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

created by mazzoleni on 23 May 2019
modified on 05 Sep 2019

[BibTeX]

Preprint

Inserted: 23 may 2019
Last Updated: 5 sep 2019

Year: 2019

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 Schr\"odinger 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.