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Abstract  In this paper we prove the existence of an optimal set for the minimization of the   k  th variational eigenvalue of the   p  ‐Laplacian among   p  ‐quasi open sets of fixed measure included in a box of finite measure. An analogous existence result is obtained for eigenvalues of the   p  ‐Laplacian associated with Schrödinger potentials. In order to deal with these nonlinear shape optimization problems, we develop a general approach which allows to treat the continuous dependence of the eigenvalues of the   p  ‐Laplacian associated with sign‐changing capacitary measures under   γ  ‐convergence.

Optimization results for the higher eigenvalues of the p ‐Laplacian associated with sign‐changing capacitary measures