The Infinity Laplacian eigenvalue problem: reformulation and a numerical scheme

TL;DR

This paper introduces a new formulation and numerical scheme for the infinity Laplacian eigenvalue problem, enabling effective approximation of eigenfunctions with proven convergence and practical computations on various domains.

Contribution

It provides a rigorous equivalence analysis and develops monotone schemes for approximating infinity Laplacian eigenfunctions, advancing numerical methods in this area.

Findings

Convergence of the numerical scheme to viscosity solutions

Successful computation of eigenfunctions on diverse domains

Numerical experiments supporting theoretical conjectures

Abstract

In this work, we present an alternative formulation of the higher eigenvalue problem associated to the infinity Laplacian, which opens the door for numerical approximation of eigenfunctions. A rigorous analysis is performed to show the equivalence of the new formulation to the traditional one. Subsequently, we present consistent monotone schemes to approximate infinity ground states and higher eigenfunctions on grids. We prove that our method converges (up to a subsequence) to a viscosity solution of the eigenvalue problem, and perform numerical experiments which investigate theoretical conjectures and compute eigenfunctions on a variety of different domains.