| Title | THE MONGE PROBLEM IN R d : VARIATIONS ON A THEME II |
| Publication Type | Journal Article |
| Year of Publication | 2011 |
| Authors | DE PASCALE, LUIGI |
| Secondary Title | Communications in Applied Analysis |
| Volume | 15 |
| Issue | 3 |
| Start Page | 325 |
| Pagination | 340 |
| Date Published | 08/2011 |
| Type of Work | scientific: mathematics |
| ISSN | 1083–2564 |
| AMS | 49J4, 49K30, 49Q20 |
| Abstract | In some recent papers [11, 12] it is proved that, under natural assumptions on the first marginal, the Monge problem in the metric space Rd equipped with a general norm admits a solution. Although the basic idea of the solution is simple the proof involves some very complex technical results. Here we will report a proof of the result in the simpler case of uniformly convex norms. Uniform convexity allow us to reduce the technical burdens while still giving the main ideas of the general proof. The proof of the density of the transport set given in this paper is original.
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| URL | http://www.acadsol.eu/en/articles/15/3/5.pdf |
| Short Title | MONGE’S PROBLEM |
| Refereed Designation | Refereed |
| Full Text | REFERENCES[1] N. Ahnad, H. K. Kim, R. McCann, Optimal transportation, topology and temporarily available on uniqueness. To appear on Bull. Math. Sci. Preprint,. http://www.math.toronto.edu/mccann/
[2] Ambrosio, L., Lecture Notes on Optimal Transportation Problems, Mathematical aspects of evolving interfaces (Funchal, 2000), Lecture Notes in Math., 1812, Springer, Berlin, 2003, 1–52. [3] 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. [4] L. Ambrosio, B. Kirchheim, A. Pratelli, Existence of optimal transport maps for crystalline norms, Duke Math. J., 125 (2004), no. 2, 207–241. [5] L. Ambrosio, A. Pratelli, Existence and stability results in the L 1 theory of optimal transportation, Optimal transportation and applications (Martina Franca, 2001), Lecture Notes in Math., 1813, Springer, Berlin, 2003, 123–160. [6] G. Anzellotti, S. Baldo, Asymptotic development by Γ-convergence, Appl. Math. Optim. 27 (1993), no. 2, 105–123. [7] H. Attouch, Viscosity solutions of minimization problems, SIAM J. Optim., 6 (1996), no. 3, 769–806. [8] S.Bianchini, L. Caravenna, On optimality of c−cyclically monotone transference plans, C. R. Math. Acad. Sci. Paris, 348 (2010), no. 11-12, 613618. [9] L.A. Caffarelli, M. Feldman, R. J. McCann, Constructing optimal maps for Monge’s transport problem as a limit of strictly convex costs, J. Amer. Math. Soc., 15 (2002), no. 1, 1–26. [10] L. Caravenna, A partial proof of Sudakov theorem via disintegration of measures, To appear on Mathematische Zeitschrift, published on line DOI: 10.1007/s00209-010-0677-6, Preprint SISSA (2008) available at http://cvgmt.sns.it. [11] T. Champion, L. De Pascale, The Monge problem for strictly convex norms in R d , J.Europ.Math.Soc., 12 (2010), no.6, 1355–1369. [12] T. Champion, L. De Pascale, The Monge problem in R d , Duke Math. J., 157 (2011), no.3, 551–572. [13] T. Champion, L. De Pascale, P. Juutinen, The ∞-Wasserstein distance: local solutions and existence of optimal transport maps, SIAM J. of Mathematical Analysis, 40 (2008), no. 1, 1-20. [14] L. C. Evans, W. Gangbo, Differential Equations Methods for the Monge–Kantorovich Mass Transfer Problem, Mem. Amer. Math. Soc., 137 (1999). [15] D. Feyel, A. S. Ustunel, Monge-Kantorovitch measure transportation and Monge-Ampre equation on Wiener space. Probab. Theory Related Fields, 128 (2004), no. 3, 347–385. [16] L.V. Kantorovich, On the translocation of masses, C.R. (Dokl.) Acad. Sci. URSS, 37 (1942), 199-201. [17] L.V. Kantorovich, On a problem of Monge (in Russian), Uspekhi Mat. Nauk., 3 (1948), 225-226. [18] G. Monge, Memoire sur la theorie des Deblais et des Remblais, Histoire de l’Academie des Sciences de Paris, 1781. [19] A. Pratelli, On the equality between Monge’s infimum and Kantorovich’s minimum in optimal mass transportation, Ann. Inst. H. Poincare Probab. Statist., 43 (2007), no. 1, 1–13. [20] A. Pratelli, On the sufficiency of c−cyclical monotonicity for optimality of transport plans. Math. Z., 258 (2008), no. 3, 677690.
[21] L. Ruschendorf, On c−optimal random variables. Statist. Probab. Lett., 27 (1996), no. 3, 267270. [22] F. Santambrogio, Absolute continuity and summability of optimal transport densities: simpler proofs and new estimates, Calc. Var. Partial Differential Equations , 36 (2009), no. 3, 343–354. [23] 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, i–v, 1–178. [24] N. S. Trudinger, X. J.Wang, On the Monge mass transfer problem, Calc. Var. Partial Differential Equations, 13 (2001), no. 1, 19–31. [25] C. Villani, Topics in optimal transportation. Graduate Studies in Mathematics, 58, American Mathematical Society (2003) [26] C. Villani, Optimal Transport, Old and New. Grundlehren der mathematischen Wissenschaften, 338, Springer Berlin Heidelberg (2008) |