caa a22 4500 605514992 CHVBK 20210128100707.0 cr unu---uuuuu 210128e20150401xx s 000 0 eng 10.1007/s00205-014-0801-6 doi (NATIONALLICENCE)springer-10.1007/s00205-014-0801-6 Lipschitz Regularity of the Eigenfunctions on Optimal Domains [Elektronische Daten] [Dorin Bucur, Dario Mazzoleni, Aldo Pratelli, Bozhidar Velichkov] We study the optimal sets $${\Omega^\ast\subseteq\mathbb{R}^d}$$ Ω * ⊆ R d for spectral functionals of the form $${F\big(\lambda_1(\Omega),\ldots,\lambda_p(\Omega)\big)}$$ F ( λ 1 ( Ω ) , ... , λ p ( Ω ) ) , which are bi-Lipschitz with respect to each of the eigenvalues $${\lambda_1(\Omega), \lambda_2(\Omega)}, \ldots, {\lambda_p(\Omega)}$$ λ 1 ( Ω ) , λ 2 ( Ω ) , ... , λ p ( Ω ) of the Dirichlet Laplacian on $${\Omega}$$ Ω , a prototype being the problem $$\min{\big\{\lambda_1(\Omega)+\cdots+\lambda_p(\Omega)\;:\;\Omega\subseteq\mathbb{R}^d,\ |\Omega|=1\big\}}.$$ min { λ 1 ( Ω ) + ⋯ + λ p ( Ω ) : Ω ⊆ R d , | Ω | = 1 } . We prove the Lipschitz regularity of the eigenfunctions $${u_1,\ldots,u_p}$$ u 1 , ... , u p of the Dirichlet Laplacian on the optimal set $${\Omega^\ast}$$ Ω * and, as a corollary, we deduce that $${\Omega^\ast}$$ Ω * is open. For functionals depending only on a generic subset of the spectrum, as for example $${\lambda_k(\Omega)}$$ λ k ( Ω ) , our result proves only the existence of a Lipschitz continuous eigenfunction in correspondence to each of the eigenvalues involved. Springer-Verlag Berlin Heidelberg, 2014 Bucur Dorin Laboratoire de Mathématiques (LAMA), Université de Savoie, Campus Scientifique, 73376, Le-Bourget-Du-Lac, France aut Mazzoleni Dario Dipartimento di Matematica, Università degli Studi di Pavia, via Ferrata, 1, 27100, Pavia, Italy aut Pratelli Aldo Department Mathematik, Friederich-Alexander Universität Erlangen-Nürnberg, Cauerstrasse, 11, 91058, Erlangen, Germany aut Velichkov Bozhidar Scuola Normale Superiore di Pisa, Piazza dei Cavalieri 7, 56126, Pisa, Italy aut Archive for Rational Mechanics and Analysis Springer Berlin Heidelberg 216/1(2015-04-01), 117-151 0003-9527 216:1<117 2015 216 205 https://doi.org/10.1007/s00205-014-0801-6 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s00205-014-0801-6 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Bucur Dorin Laboratoire de Mathématiques (LAMA), Université de Savoie, Campus Scientifique, 73376, Le-Bourget-Du-Lac, France aut NATIONALLICENCE 700 1- Mazzoleni Dario Dipartimento di Matematica, Università degli Studi di Pavia, via Ferrata, 1, 27100, Pavia, Italy aut NATIONALLICENCE 700 1- Pratelli Aldo Department Mathematik, Friederich-Alexander Universität Erlangen-Nürnberg, Cauerstrasse, 11, 91058, Erlangen, Germany aut NATIONALLICENCE 700 1- Velichkov Bozhidar Scuola Normale Superiore di Pisa, Piazza dei Cavalieri 7, 56126, Pisa, Italy aut NATIONALLICENCE 773 0- Archive for Rational Mechanics and Analysis Springer Berlin Heidelberg 216/1(2015-04-01), 117-151 0003-9527 216:1<117 2015 216 205