New results on Γ-limits of integral functionals

Ansini, Nadia; Dal Maso, Gianni; Zeppieri, Caterina Ida

Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) no. 1, pp. 185-202

For $\psi \in W^{1,p}(\Omega ;{ℝ}^m)$ and $g\in W^{-1,p}(\Omega ;{ℝ}^d)$, $1<p<+\infty$, we consider a sequence of integral functionals $F_k^{\psi ,g}:W^{1,p}(\Omega ;{ℝ}^m)\times L^p(\Omega ;{ℝ}^{d\times n})\to [0,+\infty ]$ of the form

$$F_k^{\psi ,g}(u,v)=\{\begin{array}{cc}{\int }_{\Omega }{f}_k(x,\nabla u,v)dx\hfill & \text{if}u-\psi \in W_0^{1,p}(\Omega ;{ℝ}^m)\text{and}\mathrm{div}v=g,\hfill \\ +\infty \hfill & \text{otherwise,}\hfill \end{array}$$

where the integrands $f_k$ satisfy growth conditions of order p, uniformly in k. We prove a ΓWΓ1,p

Pour tout $\psi \in W^{1,p}(\Omega ;{ℝ}^m)$ et $g\in W^{-1,p}(\Omega ;{ℝ}^d)$, $1<p<+\infty$, nous considérons une suite de fonctionnelles intégrales $F_k^{\psi ,g}:W^{1,p}(\Omega ;{ℝ}^m)\times L^p(\Omega ;{ℝ}^{d\times n})\to [0,+\infty ]$ définies par

$$F_k^{\psi ,g}(u,v)=\{\begin{array}{cc}{\int }_{\Omega }{f}_k(x,\nabla u,v)dx\hfill & \text{si}u-\psi \in W_0^{1,p}(\Omega ;{ℝ}^m)\text{et}\mathrm{div}v=g,\hfill \\ +\infty \hfill & \text{sinon},\hfill \end{array}$$

où les intégrandes $f_k$ satisfont des conditions de croissance d'ordre p, uniformément en k. Nous démontrons un résultat de Γ-compacité pour $F_k^{\psi ,g}$ par rapport à la topologie faible sur $W^{1,p}(\Omega ;{ℝ}^m)\times L^p(\Omega ;{ℝ}^{d\times n})$ et nous prouvons que sous des conditions appropriées, l'intégrande de la Γ-limite est continûment différentiable. Nous montrons également un résultat de convergence des moments pour les minima de $F_k^{\psi ,g}$.

Classification : 35D99, 35E99, 49J45
Keywords: Γ-convergence, Integral functionals, Localization method, $ (\mathrm{curl},\mathrm{div})$-quasiconvexity, Convergence of minimizers, Convergence of momenta

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