Trudinger, Neil; Wang, Xu-Jia
On the second boundary value problem for Monge-Ampère type equations and optimal transportation
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5 : Tome 8 (2009) no. 1 , p. 143-174
Zbl 1182.35134 | MR 2512204
URL stable : http://www.numdam.org/item?id=ASNSP_2009_5_8_1_143_0

Classification:  35J65,  45N60
This paper is concerned with the existence of globally smooth solutions for the second boundary value problem for certain Monge-Ampère type equations and the application to regularity of potentials in optimal transportation. In particular we address the fundamental issue of determining conditions on costs and domains to ensure that optimal mappings are smooth diffeomorphisms. The cost functions satisfy a weak form of the condition (A3), which was introduced in a recent paper with Xi-nan Ma, in conjunction with interior regularity. Our condition is optimal and includes the quadratic cost function case of Caffarelli and Urbas as well as the various examples in our previous work. The approach is through the derivation of global estimates for second derivatives of solutions.

Bibliographie

[1] L. Caffarelli, The regularity of mappings with a convex potential, J. Amer. Math. Soc. 5 (1992), 99–104. Zbl 0753.35031 | MR 1124980

[2] L. Caffarelli, Boundary regularity of maps with convex potentials II, Ann. of Math. 144 (1996), 453–496. Zbl 0916.35016 | MR 1426885

[3] L. Caffarelli, Allocation maps with general cost functions, In: “Partial Differential Equations and Applications”, Lecture Notes in Pure and Appl. Math., Vol. 177, Dekker, New York, 1996, 29–35. Zbl 0883.49030 | MR 1371577

[4] Ph. 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 8 (1991), 443–457. Numdam | Zbl 0778.35037 | | MR 1136351

[5] W. Gangbo and R. J. Mccann, The geometry of optimal transportation, Acta Math. 177 (1996), 113–161. Zbl 0887.49017 | MR 1440931

[6] L. C. Evans, Partial Differential Equations and Monge-Kantorovich mass transfer, In: “Current Developments in Mathematics”, 1997 (Cambridge, MA), Int. Press, Boston, 1999, 65–126. Zbl 0954.35011 | MR 1698853

[7] D. Gilbarg and N. S. Trudinger, “Elliptic Partial Differential Equations of Second Order”, Second Edition, Springer, Berlin, 1983. Zbl 0562.35001 | MR 737190

[8] P.F. Guan and X-J. Wang, On a Monge-Ampère equation arising in geometric optics, J. Differential Geom. 48 (1998), 205–223. Zbl 0979.35052 | MR 1630253

[9] C. E. Gutierrez and Q-B. Huang, The refractor problem in reshaping light beams, Arch. Ration. Mech. Anal., on line 13-08-08. Zbl 1173.78005 | MR 2525122

[10] Y-H. Kim and R. J. Mccann, On the cost-subdifferentials of cost-convex functions, arXiv:math/ 07061226

[11] G. Loeper, On the regularity of maps solutions of optimal transportation problems, Acta Math., to appear. Zbl 1116.35033 | MR 2506751

[12] G. M. Lieberman and N. S.Trudinger, Nonlinear oblique boundary value problems for nonlinear elliptic equations, Trans. Amer. Math. Soc. 295 (1986), 509–546. Zbl 0619.35047 | MR 833695

[13] P-L. Lions, N. S. Trudinger and J. Urbas, Neumann problem for equations of Monge-Ampère type, Comm. Pure Appl. Math. 39 (1986), 539–563. Zbl 0604.35027 | MR 840340

[14] J-K. Liu, N. S. Trudinger and X-J. Wang, Interior C 2,α regularity for potential functions in optimal transportation, in preparation.

[15] X-N. Ma, N. S. Trudinger and X-J. Wang, Regularity of potential functions of the optimal transportation problem, Arch. Ration. Mech. Anal. 177 (2005), 151–183. Zbl 1072.49035 | MR 2188047

[16] R. T. Rachev and L. Ruschendorff, “Mass Transportation Problems”, Springer, Berlin, 1998.

[17] N. S. Trudinger, On the Dirichlet problem for Hessian equations, Acta Math. 175 (1995), 151–164. Zbl 0887.35061 | MR 1368245

[18] N. S. Trudinger, “Lectures on Nonlinear Elliptic Equations of Second Order”, Lectures in Math. Sci., Vol. 9, Univ. Tokyo, 1995.

[19] N. S. Trudinger, Recent developments in elliptic partial differential equations of Monge-Ampère type, In: “ICM”, Madrid, Vol. 3, 2006, 291–302. Zbl 1130.35058 | MR 2275682

[20] N. S. Trudinger and X-J. Wang, On strict convexity and continuous differentiability of potential functions in optimal transportation, Arch. Ration. Mech. Anal., on line 15-07-08. MR 2505359

[21] N. S. Trudinger and X-J. Wang, Optimal transportation and nonlinear elliptic partial differential equations, in preparation.

[22] J. Urbas, On the second boundary value problem for equations of Monge-Ampère type, J. Reine Angew. Math. 487 (1997), 115–124. Zbl 0880.35031 | | MR 1454261

[23] J. Urbas, “Mass Transfer Problems”, Lecture Notes, Univ. of Bonn, 1998.

[24] J. Urbas, Oblique boundary value problems for equations of Monge-Ampère type, Calc. Var. Partial Differential Equations 7 (1998), 19–39. Zbl 0912.35068 | MR 1624426

[25] C. Villani, “ Topics in Optimal Transportation”, Graduate Studies in Mathematics, Vol. 58 Amer. Math. Soc., Providence, RI, 2003. Zbl 1106.90001 | MR 1964483

[26] G. Von Nessi, “On Regularity for Potentials of Optimal Transportation Problems on Spheres and Related Hessian Equations”, PhD thesis. Australian National University, 2008.

[27] X-J. Wang, On the design of a reflector antenna, Inverse Problems 12 (1996), 351–375. Zbl 0858.35142 | MR 1391544

[28] X-J. Wang, On the design of a reflector antenna II, Calc. Var. Partial Differential Equation 20 (2004), 329–341. Zbl 1065.78013 | MR 2062947