Weak convergence of Monge-Ampere measures for discrete convex mesh functions

Gerard Awanou

Published 2019-10-29Version 1

For mesh functions which satisfy a convexity condition at the discrete level, we associate the natural analogue of the Monge-Ampere measure. A discrete Aleksandrov-Bakelman-Pucci's maximum principle is derived. We use it to prove the weak convergence of Monge-Amp\`ere measures for discrete convex mesh functions, converging uniformly on compact subsets, interpolating boundary values of a continuous convex function and with Monge-Ampere masses uniformly bounded. When discrete convex mesh functions converge uniformly on the whole domain and up to the boundary, the associated Monge-Ampere measures weakly converge to the Monge-Amp\`ere measure of the limit function. The analogous result for sequences of convex functions relies on properties of convex functions and their Legendre transform. In this paper we select proofs which carry out to the discrete level. 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.