On weak convergence of Monge-Ampere measures for discrete convex mesh functions

TL;DR

This paper investigates the convergence properties of discrete convex mesh functions and their associated Monge-Ampere measures, establishing conditions under which they converge to continuous convex functions and their measures.

Contribution

It proves the weak convergence of Monge-Ampere measures for mesh functions and provides conditions for uniform convergence to convex functions, aiding discretization analysis for the Monge-Ampere equation.

Findings

Monge-Ampere measure of mesh functions equals that of their convex envelope.

Uniform convergence of mesh functions implies weak convergence of measures.

Subsequences of mesh functions converge uniformly to convex functions under certain conditions.

Abstract

To a mesh function we associate the natural analogue of the Monge-Ampere measure. The latter is shown to be equivalent to the Monge-Ampere measure of the convex envelope. We prove that the uniform convergence to a bounded convex function of mesh functions implies the uniform convergence on compact subsets of their convex envelopes and hence the weak convergence of the associated Monge-Ampere measures. We also give conditions for mesh functions to have a subsequence which converges uniformly to a convex function. Our result can be used to give alternate proofs of the convergence of some discretizations for the second boundary value problem for the Monge-Ampere equation and was used for a recently proposed discretization of the latter. For mesh functions which are uniformly bounded and satisfy a convexity condition at the discrete level, we show that there is a subsequence which converges uniformly on compact subsets to a convex function. The convex envelopes of the mesh functions of the subsequence also converge uniformly on compact subsets. If in addition they agree with a continuous convex function on the boundary, the limit function is shown to satisfy the boundary condition strongly.