New results on Γ-limits of integral functionals
Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) no. 1, p. 185-202
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Pour tout $\psi \in {W}^{1,p}\left(\Omega ;{ℝ}^{m}\right)$ et $g\in {W}^{-1,p}\left(\Omega ;{ℝ}^{d}\right)$, $1, nous considérons une suite de fonctionnelles intégrales ${F}_{k}^{\psi ,g}:{W}^{1,p}\left(\Omega ;{ℝ}^{m}\right)×{L}^{p}\left(\Omega ;{ℝ}^{d×n}\right)\to \left[0,+\infty \right]$ définies par ${F}_{k}^{\psi ,g}\left(u,v\right)=\left\{\begin{array}{cc}{\int }_{\Omega }{f}_{k}\left(x,\nabla u,v\right)\phantom{\rule{0.166667em}{0ex}}dx\hfill & \text{si}\phantom{\rule{4pt}{0ex}}u-\psi \in {W}_{0}^{1,p}\left(\Omega ;{ℝ}^{m}\right)\phantom{\rule{4pt}{0ex}}\text{et}\phantom{\rule{4pt}{0ex}}\mathrm{div}v=g,\hfill \\ +\infty \hfill & \text{sinon},\hfill \end{array}$ 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 ${W}^{1,p}\left(\Omega ;{ℝ}^{m}\right)×{L}^{p}\left(\Omega ;{ℝ}^{d×n}\right)$ 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 ${F}_{k}^{\psi ,g}$.
For $\psi \in {W}^{1,p}\left(\Omega ;{ℝ}^{m}\right)$ and $g\in {W}^{-1,p}\left(\Omega ;{ℝ}^{d}\right)$, $1, we consider a sequence of integral functionals ${F}_{k}^{\psi ,g}:{W}^{1,p}\left(\Omega ;{ℝ}^{m}\right)×{L}^{p}\left(\Omega ;{ℝ}^{d×n}\right)\to \left[0,+\infty \right]$ of the form ${F}_{k}^{\psi ,g}\left(u,v\right)=\left\{\begin{array}{cc}{\int }_{\Omega }{f}_{k}\left(x,\nabla u,v\right)\phantom{\rule{0.166667em}{0ex}}dx\hfill & \text{if}\phantom{\rule{4pt}{0ex}}u-\psi \in {W}_{0}^{1,p}\left(\Omega ;{ℝ}^{m}\right)\phantom{\rule{4pt}{0ex}}\text{and}\phantom{\rule{4pt}{0ex}}\mathrm{div}v=g,\hfill \\ +\infty \hfill & \text{otherwise,}\hfill \end{array}$ where the integrands ${f}_{k}$ satisfy growth conditions of order p, uniformly in k. We prove a ΓWΓ1,p
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},
publisher = {Elsevier},
volume = {31},
number = {1},
year = {2014},
pages = {185-202},
doi = {10.1016/j.anihpc.2013.02.005},
zbl = {1290.49024},
mrnumber = {3165285},
language = {en},
url = {http://www.numdam.org/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) no. 1, pp. 185-202. doi : 10.1016/j.anihpc.2013.02.005. http://www.numdam.org/item/AIHPC_2014__31_1_185_0/

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