Integral Functionals and the Gap Problem: Sharp Bounds for Relaxation and Energy Concentration

We consider integral functionals of the type F (u) := R Ω f(x, u, Du) dx exhibiting a gap between the coercivity and the growth exponent: L−1|Du|p ≤ f(x, u, Du) ≤ L(1 + |Du|q) 1 < p < q 1 ≤ L < +∞ . We give lower semicontinuity results and conditions ensuring that the relaxed functional F is equal to R Ω 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 F 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 & Malý and Kristensen; the energies we treat are related to strongly anisotropic settings.