Elliptic estimates in composite media with smooth inclusions: An integral equation approach

We consider a scalar elliptic equation for a composite medium consisting of homogeneous ^∂1,α0 inclusions, 0 < α₀ ≤ 1, embedded in a constant matrix phase. When the inclusions are separated and are separated from the boundary, the solution has an integral representation, in terms of potential functions defined on the boundary of each inclusion. We study the system of integral equations satisfied by these potential functions as the distance between two inclusions tends to 0. We show that the potential functions converge in ^∂0,α, 0 < α < α₀ to limiting potential functions, with which one can represent the solution when the inclusions are touching. As a consequence, we obtain uniform ∂^1,α bounds on the solution, which are independent of the inter-inclusion distances.