Boundary continuity of solutions to a basic problem in the calculus of variations

This article studies the problem of minimizing integral(Omega) F(Du(x))dx over the functions u is an element of W-1,W-1(Omega) that assume given boundary values phi on Gamma := partial derivative Omega. The Lagrangian F and the domain Omega are assumed convex but not necessarily strictly conxex. When phi is continuous and F superlinear, we prove the existence of a minimum which is continuous on the closure of Omega. We also consider the case when F is not superlinear.