Homogenization of a nonlinear monotone problem in a locally periodic domain via unfolding method

TL;DR

This paper investigates the asymptotic behavior of solutions to a nonlinear monotone boundary value problem in a locally periodic oscillating domain, employing the unfolding method to establish convergence to a limit problem.

Contribution

It introduces a novel application of the unfolding method to analyze nonlinear monotone problems in locally periodic domains, demonstrating convergence results.

Findings

Weak $L^2$-convergence of solutions and flows

Identification of the limit distributional problem

Extension of unfolding method to nonlinear boundary conditions

Abstract

In this paper, the asymptotic behavior of the solutions of a monotone problem posed in a locally periodic oscillating domain is studied. Nonlinear monotone boundary conditions are imposed on the oscillating part of the boundary whereas the Dirichlet condition is considered on the smooth separate part. Using the unfolding method, under natural hypothesis on the regularity of the domain, we prove the weak $L^2$-convergence of the zero-extended solutions of the nonlinear problem and their flows to the solutions of a limit distributional problem.