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

  • Dario Mazzoleni

    Università Cattolica Brescia, Italy
  • Aldo Pratelli

    Università di Pisa, Italy
Some estimates for the higher eigenvalues of sets close to the ball cover

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Abstract

In this paper we investigate the behavior of the eigenvalues of the Dirichlet Laplacian on sets in 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 , there is a positive constant such that for every open set with unit measure and with not excessively large one has

where is the Fraenkel asymmetry of , and where and are explicit exponents, not depending on nor on ; for the special case , a better estimate holds.

Cite this article

Dario Mazzoleni, Aldo Pratelli, Some estimates for the higher eigenvalues of sets close to the ball. J. Spectr. Theory 9 (2019), no. 4, pp. 1385–1403

DOI 10.4171/JST/280