Functionals with linear growth defined on vector valued BV functions (Q1113447)

The work is concerned with the study of the relaxation (w.r.t. \(L^ 1\) convergence) of integral functionals of the following kind \[ F(u,A)=\int_{A}f(x,u,\nabla u)dx,\quad u\in C^ 1(\Omega,{\mathbb{R}}^ k),\quad A\subset \Omega \subset {\mathbb{R}}^ n, \] where f is a continuous integrand such that f(x,u,\(\cdot)\) is convex, sublinear and 1-positively homogeneous. The authors give, in the lattice of Borel measures, an upper bound to the relaxed functional for \(u\in BV(\Omega,{\mathbb{R}}^ k)\), by means of the sum of two suitable integrals, focusing on the contribution of the ``jumps'' of the argument u. In the last section applications have been given to prove that some relaxed functionals are traces of Borel measures for any \(u\in L^ 1(\Omega,{\mathbb{R}}^ k)\). The techniques used are from real analysis and measure theory, and a fundamental point of view is the ``localization'' of a functional by considering it also depending on the domain of integration. The work could be seen as a first step towards a generalization of the results obtained by \textit{G. Dal Maso} for integrals defined on real valued functions [Manuscr. Math. 30, 387-416 (1980; Zbl 0435.49016)]. Similar results, for integrals depending on vector valued Radon measures, have been obtained by G. Bouchitte (to appear) using convex functional analysis.