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In this paper, we consider periodic soft inclusions $T_{\epsilon}$ withperiodicity $\epsilon$, where the solution, $u_{\epsilon}$, satisfiessemi-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-divergencestructure of the operator where the standard energy method based on thedivergence theorem can not be applied. The main object is developing aviscosity method to find the homogenized equation satisfied by the limit of$u_{\epsilon}$, called as $u$, as $\epsilon$ approaches to zero. We introducethe concept of a compatibility condition between the equation and the Neumanncondition on the boundary for the existence of uniformly bounded periodic firstcorrectors. The concept of second corrector has been developed to show thelimit, $u$, is the viscosity solution of a homogenized equation.

The viscosity Method for the Homogenization of soft inclusions