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Article

Keywords:
mass transport problem; measurable selections; degree theory
Summary:
In the setting of the optimal transportation problem we provide some conditions which ensure the existence and the uniqueness of the optimal map in the case of cost functions satisfying mild regularity hypothesis and no convexity or concavity assumptions.
References:
[1] Abdellaoui T., Heinich H.: Sur la distance de deux lois dans le cas vectoriel. C.R. Acad. Sci. Paris Sér. I Math. 319 (1994), 397–400. MR 1289319 | Zbl 0808.60008
[2] Alberti G., Ambrosio L.: A geometric approach to monotone functions in $\Bbb R^n$. Math. Z. (1999), 230 259–316. MR 1676726
[3] Ambrosio L.: Lecture notes on optimal transport problems. Mathematical Aspects of Evolving Interfaces (P. Colli and J.F. Rodrigues, Eds.), Lecture Notes in Mathematics, 1812, Springer, Berlin, 2003, pp. 1–52. MR 2011032 | Zbl 1047.35001
[4] Ambrosio L., Gigli N., Savaré G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures. Second Edition, Lecture Notes in Mathematics ETH Zürich, Birkhäuser, Basel, 2008. MR 2401600
[5] Ball J.M.: Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 63 (1978), 337–403. DOI 10.1007/BF00279992 | MR 0475169
[6] Ball J.M.: Global invertibility of Sobolev functions and the interpenetration of matter. Proc. Roy. Soc. Edinburgh Sect. A 88 (1981), 315–328. MR 0616782 | Zbl 0478.46032
[7] Bolton P., Dewatripont M.: Contract Theory. The MIT Press Cambridge (2005).
[8] Brenier Y.: Décomposition polaire et réarrangement monotone des champs de vecteurs. C.R. Acad. Sci. Paris Sér. I Math. 305 (1987), 805–808. MR 0923203 | Zbl 0652.26017
[9] Brenier Y.: Extended Monge-Kantorovich theory. Optimal Transportation and Applications (L.A. Caffarelli and S. Salsa, Eds.), Lecture Notes in Mathematics, 1813, Springer, Berlin, 2003, pp. 91–121. MR 2006306 | Zbl 1064.49036
[10] Brezis H.: Analyse Fonctionelle. Masson Paris (1983). MR 0697382
[11] Cannarsa P., Sinestrari C.: Semiconcave Functions, Hamilton-Jacobi Equations and Optimal Control. Progress in Nonlinear Differential Equations and their Applications, 58, Birkhäuser, Boston, 2004. MR 2041617 | Zbl 1095.49003
[12] Carlier G.: A general existence result for the principal-agent problem with adverse selection. J. Math. Econom. 35 (2001), 129–150. DOI 10.1016/S0304-4068(00)00057-4 | MR 1817791 | Zbl 0972.91068
[13] Carlier G.: Duality and existence for a class of mass transportation problems and economic applications. Adv. Math. Econom. 5 (2003), 1–21. DOI 10.1007/978-4-431-53979-7_1 | MR 2160899 | Zbl 1176.90409
[14] Carlier G., Dana R.A.: Rearrangement inequalities in non-convex insurance model. J. Math. Econom. 41 (2005), 483–503. DOI 10.1016/j.jmateco.2004.12.004 | MR 2143822
[15] Carlier G., Jimenez Ch.: On Monge's problem for Bregman-like cost function. J. Convex Anal. (2007), 14 647–656. MR 2341308
[16] Clarke F.H.: Optimization and Nonsmooth Analysis. Wiley-Interscience New York (1983). MR 0709590 | Zbl 0582.49001
[17] Cuesta-Albertos J.A., Matrán C.: Notes on the Wasserstein metric in Hilbert spaces. Ann. Probab. 17 (1989), 1264–1276. DOI 10.1214/aop/1176991269 | MR 1009457
[18] Evans L.C., Gangbo W.: Differential equation methods for the Monge-Kantorovich mass transfer problem. Mem. Amer. Math. Soc 137 (1999), 653. MR 1464149
[19] Fathi A., Figalli A.: Optimal transportation on non-compact manifold. Israel J. Math., to appear.
[20] Gangbo W., McCann R.J.: Optimal maps in Monge's mass transport problem. C.R. Acad. Sci. Paris Sér. I Math. 321 (1995), 1653–1658. MR 1367824 | Zbl 0858.49002
[21] Gangbo W., McCann R.J.: The geometry of optimal transportation. Acta Math. 177 (1996), 113–161. DOI 10.1007/BF02392620 | MR 1440931 | Zbl 0887.49017
[22] Giaquinta M., Modica G., Souček J.: Cartesian Currents in the Calculus of Variations. I Springer Berlin (1998). MR 1645086
[23] Kantorovich L.V.: On a problem of Monge. Uspekhi Mat. Nauk SSSR 3 (1948), 225–226.
[24] Kellerer H.G.: Duality theorems for marginal problems. Z. Wahrsch. Verw. Gebiete 67 (1984), 399–432. DOI 10.1007/BF00532047 | MR 0761565 | Zbl 0535.60002
[25] Laffont J.J., Matimort D.: The Theory of Incentives: The Agent-Principal Model. Princeton University Press Princeton (2001).
[26] Levin V.L.: Abstract cyclical monotonicity and Monge solutions for the general Monge-Kantorovich problem. Set-Valued Anal. 7 (1999), 7–32. DOI 10.1023/A:1008753021652 | MR 1699061 | Zbl 0934.54013
[27] Ma X.N., Trudinger N., Wang X.J.: Regularity of potential functions of the optimal transportation problem. Arch. Rational Mech. Anal. 177 (2005), 151–183. DOI 10.1007/s00205-005-0362-9 | MR 2188047 | Zbl 1072.49035
[28] Monge G.: Memoire sur la Theorie des Déblais et des Remblais. Histoire de l'Acad. des Science de Paris, 1781.
[29] Müller S., Qi T., Yan B.S.: On a new class of elastic deformations not allowing for cavitation. Ann. Inst. H. Poincaré 177 (1996), 113–161.
[30] Plakhov A.Yu.: Exact solutions of the one-dimensional Monge-Kantorovich problem. Mat. Sb. 195 9 (2004), 57–74; {\it II}, Sb. Math. 195 no. 9 (2004), 1291–1307. DOI 10.1070/SM2004v195n09ABEH000845 | MR 2122369 | Zbl 1080.49030
[31] Rachev S.T., Rüschendorf L.R.: Mass Transportation Problem. Springer Berlin (1998).
[32] Repovš D., Semenov P.V.: Continuous Selections of Multivalued Mappings. Kluver Academic Dordrecht (1998). MR 1659914
[33] Rochet J.C.: A necessary and sufficient condition for rationalizability in a quasi-linear context. J. Math. Econom. 16 (1987), 191–200. DOI 10.1016/0304-4068(87)90007-3 | MR 0902976
[34] Rüschendorf L., Uckelmann L.: Numerical and analytical results for the transportation problem of Monge-Kantorovich. Metrika 51 3 (2000), 245–258. DOI 10.1007/s001840000052 | MR 1795372
[35] Sudakov V.N.: Geometric problems in the theory of infinite-dimensional probability distributions. Proc. Steklov Inst. Math. 141 (1979), 1–178. MR 0530375
[36] Šverák V.: Regularity properties of deformation with finite energy. Arch. Rational Mech. Anal. 100 (1988), 105–127. DOI 10.1007/BF00282200 | MR 0913960
[37] Trudinger N.S., Wang X.J.: On the second boundary problem for Monge-Ampère type equations and optimal transportation. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 8 (2009), 1 143–174; archived online at arxiv.org/abs/math.AP/0601086. MR 2512204
[38] van der Putten R.: Sul lemma dei valori critici e la formula della coarea. Boll. U.M.I. (7) 6-B (1992), 561–578. MR 1191953
[39] Villani C.: Optimal Transport, Old and New. Grundlehren der Mathematischen Wissenschaften, 338, Springer, Berlin, 2009; archived online at www.umpa.ens-lyon.fr/$\sim ${\tt cvillani/Cedrif/B07D.StFlour.pdf}. MR 2459454 | Zbl 1156.53003
[40] Ziemer W.P.: Weakly Differentiable Functions. Graduate Texts in Mathematics, 120, Springer, New York, 1989. MR 1014685 | Zbl 0692.46022
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