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- W4302840005 abstract "In this paper we study the regularity of the optimal sets for the shape optimization problem [ minBig{lambda_1(Omega)+dots+lambda_k(Omega) : Omegasubsetmathbb{R}^d, text{open} , |Omega|=1Big}, ] where $lambda_1(cdot),dots,lambda_k(cdot)$ denote the eigenvalues of the Dirichlet Laplacian and $|cdot|$ the $d$-dimensional Lebesgue measure. We prove that the topological boundary of a minimizer $Omega_k^*$ is composed of a relatively open regular part which is locally a graph of a $C^{1,alpha}$ function and a closed singular part, which is empty if $d<d^*$, contains at most a finite number of isolated points if $d=d^*$ and has Hausdorff dimension smaller than $(d-d^*)$ if $d>d^*$, where the natural number $d^*in[5,7]$ is the smallest dimension at which minimizing one-phase free boundaries admit singularities. To achieve our goal, as an auxiliary result, we shall extend for the first time the known regularity theory for the one-phase free boundary problem to the vector-valued case." @default.
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- W4302840005 date "2016-09-05" @default.
- W4302840005 modified "2023-10-16" @default.
- W4302840005 title "Regularity of the optimal sets for some spectral functionals" @default.
- W4302840005 doi "https://doi.org/10.48550/arxiv.1609.01231" @default.
- W4302840005 hasPublicationYear "2016" @default.
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