On the Weak Convergence of Monge-Ampère Measures for Discrete Convex Mesh Functions

To a mesh function we associate the natural analogue of the Monge-Ampère measure. The latter is shown to be equivalent to the Monge-Ampère 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-Ampère 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-Ampère 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.