We prove that if omega is an open bounded domain with smooth and connected boundary, for every p ∈(1,+∞) 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.
On the first eigenvalue of the normalized p-Laplacian
Crasta, Graziano;Fragala', Ilaria;
2020-01-01
Abstract
We prove that if omega is an open bounded domain with smooth and connected boundary, for every p ∈(1,+∞) 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.File in questo prodotto:
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