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 and Continuous Dynamical Systems, 2019, 39 (12) : 7101-7112. doi: 10.3934/dcds.2019297
References:
[1]

S. AleskerS. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[9]

G. De PhilippisA. 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.

[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.

[11]

A. FigalliY. Jhaveri and C. Mooney, Nonlinear bounds in Hölder spaces for the Monge-Ampère equation, J. Funct. Anal., 270 (2016), 3808-3827. 

[12]

A. FigalliY.-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.

[13]

A. FigalliL. 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.

[14]

M. Gromov, Convex sets and Kähler manifolds, in Advances in Differential Geometry and Topology, World Scientific, Teaneck, NJ, 1990, 1–38.

[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.

[16]

R. J. McCann, A convexity principle for interacting gases, Adv. Math., 128 (1997), 153-179.  doi: 10.1006/aima.1997.1634.

[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.

show all references

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

References:
[1]

S. AleskerS. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[9]

G. De PhilippisA. 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.

[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.

[11]

A. FigalliY. Jhaveri and C. Mooney, Nonlinear bounds in Hölder spaces for the Monge-Ampère equation, J. Funct. Anal., 270 (2016), 3808-3827. 

[12]

A. FigalliY.-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.

[13]

A. FigalliL. 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.

[14]

M. Gromov, Convex sets and Kähler manifolds, in Advances in Differential Geometry and Topology, World Scientific, Teaneck, NJ, 1990, 1–38.

[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.

[16]

R. J. McCann, A convexity principle for interacting gases, Adv. Math., 128 (1997), 153-179.  doi: 10.1006/aima.1997.1634.

[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.

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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