The problem of distributing two conducting materials with a prescribed volume ratio in a
ball so as to minimize the first eigenvalue of an elliptic operator with Dirichlet
conditions is considered in two and three dimensions. The gap ε between the two
conductivities is assumed to be small (low contrast regime). The main result of the paper
is to show, using asymptotic expansions with respect to ε and to small geometric
perturbations of the optimal shape, that the global minimum of the first eigenvalue in low
contrast regime is either a centered ball or the union of a centered ball and of a
centered ring touching the boundary, depending on the prescribed volume ratio between the
two materials.