NoDEA 2 (1995) 371-416 1021-9722/95/030371-46 $ 1.50+0.20 @ 1995 Birkh~iuser Verlag, Basel Homogenization of integral functionals with linear growth defined on vector-valued functions* Riccardo DE ARCANGELIS Dipartimento di Matematica ed Applicazioni "R. Caccioppoli", Universit5 degli Studi di Napoli "Federico II" Complesso Monte S. Angelo - Edificio T via Cintia, 80126 Napoli, Italy Giuliano GARGIULO D.I.I.M.A. Dipartimento di Ingegneria dell'Informazione e Matematica Applicata, Facolt5 di Ingegneria, Universith di Salerno, via Ponte Don Melillo, 84084 Fisciano (Salerno), Italy 0 Introduction The present paper is concerned with a problem in homogenization theory (cf. for example [5], [7], [18] and [42] for general references and also for a bibliography, even if not exhaustive, on the subject) more exactly with the one for integral functionals with linear growth defined on vector-valued functions. In this context the homogenization problem for some elastic solid materials subjected to a change of phase in which surface energy densities appear or the one for some periodic structures in elastoplasticity can be framed. One of the results proved concerns the convergence of families of Dirichlet minimum problems and is the following (Theorem 5.1). Let us denote by R nm the set of the n x m matrices with real entries, by Y the cube ]0, 1[~ and, for every bounded open set ~2, by BV(f~;R ~) the set of the functions in LI(~2;R "~) whose components have distributional partial derivatives that are Borel measures with finite total variations on f~ (cf. [27], [33] and [46] for a general exposition on dDSu BV-functions). For every BV-function u the symbols IDu], Uu and d~ denote respectively the total variation of the R~'~-valued measure Du, the density of the *Work partially supported by the Italian M.U.R.S.T. 40% National Project "Problemi non lineari ..."