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
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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Bucur, D., Mazzoleni, D., Pratelli, A. et al. Lipschitz Regularity of the Eigenfunctions on Optimal Domains. Arch Rational Mech Anal 216, 117–151 (2015). https://doi.org/10.1007/s00205-014-0801-6
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DOI: https://doi.org/10.1007/s00205-014-0801-6