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

Gómez, Delfina; Lobo, Miguel; Pérez, Eugenia; Podolskii, Alexander V.; Shaposhnikova, Tatiana A.

ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 3, pp. 921-964.

We address homogenization problems for variational inequalities issue from unilateral constraints for the p-Laplacian posed in perforated domains of {ℝ}^n, with n\hspace{0.17em}\ge \hspace{0.17em}3 and p\in [2,n].\epsilon is a small parameter which measures the periodicity of the structure while a_{\epsilon }⩽\epsilon 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 {\beta }_{\epsilon } which may be very large, namely, {\beta }_{\epsilon }\to \infty  as \epsilon \to 0. We first consider the case where p\hspace{0.17em}<\hspace{0.17em}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,\epsilon ,a_{\epsilon } and {\beta }_{\epsilon }. 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\hspace{0.17em}=\hspace{0.17em}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.

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