Minimizing Nonconvex, Simple Integrals of Product Type

We consider the problem of minimizing simple integrals of product type, i.e. where f: →[0, ∞] is a possibly nonconvex, lower semicontinuous function with either superlinear or slow growth at infinity. Assuming that the relaxed problem ( **) obtained from ( ) by replacing f with its convex envelope f** admits a solution, we prove attainment for ( ) for every continuous, positively bounded below the coefficient g such that (i) every point t is squeezed between two intervals where g is monotone and (ii) g has no strict local minima. This shows in particular that, for those f such that the relaxed problem ( **) has a solution, the class of coefficients g that yield existence to ( ) is dense in the space of continuous, positive functions on . We discuss various instances of growth conditions on f that yield solutions to ( **) and we present examples that show that the hypotheses on g considered above for attainment are essentially sharp.