The Infinity Laplacian Eigenvalue Problem: Reformulation and a Numerical Scheme

• DOI: 10.1007/s10915-023-02425-w
• Publication Date: 2024-01-04T07:02:46Z
• Authors: Farid Bozorgnia; Leon Bungert; Daniel Tenbrinck
• Keywords: ddc:510; Mathematics - Spectral Theory; Mathematics - Analysis of PDEs; 35D40, 35P30, 65N06, 65N12, 65N25; FOS: Mathematics; Mathematics - Numerical Analysis; Numerical Analysis (math.NA); 0101 mathematics; 01 natural sciences; Spectral Theory (math.SP); Analysis of PDEs (math.AP)

AbstractIn 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.