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We consider the problem of distributing two conducting materials with aprescribed volume ratio in a given domain so as to minimize the firsteigenvalue of an elliptic operator with Dirichlet conditions. The gap betweenthe two conductivities is assumed to be small (low contrast regime). For anygeometrical configuration of the mixture, we provide a complete asymptoticexpansion of the first eigenvalue. We then consider a relaxation approach tominimize the second order approximation with respect to the mixture. We presentnumerical simulations in dimensions two and three.

Minimization of the ground state of the mixture of two conducting materials in a small contrast regime