Spectral optimization problems in ℝ^d

Abstract

The aim of this section is to study the optimal sets for functionals depending on the eigenvalues of the Dirichlet Laplacian. A typical example is the model problem

$$ \min \left\{ {{\lambda _k}\left( \Omega \right):\Omega \subset {\mathbb{R}^d},\Omega quasi - open,\left| \Omega \right| = c} \right\}, $$

((6.1))

where c > 0 is a given constant. The existence of an optimal set for the problem (6.1) was proved recently by Bucur (see [20]) and by Mazzoleni and Pratelli (see [81])two completely different techniques.