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

Abstract

Given two isotropic homogeneous materials represented by two constants $0<\alpha <\beta$ in a smooth bounded open set $\mathrm{\Omega }\subset {\mathbb{R}}^N$, and a positive number $\kappa <|\mathrm{\Omega }|$, we consider here the problem consisting in finding a mixture of these materials $\alpha {\chi }_{\omega }+\beta (1-{\chi }_{\omega })$, $\omega \subset {\mathbb{R}}^N$ measurable, with $|\omega |\le \kappa$, such that the first eigenvalue of the operator $u\in H_0^1(\mathrm{\Omega })\to -\mathrm{div}\left(\left(\alpha {\chi }_{\omega }+\beta (1-{\chi }_{\omega })\right)\nabla u\right)$ 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 $\Omega$ is a ball. Here, we show the following reciprocate result: If $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is smooth, simply connected and has connected boundary, then the problem has a solution if and only if $\Omega$ is a ball.