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Journal of Convex Analysis 14 (2007), No. 1, 069--098
Copyright 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., UniversitÓ di Cassino, Via Di Biasio, 03043 Cassino, Italy
alicandr@unicas.it

Antonio Corbo Esposito
D.A.E.I.M.I., UniversitÓ di Cassino, Via Di Biasio, 03043 Cassino, Italy
corbo@unicas.it

Chiara Leone
Dip. di Matematica "R. Caccioppoli", UniversitÓ 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

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