Abstract and Applied Analysis
Volume 3 (1998), Issue 1-2, Pages 41-64
doi:10.1155/S1085337598000438

Variational inequalities for energy functionals with nonstandard growth conditions

Abstract

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