(x))\ dx,\end{displaymath}$

where for a given function $\scriptstyle g$ , $\scriptstyle Qg$ denotes its quasiconvex envelope.

Résumé: On considère une fonction Borel mesurable $f:M^{m\times n} \rightarrow \bar\mathbb{R}$ 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,c₀>0fixés. Étant donné un ouvert borné $\Omega$ 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 Ipour 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.

Keywords and phrases: Rank-one convexity, quasiconvexity, weak lower semicontinuity.

AMS Subject Classification: 49-xx.

Article without figures.

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