Convergence of an iterative scheme for the Monge-Ampère eigenvalue problem with general initial data

2020/06/11 by Nam Q. Le, Le, Nam Q. · 1 citation
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2006.06564

openalex publication_date 2020/06/11 · openalex created_date 2020/06/19 · openalex updated_date 2026/07/28

Abstract

In this note, we revisit an iterative scheme, due to Abedin and Kitagawa (Inverse Iteration for the Monge-Ampère Eigenvalue Problem, Proc. Amer. Math. Soc. 148 (2020), no. 11, 4875--4886), to solve the Monge-Ampère eigenvalue problem on a general bounded convex domain. Using a nonlinear integration by parts, we show that the scheme converges for all convex initial data having finite and nonzero Rayleigh quotient to a nonzero Monge-Ampère eigenfunction. As an application, we obtain an energy characterization of the Monge--Ampère eigenfunctions.

Citations

• The eigenvalue problem for the Monge-Amp `ere operator on general\n bounded convex domains

Cited by

• A new approach to the Monge-Ampère eigenvalue problem

Related

• Inverse Iteration for the Monge-Ampère Eigenvalue Problem
• A Convergent Inexact Abedin-Kitagawa Iteration Method for Monge-Ampère Eigenvalue Problems
• An Iterative Approach to the Complex Monge-Ampère Eigenvalue Problem
• A new approach to the Monge-Ampère eigenvalue problem
• The eigenvalue problem for the Monge-Amp `ere operator on general\n bounded convex domains