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

In this paper we prove the existence of an optimal set for the minimization of the (Formula presented.) th variational eigenvalue of the (Formula presented.) -Laplacian among (Formula presented.) -quasi open sets of fixed measure included in a box of finite measure. An analogous existence result is obtained for eigenvalues of the (Formula presented.) -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 (Formula presented.) -Laplacian associated with sign-changing capacitary measures under (Formula presented.) -convergence.