Variational inequalities for energy functionals with nonstandard growth conditions

Abstract

We consider the obstacle problem {minimize????????I(u)=?OG(?u)dx??among functions??u:O?Rsuch?that???????u|?O=0??and??u=F??a.e. for a given function F?C2(O¯),F|?O<0 and a bounded Lipschitz domain O in Rn. The growth properties of the convex integrand G are described in terms of a N-function A:[0,8)?[0,8) with limt?8¯A(t)t-2<8. If n=3, we prove, under certain assumptions on G,C1,8-partial regularity for the solution to the above obstacle problem. For the special case where A(t)=tln(1+t) we obtain C1,a-partial regularity when n=4. One of the main features of the paper is that we do not require any power growth of G.