On the Regularity of Optimal Transports between Degenerate Densities

TL;DR

This paper investigates the regularity of optimal transport maps between degenerate densities, providing new mathematical insights into how these maps behave when densities vanish near domain boundaries.

Contribution

It establishes regularity results for optimal transports between degenerate densities on convex domains, extending understanding of the Monge--Ampère equation in degenerate cases.

Findings

Proves regularity of optimal transport maps for densities vanishing at boundaries.

Analyzes transport between measures supported on convex domains with boundary behavior.

Extends classical results to degenerate density scenarios.

Abstract

We study the most common image and informal description of the optimal transport problem for quadratic cost, also known as the second boundary value problem for the Monge--Amp\`{e}re equation -- What is the most efficient way to fill a hole with a given pile of sand? -- by proving regularity results for optimal transports between degenerate densities. In particular, our work contains an analysis of the setting in which holes and sandpiles are represented by absolutely continuous measures concentrated on bounded convex domains whose densities behave like nonnegative powers of the distance functions to the boundaries of these domains.