New results on Γ-limits of integral functionals
Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 1, p. 185-202
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For ψW 1,p (Ω; m ) and gW -1,p (Ω; d ), 1<p<+, we consider a sequence of integral functionals F k ψ,g :W 1,p (Ω; m )×L p (Ω; d×n )[0,+] of the form F k ψ,g (u,v)={ Ω f k (x,u,v)dxifu-ψW 0 1,p (Ω; m )and div v=g,+otherwise, where the integrands f k satisfy growth conditions of order p, uniformly in k. We prove a ΓWΓ1,p

Pour tout ψW 1,p (Ω; m ) et gW -1,p (Ω; d ), 1<p<+, nous considérons une suite de fonctionnelles intégrales F k ψ,g :W 1,p (Ω; m )×L p (Ω; d×n )[0,+] définies par F k ψ,g (u,v)={ Ω f k (x,u,v)dxsiu-ψW 0 1,p (Ω; m )et div v=g,+sinon, 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; m )×L p (Ω; d×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 .

DOI : https://doi.org/10.1016/j.anihpc.2013.02.005
Classification:  35D99,  35E99,  49J45
Keywords: Γ-convergence, Integral functionals, Localization method, ( curl , div )-quasiconvexity, Convergence of minimizers, Convergence of momenta
@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, Volume 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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