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

Abstract

We address homogenization problems for variational inequalities issue from unilateral con-straints for the p-Laplacian posed in perforated domains of Rn, with n 3 and p 2 [2; n]. " is a small parameter which measures the periodicity of the structure while a" " measures the size of the perforations. We impose constraints for solutions and their uxes (associated with the p-Laplacian) on the boundary of the perforations. These constraints imply that the solution is positive and that the ux is bounded from above by a negative, nonlinear monotonic function of the solution multiplied by a parameter " which may be very large, namely, " ! 1 as " ! 0. We rst 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 di erent parameters of the problem: p, n, ", a" and ". 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 the averaged constants of the problem, the perimeter of the perforations appears for any shape.