Sharp boundary ε-regularity of optimal transport maps

In this paper we develop a boundary epsilon-regularity theory for optimal transport maps between bounded open sets with C-1,C-alpha-boundary. Our main result asserts sharp C-1,C-alpha-regularity of transport maps at the boundary in form of a linear estimate under certain assumptions: The main quantitative assumptions are that the local nondimensionalized transport cost is small and that the boundaries are locally almost flat in Our method is completely variational and builds on the recently developed interior regularity theory. (C) 2021 Elsevier Inc. All rights reserved.

Automated summary

This paper develops a boundary epsilon-regularity theory for optimal transport maps between bounded open sets with C-1,C-alpha-boundary, asserting sharp C-1,C-alpha-regularity of transport maps at the boundary in form of a linear estimate under certain assumptions. The main quantitative assumptions are small local nondimensionalized transport cost and locally almost flat boundaries. The method is completely variational and builds on the recently developed interior regularity theory.