The Viscosity Method for the Homogenization of Soft Inclusions

Abstract

In this paper, we consider periodic soft inclusions T _ε with periodicity ε, where the solution, u _ε , satisfies semi-linear elliptic equations of non-divergence in ${\Omega_{\epsilon}=\Omega\setminus \overline{T}_\epsilon}$ with Neumann data on ${\partial T^{\mathfrak a}}$. The difficulty lies in the non-divergence structure of the operator where the standard energy method, which is based on the divergence theorem, cannot be applied. The main object is to develop a viscosity method to find the homogenized equation satisfied by the limit of u _ε , referred to as u, as ε 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 a second corrector is then developed to show that the limit, u, is the viscosity solution of a homogenized equation.