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In this paper we prove that the shape optimization problem $$\min \bigl\{\lambda_k(\varOmega):\ \varOmega\subset \mathbb{R}^d,\ \varOmega\ \hbox{open},\ P(\varOmega)=1,\ |\varOmega|<+\infty \bigr\}, $$ 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, α outside a closed set of Hausdorff dimension d −8. Our results are more general and apply 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.
Applied Mathematics and Optimization – Springer Journals
Published: Apr 1, 2014
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