Convergence of an iterative scheme for the Monge-Amp\`ere eigenvalue problem with general initial data

TL;DR

This paper proves the convergence of an iterative method for solving the Monge-Ampère eigenvalue problem on convex domains, establishing conditions for convergence and providing an energy characterization of solutions.

Contribution

It demonstrates convergence of a previously proposed iterative scheme for general convex initial data and introduces an energy-based characterization of Monge-Ampère eigenfunctions.

Findings

The iterative scheme converges for all convex initial data with finite, nonzero Rayleigh quotient.

The convergence leads to a nonzero Monge-Ampère eigenfunction.

An energy characterization of the eigenfunctions is established.

Abstract

In this note, we revisit an iterative scheme, due to Abedin and Kitagawa (Inverse Iteration for the Monge-Amp\`ere Eigenvalue Problem, Proc. Amer. Math. Soc. 148 (2020), no. 11, 4875--4886), to solve the Monge-Amp\`ere 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-Amp\`ere eigenfunction. As an application, we obtain an energy characterization of the Monge--Amp\`ere eigenfunctions.