Homogenization of a quasilinear elliptic problem with nonlinear Robin boundary conditions

摘要

This article is devoted to the homogenization of a quasilinear elliptic equation with oscillating coefficients in a periodically perforated domain. A nonlinear Robin condition is prescribed on the boundary of the holes, depending on a real parameter γ≥1. We suppose that the data satisfy some suitable hypotheses which ensure, as proved by the authors in Cabarrubias and Donato [B. Cabarrubias and P. Donato, Existence and uniqueness for a quasilinear elliptic problem with nonlinear Robin conditions, Carpathian J. Math. (2) (2011) (to appear)], the existence and the uniqueness of a solution of the problem. In particular, suitable growth conditions are assumed on the nonlinear boundary term, as done in Cioranescu-Donato-Zaki [D. Cioranescu, P. Donato and R. Zaki, Asymptotic behavior of elliptic problems in perforated domains with nonlinear boundary conditions, Asymptot. Anal. 53 (2007), pp. 209-235]. On the quasilinear term, some assumptions on the modulus of continuity introduced in Chipot [M. Chipot, Elliptic Equations: An Introductory Course, Birkhauser Verlag AG, Germany, 2009] and weaker than a Lipschitz condition are prescribed. We study the convergence to a limit problem, which is identified by using the periodic unfolding method. We also prove the well-posedness of the limit system. To do that, we show that the homogenized operator inherits the modulus of continuity of the initial problem. As a consequence, the uniqueness of a solution of the homogenized quasilinear problem follows.