On the regularity of the polar factorization for time dependent maps

Calculus of Variations and Partial Differential Equations

Volumn 22, Issue 3, 2005, Pages 343-374

  Loeper, G ^a,b,c  

^a UNIVERSITÉ CÔTE D'AZUR   (France)

^b FIELDS INSTITUTE   (Canada)

^c UNIVERSITY OF TORONTO   (Canada)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 13144288189     PISSN: 09442669     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00526-004-0280-y     Document Type: Article

References (23)

1. Ambrosio, L.: Transport equation and Cauchy problem for BV vector fields. to appear
2. Benamou, J.-D., Brenier, Y.: Weak existence for the semigeostrophic equations formulated as a coupled Monge-Ampère/transport problem. SIAM J. Appl. Math. 58(5), 1450-1461 (electronic), (1998)
3. Brenier, Y.: Polar factorization and monotone rearrangement of vector-valued functions. Comm. Pure Appl. Math. 44(4), 375-417 (1991)
4. 2,p estimates for solutions of the Monge-Ampère equation. Ann. of Math. (2) 131(1), 135-150 (1990)
5. Caffarelli, L.A.: A localization property of viscosity solutions to the Monge-Ampère equation and their strict convexity. Ann. of Math. (2) 131(1), 129-134 (1990)
6. Caffarelli, L.A.: Some regularity properties of solutions of Monge Ampère equation. Comm. Pure Appl. Math. 44(8-9), 965-969 (1991)
7. Caffarelli, L.A.: Boundary regularity of maps with convex potentials. Comm. Pure Appl. Math. 45(9), 1141-1151 (1992)
8. Caffarelli, L.A.: The regularity of mappings with a convex potential. J. Amer. Math. Soc. 5(1), 99-104 (1992)
9. Caffarelli, L.A., Gutiérrez, C.E.: Properties of the solutions of the linearized Monge-Ampère equation. Amer. J. Math. 119(2), 423-465 (1997)
10. Cordero-Erausquin, D.: Sur le transport de mesures périodiques. C. R. Acad. Sci. Paris Sér. I Math. 329(3), 199-202 (1999)
11. Cullen, M., Gangbo, W.: A variational approach for the 2-dimensional semi-geostrophic shallow water equations. Arch. Ration. Mech. Anal. 156(3), 241-273 (2001)
12. Cullen, M.J.P., Purser, R.J.: Properties of the Lagrangian semigeostrophic equations. J. Atmospheric Sci. 46(17): 2684-2697 (1989)
13. Delanoë, P.: 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(5), 443-457 (1991)
14. Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order, volume 224 of Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), 2nd edn. Springer, Berlin Heidelberg New York 1983
15. Lions, P.-L.: Mathematical topics in fluid mechanics. Vol. 1, vol. 3 of Oxford Lecture Series in Mathematics and its Applications. The Clarendon Press Oxford University Press, New York 1996
16. Lopes Filho, M.C., Nussenzveig Lopes, H.J.: Existence of a weak solution for the semigeostrophic equation with integrable initial data. Proc. Roy. Soc. Edinburgh Sect. A 132(2), 329-339 (2002)
17. McCann, R.J.: Polar factorization of maps on Riemannian manifolds. Geom. Funct. Anal. 11(3), 589-608(2001)
18. Murthy, M.K.V., Stampacchia, G.: Boundary value problems for some degenerate-elliptic operators. Ann. Mat. Pura Appl. (4) 80, 1-122 (1968)
19. Murthy, M.K.V., Stampacchia, G.: Errata corrige: "Boundary value problems for some degenerate-elliptic operators". Ann. Mat. Pura Appl. (4) 90, 413-414 (1971)
20. Trudinger, N.S.: Linear elliptic operators with measurable coefficients. Ann. Scuola Norm. Sup. Pisa (3) 27, 265-308 (1973)
21. Urbas, J.: On the second boundary value problem for equations of Monge-Ampère type. J. Reine Angew. Math. 487, 115-124 (1997)
22. Villani, C.: Topics in optimal transportation, vol. 58 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI 2003
23. Wang, X.J.: Remarks on the regularity of Monge-Ampère equations. In: Proceedings of the International Conference on Nonlinear P.D.E. (Hangzhou 1992)