We study the optimal sets (Formula presented.) for spectral functionals of the form (Formula presented.), which are bi-Lipschitz with respect to each of the eigenvalues (Formula presented.) of the Dirichlet Laplacian on Ω, a prototype being the problem min(Formula presented.) We prove the Lipschitz regularity of the eigenfunctions u1…up, of the Dirichlet Laplacian on the optimal set Ω* and, as a corollary, we deduce that Ω* is open. For functionals depending only on a generic subset of the spectrum, as for example λk(Ω), our result proves only the existence of a Lipschitz continuous eigenfunction in correspondence to each of the eigenvalues involved.
Lipschitz Regularity of the Eigenfunctions on Optimal Domains / Bucur, D.; Mazzoleni, D.; Pratelli, A.; Velichkov, B.. - In: ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS. - ISSN 0003-9527. - 216:1(2015), pp. 117-151. [10.1007/s00205-014-0801-6]
Lipschitz Regularity of the Eigenfunctions on Optimal Domains
Bucur D.;Mazzoleni D.;Pratelli A.;Velichkov B.
2015
Abstract
We study the optimal sets (Formula presented.) for spectral functionals of the form (Formula presented.), which are bi-Lipschitz with respect to each of the eigenvalues (Formula presented.) of the Dirichlet Laplacian on Ω, a prototype being the problem min(Formula presented.) We prove the Lipschitz regularity of the eigenfunctions u1…up, of the Dirichlet Laplacian on the optimal set Ω* and, as a corollary, we deduce that Ω* is open. For functionals depending only on a generic subset of the spectrum, as for example λk(Ω), our result proves only the existence of a Lipschitz continuous eigenfunction in correspondence to each of the eigenvalues involved.File | Dimensione | Formato | |
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