The generalized Dirichlet problem for equations of Monge-Ampère type
Annales de l'I.H.P. Analyse non linéaire, Volume 3 (1986) no. 3, p. 209-228
@article{AIHPC_1986__3_3_209_0,
     author = {Urbas, John I. E.},
     title = {The generalized Dirichlet problem for equations of Monge-Amp\`ere type},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Gauthier-Villars},
     volume = {3},
     number = {3},
     year = {1986},
     pages = {209-228},
     zbl = {0602.35038},
     mrnumber = {847307},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_1986__3_3_209_0}
}
Urbas, John I. E. The generalized Dirichlet problem for equations of Monge-Ampère type. Annales de l'I.H.P. Analyse non linéaire, Volume 3 (1986) no. 3, pp. 209-228. http://www.numdam.org/item/AIHPC_1986__3_3_209_0/

[1] A.D. Aleksandrov, Die innere Geometrie der konvexen Flächen, Akademie-Verlag, Berlin, 1955. | MR 71041 | Zbl 0065.15102

[2] A.D. Aleksandrov, Dirichlet's problem for the equation Det ∥zij∥ = φ(z1, ..., zn, z, x1, ..., xn), Vestnik Leningrad Univ., t. 13, 1958, p. 5-24, (Russian). | Zbl 0114.30202

[3] I.Ya. Bakel'Man, Generalized solutions of the Monge-Ampère equations, Dokl. Akad. Nauk SSSR, t. 114, 1957, p. 1143-1145, (Russian). | MR 95481 | Zbl 0114.29602

[4] I.Ya. Bakel'Man, Theory of quasilinear elliptic equations, Sibirsk. Mat. Ž., t. 2, 1961, p. 179-186, (Russian). | MR 126604 | Zbl 0100.30503

[5] I.Ya. Bakel'Man, The Dirichlet problem for the elliptic n-dimensional Monge-Ampère equations and related problems in the theory of quasilinear equations, Proceedings of Seminar on Monge-Ampère Equations and Related Topics, (Firenze 1980), Instituto Nazionale di Alta Matematica, Roma, 1982, p. 1-78. | Zbl 0524.53039

[6] T. Bonnesen, W. Fenchel, Theorie der konvexen Körper, Springer, Berlin, 1934. | JFM 60.0673.01 | MR 344997 | Zbl 0008.07708

[7] H. Busemann, Convex surfaces, Interscience, New York, 1958. | MR 105155 | Zbl 0196.55101

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

[9] S.-Y. Cheng, S.-T. Yau, On the regularity of the Monge-Ampère equation det (∂2u/∂xi∂xj) = F(x, u), Comm. Pure Appl. Math., t. 30, 1977, p. 41-68. | MR 437805 | Zbl 0347.35019

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

[11] M. Giaquinta, On the Dirichlet problem for surfaces of prescribed mean curvature, Manuscripta Math., t. 12, 1974, p. 73-86. | MR 336532 | Zbl 0276.35038

[12] D. Gilbarg, N.S. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, Second Edition, 1983. | MR 737190 | Zbl 0562.35001

[13] N.M. Ivochkina, Classical solvability of the Dirichlet problem for the Monge-Ampère equation, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), t. 131, 1983, p. 72-79. | MR 718679 | Zbl 0522.35028

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

[15] P.-L. Lions, Sur les équations de Monge-Ampère I, Manuscripta Math., t. 41, 1983, p. 1-44. | MR 689131 | Zbl 0509.35036

[16] P.-L. Lions, Sur les équations de Monge-Ampère II, Arch. Rational Mech. Anal. (to appear). | MR 786541 | Zbl 0579.35027

[17] A.V. Pogorelov, Monge-Ampère equations of elliptic type, Noordhoff, Gröningen, 1964. | MR 180763 | Zbl 0133.04902

[18] A.V. Pogorelov, On the regularity of generalized solutions of the equation det (∂2u/∂xi∂xj) = φ(x1, ..., xn) > 0, Dokl. Akad. Nauk SSSR, t. 200, 1971, p. 543-547, (Russian). English translation in Soviet Math. Dokl., t. 12, 1971, p. 1436- 1440. | MR 293227 | Zbl 0246.35014

[19] A.V. Pogorelov, The Dirichlet problem for the n-dimensional analogue of the Monge-Ampère equation, Dokl. Akad. Nauk SSSR, t. 201, 1971, p. 790-793, (Russian). English translation in Soviet Math. Dokl, t. 12, 1971, p. 1727-1731. | MR 293228 | Zbl 0238.35071

[20] A.V. Pogorelov, The Minkowski multidimensional problem, J. Wiley, New York, 1978.

[21] J. Rauch, B.A. Taylor, The Dirichlet problem for the multidimensional Monge-Ampère equation, Rocky Mountain J. Math., t. 7, 1977, p. 345-364. | MR 454331 | Zbl 0367.35025

[22] I.Kh. Sabitov, The regularity of conver regions with a metric that is regular in the Hölmer classes, Sibirsk. Mat. Ž., t. 17, 1976, p.907-915, (Russian). English translation in Siberian Math. J., t. 17, 1976, p. 681-687. | MR 425854 | Zbl 0356.53017

[23] F. Schulz, Über die Differentialgleichung rt - s2 = f und das Weylsche Einbettungsproblem, Math. Z., t. 179, 1982, p. 1-10. | MR 643043 | Zbl 0535.35021

[24] N.S. Trudinger, Fully nonlinear, uniformly elliptic equations under natrual structure conditions, Trans. Amer. Math. Soc., t. 278, 1983, p. 751-770. | MR 701522 | Zbl 0518.35036

[25] N.S. Trudinger, J.I.E. Urbas, The Dirichlet problem for the equation of prescribed Gauss curvature, Bull. Austral. Math. Soc., t. 28, 1983, p. 217-231. | MR 729009 | Zbl 0524.35047

[26] N.S. Trudinger, J.I.E. Urbas, On second derivative estimates for equations of Monge-Ampère type, Bull. Austral. Math. Soc., t. 30, 1984, p. 321-334. | MR 766792 | Zbl 0557.35054

[27] J.I.E. Urbas, Elliptic equations of Monge-Ampère type, Thesis, Australian National University, 1984.

[28] J.I.E. Urbas, The equation of prescribed Gauss curvature without boundary conditions, J. Differential Geometry, t. 20, 1984, p. 311-327. | MR 788283 | Zbl 0566.53013

[29] J.I.E. Urbas, Global Hölder estimates for equation of Monge-Ampère type, (to appear). | MR 918234 | Zbl 0674.35026