\(L^p\) bounds for boundary-to-boundary transport densities, and \(W^{1,p}\) bounds for the BV least gradient problem in 2D

  • Samer Dweik
  • Filippo SantambrogioEmail author


The least gradient problem (minimizing the total variation with given boundary data) is equivalent, in the plane, to the Beckmann minimal-flow problem with source and target measures located on the boundary of the domain, which is in turn related to an optimal transport problem. Motivated by this fact, we prove \(L^p\) summability results for the solution of the Beckmann problem in this setting, which improve upon previous results where the measures were themselves supposed to be  \(L^p\). In the plane, we carry out all the analysis for general strictly convex norms, which requires to first introduce the corresponding optimal transport tools. We then obtain results about the \(W^{1,p}\) regularity of the solution of the anisotropic least gradient problem in uniformly convex domains.

Mathematics Subject Classification

49J45 49Q20 35B65 



  1. 1.
    Ambrosio, L.: Lecture notes on optimal transport problems. In: Mathematical Aspects of Evolving Interfaces, Lecture Notes in Mathematics (1812), pp. 1–52. Springer, New York (2003)Google Scholar
  2. 2.
    Ambrosio, L., Pratelli, A.: Existence and stability results in the \(L^1\) theory of optimal transportation. In: Caffarelli, L.A., Salsa, S. (eds.) Optimal Transportation and Applications, Lecture Notes in Mathematics (CIME Series, Martina Franca, 2001) 1813, pp. 123–160 (2003)Google Scholar
  3. 3.
    Beckmann, M.: A continuous model of transportation. Econometrica 20, 643–660 (1952)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bombieri, E., De Giorgi, E., Giusti, E.: Minimal cones and the Bernstein problem. Invent. Math. 7(3), 243–268 (1969)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bouchitté, G., Buttazzo, G.: Characterization of optimal shapes and masses through Monge-Kantorovich equation. J. Eur. Math. Soc. 3(2), 139–168 (2001)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bouchitté, G., Buttazzo, G., Seppecher, P.: Shape optimization solutions via Monge-Kantorovich equation. C. R. Acad. Sci. Paris Sér. I Math. 324(10), 1185–1191 (1997)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bouchitté, G., Champion, T., Jimenez, C.: Completion of the space of measures in the Kantorovitch norm In: Acerbi, E.D., Mingione, G.R. (eds.) Proceedings of “Trends in the Calculus of Variations”, Parma, 2004, Rivista di Matematica della Università di Parma, serie 7 (4*), pp. 127–139 (2004)Google Scholar
  8. 8.
    Brasco, L., Carlier, G., Santambrogio, F.: Congested traffic dynamics, weak flows and very degenerate elliptic equations. J. Math. Pures Appl. 93(6), 652–671 (2010)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Buckley, J.J.: Graphs of measurable functions. Proc. Am. Math. Soc. 44, 78–80 (1974)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Caffarelli, L., Feldman, M., McCann, R.: Constructing optimal maps for Monge’s transport problem as a limit of strictly convex costs. J. Am. Math. Soc. 15(1), 1–26 (2002)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Caravenna, L.: A proof of Sudakov theorem with strictly convex norms. Math. Z. 268, 371–407 (2011)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Carlier, G., Jimenez, C., Santambrogio, F.: Optimal transportation with traffic congestion and Wardrop equilibria. SIAM J. Control Optim. 47, 1330–1350 (2008)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Cellina, A.: On the bounded slope condition and the validity of the Euler Lagrange equation. SIAM J. Control Optim. 40(4), 1270–1279 (2001)MathSciNetCrossRefGoogle Scholar
  14. 14.
    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)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Champion, T., De Pascale, L.: The Monge problem in \(R^d\). Duke Math. J. 157(3), 551–572 (2011)MathSciNetCrossRefGoogle Scholar
  16. 16.
    De Pascale, L., Evans, L.C., Pratelli, A.: Integral estimates for transport densities. Bull. Lond. Math. Soc. 36(3), 383–395 (2004)MathSciNetCrossRefGoogle Scholar
  17. 17.
    De Pascale, L., Pratelli, A.: Regularity properties for Monge transport density and for solutions of some shape optimization problem. Calc. Var. Partial Differ. Equ. 14(3), 249–274 (2002)MathSciNetCrossRefGoogle Scholar
  18. 18.
    De Pascale, L., Pratelli, A.: Sharp summability for Monge transport density via interpolation. ESAIM Control Optim. Calc. Var. 10(4), 549–552 (2004)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Dweik, S., Santambrogio, F.: Summability estimates on transport densities with Dirichlet regions on the boundary via symmetrization techniques. ESAIM Control Optim. Calc. Var. 24(3), 1167–1180 (2018)CrossRefGoogle Scholar
  20. 20.
    Evans, L.C., Gangbo, W.: Differential equations methods for the Monge-Kantorovich mass transfer problem. Mem. Amer. Math. Soc., 137, 653 (1999)Google Scholar
  21. 21.
    Feldman, M., McCann, R.: Uniqueness and transport density in Monge’s mass transportation problem. Calc. Var. Partial Differ. Equ. 15(1), 81–113 (2002)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Gangbo, W., McCann, R.J.: Shape recognition via Wasserstein distance. Quart. Appl. Math. 58, 705–737 (2000)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Górny, W.: Planar least gradient problem: existence, regularity and anisotropic case arXiv:1608.02617
  24. 24.
    Górny, W., Rybka, P., Sabra, A.: Special cases of the planar least gradient problem. Nonlinear Anal. 151, 66–95 (2017)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Hartman, P.: On the bounded slope condition. Pacific J. Math. 18(3), 495–511 (1966)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Mercier, G.: Continuity results for TV-minimizers. Indiana University Math. J. 67, 1499–1545 (2018)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Moradifam, A., Nachman, A., Tamasan, A.: Uniqueness of minimizers of weighted least gradient problems arising in conductivity imaging. Calc. Var. Partial Differ. Equ 57, 6 (2018)CrossRefGoogle Scholar
  28. 28.
    Mazon, J.M.: The Euler-Lagrange equation for the anisotropic least gradient problem. Nonlinear Anal. Real World Appl. 31, 452–472 (2016)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Mazon, J.M., Rossi, J.D., de Leon, S.S.: Functions of least gradient and 1-harmonic functions. Indiana Univ. J. Math. 63:1067–1084Google Scholar
  30. 30.
    McCann, R., Sosio, M.: Hölder continuity for optimal multivalued mappings. SIAM J. Math. Anal. 43, 1855–1871 (2011)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Monge, G.: Mémoire sur la théorie des déblais et des remblais, Histoire de l’Académie Royale des Sciences de Paris, pp. 666–704 (1781)Google Scholar
  32. 32.
    Lellmann, J., Lorenz, D.A., Schoenlieb, C., Valkonen, T.: Imaging with Kantorovich–Rubinstein discrepancy. SIAM J. Imaging Sci. 7(4), 2833–2859 (2014)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Santambrogio, F.: Absolute continuity and summability of transport densities: simpler proofs and new estimates. Calc. Var. Partial Differ. Equ. 36, 343–354 (2009)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Santambrogio, F.: Optimal transport for applied mathematicians. In: Progress in Nonlinear Differential Equations and their Applications 87, Birkhäuser Basel (2015)Google Scholar
  35. 35.
    Trudinger, N., Wang, X.-J.: On the Monge mass transfer problem. Calc. Var. Partial Differ. Equ. 13, 19–31 (2001)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Spradlin, G., Tamasan, A.: Not all traces on the circle come from functions of least gradient in the disk. Indiana Uni. Math. J. 63, 1819–1837 (2014)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Stampacchia, G.: On some regular multiple integral problems in the calculus of variations. Commun. Pure Appl. Math. 16, 383–421 (1963)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Sternberg, P., Williams, G., Ziemer, W.P.: Existence, uniqueness, and regularity for functions of least gradient. J. Reine Angew. Math. 430, 35–60 (1992)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques d’OrsayUniv. Paris-Sud, CNRS, Université Paris-SaclayOrsay CedexFrance
  2. 2.Institut Camille JordanUniversité Claude Bernard Lyon 1Villeurbanne CedexFrance

Personalised recommendations