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(\triangledown 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 ≥ 3 . The main feature of the paper is that we do not require any power growth of G .