On the exterior Dirichlet problem for Hessian equations

Bao, Jiguang; Li, Haigang; Li, Yanyan

doi:10.1090/S0002-9947-2014-05867-4

On the exterior Dirichlet problem for Hessian equations
HTML articles powered by AMS MathViewer

by Jiguang Bao, Haigang Li and Yanyan Li PDF

Trans. Amer. Math. Soc. 366 (2014), 6183-6200 Request permission

Abstract:

In this paper, we establish a theorem on the existence of the solutions of the exterior Dirichlet problem for Hessian equations with prescribed asymptotic behavior at infinity. This extends a result of Caffarelli and Li (2003) for the Monge-Ampère equation to Hessian equations.

References

L. A. Caffarelli, Topics in PDEs: The Monge-Ampère equation. Graduate course, Courant Institute, New York University, 1995.
Luis A. Caffarelli and Xavier Cabré, Fully nonlinear elliptic equations, American Mathematical Society Colloquium Publications, vol. 43, American Mathematical Society, Providence, RI, 1995. MR 1351007, DOI 10.1090/coll/043
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, 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
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
Michael G. Crandall, Hitoshi Ishii, and Pierre-Louis Lions, User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 1, 1–67. MR 1118699, DOI 10.1090/S0273-0979-1992-00266-5
Limei Dai, Existence of solutions with asymptotic behavior of exterior problems of Hessian equations, Proc. Amer. Math. Soc. 139 (2011), no. 8, 2853–2861. MR 2801627, DOI 10.1090/S0002-9939-2011-10833-5
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
Hitoshi Ishii, On uniqueness and existence of viscosity solutions of fully nonlinear second-order elliptic PDEs, Comm. Pure Appl. Math. 42 (1989), no. 1, 15–45. MR 973743, DOI 10.1002/cpa.3160420103
Robert Jensen, The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations, Arch. Rational Mech. Anal. 101 (1988), no. 1, 1–27. MR 920674, DOI 10.1007/BF00281780
Huaiyu Jian, Hessian equations with infinite Dirichlet boundary value, Indiana Univ. Math. J. 55 (2006), no. 3, 1045–1062. MR 2244597, DOI 10.1512/iumj.2006.55.2728
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
Norman Meyers and James Serrin, The exterior Dirichlet problem for second order elliptic partial differential equations, J. Math. Mech. 9 (1960), 513–538. MR 0117421, DOI 10.1512/iumj.1960.9.59029
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, The Dirichlet problem for the prescribed curvature equations, Arch. Rational Mech. Anal. 111 (1990), no. 2, 153–179. MR 1057653, DOI 10.1007/BF00375406
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, 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
Chong Wang and Jiguang Bao, Necessary and sufficient conditions on existence and convexity of solutions for Dirichlet problems of Hessian equations on exterior domains, Proc. Amer. Math. Soc. 141 (2013), no. 4, 1289–1296. MR 3008876, DOI 10.1090/S0002-9939-2012-11738-1