Abstract
In this paper, we study the optimal mass transportation problem in \({\mathbb{R}^{d}}\) for a class of cost functions that we call relativistic cost functions. Consider as a typical example, the cost function c(x, y) = h(x − y) being the restriction of a strictly convex and differentiable function to a ball and infinite outside this ball. We show the existence and uniqueness of the optimal map given a relativistic cost function and two measures with compact support, one of the two being absolutely continuous with respect to the Lebesgue measure. With an additional assumption on the support of the initial measure and for supercritical speed of propagation, we also prove the existence of a Kantorovich potential and study the regularity of this map. Besides these general results, a particular attention is given to a specific cost because of its connections with a relativistic heat equation as pointed out by Brenier (Extended Monge–Kantorovich Theory. Optimal Transportation and Applications, 2003).
Similar content being viewed by others
References
Agueh M.: Existence of solutions to degenerate parabolic equations via the Monge–Kantorovich theory. Adv. Differ. Equ. 10(3), 309–360 (2005)
Ambrosio, L.: Lecture Notes on Optimal Transport Problems. Mathematical Aspects of Evolving Interfaces (Funchal, 2000). Lecture Notes in Math., vol. 1812, pp. 1–52. Springer, Berlin (2003)
Ambrosio, L.: Steepest Descent Flows and Applications to Spaces of Probability Measures. Lectures notes. Santander (July 2004)
Ambrosio, L., Pratelli, A.: Existence and Stability Results in the L 1 Theory of Optimal Transportation. Lecture Notes in Math., vol. 1813, pp. 123–160. Springer, Berlin (2003)
Ambrosio L., Fusco N., Pallara D.: Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York (2000)
Ambrosio L., Kirchheim B., Pratelli A.: Existence of optimal transport maps for crystalline norms. Duke Math. J. 125(2), 207–241 (2004)
Ambrosio L., Gigli N., Savaré G.: Gradient Flows in Metric Spaces and in the Spaces of Probability Measures. Lectures in Mathematics ETH Zurich. Birkauser Verlag, Basel (2005)
Ambrosio L., Colesanti A., Villa E.: Outer Minkowski content for some classes of closed sets. Math. Ann. 342(4), 727–748 (2008)
Andreu F., Caselles V., Mazón J.M.: Existence and uniqueness of solution for a parabolic quasilinear problem for linear growth functionals with L 1 data. Math. Ann. 322, 139–206 (2002)
Andreu F., Caselles V., Mazón J.M. (2004). Parabolic Quasilinear Equations Minimizing Linear Growth Functionals. Progress in Mathematics, vol. 223. Birkhäuser Verlag
Andreu F., Caselles V., Mazón J.M.: A strongly degenerate quasilinear equation: the elliptic case. Ann. Sc. Norm. Super. Pisa Cl. Sci. 3(3), 555–587 (2004)
Andreu F., Caselles V., Mazón J.M.: strong degenerate quasilinear equation: the parabolic case. Arch Rational Mech. Anal. 176(3), 415–453 (2005)
Andreu F., Caselles V., Mazón J.M.: A strongly degenerate quasilinear elliptic equation. Nonlinear Anal. 61, 637–669 (2005)
Andreu F., Caselles V., Mazón J.M.: The Cauchy problem for a strong degenerate quasilinear equation. J. Eur. Math. Soc. 7, 361–393 (2005)
Andreu F., Caselles V., Mazón J.M., Moll S.: Finite propagation speed for limited flux diffusion equations. Arch. Ration. Mech. Anal. 182(2), 269–297 (2006)
Brenier Y.: Décomposition polaire et réarrangement monotone des champs de vecteurs. C. R. Acad. Sci. Paris Sér. I Math. 305(19), 805–808 (1987)
Brenier Y.: Polar factorization and monotone rearrangement of vector-valued functions. Commun. Pure Appl. Math. 44(4), 375–417 (1991)
Brenier, Y.: Extended Monge–Kantorovich Theory. Optimal Transportation and Applications (Martina Franca, 2001). Lecture Notes in Math., vol. 1813, pp. 91–121. Springer, Berlin (2003)
Brezis, H.: Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. (French) North-Holland Mathematics Studies, No. 5. Notas de Matemà àtica (50). North-Holland Publishing Co., Amsterdam–London; American Elsevier Publishing Co., Inc., New York (1973)
Caffarelli L.A., Feldman M., McCann R.J: Constructing optimal maps for Monge’s transport problem as a limit of strictly convex costs. J. Am. Math. Soc. 15(1), 1–26 (2002)
Caravenna L.: A proof of Sudakov theorem with strictly convex norms. Math. Z. 268(1–2), 371–407 (2011)
Carlier G., De Pascale L., Santambrogio F.: A strategy for non-strictly convex transport costs and the example of ||x − y||p in \({\mathbb{R}^2}\) . Commun. Math. Sci. 8(4), 931–941 (2010)
Caselles V.: Convergence of the “relativistic” heat equation to the heat equation as \({c\to\infty}\) . Publicaciones Matemà àtiques 51(1), 121–142 (2007)
Champion T., De Pascale L.: The Monge problem for strictly convex norms in \({\mathbb{R}^d}\) . J. Eur. Math. Soc. 12(6), 1355–1369 (2010)
Champion T., De Pascale L.: The Monge problem in \({\mathbb{R}^d}\) . Duke Math. J. 157(3), 551–572 (2011)
Champion T., De Pascale L., Juutinen P.: The ∞-Wasserstein distance: local solutions and existence of optimal transport maps. SIAM J. Math. Anal. 40(1), 1–20 (2008)
Evans, L., Gangbo, W.: Differential equation methods for the Monge kantorovich mass transfer. Memoirs AMS, 653 (1999)
Gangbo W., McCann R.J.: Optimal maps in Monge’s mass transport problem. C. R. Acad. Sci. Paris Sér. I Math. 321(12), 1653–1658 (1995)
Gangbo W., McCann R.J.: The geometry of optimal transportation. Acta Math. 177(2), 113–161 (1996)
Jimenez, C., Santambrogio, F.: Optimal transportation for a quadratic cost with convex constraints and applications. Journal de Mathématiques Pures et Appliquées, accepted
Jordan R., Kinderlehrer D., Otto F.: The variational formulation of the Fokker–Planck equation. SIAM J. Math. Anal. 29(1), 1–17 (1998)
Kantorovitch L.: On the translocation of masses. C. R. (Doklady) Acad. Sci. URSS (N.S.) 37, 199–201 (1942)
McCann R., Puel M.: Constructing a relativistic heat flow by transport time steps. Ann. Inst. H. Poincaré Anal. Non Linéaire 26(6), 2539–2580 (2009)
Mihalas, D., Mihalas, B.: Foundations of Radiation Hydrodynamics. Oxford University Press (1984)
Otto, F.: Doubly degenerate diffusion equations as steepest descent (preprint 1996)
Rockafellar R.T.v: Convex analysis. Princeton Mathematical Series No. 28. Princeton University Press, Princeton (1970)
Rosenau P.: Tempered diffusion: a transport process with propagating fronts and initial delay. Phys. Rev. A 46, 7371–7374 (1992)
Schneider R.: Convex Bodies: The Brunn–Minkowski Theory. Encyclopedia of Mathematics and its Applications, vol. 44. Cambridge University Press, Cambridge (1993)
Smith C.S., Knott M.: Note on the optimal transportation of distributions. J. Optim. Theory Appl. 52(2), 323–329 (1987)
Trudinger N.S., Wang X.-J.: On the Monge mass transfer problem. Calc. Var. Partial Differ. Equ. 13(1), 19–31 (2001)
Villani, C.: Topics in Optimal Transportation. Graduate Studies in Math., vol. 58. AMS (2003)
Villani, C.: Optimal Transport. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 338. Springer-Verlag, Berlin (2009); old and new
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by L. Ambrosio.
Rights and permissions
About this article
Cite this article
Bertrand, J., Puel, M. The optimal mass transport problem for relativistic costs. Calc. Var. 46, 353–374 (2013). https://doi.org/10.1007/s00526-011-0485-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00526-011-0485-9