Minimization of the \(k\)-th eigenvalue of the Dirichlet Laplacian (Q1762432)

The author studies the minimization problem \[ A\to \lambda_k(A)+|A|\leqno(1) \] in the family of all quasi-open sets of \(\mathbb{R}^N\), which is equivalent to \(\min\{\lambda_k(A),\,\,\,A\subseteq \mathbb{R}^N,\,\,\,|A|=c\}\), where \(\lambda_k\) is the \(k\)-th eigenvalue of the Dirichlet Laplacian. He proves the existence of solutions \(A\) of problem (1) and shows that every solution is a local shape subsolution for an energy minimizing free boundary problem. Besides, every minimizer is bounded, has finite perimeter and its fine interior has the same measure as \(A\).