# Regularity of monotone transport maps between unbounded domains

## TL;DR

This paper extends the regularity results of monotone transport maps to cases where the involved measures have unbounded supports, broadening applicability in PDE and geometric contexts.

## Contribution

It demonstrates that regularity of monotone transport maps can be achieved without the compact support assumption, under certain conditions.

## Key findings

- Regularity holds for measures with unbounded supports in specific scenarios
- Extends classical results beyond compactly supported measures
- Provides new insights for applications in PDE and geometry

## Abstract

The regularity of monotone transport maps plays an important role in several applications to PDE and geometry. Unfortunately, the classical statements on this subject are restricted to the case when the measures are compactly supported. In this note we show that, in several situations of interest, one can to ensure the regularity of monotone maps even if the measures may have unbounded supports.