Archive for Rational Mechanics and Analysis

, Volume 216, Issue 1, pp 117–151

Lipschitz Regularity of the Eigenfunctions on Optimal Domains

  • Dorin Bucur
  • Dario Mazzoleni
  • Aldo Pratelli
  • Bozhidar Velichkov
Article

Abstract

We study the optimal sets \({\Omega^\ast\subseteq\mathbb{R}^d}\) for spectral functionals of the form \({F\big(\lambda_1(\Omega),\ldots,\lambda_p(\Omega)\big)}\), which are bi-Lipschitz with respect to each of the eigenvalues \({\lambda_1(\Omega), \lambda_2(\Omega)}, \ldots, {\lambda_p(\Omega)}\) of the Dirichlet Laplacian on \({\Omega}\), a prototype being the problem
$$\min{\big\{\lambda_1(\Omega)+\cdots+\lambda_p(\Omega)\;:\;\Omega\subseteq\mathbb{R}^d,\ |\Omega|=1\big\}}.$$
We prove the Lipschitz regularity of the eigenfunctions \({u_1,\ldots,u_p}\) of the Dirichlet Laplacian on the optimal set \({\Omega^\ast}\) and, as a corollary, we deduce that \({\Omega^\ast}\) is open. For functionals depending only on a generic subset of the spectrum, as for example \({\lambda_k(\Omega)}\), our result proves only the existence of a Lipschitz continuous eigenfunction in correspondence to each of the eigenvalues involved.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Dorin Bucur
    • 1
  • Dario Mazzoleni
    • 2
  • Aldo Pratelli
    • 3
  • Bozhidar Velichkov
    • 4
  1. 1.Laboratoire de Mathématiques (LAMA)Université de SavoieLe-Bourget-Du-LacFrance
  2. 2.Dipartimento di MatematicaUniversità degli Studi di PaviaPaviaItaly
  3. 3.Department MathematikFriederich-Alexander Universität Erlangen-NürnbergErlangenGermany
  4. 4.Scuola Normale Superiore di PisaPisaItaly

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