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On the first eigenvalue of the normalized $p$-Laplacian
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by Graziano Crasta, Ilaria Fragalà and Bernd Kawohl

Proc. Amer. Math. Soc. 148 (2020), 577-590

DOI: https://doi.org/10.1090/proc/14823

Published electronically: November 6, 2019

Abstract:

We prove that if $\Omega$ is an open bounded domain with smooth and connected boundary, for every $p \in (1, + \infty )$ the first Dirichlet eigenvalue of the normalized $p$-Laplacian is simple in the sense that two positive eigenfunctions are necessarily multiple of each other. We also give a (nonoptimal) lower bound for the eigenvalue in terms of the measure of $\Omega$, and we address the open problem of proving a Faber–Krahn-type inequality with balls as optimal domains.

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