On $p$-Laplacian reaction-diffusion problems with dynamical boundary conditions in perforated media

TL;DR

This paper studies the homogenization of nonlinear p-Laplacian reaction-diffusion equations in perforated domains with dynamical boundary conditions, extending previous results from the linear case to the nonlinear setting.

Contribution

It generalizes earlier homogenization results from the Laplacian to the p-Laplacian, incorporating nonlinear boundary effects in perforated media.

Findings

Proves convergence of solutions to a nonlinear p-Laplacian equation without holes.

Derives effective boundary terms from nonlinear dynamical boundary conditions.

Extends homogenization theory to nonlinear reaction-diffusion problems.

Abstract

This paper deals with the homogenization of the $p$-Laplacian reaction-diffusion problems in a domain containing periodically distributed holes of size $\varepsilon$, with a dynamical boundary condition of pure-reactive type. We generalize our previous results established in the case where the diffusion is modeled by the Laplacian operator, i.e., with $p=2$. We prove the convergence of the homogenization process to a nonlinear $p$-Laplacian reaction-diffusion equation defined on a unified domain without holes with zero Dirichlet boundary condition and with extra terms coming from the influence of the nonlinear dynamical boundary conditions.