  We consider the problem of distributing two conducting materials with a
prescribed volume ratio in a given domain so as to minimize the first
eigenvalue of an elliptic operator with Dirichlet conditions. The gap between
the two conductivities is assumed to be small (low contrast regime). For any
geometrical configuration of the mixture, we provide a complete asymptotic
expansion of the first eigenvalue. We then consider a relaxation approach to
minimize the second order approximation with respect to the mixture. We present
numerical simulations in dimensions two and three.
