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On the exterior Dirichlet problem for Hessian equations


Authors: Jiguang Bao, Haigang Li and Yanyan Li
Journal: Trans. Amer. Math. Soc. 366 (2014), 6183-6200
MSC (2010): Primary 35J60, 35J67
DOI: https://doi.org/10.1090/S0002-9947-2014-05867-4
Published electronically: June 16, 2014
MathSciNet review: 3267007
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Abstract | References | Similar Articles | Additional Information

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.


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  • [1] L. A. Caffarelli, Topics in PDEs: The Monge-Ampère equation. Graduate course, Courant Institute, New York University, 1995.
  • [2] 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 (96h:35046)
  • [3] 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 (2004c:35116), https://doi.org/10.1002/cpa.10067
  • [4] 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 (87f:35098), https://doi.org/10.1007/BF02392544
  • [5] 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 0106487 (21 #5219)
  • [6] 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 (87k:53127), https://doi.org/10.1002/cpa.3160390606
  • [7] 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 (2002e:35072), https://doi.org/10.1002/cpa.1016
  • [8] 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 (92j:35050), https://doi.org/10.1090/S0273-0979-1992-00266-5
  • [9] 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 (2012d:35091), https://doi.org/10.1090/S0002-9939-2011-10833-5
  • [10] 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, https://doi.org/10.1007/s11464-011-0109-x
  • [11] Philippe Delanoë, Partial decay on simple manifolds, Ann. Global Anal. Geom. 10 (1992), no. 1, 3-61. MR 1172619 (93h:58144), https://doi.org/10.1007/BF00128337
  • [12] 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 (2001d:53010), https://doi.org/10.1007/PL00004700
  • [13] 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 (2001e:53013), https://doi.org/10.1007/s006050050084
  • [14] 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 (89m:35070), https://doi.org/10.1002/cpa.3160420103
  • [15] 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 (89a:35038), https://doi.org/10.1007/BF00281780
  • [16] Huaiyu Jian, Hessian equations with infinite Dirichlet boundary value, Indiana Univ. Math. J. 55 (2006), no. 3, 1045-1062. MR 2244597 (2008f:35120), https://doi.org/10.1512/iumj.2006.55.2728
  • [17] Konrad Jörgens, Über die Lösungen der Differentialgleichung $ rt-s^2=1$, Math. Ann. 127 (1954), 130-134 (German). MR 0062326 (15,961e)
  • [18] 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 (2002i:53070)
  • [19] 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 (22 #8200)
  • [20] A. V. Pogorelov, On the improper convex affine hyperspheres, Geometriae Dedicata 1 (1972), no. 1, 33-46. MR 0319126 (47 #7672)
  • [21] Neil S. Trudinger, The Dirichlet problem for the prescribed curvature equations, Arch. Rational Mech. Anal. 111 (1990), no. 2, 153-179. MR 1057653 (91g:35118), https://doi.org/10.1007/BF00375406
  • [22] Neil S. Trudinger, On the Dirichlet problem for Hessian equations, Acta Math. 175 (1995), no. 2, 151-164. MR 1368245 (96m:35113), https://doi.org/10.1007/BF02393303
  • [23] Neil S. Trudinger, Weak solutions of Hessian equations, Comm. Partial Differential Equations 22 (1997), no. 7-8, 1251-1261. MR 1466315 (99a:35077), https://doi.org/10.1080/03605309708821299
  • [24] Neil S. Trudinger and Xu-Jia Wang, The Bernstein problem for affine maximal hypersurfaces, Invent. Math. 140 (2000), no. 2, 399-422. MR 1757001 (2001h:53016), https://doi.org/10.1007/s002220000059
  • [25] 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 (92h:35074), https://doi.org/10.1512/iumj.1990.39.39020
  • [26] C. Wang and J. G. 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

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Additional Information

Jiguang Bao
Affiliation: School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, People’s Republic of China
Email: jgbao@bnu.edu.cn

Haigang Li
Affiliation: School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, People’s Republic of China
Email: hgli@bnu.edu.cn

Yanyan Li
Affiliation: Department of Mathematics, Rutgers University, 110 Frelinghuysen Road, Piscataway, New Jersey 08854
Email: yyli@math.rutgers.edu

DOI: https://doi.org/10.1090/S0002-9947-2014-05867-4
Received by editor(s): April 12, 2012
Published electronically: June 16, 2014
Additional Notes: The second author was the corresponding author
Article copyright: © Copyright 2014 American Mathematical Society

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