Principal eigenvalue problem for infinity Laplacian in metric spaces

TL;DR

This paper develops a direct PDE method to find the principal eigenvalue and eigenfunctions of the infinity Laplacian in metric spaces, avoiding measure assumptions and standard variational approaches.

Contribution

It introduces a novel PDE approach and solution concept for the infinity Laplacian eigenvalue problem in metric spaces, expanding analysis beyond Euclidean settings.

Findings

Established existence of eigenfunctions in geodesic metric spaces.

Developed a measure-free PDE framework for the infinity Laplacian.

Provided a new approach differing from classical variational methods.

Abstract

This paper is concerned with the Dirichlet eigenvalue problem associated to the $\infty$-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 $\infty$-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.