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.
Categories: math.AP