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

Abstract

The present paper deals with the asymptotic behavior of equi-coercive sequences {Fn} of nonlinear functionals defined over vector-valued functions in W1,p 0 (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 L r(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 hyper-elastic energies.