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

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 $\gamma$-convergence.

This is an extended version of the published paper (on J. London Math. Soc.). In particular, the reader can find an extended Section 2 of preliminaries and the detailed proofs of Theorem 3.1, Proposition 5.9 and Corollary 5.26, which have been omitted or shortened in the published version. The numbering of Sections, Theorems, Lemmas, etc is unchanged