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

### Juan Casado-Díaz

Dpto. de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Spain

## Abstract

Given two isotropic homogeneous materials represented by two constants $0<α<β$ in a smooth bounded open set $Ω⊂R_{N}$, and a positive number $κ<∣Ω∣$, we consider here the problem consisting in finding a mixture of these materials $αχ_{ω}+β(1−χ_{ω})$, $ω⊂R_{N}$ measurable, with $∣ω∣≤κ$, such that the first eigenvalue of the operator $u∈H_{0}(Ω)→−div((αχ_{ω}+β(1−χ_{ω}))∇u)$ 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 $Ω⊂R_{N}$ is smooth, simply connected and has connected boundary, then the problem has a solution if and only if Ω is a ball.

## Cite this article

Juan Casado-Díaz, A characterization result for the existence of a two-phase material minimizing the first eigenvalue. Ann. Inst. H. Poincaré Anal. Non Linéaire 34 (2017), no. 5, pp. 1215–1226

DOI 10.1016/J.ANIHPC.2016.09.006