Advanced Search
Article Contents
Article Contents

Regularity of monotone transport maps between unbounded domains

  • * Corresponding author: Alessio Figalli

    * Corresponding author: Alessio Figalli

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)"

Abstract Full Text(HTML) Figure(1) Related Papers Cited by
  • 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.

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


    \begin{equation} \\ \end{equation}
  • 加载中
  • 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

  • [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.
  • 加载中
Open Access Under a Creative Commons license



Article Metrics

HTML views(528) PDF downloads(300) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint