Calculus of Variations and Geometric Measure Theory
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D. Mazzoleni - A. Pratelli

Some estimates for the higher eigenvalues of sets close to the ball

created by mazzoleni on 10 May 2017

[BibTeX]

Preprint

Inserted: 10 may 2017
Last Updated: 10 may 2017

Year: 2017

Abstract:

In this paper we investigate the behavior of the eigenvalues of the Dirichlet Laplacian on sets in $\R^N$ whose first eigenvalue is close to the one of the ball with the same volume. In particular in our main Theorem we prove that, for all $k\in\N$, there is a positive constant $C=C(k,N)$ such that for every open set $\Omega\subseteq \R^N$ with unit measure and with $\lambda_1(\Omega)$ not excessively large one has \[
\lambda_k(\Omega)-\lambda_k(B)
\leq C (\lambda_1(\Omega)-\lambda_1(B))^\beta\,, \qquad \lambda_k(B)-\lambda_k(\Omega)\leq Cd(\Omega)^{\beta'}\,, \] 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.


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