A CHARACTERIZATION OF C1-CONVEX SETS IN SOBOLEV SPACES

In this paper we prove that, if K is a closed subset of W0(1,p)(OMEGA, R(m)) with 1 < p < +infinity and m greater-than-or-equal-to 1, then K is stable under convex combinations with C1 coefficients if and only if there exists a closed and convex valued multifunction from OMEGA to R(m) such that (*) K = {u is-an-element-to W0(1,p)(OMEGA, R(m)) : u(x) is-an-element-of K(x) for q.e. x is-an-element-of OMEGA}. The case m = 1 has already been studied by using truncation arguments which rely on the order structure of R (see [2]). In the case m > 1 a different approach is needed, and new techniques involving suitable Lipschitz projections onto convex sets are developed. Our results are used to prove the stability, with respect to the convergence in the sense of Mosco, of the class of convex sets of the form (*); this property may be useful in the study of the limit behaviour of a sequence of variational problems of obstacle type.