Relaxation in BV of Integral Functionals Defined on Sobolev Functions with Values in the Unit Sphere

R. Alicandro ; A. Corbo Esposito ; C. Leone

Journal of convex analysis, Tome 14 (2007) no. 1, pp. 69-98

Résumé

We study the relaxation with respect to the $L^1$ norm of integral functionals of the type $$ F(u)=\int_\Omega f(x,u,\nabla u)\,dx\ ,\quad u\in W^{1,1}(\Omega;S^{d-1}) $$ where $\Omega$ is a bounded open set of $ R^N$, $S^{d-1}$ denotes the unite sphere in $ R^d$, $N$ and $d$ being any positive integers, and $f$ satisfies linear growth conditions in the gradient variable. In analogy with the unconstrained case, we show that, if, in addition, $f$ is quasiconvex in the gradient variable and satisfies some technical continuity hypotheses, then the relaxed functional $\overline F$ has an integral representation on $BV(\Omega;S^{d-1})$ of the type $$ \bar F(u)=\int_{\Omega}f(x,u,\nabla u)\,dx+\int_{S(u)}K(x,u^-,u^+,\nu_u)\,d{\cal H}^{N-1} + \int_\Omega f^\infty (x,u,d C(u)), $$ where the suface energy density $K$ is defined by a suitable Dirichlet-type problem.

Détail

Classification : 49J45,74Q99
Mots-clés : Relaxation, unit sphere, BV-functions