Homogenization of elliptic problems with quadratic growth and nonhomogenous Robin conditions in perforated domains

Abstract

This paper deals with the homogenization of a class of nonlinear elliptic problems with quadratic growth in a periodically perforated domain. The authors prescribe a Dirichlet condition on the exterior boundary and a nonhomogeneous nonlinear Robin condition on the boundary of the holes. The main difficulty, when passing to the limit, is that the solution of the problems converges neither strongly in L ²(Ω) nor almost everywhere in Ω. A new convergence result involving nonlinear functions provides suitable weak convergence results which permit passing to the limit without using any extension operator. Consequently, using a corrector result proved in [Chourabi, I. and Donato, P., Homogenization and correctors of a class of elliptic problems in perforated domains, Asymptotic Analysis, 92(1), 2015, 1–43, DOI: 10.3233/ASY-151288], the authors describe the limit problem, presenting a limit nonlinearity which is different for the two cases, that of a Neumann datum with a nonzero average and with a zero average.

Keywords

Homogenization / Elliptic problems / Quadratic growth / Nonhomogeneous Robin boundary conditions / Perforated domains

References

[1] Artola M., Duvaut G.. Homogénéisation d’une classe de problèmes non linéaires. C. R. Acad. Paris Série A, 1979, 288: 775-778

[2] Artola, M. and Duvaut, G., Un résultat d’homogénéisation pour une classe de problèmes de diffusion non linéaires stationnaires, Ann. Faculte Sci. Toulouse, IV,1982, 1–27.

[3] Bendib S.. Homogénéisation d’une classe de problèmes non linéaires avec des conditions de Fourier dans des ouverts perforés, 2004

[4] Bensoussan A., Boccardo L., Dall’Aglio A., Murat F.. H-convergence for quasilinear elliptic equations under natural hypotheses on the correctors. Proceedings of the Conference Composite Media and Homogenization Theory, 1993

[5] Bensoussan A., Boccardo L., Murat F.. H-convergence for quasi-linear elliptic equations with quadratic growth. Appl. Math. Optim., 1992, 26: 253-272

[6] Bensoussan A., Lions J. L., Papanicolaou G.. Asymptotic Analysis for Periodic Structures, 1978, Amsterdam: AMS Chelsea Publishing

[7] Boccardo L., Murat F.. Homogénéisation de Problèmes Quasi linéaires, Atti del convergno, Studio di Problemi-Limite dell’Analisi Funzionale (Bressanone 7–9 sett. 1981), 1982 13-51

[8] Cabarrubias B., Donato P.. Homogenization of a quasilinear elliptic problem with nonlinear Robin boundary conditions. Applicable Analysis, 2012, 91(6): 1111-1127

[9] Chourabi I., Donato P.. Bounded solutions for a quasilinear singular problem with nonlinear Robin boundary conditions. Differential and Integral Equations, 2013, 26(9–10): 975-1008

[10] Chourabi I., Donato P.. Homogenization and correctors of a class of elliptic problems in perforated domains. Asymptotic Analysis, 2015, 92(1): 1-43

[11] Cioranescu, D., Damlamian, A., Donato, P., et al., The periodic unfolding method in domains with holes, SIAM J. Math. Anal., 44(2), 718–760.

[12] Cioranescu D., Damlamian A., Griso G.. Periodic unfolding and homogenization, C. R. Acad. Sci. Paris. Ser. I, 2002, 335: 99-104

[13] Cioranescu D., Damlamian A., Griso G.. The periodic unfolding and homogenization. SIAM J. Math. Anal., 2008, 40(4): 1585-1620

[14] Cioranescu D., Donato P., Zaki R.. The periodic unfolding and Robin problems in perforated domains, C. R. Acad. Sci. Paris. Ser. I, 2006, 342: 467-474

[15] Cioranescu D., Donato P.. Homogénéisation du probléme de Neumann non homogène dans des ouverts perforés. Asymptot. Anal., 1988, 1: 115-138

[16] Cioranescu D., Donato P., Zaki R.. The periodic unfolding method in perforated domains. Portugal Math., 2006, 63(4): 476-496

[17] Cioranescu D., Donato P., Zaki R.. Asymptotic behavior of elliptic problems in perforated domains with nonlinear boundary conditions. Asymptot. Anal., 2007, 53: 209-235

[18] Cioranescu D., Saint Jean Paulin J.. Homogenization in open sets with holes. J. Math. Anal. Appl., 1979, 71: 590-607

[19] Donato P., Gaudiello A., Sgambati L.. Homogenization of bounded solutions of elliptic equations with quadratic growth in periodically perforated domains. Asymptotic Anal., 1998, 16(3–4): 223-243

[20] Donato P., Giachetti D.. Homogenization of some singular nonlinear elliptic problems. International Journal of Pure and Applied Mathematics, 2011, 73(3): 349-378

[21] Donato P., Yang Z. Y.. The period unfolding method for the wave equations in domains with holes. Advances in Mathematical Sciences and Applications, 2012, 22(2): 521-551

[22] Meyers N. G.. An Lp-estimate for the gradient of solution of second order elliptic divergence equations. Ann. Sc. Norm. Sup., 1963, 17: 189-206

[23] Murat F.. H-convergence, Séminaire d’analyse fonctionnelle et numérique, 1978

[24] Stampacchia G.. Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus. Ann. Inst. Fourier, 1965, 15(1): 189-258