The viscosity Method for the Homogenization of soft inclusions

TL;DR

This paper develops a viscosity-based homogenization method for semi-linear elliptic equations with soft inclusions, addressing challenges posed by non-divergence operators and boundary conditions.

Contribution

It introduces a novel viscosity approach and compatibility conditions for homogenizing semi-linear elliptic equations with soft inclusions and Neumann boundary conditions.

Findings

Established a viscosity method for homogenization.

Defined compatibility conditions for boundary and equations.

Proved the limit function is a viscosity solution of the homogenized equation.

Abstract

In this paper, we consider periodic soft inclusions $T_{\epsilon}$ with periodicity $\epsilon$, where the solution, $u_{\epsilon}$, satisfies semi-linear elliptic equations of non-divergence in $\Omega_{\epsilon}=\Omega\setminus \bar{T}_\epsilon$ with a Neumann data on $\partial T^{\mathfrak a}$ . The difficulty lies in the non-divergence structure of the operator where the standard energy method based on the divergence theorem can not be applied. The main object is developing a viscosity method to find the homogenized equation satisfied by the limit of $u_{\epsilon}$, called as $u$, as $\epsilon$ approaches to zero. We introduce the concept of a compatibility condition between the equation and the Neumann condition on the boundary for the existence of uniformly bounded periodic first correctors. The concept of second corrector has been developed to show the limit, $u$, is the viscosity solution of a homogenized equation.