Spectral optimization problems in ℝ d
Bozhidar Velichkov · Scuola Normale Superiore eBooks · 2015
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 (6.1) min{ λ k ( Ω ):Ω⊂ ℝ d ,Ωquasi−open,| Ω |=c }, $$ \min \left\{ {{\lambda _k}\left( \Omega \right):\Omega \subset {\mathbb{R}^d},\Omega quasi - open,\left| \Omega \right| = c} \right\}, $$ 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.