| 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}=Omegasetminus 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. |
