Necessary and sufficient conditions on existence and convexity of solutions for Dirichlet problems of Hessian equations on exterior domains

Wang, Chong; Bao, Jiguang

doi:10.1090/S0002-9939-2012-11738-1

Necessary and sufficient conditions on existence and convexity of solutions for Dirichlet problems of Hessian equations on exterior domains
HTML articles powered by AMS MathViewer

by Chong Wang and Jiguang Bao PDF

Proc. Amer. Math. Soc. 141 (2013), 1289-1296 Request permission

Abstract:

In this paper, we are concerned with the Dirichlet problems of Hessian equations on exterior domains with prescribed asymptotic behavior at infinity, and we obtain the necessary and sufficient conditions on existence and convexity of radial solutions.

References

L. A. Caffarelli, Topics in PDEs: The Monge-Ampère equation, Graduate course, Courant Institute, New York University, 1995.
L. Caffarelli and Yanyan Li, An extension to a theorem of Jörgens, Calabi, and Pogorelov, Comm. Pure Appl. Math. 56 (2003), no. 5, 549–583. MR 1953651, DOI 10.1002/cpa.10067
L. Caffarelli and YanYan Li, Some multi-valued solutions to Monge-Ampère equations, Comm. Anal. Geom. 14 (2006), no. 3, 411–441. MR 2260718, DOI 10.4310/CAG.2006.v14.n3.a1
L. Caffarelli, L. Nirenberg, and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations. III. Functions of the eigenvalues of the Hessian, Acta Math. 155 (1985), no. 3-4, 261–301. MR 806416, DOI 10.1007/BF02392544
Eugenio Calabi, Improper affine hyperspheres of convex type and a generalization of a theorem by K. Jörgens, Michigan Math. J. 5 (1958), 105–126. MR 106487
Andrea Colesanti and Paolo Salani, Hessian equations in non-smooth domains, Nonlinear Anal. 38 (1999), no. 6, Ser. A: Theory Methods, 803–812. MR 1710156, DOI 10.1016/S0362-546X(98)00121-7
Shiu Yuen Cheng and Shing-Tung Yau, Complete affine hypersurfaces. I. The completeness of affine metrics, Comm. Pure Appl. Math. 39 (1986), no. 6, 839–866. MR 859275, DOI 10.1002/cpa.3160390606
Kai-Seng Chou and Xu-Jia Wang, A variational theory of the Hessian equation, Comm. Pure Appl. Math. 54 (2001), no. 9, 1029–1064. MR 1835381, DOI 10.1002/cpa.1016
Limei Dai and Jiguang Bao, On uniqueness and existence of viscosity solutions to Hessian equations in exterior domains, Front. Math. China 6 (2011), no. 2, 221–230. MR 2780888, DOI 10.1007/s11464-011-0109-x
Philippe Delanoë, Partial decay on simple manifolds, Ann. Global Anal. Geom. 10 (1992), no. 1, 3–61. MR 1172619, DOI 10.1007/BF00128337
L. Ferrer, A. Martínez, and F. Milán, An extension of a theorem by K. Jörgens and a maximum principle at infinity for parabolic affine spheres, Math. Z. 230 (1999), no. 3, 471–486. MR 1679973, DOI 10.1007/PL00004700
L. Ferrer, A. Martínez, and F. Milán, The space of parabolic affine spheres with fixed compact boundary, Monatsh. Math. 130 (2000), no. 1, 19–27. MR 1762061, DOI 10.1007/s006050050084
Bo Guan, The Dirichlet problem for a class of fully nonlinear elliptic equations, Comm. Partial Differential Equations 19 (1994), no. 3-4, 399–416. MR 1265805, DOI 10.1080/03605309408821022
Konrad Jörgens, Über die Lösungen der Differentialgleichung $rt-s^2=1$, Math. Ann. 127 (1954), 130–134 (German). MR 62326, DOI 10.1007/BF01361114
J. Jost and Y. L. Xin, Some aspects of the global geometry of entire space-like submanifolds, Results Math. 40 (2001), no. 1-4, 233–245. Dedicated to Shiing-Shen Chern on his 90th birthday. MR 1860371, DOI 10.1007/BF03322708
A. V. Pogorelov, On the improper convex affine hyperspheres, Geometriae Dedicata 1 (1972), no. 1, 33–46. MR 319126, DOI 10.1007/BF00147379
Neil S. Trudinger, On the Dirichlet problem for Hessian equations, Acta Math. 175 (1995), no. 2, 151–164. MR 1368245, DOI 10.1007/BF02393303
Neil S. Trudinger, Weak solutions of Hessian equations, Comm. Partial Differential Equations 22 (1997), no. 7-8, 1251–1261. MR 1466315, DOI 10.1080/03605309708821299
Neil S. Trudinger and Xu-Jia Wang, Hessian measures. II, Ann. of Math. (2) 150 (1999), no. 2, 579–604. MR 1726702, DOI 10.2307/121089
Neil S. Trudinger and Xu-Jia Wang, The Bernstein problem for affine maximal hypersurfaces, Invent. Math. 140 (2000), no. 2, 399–422. MR 1757001, DOI 10.1007/s002220000059
John I. E. Urbas, On the existence of nonclassical solutions for two classes of fully nonlinear elliptic equations, Indiana Univ. Math. J. 39 (1990), no. 2, 355–382. MR 1089043, DOI 10.1512/iumj.1990.39.39020