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

journal contribution
posted on 30.03.2022, 18:41 by Gerard AwanouGerard Awanou
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.

Funding

OP: Variational Principles, Minimization Diagrams, and Mixed Finite Elements in Computational Geometric Optics | Funder: Directorate for Mathematical & Physical Sciences | Grant ID: 1720276

Mixed Finite Elements, Monge-Ampere equation and Optimal Transportation | Funder: Directorate for Mathematical & Physical Sciences | Grant ID: 1319640

History

Citation

Awanou, G. (2021). On the Weak Convergence of Monge-Ampère Measures for Discrete Convex Mesh Functions. Acta Applicandae Mathematicae, 172(1), 6-. https://doi.org/10.1007/s10440-021-00400-x

Publisher

Springer Science and Business Media LLC

Language

en

issn

0167-8019