Boundary regularity of optimal transport maps on convex domains

TL;DR

This paper investigates the boundary regularity of optimal transport maps between convex domains with quadratic cost, establishing new regularity results under various smoothness conditions and characterizing pointwise regularity for planar polytopes.

Contribution

It provides new regularity results for optimal transport potentials up to the boundary, including $C^{1, 1- ext{epsilon}}$ and $C^{2, ext{alpha}}$ regularity, and introduces a monotonicity formula as a key technical tool.

Findings

Proves $C^{1, 1- ext{epsilon}}$ regularity for potentials up to the boundary.

Improves to $C^{2, ext{alpha}}$ regularity when boundary is $C^{1, ext{alpha}}$.

Characterizes pointwise $C^{1, 1}$-regularity for planar polytopes.

Abstract

We study the regularity of optimal transport maps between convex domains with quadratic cost. For nondegenerate $C^{\alpha}$-densities, we prove $C^{1, 1-\varepsilon}$-regularity of the potentials up to the boundary. If in addition the boundary is $C^{1, \alpha}$, we improve this to $C^{2, \alpha}$-regularity. We also investigate pointwise $C^{1, 1}$-regularity at boundary points. We obtain a complete characterization of pointwise $C^{1, 1}$-regularity for planar polytopes in terms of the geometry of tangent cones. Furthermore, we study the regularity of optimal transport maps with degenerate densities on cones, which arise from recent developments in K\"ahler geometry. The main new technical tool we introduce is a monotonicity formula for optimal transport maps on convex domains which characterizes the homogeneity of blow-ups.