Existence and regularity of optimal shapes for spectral functionals with Robin boundary conditions

Abstract: We establish the existence and find some qualitative properties of open sets that minimize functionals of the form $ F(\lambda_1(\Omega;\beta),\dots,\lambda_k(\Omega;\beta))$ under measure constraint on $\Omega$, where $\lambda_i(\Omega;\beta)$ designates the $i$-th eigenvalue of the Laplace operator on $\Omega$ with Robin boundary conditions of parameter $\beta>0$. Moreover, we show that minimizers of $\lambda_k(\Omega;\beta)$ for $k\geq 2$ verify the conjecture $\lambda_k(\Omega;\beta)=\lambda_{k-1}(\Omega;\beta)$ in dimension three and more.