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Abstract 

We consider a scalar elliptic equation for a composite medium consisting of
homogeneous smooth inclusions, 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 C^{0,alpha} to limiting
potential functions, with which one can represent the solution when the inclusions
are touching. As a consequence, we obtain uniform C^{1,alpha} bounds on the solution,
which are independent of the interinclusion distances.
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