Some estimates for the higher eigenvalues of sets close to the ball
Dario Mazzoleni, Aldo M. Pratelli · Journal of Spectral Theory · 2019
In this paper we investigate the behavior of the eigenvalues of the Dirichlet Laplacian on sets in \mathbb R^N whose first eigenvalue is close to the one of the ball with the same volume. In particular in our main Theorem 1.1 we prove that, for all k\in\mathbb N , there is a positive constant C=C(k,N) such that for every open set \Omega\subseteq \mathbb R^N with unit measure and with \lambda_1(\Omega) not excessively large one has |\lambda_k(\Omega)-\lambda_k(B)|\leq C (\lambda_1(\Omega)-\lambda_1(B))^\beta, \:\:\: \lambda_k(B)-\lambda_k(\Omega)\leq Cd(\Omega)^{\beta'}\, where d(\Omega) is the Fraenkel asymmetry of \Omega , and where \beta and \beta' are explicit exponents, not depending on k nor on N ; for the special case N=2 , a better estimate holds.