Integral Functionals and the Gap Problem: Sharp Bounds for Relaxation and Energy Concentration
Giuseppe Mingione, Domenico Mucci · SIAM Journal on Mathematical Analysis · 2005
We consider integral functionals of the type $F(u):=\int_{\Omega} f(x,u,Du)\ dx$ exhibiting a gap between the coercivity and the growth exponent $$L^{-1}|Du|^p\leq f(x,u,Du)\leq L(1+|Du|^q), 1 < p < q,1\leq L < + \infty\,.$$ We give lower semicontinuity results and conditions ensuring that the relaxed functional $\ol{F}$ 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 $\ol{F}$ with a nonzero singular part, which is explicitly computed. The main point in our analysis is that such relaxation results depend in a 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 nonconvex integrals due to Fonseca and Malý and Kristensen; the energies we treat are related to strongly anisotropic settings.