New results on Gamma-limits of integral functionals

For ψâ̂̂W1,p(Ω;Rm) and gâ̂̂W-1,p(Ω;Rd), 1<p<+, we consider a sequence of integral functionals Fkψ,g:W1 ,p(Ω;Rm)×Lp(Ω;Rd× n)→[0,+] of the formFkψ,g(u,v)={∫Ω fk(x,u,v)dxif u-ψâ̂̂W01,p(Ω;Rm) and divv=g,+otherwise, where the integrands fk satisfy growth conditions of order p, uniformly in k. We prove a Γ-compactness result for Fkψ,g with respect to the weak topology of W1 ,p(Ω;Rm)×Lp(Ω;Rd× n) and we show that under suitable assumptions the integrand of the Γ-limit is continuously differentiable. We also provide a result concerning the convergence of momenta for minimizers of Fkψ,g. © 2013 Elsevier Masson SAS. All rights reserved