Principal eigenvalue problem for infinity Laplacian in metric spaces

2021/09/18 by Qing Liu, Liu, Qing, Ayato Mitsuishi +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Metric Geometry (math.MG) #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2109.08897

openalex publication_date 2021/09/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

This paper is concerned with the Dirichlet eigenvalue problem associated to the ∞-Laplacian in metric spaces. We establish a direct PDE approach to find the principal eigenvalue and eigenfunctions in a proper geodesic space without assuming any measure structure. We provide an appropriate notion of solutions to the ∞-eigenvalue problem and show the existence of solutions by adapting Perron's method. Our method is different from the standard limit process via the variational eigenvalue formulation for p-Laplacian in the Euclidean space.

Citations

• Cheeger constant, p-Laplacian, and Gromov-Hausdorff convergence
• User’s guide to viscosity solutions of second order partial differential equations
• Certain min-max values related to the p-energy and packing radii of Riemannian manifolds and metric measure spaces

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