Principal eigenvalue problem for infinity Laplacian in metric spaces

This article is concerned with the Dirichlet eigenvalue problem associated with the infinity-Laplacian in metric spaces. We establish a direct partial differential equation 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 infinity-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.