Volume 24, Number 3, July–September 2018
|Page(s)||921 - 964|
|Published online||07 June 2018|
Unilateral problems for the p-Laplace operator in perforated media involving large parameters★
Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria, Avenida de los Castros s/n,
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
a Corresponding author: firstname.lastname@example.org
Revised: 7 February 2017
Accepted: 17 March 2017
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 aε ≪ 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 βε which may be very large, namely, βε → ∞ 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, aε and βε. 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.
Mathematics Subject Classification: 35B27 / 35J60 / 35J87 / 35B25
Key words: Nonlinear homogenization / perforated media / variational inequalities / critical relations for parameter
© EDP Sciences, SMAI 2018
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