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

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.

v4: Question 5.1 is now completely answered in Theorem 1.8; to be published in Advances in Continuous and Discrete Models: Theory and Applications