Lipschitz Regularity of the Eigenfunctions on Optimal Domains

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• 作者：Dorin Bucur (1)
  Dario Mazzoleni (2)
  Aldo Pratelli (3)
  Bozhidar Velichkov (4)
  
  1. Laboratoire de Math茅matiques (LAMA) ; Universit茅 de Savoie ; Campus Scientifique ; 73376 ; Le-Bourget-Du-Lac ; France
  2. Dipartimento di Matematica ; Universit脿 degli Studi di Pavia ; via Ferrata ; 1 ; 27100 ; Pavia ; Italy
  3. Department Mathematik ; Friederich-Alexander Universit盲t Erlangen-N眉rnberg ; Cauerstrasse ; 11 ; 91058 ; Erlangen ; Germany
  4. Scuola Normale Superiore di Pisa ; Piazza dei Cavalieri 7 ; 56126 ; Pisa ; Italy
  
• 刊名：Archive for Rational Mechanics and Analysis
• 出版年：2015
• 出版时间：April 2015
• 年：2015
• 卷：216
• 期：1
• 页码：117-151
• 全文大小：395 KB
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  23. Mazzoleni, D., Pratelli, A.: Existence of minimizers for spectral problems. / J. Math. Pures Appl. 100(3), 433鈥?53 (2013) 10.1016/j.matpur.2013.01.008" target="_blank" title="It opens in new window">CrossRef
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• 刊物类别：Physics and Astronomy
• 刊物主题：Physics
  Mechanics
  Electromagnetism, Optics and Lasers
  Mathematical and Computational Physics
  Complexity
  Fluids
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-0673

文摘

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.