Convergence Rate Estimates for Aleksandrov's Solution to the Monge--Ampère Equation

Abstract

In this paper, we establish convergence rate estimates for convex solutions to the Dirichlet problem of the Monge--Ampère equation det D^2u=f in Omega, where f is a positive and continuous function and Omega is a bounded convex domain in the Euclidean space mathbb{R}^n. We approximate the solution u by a sequence of convex polyhedra v_h, which are generalized solutions to the Monge--Ampère equation in the sense of Aleksandrov, and the associated Monge--Ampère measures nu_h are supported on a properly chosen grid in Omega. We will derive the convergence rate estimates for the cases when f is smooth, Hölder continuous, and merely continuous.