Abstract
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.
Similar content being viewed by others
References
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)
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)
Beckmann, M.: A continuous model of transportation. Econometrica 20, 643–660 (1952)
Bombieri, E., De Giorgi, E., Giusti, E.: Minimal cones and the Bernstein problem. Invent. Math. 7(3), 243–268 (1969)
Bouchitté, G., Buttazzo, G.: Characterization of optimal shapes and masses through Monge-Kantorovich equation. J. Eur. Math. Soc. 3(2), 139–168 (2001)
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)
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)
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)
Buckley, J.J.: Graphs of measurable functions. Proc. Am. Math. Soc. 44, 78–80 (1974)
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)
Caravenna, L.: A proof of Sudakov theorem with strictly convex norms. Math. Z. 268, 371–407 (2011)
Carlier, G., Jimenez, C., Santambrogio, F.: Optimal transportation with traffic congestion and Wardrop equilibria. SIAM J. Control Optim. 47, 1330–1350 (2008)
Cellina, A.: On the bounded slope condition and the validity of the Euler Lagrange equation. SIAM J. Control Optim. 40(4), 1270–1279 (2001)
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 \(R^d\). Duke Math. J. 157(3), 551–572 (2011)
De Pascale, L., Evans, L.C., Pratelli, A.: Integral estimates for transport densities. Bull. Lond. Math. Soc. 36(3), 383–395 (2004)
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)
De Pascale, L., Pratelli, A.: Sharp summability for Monge transport density via interpolation. ESAIM Control Optim. Calc. Var. 10(4), 549–552 (2004)
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)
Evans, L.C., Gangbo, W.: Differential equations methods for the Monge-Kantorovich mass transfer problem. Mem. Amer. Math. Soc., 137, 653 (1999)
Feldman, M., McCann, R.: Uniqueness and transport density in Monge’s mass transportation problem. Calc. Var. Partial Differ. Equ. 15(1), 81–113 (2002)
Gangbo, W., McCann, R.J.: Shape recognition via Wasserstein distance. Quart. Appl. Math. 58, 705–737 (2000)
Górny, W.: Planar least gradient problem: existence, regularity and anisotropic case arXiv:1608.02617
Górny, W., Rybka, P., Sabra, A.: Special cases of the planar least gradient problem. Nonlinear Anal. 151, 66–95 (2017)
Hartman, P.: On the bounded slope condition. Pacific J. Math. 18(3), 495–511 (1966)
Mercier, G.: Continuity results for TV-minimizers. Indiana University Math. J. 67, 1499–1545 (2018)
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)
Mazon, J.M.: The Euler-Lagrange equation for the anisotropic least gradient problem. Nonlinear Anal. Real World Appl. 31, 452–472 (2016)
Mazon, J.M., Rossi, J.D., de Leon, S.S.: Functions of least gradient and 1-harmonic functions. Indiana Univ. J. Math. 63:1067–1084
McCann, R., Sosio, M.: Hölder continuity for optimal multivalued mappings. SIAM J. Math. Anal. 43, 1855–1871 (2011)
Monge, G.: Mémoire sur la théorie des déblais et des remblais, Histoire de l’Académie Royale des Sciences de Paris, pp. 666–704 (1781)
Lellmann, J., Lorenz, D.A., Schoenlieb, C., Valkonen, T.: Imaging with Kantorovich–Rubinstein discrepancy. SIAM J. Imaging Sci. 7(4), 2833–2859 (2014)
Santambrogio, F.: Absolute continuity and summability of transport densities: simpler proofs and new estimates. Calc. Var. Partial Differ. Equ. 36, 343–354 (2009)
Santambrogio, F.: Optimal transport for applied mathematicians. In: Progress in Nonlinear Differential Equations and their Applications 87, Birkhäuser Basel (2015)
Trudinger, N., Wang, X.-J.: On the Monge mass transfer problem. Calc. Var. Partial Differ. Equ. 13, 19–31 (2001)
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)
Stampacchia, G.: On some regular multiple integral problems in the calculus of variations. Commun. Pure Appl. Math. 16, 383–421 (1963)
Sternberg, P., Williams, G., Ziemer, W.P.: Existence, uniqueness, and regularity for functions of least gradient. J. Reine Angew. Math. 430, 35–60 (1992)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by L. Ambrosio.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Dweik, S., Santambrogio, F. \(L^p\) bounds for boundary-to-boundary transport densities, and \(W^{1,p}\) bounds for the BV least gradient problem in 2D. Calc. Var. 58, 31 (2019). https://doi.org/10.1007/s00526-018-1474-z
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00526-018-1474-z