Sharp boundary $\epsilon$-regularity of optimal transport maps

Abstract

In this paper we develop a boundary $\epsilon$-regularity theory for optimal transport maps between bounded open sets with $C^{1,\alpha }$-boundary. Our main result asserts sharp $C^{1,\alpha }$-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 $C^{1,\alpha }$. Our method is completely variational and builds on the recently developed interior regularity theory.

52 pages, 6 figures, final version

References in corpus (2)

• Quantitative linearization results for the Monge-Ampère equation
• Variational approach to regularity of optimal transport maps: general cost functions

Cited by in corpus (1)

• Variational approach to regularity of optimal transport maps: general cost functions