Existence and regularity of minimizers of nonconvex functionals depending on u and delu

We consider variational problems of the form min{integral(Omega)[f(del u(x)) + g(x,u(x))]dx:u is an element of u(0) + H-0(1)(Omega)}, where f: R-N --> [0,infinity] is a possibly nonconvex function with quadratic growth at infinity and g(x, u) is Lipschitz continuous and strictly increasing (decreasing;) in Ir. We prove the existence and local Lipschitz regularity of solutions for every boundary datum u(0) is an element of H-1(Omega) boolean AND L-infinity(Omega) on the basis of the structure of the epigraph of the convex envelope of f. (C) 1999 Academic Press.