# American Institute of Mathematical Sciences

December  2019, 39(12): 7101-7112. doi: 10.3934/dcds.2019297

## Regularity of monotone transport maps between unbounded domains

 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

A Luis A. Caffarelli en su 70 años, con amistad y admiración

Received  November 2018 Published  September 2019

Fund Project: 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)".

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.

Citation: Dario Cordero-Erausquin, Alessio Figalli. Regularity of monotone transport maps between unbounded domains. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 7101-7112. doi: 10.3934/dcds.2019297
##### References:
 [1] S. Alesker, S. Dar and V. Milman, A remarkable measure preserving diffeomorphism between two convex bodies in ${\mathbb{R}}^n$, Geom. Dedicata, 74 (1999), 201-212.  doi: 10.1023/A:1005087216335.  Google Scholar [2] L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Second edition. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2008. x+334 pp.  Google Scholar [3] L. A. Caffarelli, A localization property of viscosity solutions to the Monge-Ampère equation and their strict convexity, Ann. of Math., 131 (1990), 129-134.  doi: 10.2307/1971509.  Google Scholar [4] L. A. Caffarelli, Some regularity properties of solutions of Monge Ampère equation, Comm. Pure Appl. Math., 44 (1991), 965-969.  doi: 10.1002/cpa.3160440809.  Google Scholar [5] L. A. Caffarelli, Interior $W^{2, p}$ estimates for solutions of the Monge-Ampère equation, Ann. of Math. (2), 131 (1990), 135-150.  doi: 10.2307/1971510.  Google Scholar [6] L. A. Caffarelli, The regularity of mappings with a convex potential, J. Amer. Math. Soc., 5 (1992), 99-104.  doi: 10.1090/S0894-0347-1992-1124980-8.  Google Scholar [7] G. De Philippis and A. Figalli, $W^{2, 1}$ regularity for solutions of the Monge-Ampère equation, Invent. Math., 192 (2013), 55-69.  doi: 10.1007/s00222-012-0405-4.  Google Scholar [8] G. De Philippis and A. Figalli, The Monge-Ampère equation and its link to optimal transportation, Bull. Amer. Math. Soc. (N.S.), 51 (2014), 527-580.  doi: 10.1090/S0273-0979-2014-01459-4.  Google Scholar [9] G. De Philippis, A. Figalli and O. Savin, A note on interior $W^{2, 1+\epsilon}$ estimates for the Monge-Ampère equation, Math. Ann., 357 (2013), 11-22.  doi: 10.1007/s00208-012-0895-9.  Google Scholar [10] A. Figalli, The Monge-Ampère Equation and Its Applications, Zürich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich, 2017. x+200 doi: 10.4171/170.  Google Scholar [11] A. Figalli, Y. Jhaveri and C. Mooney, Nonlinear bounds in Hölder spaces for the Monge-Ampère equation, J. Funct. Anal., 270 (2016), 3808-3827.   Google Scholar [12] A. Figalli, Y.-H. Kim and R. J. McCann, Hölder continuity and injectivity of optimal maps, Arch. Ration. Mech. Anal., 209 (2013), 747-795.  doi: 10.1007/s00205-013-0629-5.  Google Scholar [13] A. Figalli, L. Rifford and C. Villani, Necessary and sufficient conditions for continuity of optimal transport maps on Riemannian manifolds, Tohoku Math. J. (2), 63 (2011), 855-876.  doi: 10.2748/tmj/1325886291.  Google Scholar [14] M. Gromov, Convex sets and Kähler manifolds, in Advances in Differential Geometry and Topology, World Scientific, Teaneck, NJ, 1990, 1–38.  Google Scholar [15] R. J. McCann, Existence and uniqueness of monotone measure-preserving maps, Duke Math. J., 80 (1995), 309-323.  doi: 10.1215/S0012-7094-95-08013-2.  Google Scholar [16] R. J. McCann, A convexity principle for interacting gases, Adv. Math., 128 (1997), 153-179.  doi: 10.1006/aima.1997.1634.  Google Scholar [17] C. Mooney, Partial regularity for singular solutions to the Monge-Ampère equation, Comm. Pure Appl. Math., 68 (2015), 1066-1084.  doi: 10.1002/cpa.21534.  Google Scholar

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##### References:
 [1] S. Alesker, S. Dar and V. Milman, A remarkable measure preserving diffeomorphism between two convex bodies in ${\mathbb{R}}^n$, Geom. Dedicata, 74 (1999), 201-212.  doi: 10.1023/A:1005087216335.  Google Scholar [2] L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Second edition. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2008. x+334 pp.  Google Scholar [3] L. A. Caffarelli, A localization property of viscosity solutions to the Monge-Ampère equation and their strict convexity, Ann. of Math., 131 (1990), 129-134.  doi: 10.2307/1971509.  Google Scholar [4] L. A. Caffarelli, Some regularity properties of solutions of Monge Ampère equation, Comm. Pure Appl. Math., 44 (1991), 965-969.  doi: 10.1002/cpa.3160440809.  Google Scholar [5] L. A. Caffarelli, Interior $W^{2, p}$ estimates for solutions of the Monge-Ampère equation, Ann. of Math. (2), 131 (1990), 135-150.  doi: 10.2307/1971510.  Google Scholar [6] L. A. Caffarelli, The regularity of mappings with a convex potential, J. Amer. Math. Soc., 5 (1992), 99-104.  doi: 10.1090/S0894-0347-1992-1124980-8.  Google Scholar [7] G. De Philippis and A. Figalli, $W^{2, 1}$ regularity for solutions of the Monge-Ampère equation, Invent. Math., 192 (2013), 55-69.  doi: 10.1007/s00222-012-0405-4.  Google Scholar [8] G. De Philippis and A. Figalli, The Monge-Ampère equation and its link to optimal transportation, Bull. Amer. Math. Soc. (N.S.), 51 (2014), 527-580.  doi: 10.1090/S0273-0979-2014-01459-4.  Google Scholar [9] G. De Philippis, A. Figalli and O. Savin, A note on interior $W^{2, 1+\epsilon}$ estimates for the Monge-Ampère equation, Math. Ann., 357 (2013), 11-22.  doi: 10.1007/s00208-012-0895-9.  Google Scholar [10] A. Figalli, The Monge-Ampère Equation and Its Applications, Zürich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich, 2017. x+200 doi: 10.4171/170.  Google Scholar [11] A. Figalli, Y. Jhaveri and C. Mooney, Nonlinear bounds in Hölder spaces for the Monge-Ampère equation, J. Funct. Anal., 270 (2016), 3808-3827.   Google Scholar [12] A. Figalli, Y.-H. Kim and R. J. McCann, Hölder continuity and injectivity of optimal maps, Arch. Ration. Mech. Anal., 209 (2013), 747-795.  doi: 10.1007/s00205-013-0629-5.  Google Scholar [13] A. Figalli, L. Rifford and C. Villani, Necessary and sufficient conditions for continuity of optimal transport maps on Riemannian manifolds, Tohoku Math. J. (2), 63 (2011), 855-876.  doi: 10.2748/tmj/1325886291.  Google Scholar [14] M. Gromov, Convex sets and Kähler manifolds, in Advances in Differential Geometry and Topology, World Scientific, Teaneck, NJ, 1990, 1–38.  Google Scholar [15] R. J. McCann, Existence and uniqueness of monotone measure-preserving maps, Duke Math. J., 80 (1995), 309-323.  doi: 10.1215/S0012-7094-95-08013-2.  Google Scholar [16] R. J. McCann, A convexity principle for interacting gases, Adv. Math., 128 (1997), 153-179.  doi: 10.1006/aima.1997.1634.  Google Scholar [17] C. Mooney, Partial regularity for singular solutions to the Monge-Ampère equation, Comm. Pure Appl. Math., 68 (2015), 1066-1084.  doi: 10.1002/cpa.21534.  Google Scholar
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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