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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