Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Previous Article Journal of Convex Analysis 14 (2007), No. 1, 069--098Copyright Heldermann Verlag 2007 Relaxation in BV of Integral Functionals Defined on Sobolev Functions with Values in the Unit Sphere Roberto Alicandro D.A.E.I.M.I., Università di Cassino, Via Di Biasio, 03043 Cassino, Italy alicandr@unicas.it Antonio Corbo Esposito D.A.E.I.M.I., Università di Cassino, Via Di Biasio, 03043 Cassino, Italy corbo@unicas.it Chiara Leone Dip. di Matematica "R. Caccioppoli", Università di Napoli, Via Cintia, 80126 Napoli, Italy chileone@unina.it [Abstract-pdf] 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. Keywords: Relaxation, unit sphere, BV-functions. MSC: 49J45,74Q99 [ Fulltext-pdf  (243  KB)] for subscribers only.