More Like this

1. Regularity of the Solution to a Real Monge–Ampère Equation on the Boundary of a Simplex

   https://doi.org/10.1093/imrn/rnaf013  

   Andreasson, Rolf; Hultgren, Jakob; Jonsson, Mattias; Mazzon, Enrica; McCleerey, Nicholas (February 2025, International Mathematics Research Notices)

   Abstract Motivated by conjectures in Mirror Symmetry, we continue the study of the real Monge–Ampère operator on the boundary of a simplex. This can be formulated in terms of optimal transport, and we consider, more generally, the problem of optimal transport between symmetric probability measures on the boundary of a simplex and of the dual simplex. For suitably regular measures, we obtain regularity properties of the transport map, and of its convex potential. To do so, we exploit boundary regularity results for optimal transport maps by Caffarelli, together with the symmetries of the simplex. 

   more » « less
2. Chapter 2 - The Monge-Ampere equation

   https://doi.org/10.1016/bs.hna.2019.05.003  

   Neilan, Michael; Salgado, Abner J.; Zhang, Wujun (January 2020, Handbook of numerical analysis)

   We review recent advances in the numerical analysis of the Monge–Ampère equation. Various computational techniques are discussed including wide stencil finite difference schemes, two-scaled methods, finite element methods, and methods based on geometric considerations. Particular focus is the development of appropriate stability and consistency estimates which lead to rates of convergence of the discrete approximations. Finally we present numerical experiments which highlight each method for a variety of test problem with different levels of regularity. 

   more » « less
3. Singular Abreu equations and linearized Monge–Ampère equations with drifts

   https://doi.org/10.4171/JEMS/1548  

   Kim, Young Ho; Le, Nam Q; Wang, Ling; Zhou, Bin (October 2024, Journal of the European Mathematical Society)

   We study the solvability of singular Abreu equations which arise in the approximation of convex functionals subject to a convexity constraint. Previous works established the solvability of their second boundary value problems either in two dimensions, or in higher dimensions under either a smallness condition or a radial symmetry condition. Here, we solve the higher-dimensional case by transforming singular Abreu equations into linearized Monge–Ampère equations with drifts. We establish global Hölder estimates for linearized Monge–Ampère equations with drifts under suitable hypotheses, and then apply them to prove the regularity and solvability of the second boundary value problem for singular Abreu equations in higher dimensions. Many cases with general right-hand side are also discussed. 

   more » « less
4. Singular structures in solutions to the Monge-Ampère equation with point masses

   https://doi.org/10.3934/mine.2023083  

   Mooney, Connor; Rakshit, Arghya (January 2023, Mathematics in Engineering)

   We construct new examples of Monge-Ampère metrics with polyhedral singular structures, motivated by problems related to the optimal transport of point masses and to mirror symmetry. We also analyze the stability of the singular structures under small perturbations of the data given in the problem under consideration. 

   more » « less
5. Optimal boundary regularity for some singular Monge-Ampère equations on bounded convex domains

   https://doi.org/10.3934/dcds.2021188  

   Le, Nam Q. (January 2022, Discrete & Continuous Dynamical Systems)

   By constructing explicit supersolutions, we obtain the optimal global Hölder regularity for several singular Monge-Ampère equations on general bounded open convex domains including those related to complete affine hyperbolic spheres, and proper affine hyperspheres. Our analysis reveals that certain singular-looking equations, such as \begin{document}$$ \det D^2 u = |u|^{-n-2-k} (x\cdot Du -u)^{-k} $$\end{document} with zero boundary data, have unexpected degenerate nature. 

   more » « less