A sharp attainment result for nonconvex variational problems

We consider the problem of minimizing autonomous, multiple integrals like (P) min {integral(Omega) f(u, delu) dx : u is an element of u(0) + W-0(1,p)(Omega)} where f: R x R-N --> [0, infinity) is a continuous, possibly nonconvex function of the gradient variable delu. Assuming that the bipolar function f** of f is affine as a function of the gradient delu on each connected component of the sections of the detachment set D = {f ** < f}, we prove attainment for (P) under mild assumptions on f and f **. We present examples that show that the hypotheses on f and f ** considered here for attainment are essentially sharp.