# 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

[BibTeX]

Published Paper

Inserted: 28 sep 2004
Last Updated: 2 apr 2007

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

Abstract:

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