Convergence of an iterative scheme for the Monge-Ampère eigenvalue problem with general initial data
Nam Q. Le
TL;DR
This work proves that an inverse-iteration scheme for the Monge–Ampère eigenvalue problem converges for all nonzero convex initial data with finite Rayleigh quotient on any bounded convex domain, producing a nontrivial eigenfunction and driving the Rayleigh quotient to the eigenvalue. The authors develop a reverse Aleksandrov estimate via nonlinear integration by parts, establish eventual smoothness of iterates, and derive a new monotonicity formula that underpins uniform convergence of the sequence. They also provide a rate-bound linking the convergence of the Rayleigh quotient to the distance between iterates and the limit eigenfunction. As an application, they obtain an energy characterization: equality of the Monge–Ampère energy with the eigenvalue enforces that the function is a Monge–Ampère eigenfunction, tying variational energy to eigenstructure and extending the method’s theoretical reach.
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.