Abstract

We study variational problems for the functional F(u) = ??f(x, u(x), Lu(x)) dx where u ? u0 + V, with V being any closed linear subspace of W2,p(?) containing W0 2,p(?), ? is a bounded open set, p > 1, L is a differential operator of second order. We determine the greatest lower semicontinuous function majorised by F for the weak topology of W2,p, for its sequential version if f satisfies no coercivity assumption, showing that in both cases the relaxed functional is expressed in terms of the function ? ? f**(x, u, ?). Finally, an existence result in case f (not necessarily convex) depending only on the Laplacian, is given.