Existence and regularity of minimizers for some spectral functionals with perimeter constraint (1303.0968v2)

Abstract: In this paper we prove that the shape optimization problem $$\min\left{\lambda_k(\Omega):\ \Omega\subset\R^d,\ \Omega\ \hbox{open},\ P(\Omega)=1,\ |\Omega|<+\infty\right},$$ has a solution for any $k\in\N$ 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 apply to spectral functionals of the form $f(\lambda_{k_1}(\Omega),\dots,\lambda_{k_p}(\Omega))$, for increasing functions $f$ satisfying some suitable bi-Lipschitz type condition.