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arXiv · 1905.09563

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

Marco Degiovanni·Dario Mazzoleni
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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.

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Marco Degiovanni, Dario Mazzoleni. 2021-01-14. Optimization results for the higher eigenvalues of the $p$-Laplacian associated with sign-changing capacitary measures. https://doi.org/10.1112/jlms.12425

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