In this talk, we consider periodic Soft inclusion problems of elliptic and parabolic linear equation of non-divergence form. Usually, it is called Soft inclusion problems to find effective conductivity of composites consisting of a medium with non-conducting grains.
Mathematically, non-conducting grains are described by union of disjoint holes with periodicity epsilon. For each epsilon, the unique current density function (the solution of epsilon problem) u-epsilon exists for a given boundary data. We note that, at the boundary of grains, the Neumann data of u-epsilon vanishes.