Minimizing nonconvex, simple integrals of product type

We consider the problem of minimizing simple integrals of product type, i.e. min {integral (T)(0) g(x(t))f(x ´ (t)) dt: x is an element of AC([0, T]), x(0) = x(0), x(T) = x(T)}. where f:R --> [0, proportional to] is a possibly nonconvex, lower semicontinuous function with either superlinear or slow growth at infinity. Assuming that the relaxed problem (P**) obtained from (P) by replacing f with its convex envelope f** admits a solution. we prove attainment for (P) for every continuous, positively bounded below the coefficient g such that (i) every point t is an element ofR 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 (P**) has a solution, the class of coefficients g that yield existence to (P) is dense in the space of continuous, positive Functions on R. We discuss various instances of growth conditions on f that yield solutions to (P**) and we present examples that show that the hypotheses on g considered above for attainment are essentially sharp.