Principal eigenvalue problem for infinity Laplacian in metric spaces

Abstract

This article is concerned with the Dirichlet eigenvalue problem associated with the ∞\infty -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 ∞\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 pp-Laplacian in the Euclidean space.

Keywords

• eigenvalue problems
• metric spaces
• infinity laplacian
• eikonal equation
• viscosity solutions
• 35p30
• 35r02
• 35d40