Abstract
In this paper we study the regularity of the optimal sets for the shape optimization problem
where \({\lambda_{1}(\cdot),\ldots,\lambda_{k}(\cdot)}\) 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 \({C^{\infty}}\) 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.
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Mazzoleni, D., Terracini, S. & Velichkov, B. Regularity of the optimal sets for some spectral functionals. Geom. Funct. Anal. 27, 373–426 (2017). https://doi.org/10.1007/s00039-017-0402-2
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DOI: https://doi.org/10.1007/s00039-017-0402-2
Keywords and phrases
- Shape optimization
- Dirichlet eigenvalues
- optimality conditions
- regularity of free boundaries
- viscosity solutions