Chin. Ann. Math. 37B(6), 2016, 833–852 DOI: 10.1007/s11401-016-1008-y Chinese Annals of Mathematics, Series B c  The Editorial Office of CAM and Springer-Verlag Berlin Heidelberg 2016 Homogenization of Elliptic Problems with Quadratic Growth and Nonhomogenous Robin Conditions in Perforated Domains Imen CHOURABI 1 Patrizia DONATO 2 Abstract This paper deals with the homogenization of a class of nonlinear elliptic prob- lems with quadratic growth in a periodically perforated domain. The authors prescribe a Dirichlet condition on the exterior boundary and a nonhomogeneous nonlinear Robin con- dition 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 con- vergence results which permit passing to the limit without using any extension operator. Consequently, using a corrector result proved in [Chourabi, I. and Donato, P., Homoge- nization 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, Nonhomoge- neous Robin boundary conditions, Perforated domains 2000 MR Subject Classification 17B40, 17B50 1 Introduction In this paper, we study the homogenization of a class of a nonlinear elliptic problems con- taining a nonlinear term depending on the solution u ε and on its gradient ∇u ε with quadratic growth. The problem is posed in the perforated domain Ω ∗ ε =Ω\T ε obtained by removing from an open bounded set Ω of R N (N ≥ 2) a closed set T ε representing a set of ε-periodic holes of size ε. We prescribe a Dirichlet condition on the exterior boundary Γ ε 0 and a nonhomoge- neous nonlinear Robin condition on the boundary Γ ε 1 of the holes. More precisely, we study the asymptotic behavior, as ε tends to zero, of the bounded solution u ε of the following problem: ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ −div(A ε (x, u ε )∇u ε )+ λu ε = b ε (x, u ε , ∇u ε )+ f in Ω ∗ ε , (A ε (x, u ε )∇u ε ) · ν + ε γ ρ ε (x)h(u ε )= g ε on Γ ε 1 , u ε =0 on Γ ε 0 , (1.1) where λ ≥ 0 and ν is the unit external normal vector with respect to Ω ∗ ε . Manuscript received July 25, 2015. 1 Universit´e de Monastir, UR Analysis and Control of PDE, UR13ES64, Department of Mathematics Faculty of Sciences of Monastir 5019 Monastir, Tunisia. E-mail: ichourabi@yahoo.fr 2 Normandie Universit´e, Universit´e de Rouen, Laboratoire de Math´ematiques Rapha¨el Salem, UMR CNRS 6085, Avenue de l’Universit´e, BP 12, 76801 Saint Etienne de Rouvray, France. E-mail: Patrizia.Donato@univ-rouen.fr