A characterization result for the existence of a two-phase material minimizing the first eigenvalue
Juan Casado‐Díaz · Annales de l Institut Henri Poincaré C Analyse Non Linéaire · 2016
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 | \leq \kappa , such that the first eigenvalue of the operator u \in H_{0}^{1}(\mathrm{\Omega })\rightarrow −\mathrm{div}\left(\left(\alpha \chi _{\omega } + \beta (1−\chi _{\omega })\right)\mathrm{∇}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 Ω 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 Ω is a ball.