Homogenization of a class of singular elliptic problems in perforated domains. (August 2018)

• Record Type: Journal Article
• Authors: Donato, Patrizia; Monsurrò, Sara; Raimondi, Federica
• Is Part Of: Nonlinear analysis. Volume 173(2018)
• Page Start: 180
• Page End: 208
• Publication Date: 2018-08
• Subjects: 35B27 -- 35J62 -- 35J66 -- 35J75; Perforated domains -- Homogenization -- Periodic unfolding method -- Quasilinear elliptic equations -- Singular equations -- Nonlinear boundary conditions; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2018.04.005
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this work we study the asymptotic behavior of a class of quasilinear elliptic problems posed in a domain perforated by ε -periodic holes of size ε . The quasilinear equations present a nonlinear singular lower order term f ζ ( u ε ), where u ε is the solution of the problem at ε -level, ζ is a continuous function singular in zero and f a function whose summability depends on the growth of ζ near its singularity. We prescribe a nonlinear Robin condition on the boundary of the holes contained in Ω and a homogeneous Dirichlet condition on the exterior boundary. The particular case of a Neumann boundary condition on the holes is already new. The main tool in the homogenization process consists in proving a suitable convergence result, which shows that the gradient of u ε behaves like that of the solution of a suitable linear problem associated with a weak cluster point of the sequence { u ε }, as ε → 0 . This allows us not only to pass to the limit in the quasilinear term, but also to study the singular term near its singularity, via an accurate a priori estimate. We also get a corrector result for our problem. The main novelty of this work is that for the first time the unfolding method is used to treat a singular term as f ζ ( u ε ) . This plays an essential role in order to get an almost everywhere convergence of the solution u ε, needed in the study the asymptotic behavior of the problem.