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<β in a smooth bounded open set Ω⊂RN, and a positive number κ<|Ω|, 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∈H01(Ω)→−div((αχω+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 Ω is a ball. Here, we show the following reciprocate result: If Ω⊂RN is smooth, simply connected and has connected boundary, then the problem has a solution if and only if Ω is a ball.