Unilateral problems for the p-Laplace operator in perforated media involving large parameters^★

¹ Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria, Avenida de los Castros s/n, 39005 Santander, Spain
gomezdel@unican.es; miguel.lobo@unican.es
² Departamento de Matemática Aplicada y Ciencias de la Computación, Universidad de Cantabria, Avenida de los Castros s/n, 39005 Santander, Spain
³ Department of Differential Equations, Moscow State University, Leninskie Gory, 119991 GSP-2, Moscow, Russia
originalea@ya.ru; shaposh.tan@mail.ru

^a Corresponding author: meperez@unican.es

Received: 31 May 2016
Revised: 7 February 2017
Accepted: 17 March 2017

Abstract

We address homogenization problems for variational inequalities issue from unilateral constraints for the p-Laplacian posed in perforated domains of ℝⁿ, with n ≥ 3 and p ∈ [2, n]. ε is a small parameter which measures the periodicity of the structure while a_ε ≪ ε measures the size of the perforations. We impose constraints for solutions and their fluxes (associated with the p-Laplacian) on the boundary of the perforations. These constraints imply that the solution is positive and that the flux is bounded from above by a negative, nonlinear monotonic function of the solution multiplied by a parameter β_ε which may be very large, namely, β_ε → ∞ as ε → 0. We first consider the case where p < n and the domains periodically perforated by tiny balls and we obtain homogenized problems depending on the relations between the different parameters of the problem: p, n, ε, a_ε and β_ε. Critical relations for parameters are obtained which mark important changes in the behavior of the solutions. Correctors which provide improved convergence are also computed. Then, we extend the results for p = n and the case of non periodically distributed isoperimetric perforations. We make it clear that in the averaged constants of the problem, the perimeter of the perforations appears for any shape.