A characterization result for the existence of a two-phase material minimizing the first eigenvalue

Given two isotropic homogeneous materials represented by two constants 0 <α< | |, we consider here the problem consisting in finding a mixture of these materials αχω + β(1 − χω), ω ⊂ RN measurable, with |ω| ≤ κ, such that the first eigenvalue of the operator u ∈ H1 0 ( ) → −divαχω + β(1 − χω) ∇u reaches the minimum value. In a recent paper, [6], we have proved that this problem has not solution in general. On the other hand, it was proved in [1] that it has solution if is a ball. Here, we show the following reciprocate result: If ⊂ RN is smooth, simply connected and has connected boundary, then the problem has a solution if and only if is a ball.