On existence of energy minimizing configurations for mixtures of two imperfectly bonded conductors

We consider a domain filled with a suspension of heat conducting spheres of conductivity sigma[sub p] embedded in a matrix of lesser conductivity sigma[sub m]. It is assumed that there exists a thermal contact resistance at the sphere - matrix interface.The contact resistance is characterized by a scalar Beta, which has dimensions of conductivity per unit lenght. A current flux is prescribed on the domain boundary and we seek the energy minimizing configuration among all suspensions satisfying a reource constraint on the total volume of spheres. We establish the existence of an energy minimizing configuration within the class of polydisperse suspensions of spheres. The optimal suspension depends upon the size of the domain and consists of spheres of radii greater than or equal to R[sub cr] = Beta[...] or no spheres at all. Here R[sub cr] is the ratio between the interfacial resistance and the mismatch between the resistivity of each phase