Regularity of the optimal sets for some spectral functionals

By Dario Mazzoleni, S. Terracini, Bozhidar Velichkov. 2016

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AbstractIn this paper we study the regularity of the optimal sets for the shape optimization problem $$\min\Big\{\lambda_{1}(\Omega)+\dots+\lambda_{k}(\Omega) : \Omega \subset {\mathbb {R}}^{d} {\rm open},\quad |\Omega| = 1\Big\},$$min{λ1(Ω)+⋯+λk(Ω):Ω⊂Rdopen,|Ω|=1},where $${\lambda_{1}(\cdot),\ldots,\lambda_{k}(\cdot)}$$λ1(·),…,λk(·) denote the eigenvalues of the Dirichlet Laplacian and $${|\cdot|}$$|·| the d-dimensional Lebesgue measure. We prove that the topological boundary of a minimizer $${\Omega_{k}^{*}}$$Ωk∗ is composed of a relatively open regular part which is locally a graph of a $${C^{\infty}}$$C∞ function and a closed singular part, which is empty if $${d < d^{*}}$$d<d∗, contains at most a finite number of isolated points if $${d = d^{*}}$$d=d∗ and has Hausdorff dimension smaller than $${(d-d^{*})}$$(d-d∗) if $${d > d^{*}}$$d>d∗, where the natural number $${d^{*} \in [5,7]}$$d∗∈[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.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/814194bd8607463bb4a89dc2f218f52d4bfe7492