On the first eigenvalue of the normalized p-Laplacian

2018/11/25 by Graziano Crasta, Crasta, Graziano, Ilaria Fragalà +3
Computer Science · Mathematics · #35J60 #47J10 #49K20 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics

paper · doi:10.48550/arxiv.1811.10024

openalex publication_date 2018/11/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that, if Ω 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 (non-optimal) lower bound for the eigenvalue in terms of the measure of Ω, and we address the open problem of proving a Faber-Krahn type inequality with balls as optimal domains.

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