On the numerical solution of nonlinear eigenvalue problems for the Monge-Ampère operator^*

¹ Department of Mathematics, University of Houston, 4800 Calhoun Road, Houston, TX 77204, USA.
² Department of Mathematics, the Hong Kong Baptist University, Hong Kong.
³ Department of Mathematics, the Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong.
⁴ School of Mathematics, Georgia Institute of Technology, 686 Cherry Street, Atlanta, GA 30332, USA.
⁵ Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA.

^** Corresponding author: hao.liu@math.gatech.edu

Received: 20 August 2020
Accepted: 19 October 2020

Abstract

In this article, we report the results we obtained when investigating the numerical solution of some nonlinear eigenvalue problems for the Monge-Ampère operator v → det D²v. The methodology we employ relies on the following ingredients: (i) a divergence formulation of the eigenvalue problems under consideration. (ii) The time discretization by operator-splitting of an initial value problem (a kind of gradient flow) associated with each eigenvalue problem. (iii) A finite element approximation relying on spaces of continuous piecewise affine functions. To validate the above methodology, we applied it to the solution of problems with known exact solutions: The results we obtained suggest convergence to the exact solution when the space discretization step h → 0. We considered also test problems with no known exact solutions.