×

zbMATH — the first resource for mathematics

Free boundaries in optimal transport and Monge-Ampère obstacle problems. (English) Zbl 1196.35231
Let the functions \(0 \leq f,g \in L^{1}({\mathbb R}_n)\) be compactly supported. The authors investigate the problem of transporting a fraction \(m \leq \min \{\|f\|_{L_1}, \|g\|_{L_1}\}\) of the mass of \(f\) onto \(g\) as cheaply as possible, where cost per unit mass transported is given by a cost function \(c\) which is quadratic, \(c\,({\mathbf x}, {\mathbf y}) = |{\mathbf x} - {\mathbf y}|^{2}/2\). It can be shown that this problem is equivalent to a double obstacle problem for the Monge-Ampère equation for which sufficient conditions are given to guarantee uniqueness of the solution, such as \(f\) vanishing on \(\text{supp} \, g\) in the quadratic sense. The Lagrange multiplier \(\lambda\) controlling the optimal mass parametrizes the distance between the upper and lower obstacles.
In a further section, the authors show the monotone dependence of the active domains \(U\) and \(V\) on the amount \(m\) of mass transferred. For the quadratic cost function, the semiconcavity of the free boundary is proved when \(f\) and \(g\) are compactly supported on opposite sides of a hyperplane. In the last two sections, interior and boundary regularity for the optimal mapping are shown under certain assumptions on \(f\) and \(g\). An appendix included makes the boundary regularity analysis self-contained. The bibliography contains 83 items.

MSC:
35R35 Free boundary problems for PDEs
35B65 Smoothness and regularity of solutions to PDEs
35J87 Unilateral problems for nonlinear elliptic equations and variational inequalities with nonlinear elliptic operators
35J96 Monge-Ampère equations
49J20 Existence theories for optimal control problems involving partial differential equations
PDF BibTeX XML Cite
Full Text: DOI Link
References:
[1] T. Abdellaoui and H. Heinich, ”Sur la distance de deux lois dans le cas vectoriel,” C. R. Acad. Sci. Paris Sér. I Math., vol. 319, iss. 4, pp. 397-400, 1994. · Zbl 0808.60008
[2] M. Agueh, ”Existence of solutions to degenerate parabolic equations via the Monge-Kantorovich theory,” Adv. Differential Equations, vol. 10, iss. 3, pp. 309-360, 2005. · Zbl 1103.35051
[3] N. Ahmad, ”The geometry of shape recognition via a Monge-Kantorovich optimal transport problem,” PhD Thesis , Brown University, 2003.
[4] A. Alexandroff, ”Smoothness of the convex surface of bounded Gaussian curvature,” C. R. \((\)Doklady\()\) Acad. Sci. URSS, vol. 36, pp. 195-199, 1942. · Zbl 0061.37605
[5] L. Ambrosio, ”Lecture notes on optimal transport problems,” in Mathematical Aspects of Evolving Interfaces (Funchal, 2000), New York: Springer-Verlag, 2003, pp. 1-52. · Zbl 1047.35001
[6] L. Ambrosio, N. Gigli, and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Basel: Birkhäuser, 2005. · Zbl 1090.35002
[7] L. Ambrosio, B. Kirchheim, and A. Pratelli, ”Existence of optimal transport maps for crystalline norms,” Duke Math. J., vol. 125, iss. 2, pp. 207-241, 2004. · Zbl 1076.49022
[8] L. Ambrosio and A. Pratelli, ”Existence and stability results in the \(L^1\) theory of optimal transportation,” in Optimal Transportation and Applications (Martina Franca, 2001), New York: Springer-Verlag, 2003, pp. 123-160. · Zbl 1065.49026
[9] L. Ambrosio and S. Rigot, ”Optimal mass transportation in the Heisenberg group,” J. Funct. Anal., vol. 208, iss. 2, pp. 261-301, 2004. · Zbl 1076.49023
[10] G. Bouchitté, G. Buttazzo, and P. Seppecher, ”Shape optimization solutions via Monge-Kantorovich equation,” C. R. Acad. Sci. Paris Sér. I Math., vol. 324, iss. 10, pp. 1185-1191, 1997. · Zbl 0884.49023
[11] Y. Brenier, ”Décomposition polaire et réarrangement monotone des champs de vecteurs,” C. R. Acad. Sci. Paris Sér. I Math., vol. 305, iss. 19, pp. 805-808, 1987. · Zbl 0652.26017
[12] Y. Brenier, ”Polar factorization and monotone rearrangement of vector-valued functions,” Comm. Pure Appl. Math., vol. 44, iss. 4, pp. 375-417, 1991. · Zbl 0738.46011
[13] G. Buttazzo, A. Pratelli, and E. Stepanov, ”Optimal pricing policies for public transportation networks,” SIAM J. Optim., vol. 16, iss. 3, pp. 826-853, 2006. · Zbl 1093.49030
[14] L. A. Caffarelli, ”Some regularity properties of solutions of Monge Ampère equation,” Comm. Pure Appl. Math., vol. 44, iss. 8-9, pp. 965-969, 1991. · Zbl 0761.35028
[15] L. A. Caffarelli, ”The regularity of mappings with a convex potential,” J. Amer. Math. Soc., vol. 5, iss. 1, pp. 99-104, 1992. · Zbl 0753.35031
[16] L. A. Caffarelli, ”Boundary regularity of maps with convex potentials,” Comm. Pure Appl. Math., vol. 45, iss. 9, pp. 1141-1151, 1992. · Zbl 0778.35015
[17] L. A. Caffarelli, ”Boundary regularity of maps with convex potentials. II,” Ann. of Math., vol. 144, iss. 3, pp. 453-496, 1996. · Zbl 0916.35016
[18] L. A. Caffarelli, ”Allocation maps with general cost functions,” in Partial Differential Equations and Applications, New York: Dekker, 1996, pp. 29-35. · Zbl 0883.49030
[19] L. A. Caffarelli, M. Feldman, and R. J. McCann, ”Constructing optimal maps for Monge’s transport problem as a limit of strictly convex costs,” J. Amer. Math. Soc., vol. 15, iss. 1, pp. 1-26, 2002. · Zbl 1053.49032
[20] G. Carlier, ”A general existence result for the principal-agent problem with adverse selection,” J. Math. Econom., vol. 35, iss. 1, pp. 129-150, 2001. · Zbl 0972.91068
[21] G. Carlier, ”Duality and existence for a class of mass transportation problems and economic applications,” in Advances in Mathematical Economics, New York: Springer-Verlag, 2003, vol. 5, pp. 1-21. · Zbl 1176.90409
[22] G. Carlier and I. Ekeland, ”Equilibrium structure of a bidimensional asymmetric city,” Nonlinear Anal. Real World Appl., vol. 8, iss. 3, pp. 725-748, 2007. · Zbl 1136.91023
[23] G. Carlier and T. Lachand-Robert, ”Regularity of solutions for some variational problems subject to a convexity constraint,” Comm. Pure Appl. Math., vol. 54, iss. 5, pp. 583-594, 2001. · Zbl 1035.49034
[24] K. S. Chou and Y. D. Wang, ”An obstacle problem for the Monge-Ampère equation,” Comm. Partial Differential Equations, vol. 18, iss. 5-6, pp. 1069-1084, 1993. · Zbl 0788.35054
[25] J. A. Cuesta and C. Matrán, ”Notes on the Wasserstein metric in Hilbert spaces,” Ann. Probab., vol. 17, iss. 3, pp. 1264-1276, 1989. · Zbl 0688.60011
[26] J. A. Cuesta-Albertos, C. Matrán, and A. Tuero-D’iaz, ”On the monotonicity of optimal transportation plans,” J. Math. Anal. Appl., vol. 215, iss. 1, pp. 86-94, 1997. · Zbl 0892.60020
[27] M. Cullen and W. Gangbo, ”A variational approach for the 2-dimensional semi-geostrophic shallow water equations,” Arch. Ration. Mech. Anal., vol. 156, iss. 3, pp. 241-273, 2001. · Zbl 0985.76008
[28] M. J. P. Cullen and R. J. Purser, ”An extended Lagrangian theory of semigeostrophic frontogenesis,” J. Atmospheric Sci., vol. 41, iss. 9, pp. 1477-1497, 1984.
[29] P. Delanoë, ”Classical solvability in dimension two of the second boundary-value problem associated with the Monge-Ampère operator,” Ann. Inst. H. Poincaré Anal. Non Linéaire, vol. 8, iss. 5, pp. 443-457, 1991. · Zbl 0778.35037
[30] L. De Pascale, L. C. Evans, and A. Pratelli, ”Integral estimates for transport densities,” Bull. London Math. Soc., vol. 36, iss. 3, pp. 383-395, 2004. · Zbl 1068.35170
[31] J. Dolbeault and R. Monneau, ”Convexity estimates for nonlinear elliptic equations and application to free boundary problems,” Ann. Inst. H. Poincaré Anal. Non Linéaire, vol. 19, iss. 6, pp. 903-926, 2002. · Zbl 1034.35047
[32] E. Ekeland, ”Existence, uniqueness and efficiency of equilibrium in hedonic markets with multidimensional types,” , preprint , 2005. · Zbl 1203.91153
[33] L. C. Evans and W. Gangbo, ”Differential Equations Methods for the Monge-Kantorovich Mass Transfer Problem,” Mem. Amer. Math. Soc., vol. 137, p. viii, 1999. · Zbl 0920.49004
[34] H. Federer, Geometric measure theory, New York: Springer-Verlag, 1969, vol. 153. · Zbl 0176.00801
[35] M. Feldman and R. J. McCann, ”Monge’s transport problem on a Riemannian manifold,” Trans. Amer. Math. Soc., vol. 354, iss. 4, pp. 1667-1697, 2002. · Zbl 1038.49041
[36] M. Feldman and R. J. McCann, ”Uniqueness and transport density in Monge’s mass transportation problem,” Calc. Var. Partial Differential Equations, vol. 15, iss. 1, pp. 81-113, 2002. · Zbl 1003.49031
[37] W. Gangbo, ”An elementary proof of the polar factorization of vector-valued functions,” Arch. Rational Mech. Anal., vol. 128, iss. 4, pp. 381-399, 1994. · Zbl 0828.57021
[38] W. Gangbo, ”Habilitation thesis,” 1995.
[39] W. Gangbo and R. J. McCann, ”Optimal maps in Monge’s mass transport problem,” C. R. Acad. Sci. Paris Sér. I Math., vol. 321, iss. 12, pp. 1653-1658, 1995. · Zbl 0858.49002
[40] W. Gangbo and R. J. McCann, ”The geometry of optimal transportation,” Acta Math., vol. 177, iss. 2, pp. 113-161, 1996. · Zbl 0887.49017
[41] W. Gangbo and R. J. McCann, ”Shape recognition via Wasserstein distance,” Quart. Appl. Math., vol. 58, iss. 4, pp. 705-737, 2000. · Zbl 1039.49038
[42] W. Gangbo and A. Świcech, ”Optimal maps for the multidimensional Monge-Kantorovich problem,” Comm. Pure Appl. Math., vol. 51, iss. 1, pp. 23-45, 1998. · Zbl 0889.49030
[43] U. Gianazza, G. Savaré, and G. Toscani, ”The Wasserstein gradient flow of the Fisher information and the quantum drift-diffusion equation,” Arch. Rational Mech. Anal, vol. 194, pp. 133-220, 2009. · Zbl 1223.35264
[44] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Second ed., New York: Springer-Verlag, 1983. · Zbl 0562.35001
[45] C. E. Gutiérrez, The Monge-Ampère equation, Boston, MA: Birkhäuser, 2001, vol. 44. · Zbl 0989.35052
[46] C. E. Gutiérrez and Q. Huang, ”Geometric properties of the sections of solutions to the Monge-Ampère equation,” Trans. Amer. Math. Soc., vol. 352, iss. 9, pp. 4381-4396, 2000. · Zbl 0958.35043
[47] F. John, ”Extremum problems with inequalities as subsidiary conditions,” in Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948, New York, N. Y.: Interscience Publishers, 1948, pp. 187-204. · Zbl 0034.10503
[48] R. Jordan, D. Kinderlehrer, and F. Otto, ”The variational formulation of the Fokker-Planck equation,” SIAM J. Math. Anal., vol. 29, iss. 1, pp. 1-17, 1998. · Zbl 0915.35120
[49] L. Kantorovitch, ”On the translocation of masses,” C. R. \((\)Doklady\()\) Acad. Sci. URSS, vol. 37, pp. 199-201, 1942. · Zbl 0061.09705
[50] H. G. Kellerer, ”Duality theorems for marginal problems,” Z. Wahrsch. Verw. Gebiete, vol. 67, iss. 4, pp. 399-432, 1984. · Zbl 0535.60002
[51] A. Figalli, Y. -H. Kim, and R. J. McCann, ”Work in progress,”.
[52] M. Knott and C. S. Smith, ”On the optimal mapping of distributions,” J. Optim. Theory Appl., vol. 43, iss. 1, pp. 39-49, 1984. · Zbl 0519.60010
[53] K. Lee, ”The obstacle problem for Monge-Ampère equation,” Comm. Partial Differential Equations, vol. 26, iss. 1-2, pp. 33-42, 2001. · Zbl 0982.35039
[54] G. Loeper, ”On the regularity of solutions of optimal transportation problems,” Acta Math., vol. 202, pp. 133-220, 2009. · Zbl 1219.49038
[55] X. Ma, N. S. Trudinger, and X. Wang, ”Regularity of potential functions of the optimal transportation problem,” Arch. Ration. Mech. Anal., vol. 177, iss. 2, pp. 151-183, 2005. · Zbl 1072.49035
[56] R. J. McCann, ”Existence and uniqueness of monotone measure-preserving maps,” Duke Math. J., vol. 80, iss. 2, pp. 309-323, 1995. · Zbl 0873.28009
[57] R. J. McCann, ”A convexity principle for interacting gases,” Adv. Math., vol. 128, iss. 1, pp. 153-179, 1997. · Zbl 0901.49012
[58] R. J. McCann, ”Equilibrium shapes for planar crystals in an external field,” Comm. Math. Phys., vol. 195, iss. 3, pp. 699-723, 1998. · Zbl 0936.74029
[59] R. J. McCann, ”Exact solutions to the transportation problem on the line,” R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., vol. 455, iss. 1984, pp. 1341-1380, 1999. · Zbl 0947.90010
[60] R. J. McCann, ”Polar factorization of maps on Riemannian manifolds,” Geom. Funct. Anal., vol. 11, iss. 3, pp. 589-608, 2001. · Zbl 1011.58009
[61] E. Milakis, ”On the regularity of optimal sets in mass transfer problems,” Comm. Partial Differential Equations, vol. 31, iss. 4-6, pp. 817-826, 2006. · Zbl 1117.49034
[62] G. Monge, ”Mémoire sur la théorie des déblais et de remblais,” in Histoire de l’Académie Royale des Sciences de Paris, avec les Mémoires de Mathématique et de Physique pour la même Année, , 1781, pp. 666-704.
[63] J. R. Munkres, Topology, Second ed., Englewood Cliffs, NJ: Prentice Hall, 2000. · Zbl 0951.54001
[64] F. Otto, ”Doubly degenerate diffusion equations as steepest descent,” , preprint , 1996.
[65] Y. A. Plakhov, ”Exact solutions of the one-dimensional Monge-Kantorovich problem,” Mat. Sb., vol. 195, iss. 9, pp. 57-74, 2004. · Zbl 1080.49030
[66] S. T. Rachev and L. Rüschendorf, Mass Transportation Problems, New York: Springer-Verlag, 1998, vol. I. · Zbl 0990.60500
[67] J. -C. Rochet and P. Choné, ”Ironing, sweeping and multidimensional screening,” Econometrica, vol. 66, pp. 783-826, 1998. · Zbl 1015.91515
[68] R. Rockafellar, Convex Analysis, Princeton, N.J.: Princeton Univ. Press, 1970, vol. 28. · Zbl 0193.18401
[69] L. Rüschendorf and S. T. Rachev, ”A characterization of random variables with minimum \(L^2\)-distance,” J. Multivariate Anal., vol. 32, iss. 1, pp. 48-54, 1990. · Zbl 0688.62034
[70] Rüschendorf, L. and Uckelmann, L., Distributions with Given Marginals and Moment ProblemsDordrecht: Kluwer Academic Publishers, 1997.
[71] O. Savin, ”A free boundary problem with optimal transportation,” Comm. Pure Appl. Math., vol. 57, iss. 1, pp. 126-140, 2004. · Zbl 1065.49030
[72] O. Savin, ”The obstacle problem for Monge Ampere equation,” Calc. Var. Partial Differential Equations, vol. 22, iss. 3, pp. 303-320, 2005. · Zbl 1084.35026
[73] R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Cambridge: Cambridge Univ. Press, 1993. · Zbl 1143.52002
[74] C. S. Smith and M. Knott, ”Note on the optimal transportation of distributions,” J. Optim. Theory Appl., vol. 52, iss. 2, pp. 323-329, 1987. · Zbl 0586.49005
[75] V. N. Sudakov, ”Geometric problems in the theory of infinite-dimensional probability distributions,” Proc. Steklov Inst. Math., iss. 2, p. i-v, 1, 1979. · Zbl 0409.60005
[76] N. S. Trudinger and X. Wang, ”On the Monge mass transfer problem,” Calc. Var. Partial Differential Equations, vol. 13, iss. 1, pp. 19-31, 2001. · Zbl 1010.49030
[77] N. S. Trudinger and X. Wang, ”On the second boundary value problem for Monge-Ampère type equations and optimal transportation,” Ann. Sc. Norm. Super. Pisa Cl. Sci., vol. 8, iss. 1, pp. 143-174, 2009. · Zbl 1182.35134
[78] L. Uckelmann, ”Optimal couplings between one-dimensional distributions,” in Distributions with Given Marginals and Moment Problems, Dordrecht: Kluwer Acad. Publ., 1997, pp. 275-281. · Zbl 0907.60022
[79] J. Urbas, ”On the second boundary value problem for equations of Monge-Ampère type,” J. Reine Angew. Math., vol. 487, pp. 115-124, 1997. · Zbl 0880.35031
[80] C. Villani, Topics in Optimal Transportation, Providence, RI: Amer. Math. Soc., 2003. · Zbl 1106.90001
[81] C. Villani, Optimal Transport, Old and New, New York: Springer-Verlag, 2009. · Zbl 1156.53003
[82] J. G. Wolfson, ”Minimal Lagrangian diffeomorphisms and the Monge-Ampère equation,” J. Differential Geom., vol. 46, iss. 2, pp. 335-373, 1997. · Zbl 0926.53032
[83] Q. Xia, ”The formation of a tree leaf,” ESAIM Control Optim. Calc. Var., vol. 13, iss. 2, pp. 359-377, 2007. · Zbl 1114.92048
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.