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, pp. 143-174.

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.

Classification : 35J65, 45N60
Trudinger, Neil 1 ; Wang, Xu-Jia 1

1 Centre for Mathematics and Its Applications, The Australian National University, Canberra, ACT 0200, Australia
@article{ASNSP_2009_5_8_1_143_0,
     author = {Trudinger, Neil and Wang, Xu-Jia},
     title = {On the second boundary value problem for {Monge-Amp\`ere} type equations and optimal transportation},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     pages = {143--174},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 8},
     number = {1},
     year = {2009},
     mrnumber = {2512204},
     zbl = {1182.35134},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2009_5_8_1_143_0/}
}
TY  - JOUR
AU  - Trudinger, Neil
AU  - Wang, Xu-Jia
TI  - On the second boundary value problem for Monge-Ampère type equations and optimal transportation
JO  - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY  - 2009
SP  - 143
EP  - 174
VL  - 8
IS  - 1
PB  - Scuola Normale Superiore, Pisa
UR  - http://www.numdam.org/item/ASNSP_2009_5_8_1_143_0/
LA  - en
ID  - ASNSP_2009_5_8_1_143_0
ER  - 
%0 Journal Article
%A Trudinger, Neil
%A Wang, Xu-Jia
%T On the second boundary value problem for Monge-Ampère type equations and optimal transportation
%J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
%D 2009
%P 143-174
%V 8
%N 1
%I Scuola Normale Superiore, Pisa
%U http://www.numdam.org/item/ASNSP_2009_5_8_1_143_0/
%G en
%F ASNSP_2009_5_8_1_143_0
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, pp. 143-174. http://www.numdam.org/item/ASNSP_2009_5_8_1_143_0/

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

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

[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. | MR | Zbl

[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. | EuDML | Numdam | MR | Zbl

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

[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. | MR | Zbl

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

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

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

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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

[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. | EuDML | MR | Zbl

[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. | MR | Zbl

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

[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. | MR | Zbl

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