Variational inequalities for energy functionals with nonstandard growth conditions

Abstract

We consider the obstacle problem $$\{\begin{array}{l}\text{minimize}I\left(u\right)={\int }_{\Omega }G\left(\nabla u\right)dx\text{among functions}u:\Omega \to R\\ \text{such}\text{that}u{|}_{\partial \Omega }=0\text{and}u\ge \Phi \text{a}\text{.e}\text{.}\end{array}$$ for a given function and a bounded Lipschitz domain $\Omega$ in ${\mathbf{R}}^n$. The growth properties of the convex integrand $G$ are described in terms of a $N$-function $A:\left[0,\infty \right)\to \left[0,\infty \right)$ with . If $n\le 3$, we prove, under certain assumptions on $G,C^{1,\infty }$-partial regularity for the solution to the above obstacle problem. For the special case where $A\left(t\right)=tln\left(1+t\right)$ we obtain $C^{1,\alpha }$-partial regularity when $n\le 4$. One of the main features of the paper is that we do not require any power growth of $G$.