Existence and regularity of minimizers for some spectral functionals with perimeter constraint. (English) Zbl 1297.49077
Summary: In this paper we prove that the shape optimization problem
\[
\min \bigl\{\lambda_k(\varOmega):\;\varOmega\subset \mathbb{R}^d,\;\varOmega\;\text{ open},\;P(\varOmega)=1,\;|\varOmega|<+\infty \bigr\},
\]
has a solution for any \(k\in \mathbb{N}\) and dimension \(d\). Moreover, every solution is a bounded connected open set with boundary which is \(C^{1,\alpha}\) outside a closed set of Hausdorff dimension \(d-8\). Our results are more general and can be applied to spectral functionals of the form \(f(\lambda_{k_{1}}(\varOmega),\dots,\lambda_{k_{p}}(\varOmega))\), for increasing functions \(f\) satisfying some suitable bi-Lipschitz type condition.
MSC:
| 49Q10 | Optimization of shapes other than minimal surfaces |
| 49N60 | Regularity of solutions in optimal control |
Keywords:
shape optimization; eigenvalues; Dirichlet Laplacian; \(C^{1,\alpha}\)-boundary; bi-Lipschitz type conditionReferences:
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