Existence and Regularity of Minimizers of Nonconvex Functionals Depending on u and =u

We consider variational problems of the form min Hwf =u x..qg x,u x..xdx: ugu qH1 V. , 0 0 5 V where f : RNªw0, `xis a possibly nonconvex function with quadratic growth at infinity and g x, u.is Lipschitz continuous and strictly increasing decreasing.in u. We prove the existence and local Lipschitz regularity of solutions for every boundary datum u0gH1 V.lL` V. on the basis of the structure of the epigraph of the convex envelope of f.