Variational Integrals on Orlicz–Sobolev Spaces

Abstract

We consider vector functions $u:{\mathbb{R}}^n\supset \Omega \to {\mathbb{R}}^N$ minimizing variational integrals of the form ${\int }_{\Omega }G(▽u)dx$ with convex density $G$ whose growth properties are described in terms of an $N$-function $A:(0,\infty )\to (0,\infty )$ with limsup${}_{t\to \infty }A(t)t^{–2}<\infty$. We then prove - under certain technical assumptions on $G$ - full regularity of $u$ provided that $n=2$, and partial $C^1$-regularity in the case $n\ge 3$. The main feature of the paper is that we do not require any power growth of $G$.