A Liouville theorem for solutions of the Monge-Ampère equation with periodic data
Caffarelli, L ; Li, Yan Yan
Annales de l'I.H.P. Analyse non linéaire, Tome 21 (2004), p. 97-120 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU) 
@article{AIHPC_2004__21_1_97_0,
     author = {Caffarelli, Luis and Li, Yan Yan},
     title = {A Liouville theorem for solutions of the Monge-Amp\`ere equation with periodic data},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     volume = {21},
     year = {2004},
     pages = {97-120},
     doi = {10.1016/j.anihpc.2003.01.005},
     zbl = {1108.35051},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_2004__21_1_97_0}
}

Caffarelli, L; Li, Yan Yan. A Liouville theorem for solutions of the Monge-Ampère equation with periodic data. Annales de l'I.H.P. Analyse non linéaire, Tome 21 (2004) pp. 97-120. doi : 10.1016/j.anihpc.2003.01.005. http://gdmltest.u-ga.fr/item/AIHPC_2004__21_1_97_0/

[1] Caffarelli L., A localization property of viscosity solutions to the Monge-Ampère equation and their strict convexity, Ann. of Math. 13 (1990) 129-134. | MR 1038359 | Zbl 0704.35045

[2] Caffarelli L., Interior W2,p estimates for solutions of the Monge-Ampère equation, Ann. of Math. 131 (1990) 135-150. | MR 1038360 | Zbl 0704.35044

[3] Caffarelli L., Graduate Course at the Courant Institute, New York University, New York, 1995.

[4] Caffarelli L., Monotonicity properties of optimal transportation and the FKG and related inequalities, Comm. Math. Phys. 214 (2000) 547-563. | MR 1800860 | Zbl 0978.60107

[5] Caffarelli L., Cabre X., Fully Nonlinear Elliptic Equations, American Mathematical Society Colloquium Publications, vol. 43, American Mathematical Society, Providence, RI, 1995. | MR 1351007 | Zbl 0834.35002

[6] Caffarelli L., Gutiérrez C., Properties of the solutions of the linearized Monge-Ampère equation, Amer. J. Math. 119 (1997) 423-465. | MR 1439555 | Zbl 0878.35039

[7] Caffarelli L., Li Y.Y., An extension to a theorem of Jörgens, Calabi, and Pogorelov, Comm. Pure Appl. Math. 56 (2003) 549-583. | MR 1953651 | Zbl 01981600

[8] Caffarelli L., Nirenberg L., Spruck J., The Dirichlet problem for nonlinear second-order elliptic equations, I. Monge-Ampère equation, Comm. Pure Appl. Math. 37 (1984) 369-402. | MR 739925 | Zbl 0598.35047

[9] Caffarelli L., Viaclovsky J., On the regularity of solutions to Monge-Ampère equations on Hessian manifolds, Comm. Partial Differential Equations 26 (2001) 2339-2351. | MR 1876421 | Zbl 0990.35029

[10] Calabi E., Improper affine hypersurfaces of convex type and a generalization of a theorem by K. Jörgens, Michigan Math. J. 5 (1958). | MR 106487 | Zbl 0113.30104

[11] Cheng S.Y., Yau S.T., The real Monge-Ampère equation and affine flat structures, in: Proceedings of the Symposium on Differential Geometry and Differential Equations, vols. 1-3, Beijing, 1980, Science Press, Beijing, 1982, pp. 339-370. | MR 714338 | Zbl 0517.35020

[12] Cheng S.Y., Yau S.T., Complete affine hypersurfaces. I. The completeness of affine metrics, Comm. Pure Appl. Math. 39 (1986) 839-866. | MR 859275 | Zbl 0623.53002

[13] Chou K.-S., Wang X.-J., A variational theory of the Hessian equation, Comm. Pure Appl. Math. 54 (2001) 1029-1064. | MR 1835381 | Zbl 1035.35037

[14] De Guzman M., Differentiation of Integrals in Rn, Lecture Notes, vol. 481, Springer-Verlag, Berlin, 1976. | Zbl 0327.26010

[15] Evans L.C., Classical solutions of fully nonlinear, convex, second-order elliptic equations, Comm. Pure Appl. Math. 35 (1982) 333-363. | MR 649348 | Zbl 0469.35022

[16] Gilbarg D., Trudinger N., Elliptic Partial Differential Equations of Second Order, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. | MR 737190 | Zbl 0562.35001

[17] Jörgens K., Über die Lösungen der Differentialgleichung rt−s2=1, Math. Ann. 127 (1954) 130-134.

[18] Krylov N.V., Boundedly inhomogeneous elliptic and parabolic equation in a domain, Izv. Akad. Nauk SSSR 47 (1983) 75-108. | MR 688919 | Zbl 0578.35024

[19] Krylov N.V., Safonov M.V., An estimate of the probability that a diffusion process hits a set of positive measure, Dokl. Akad. Nauk. SSSR 245 (1979) 253-255, English translation in: , Soviet Math. Dokl. 20 (1979) 253-255. | MR 525227 | Zbl 0459.60067

[20] Li Y.Y., Some existence results of fully nonlinear elliptic equations of Monge-Ampère type, Comm. Pure Appl. Math. 43 (1990) 233-271. | MR 1038143 | Zbl 0705.35038

[21] Pogorelov A.V., On the improper affine hypersurfaces, Geom. Dedicata 1 (1972) 33-46. | MR 319126 | Zbl 0251.53005

[22] Trudinger N., Wang X., The Bernstein problem for affine maximal hypersurfaces, Invent. Math. 140 (2000) 399-422. | MR 1757001 | Zbl 0978.53021