Spectral properties of symmetrized AMV operators

2024/11/15 by Manuel Dias, Dias, Manuel, David Tewodrose +1 · 1 citation
Mathematics · #30L99 #35J05 #35P05 #58J50 #Analysis of PDEs (math.AP) #FOS: Mathematics #Holomorphic and Operator Theory #Metric Geometry (math.MG) #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2411.10202

openalex publication_date 2024/11/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

The symmetrized Asymptotic Mean Value Laplacian Δ, obtained as limit of approximating operators Δ_r, is an extension of the classical Euclidean Laplace operator to the realm of metric measure spaces. We show that, as r \downarrow 0, the operators Δ_r eventually admit isolated eigenvalues defined via min-max procedure on any compact locally Ahlfors regular metric measure space. Then we prove L² and spectral convergence of Δ_r to the Laplace--Beltrami operator of a compact Riemannian manifold, imposing Neumann conditions when the manifold has a non-empty boundary.

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