Integral functionals and the gap problem: sharp bounds for relaxation and energy concentration

We consider integral functionals of the type F(u):=\int_{\Omega} f(x,u,Du)\ dx exhibiting a gap of type p-q between the coercivity and the growth exponent. We give lower semicontinuity results and conditions ensuring that the relaxed functional is equal to \int_{\Omega} Qf(x,u,Du)\ dx, where Qf denotes the usual quasi-convex envelope; our conditions are sharp. Indeed we also provide counterexamples where such an integral representation fails, showing that energy concentrations appear in the relaxation procedure leading to a measure representation of the relaxed functional with a non zero singular part, which is explicitly computed. The main point in our analysis is that such relaxation results depend in subtle way on the interaction between the ratio q/p and the degree of regularity of the integrand f with respect to the variable x. Our results extend theorems for non-convex integrals due to Fonseca-Maly and Kristensen; the energies we treat are related to strongly anisotropic settings.