Minimization of the third eigenvalue of the Dirichlet Laplacian

Abstract

In this paper, we prove the existence of a domain which minimizes the third Dirichlet eigenvalue λ3 of the Laplacian (–Δ) in the family of domains of RNof given volume. Some remarks are made on the existence of a minimizer for λk, k ⪖ 4 in the same family, and for a class of functionals depending on (λ1,λ2).