Prof. CHEN Shibing from University of Science and Technology of China (USTC) of the Chinese Academy of Sciences and his collaborators have studied the global regularity for the Monge-Ampère equation with natural boundary condition. The study was published in Annals of Mathematics.
The optimal transport problem is about finding the most economical way to transfer mass from one location to another location. When the densities are continuous, the problem can be reduced to a natural boundary value problem of Monge-Ampère equation. The Monge-Ampère equation, which has been studied by many famous mathematicians, is a well-known equation in the field of nonlinear partial differential equations.
The regularity of optimal transport map is a fundamental question in the area. In 1996, Caffarelli proved, in his landmark work, that the optimal transport map is smooth when the two domains are uniformly convex and the density functions are smooth. For more than 20 years, experts in the field have believed that these conditions, especially the uniform convexity of the domains, were essential.
In this study, the researchers proved the global smoothness of the solution to the boundary value problem of Monge-Ampère equation derived from the optimal transport problem.
They removed the uniform convexity condition and reduced the regularity assumption on domains in the Euclidean space.
Besides, they developed a new technology which turns out to be an important tool for the free boundary problem arising in optimal transportation.
This study has a potential impact on the fields such as image processing and machine learning.
Annals of Mathematics
Global regularity for the Monge-Ampère equation with natural boundary condition