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 -th variational eigenvalue of the -Laplacian among -quasi open sets of fixed measure included in a box of finite measure. An analogous existence result is obtained for eigenvalues of the -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 -Laplacian associated with sign-changing capacitary measures under -convergence.
Keywords
Cite
@article{arxiv.1905.09563,
title = {Optimization results for the higher eigenvalues of the $p$-Laplacian associated with sign-changing capacitary measures},
author = {Marco Degiovanni and Dario Mazzoleni},
journal= {arXiv preprint arXiv:1905.09563},
year = {2021}
}
Comments
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