Limits of minimum problems with convex

Abstract

This work deals with the asymptotic behaviour of a sequence of minimum

problems on a Sobolev space of vector valued functions subject to constraints of obstacle type. We consider sequences of the formMin{ n W ( x , Du(z)) dr : u é E H^1> _O,(Ω R^m), u(x) é K_n ( x ) for p.e. x ´ A}, (∗) where Ω is an open subset of Rⁿ, m 2 1, W is quadratic in the second variable and non-negative, A is an open subset of R, and Kh (h ´ N) is a closed and convex valued multifunction from Ω to Rm.The well-known relaxation phenomenon of the scalar case still takes place for (∗); this is obtained by proving a compactness result for a general class of constraint functionals. Applications are given to Dirichlet problems in perforated domains for the usual energy functional of linearized elasticity.