On the minimum problem for nonconvex, multiple integrals of product type

We consider the problem of minimizing multiple integrals of product type, i.e. (P) min [GRAPHICS] where Omega is a bounded, open set in R-N, f: R-N --> [0, infinity) is a possibly nonconvex, lower semicontinuous function with p-growth at infinity for some 1 < p < infinity and the boundary datum u(0) is in W-1,W-p(Omega) boolean AND L-infinity(Omega) (or simply in W-1,W-p(Omega) if N < p < infinity). Assuming that the convex envelope f** of f is affine on each connected component of the set {f** < f}, we prove attainment for (P) for every continuous, positively bounded below function g such that (i) every point t <is an element of> R is squeezed between two intervals where g is monotone and (ii) g has no strict local minima. This shows in particular that the class of coefficents g that yield existence to (P) is dense in the space of continuous, positive functions on R. We present examples which show that these conditions for attainment are essentially sharp.