Homogenization of a nonlinear elliptic problem with large nonlinear potential

TL;DR

This paper investigates the asymptotic behavior of nonlinear elliptic equations with large potentials, deriving a two-scale homogenized system that describes the limit of solutions as the scale parameter tends to zero.

Contribution

It introduces a novel homogenization approach for nonlinear elliptic problems with large potentials, explicitly characterizing the limit system via two-scale convergence.

Findings

Weak convergence of solutions to a homogenized limit

Explicit characterization of the two-scale limit system

Introduction of a new convergence result for nonlinear problems

Abstract

Homogenization is studied for a nonlinear elliptic boundary-value problem with a large nonlinear potential. More specifically we are interested in the asymptotic behavior of a sequence of p-Laplacians of the form $$ -\text{div}(a(\frac{x}{\varepsilon})|Du_\varepsilon|^{p-2}Du_\varepsilon) +\frac{1}{\varepsilon}V(\frac{x}{\varepsilon})|u_\varepsilon|^{p-2}u_\varepsilon=f. $$ It is shown that, under a centring condition on the potential $V$, there exists a two-scale homogenized system with solution $(u, u_1)$ such that the sequence $u_\varepsilon$ of solutions converges weakly to $u$ in $W^{1,p}$ and the gradients $D_x u_\varepsilon$ two-scale converges weakly to $D_x u+ D_y u_1$ in $L^p$, respectively. We characterize the limit system explicitly by means of two-scale convergence and a new convergence result.