Regularity of monotone transport maps between unbounded domains

Dario Cordero-Erausquin; Alessio Figalli

doi:10.3934/dcds.2019297

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2019, Volume 39, Issue 12: 7101-7112. Doi: 10.3934/dcds.2019297 open access

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Regularity of monotone transport maps between unbounded domains

Dario Cordero-Erausquin^1, and
Alessio Figalli^2, ,

1.
Institut de Mathematiques de Jussieu, Sorbonne Université - UPMC (Paris 6), 4 Place Jussieu, 75005 Paris, France
2.
ETH Zürich, Mathematics Department, Rämistrasse 101, 8092 Zürich, Switzerland

^* Corresponding author: Alessio Figalli
^* Corresponding author: Alessio Figalli

Received: October 31, 2018

Published: September 2019

The second author has received funding from the European Research Council under the Grant Agreement No. 721675 "Regularity and Stability in Partial Differential Equations (RSPDE)"

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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.

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Mathematics Subject Classification: Primary: 35J60, 49Q20; Secondary: 35J96, 49N60.

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Figure 1. We subtract the affine function $ \ell_\varepsilon(z): = \varepsilon(z_1+\eta_0) $ from $ u $. Because $ \mathbf{0} \in \partial ({\rm dom}(u)) $, $ u_\varepsilon|_{\partial S_\varepsilon} $ contains some vertical segments in its graph

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