Duke Mathematical Journal

The Monge problem in Rd

Thierry Champion and Luigi De Pascale

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Abstract

We first consider the Monge problem in a convex bounded subset of Rd. The cost is given by a general norm, and we prove the existence of an optimal transport map under the classical assumption that the first marginal is absolutely continuous with respect to the Lebesgue measure. In the final part of the paper we show how to extend this existence result to a general open subset of Rd.

Article information

Source
Duke Math. J. Volume 157, Number 3 (2011), 551-572.

Dates
First available in Project Euclid: 1 April 2011

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1301678733

Digital Object Identifier
doi:10.1215/00127094-1272939

Mathematical Reviews number (MathSciNet)
MR2785830

Subjects
Primary: 49J45: Methods involving semicontinuity and convergence; relaxation
Secondary: 49K30: Optimal solutions belonging to restricted classes 49Q20: Variational problems in a geometric measure-theoretic setting

Citation

Champion, Thierry; De Pascale, Luigi. The Monge problem in ${\mathbb R}^d$. Duke Math. J. 157 (2011), no. 3, 551--572. doi:10.1215/00127094-1272939. https://projecteuclid.org/euclid.dmj/1301678733


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References

  • L. Ambrosio, “ Lecture notes on optimal transportation problems” in Mathematical Aspects of Evolving Interfaces (Funchal, Portugal, 2000), Lecture Notes in Math. 1812, Springer, Berlin, 2003, 1–52.
  • L. Ambrosio, B. Kirchheim, and A. Pratelli, Existence of optimal transport maps for crystalline norms, Duke Math. J. 125 (2004), 207–241.
  • L. Ambrosio and A. Pratelli, “ Existence and stability results in the $L\sp 1$ theory of optimal transportation” in Optimal Transportation and Applications (Martina Franca, Italy, 2001), Lecture Notes in Math. 1813, Springer, Berlin, 2003, 123–160.
  • G. Anzellotti and S. Baldo, Asymptotic development by $\Gamma$-convergence, Appl. Math. Optim. 27 (1993), 105–123.
  • H. Attouch, Viscosity solutions of minimization problems, SIAM J. Optim. 6 (1996), 769–806.
  • L. A. Caffarelli, M. Feldman, and R. J. Mccann, Constructing optimal maps for Monge's transport problem as a limit of strictly convex costs, J. Amer. Math. Soc. 15 (2002), 1–26.
  • L. Caravenna, A proof of Sudakov theorem with strictly convex norms, Math. Zeitschift (2010), DOI 10.1007/S00209-010-0677-6
  • T. Champion and L. De Pascale, The Monge problem for strictly convex norms in $\R^d$, J. Eur. Math. Soc. 12 (2010), 1355–1369.
  • T. Champion, L. De Pascale, and P. Juutinen, The $\infty$-Wasserstein distance: Local solutions and existence of optimal transport maps, SIAM J. Math. Anal. 40 (2008), 1–20.
  • L. De Pascale, L. C. Evans, and A. Pratelli, Integral estimates for transport densities, Bull. London Math. Soc. 36 (2004), 383–396.
  • L. De Pascale and A. Pratelli, Regularity properties for Monge transport density and for solutions of some shape optimization problem, Calc. Var. Partial Differential Equations 14 (2002), 249–274.
  • —, Sharp summability for Monge transport density via interpolation, ESAIM Control Optim. Calc. Var. 10 (2004), 549–552.
  • L. C. Evans and W. Gangbo, Differential equations methods for the Monge-Kantorovich mass transfer problem, Mem. Amer. Math. Soc. 137 (1999), no. 653.
  • L.V. Kantorovich, On the translocation of masses, C.R. (Dokl.) Acad. Sci. URSS 37 (1942), 199–201.
  • —, On a problem of Monge, C. R. (Dokl.) Acad. Sci. URSS (N.S.) 3 (1948), 225–226; English translation in J. Math. Sci. (N.Y.) 133 (2006), 1383.
  • G. Monge, Mémoire sur la théorie des Déblais et des Remblais, Histoire de l'Académie des Sciences de Paris, 1781.
  • A. Pratelli, On the equality between Monge's infimum and Kantorovich's minimum in mass transportation, Ann. Inst. H. Poincaré Probab. Statist 43 (2007), 1–13.
  • F. Santambrogio, Absolute continuity and summability of transport densities: Simpler proofs and new estimates Calc. Var. Partial Differential Equations 36 (2009), 343–354.
  • V. N. Sudakov, Geometric problems in the theory of infinite-dimensional probability distributions, cover-to-cover translation of Trudy Mat. Inst. Steklov 141 (1976), Proc. Steklov Inst. Math. (1979), no. 2, 1–178.
  • N. S. Trudinger and X.-J. Wang, On the Monge mass transfer problem, Calc. Var. Partial Differential Equations 13 (2001), 19–31.
  • C. Villani, Topics in optimal transportation, Grad. Stud. Math. 58, Amer. Math. Soc., Providence, 2003.
  • —, Optimal Transport, Old and New, Grundlehren Math. Wiss. 338, Springer, Berlin, 2008.