Γ-convergence of equi-coercive nonlinear energies defined on vector-valued functions, with non-uniformly bounded coefficients

The present paper deals with the asymptotic behavior of equi-coercive sequences {ℱn} of nonlinear functionals defined over vector-valued functions in W01,p(Ω)M, where p>1, M≥1, and Ω is a bounded open set of RN, N≥2. The strongly local energy density Fn(⋅,Du) of the functional ℱn satisfies a Lipschitz condition with respect to the second variable, which is controlled by a positive sequence {an} which is only bounded in some suitable space Lr(Ω). We prove that the sequence {ℱn}Γ-converges for the strong topology of Lp(Ω)M to a functional ℱ which has a strongly local density F(⋅,Du) for sufficiently regular functions  u. This compactness result extends former results on the topic, which are based either on maximum principle arguments in the nonlinear scalar case, or adapted div-curl lemmas in the linear case. Here, the vectorial character and the nonlinearity of the problem need a new approach based on a careful analysis of the asymptotic minimizers associated with the functional ℱn. The relevance of the conditions which are imposed to the energy density Fn(⋅,Du), is illustrated by several examples including some classical hyperelastic energies.