Global minimizer of the ground state for two phase conductors in low contrast regime^∗,∗∗

Technische Universität Berlin, Sekretariat MA 4-5, Straße des 17. Juni 136, 10623 Berlin, Germany
laurain@math.tu-berlin.de

Received: 2 April 2013
Revised: 24 June 2013

Abstract

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.