New results on Γ-limits of integral functionals

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

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

Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014), p. 185-202 / Harvested from Numdam

Résumé
Abstract

Pour tout $ψ \in W^{1, p} (Ω; ℝ^{m})$ et $g \in W^{- 1, p} (Ω; ℝ^{d})$ , $1 < p < + \infty$ , nous considérons une suite de fonctionnelles intégrales $F_{k}^{ψ, g} : W^{1, p} (Ω; ℝ^{m}) \times L^{p} (Ω; ℝ^{d \times n}) \to [0, + \infty]$ définies par $F_{k}^{ψ, g} (u, v) = {\begin{matrix} \int_{Ω} f_{k} (x, \nabla u, v) d x & si u - ψ \in W_{0}^{1, p} (Ω; ℝ^{m}) et div v = g, \\ + \infty & sinon, \end{matrix}$ où les intégrandes $f_{k}$ satisfont des conditions de croissance d'ordre p, uniformément en k. Nous démontrons un résultat de Γ-compacité pour $F_{k}^{ψ, g}$ par rapport à la topologie faible sur $W^{1, p} (Ω; ℝ^{m}) \times L^{p} (Ω; ℝ^{d \times n})$ 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 $F_{k}^{ψ, g}$ .

For $ψ \in W^{1, p} (Ω; ℝ^{m})$ and $g \in W^{- 1, p} (Ω; ℝ^{d})$ , $1 < p < + \infty$ , we consider a sequence of integral functionals $F_{k}^{ψ, g} : W^{1, p} (Ω; ℝ^{m}) \times L^{p} (Ω; ℝ^{d \times n}) \to [0, + \infty]$ of the form $F_{k}^{ψ, g} (u, v) = {\begin{matrix} \int_{Ω} f_{k} (x, \nabla u, v) d x & if u - ψ \in W_{0}^{1, p} (Ω; ℝ^{m}) and div v = g, \\ + \infty & otherwise, \end{matrix}$ where the integrands $f_{k}$ satisfy growth conditions of order p, uniformly in k. We prove a ΓWΓ1,p

Publié le : 2014-01-01

MR 3165285 | Zbl 1290.49024

DOI : https://doi.org/10.1016/j.anihpc.2013.02.005
Classification: 35D99, 35E99, 49J45

@article{AIHPC_2014__31_1_185_0,
     author = {Ansini, Nadia and Dal Maso, Gianni and Zeppieri, Caterina Ida},
     title = {New results on $\Gamma$-limits of integral functionals},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     volume = {31},
     year = {2014},
     pages = {185-202},
     doi = {10.1016/j.anihpc.2013.02.005},
     mrnumber = {3165285},
     zbl = {1290.49024},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_2014__31_1_185_0}
}

Ansini, Nadia; Dal Maso, Gianni; Zeppieri, Caterina Ida. New results on Γ-limits of integral functionals. Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) pp. 185-202. doi : 10.1016/j.anihpc.2013.02.005. http://gdmltest.u-ga.fr/item/AIHPC_2014__31_1_185_0/

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