Calculus of Variations and Geometric Measure Theory

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

Relaxation in BV of integral functionals defined on Sobolev functions with values in the unit sphere

created on 28 Sep 2004
modified by alicandr on 02 Apr 2007


Published Paper

Inserted: 28 sep 2004
Last Updated: 2 apr 2007

Journal: J. Convex Anal.
Volume: 14
Number: 1
Pages: 69-98
Year: 2007


In this paper 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 $$ \bar F(u)=\int{\Omega}f(x,u,\nabla u)\,dx+\int{S(u)}K(x,u-,u+,\nuu)\,d{\cal H}{N-1} + \int\Omega f\infty (x,u,d C(u)), $$ where the suface energy density $K$ is defined by a suitable Dirichlet-type problem.

Keywords: relaxation, BV functions, unit sphere