The Infinity Laplacian eigenvalue problem: reformulation and a numerical scheme

TL;DR

This study tackles the challenging infinity Laplacian eigenvalue problem by reformulating it into a single, sign-agnostic framework using the operator $H_\Lambda$, and proves its equivalence to the classical $F_\Lambda$ formulation in the viscosity-solution sense. Building on this, the authors develop consistent monotone grid schemes for ground states and higher eigenfunctions, establishing subsequential convergence to viscosity solutions via a Barles–Souganidis-type argument and without relying on a comparison principle. They implement a fixed-point (Euler) iteration to solve the discrete system and demonstrate robust numerical performance on diverse domains, including clear evidence that the infinity ground state generally differs from the infinity harmonic potential on a square. The work advances the numerical study of infinity-Laplacian eigenfunctions, enabling systematic exploration of conjectures and geometric characterizations, while also highlighting open questions such as second eigenvalues on general domains and convergence rates. Overall, the method provides a practical tool for computing infinity-Laplacian eigenfunctions and investigates fundamental qualitative properties of these nonlinear, non-unique objects.

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.

Figures (9)

• Figure 1: From left to right: level lines of infinity ground state on the square for stencils of size $3\times 3$, $5\times 5$, $7\times 7$, and $11\times 11$. Smoothness of the level lines increases with larger neighborhoods.
• Figure 2: Infinity ground state (left), infinity harmonic (center) function, and difference (right) on the square.
• Figure 3: Infinity ground states on different domains. All results were initialized with the distance function of the domain.
• Figure 4: Level lines of the dumbbell ground state (cf. bottom center in \ref{['fig:domains']}). The gradient looks singular between the two maxima.
• Figure 5: Discrete non-uniqueness of ground states on the rectangle for different grid resolutions. Surface and contour plots of computed results, initialized with distance function (left) and zero (right). The pointwise difference at a low resolution is substantial (top rows). For high resolutions they are less different (bottom rows).

Theorems & Definitions (24)

• Definition 2.1
• Example 2.2
• Theorem 2.3: Corollary 3.14 in jensen1993uniqueness
• Theorem 2.4
• Definition 2.5: Variational ground states
• Remark 1: (Non-)uniqueness
• Theorem 2.6: Theorem 4.1 in juutinen2005higher
• Remark 2
• Theorem 3.1: Equivalent formulation of the eigenvalue problem
• proof