On the exterior Dirichlet problem for special Lagrangian equations
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Abstract:
In this paper, we establish the existence and uniqueness theorem of the exterior Dirichlet problem for special Lagrangian equations with prescribed asymptotic behavior at infinity.References
- Jiguang Bao and Haigang Li, The exterior Dirichlet problem for special Lagrangian equations in dimensions $n\leq 4$, Nonlinear Anal. 89 (2013), 219–229. MR 3073326, DOI 10.1016/j.na.2013.05.013
- Jiguang Bao, Haigang Li, and Yanyan Li, On the exterior Dirichlet problem for Hessian equations, Trans. Amer. Math. Soc. 366 (2014), no. 12, 6183–6200. MR 3267007, DOI 10.1090/S0002-9947-2014-05867-4
- 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
- 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
- Jingyi Chen, Micah Warren, and Yu Yuan, A priori estimate for convex solutions to special Lagrangian equations and its application, Comm. Pure Appl. Math. 62 (2009), no. 4, 583–595. MR 2492708, DOI 10.1002/cpa.20261
- Vicente Cortés, Christoph Mayer, Thomas Mohaupt, and Frank Saueressig, Special geometry of Euclidean supersymmetry. I. Vector multiplets, J. High Energy Phys. 3 (2004), 028, 73. MR 2061551, DOI 10.1088/1126-6708/2004/03/028
- 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
- Lei Fu, An analogue of Bernstein’s theorem, Houston J. Math. 24 (1998), no. 3, 415–419. MR 1686614
- Reese Harvey and H. Blaine Lawson Jr., Calibrated geometries, Acta Math. 148 (1982), 47–157. MR 666108, DOI 10.1007/BF02392726
- G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, first edition, Cambridge University Press, Cambridge, 1934 (second edition, 1952).
- 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
- Haigang Li and Jiguang Bao, The exterior Dirichlet problem for fully nonlinear elliptic equations related to the eigenvalues of the Hessian, J. Differential Equations 256 (2014), no. 7, 2480–2501. MR 3160451, DOI 10.1016/j.jde.2014.01.011
- Haigang Li and Limei Dai, The exterior Dirichlet problem for Hessian quotient equations, J. Math. Anal. Appl. 393 (2012), no. 2, 534–543. MR 2921696, DOI 10.1016/j.jmaa.2012.03.034
- Dongsheng Li and Zhisu Li, On the exterior Dirichlet problem for Hessian quotient equations, J. Differential Equations 264 (2018), no. 11, 6633–6662. MR 3771820, DOI 10.1016/j.jde.2018.01.047
- Yu Yuan, A Bernstein problem for special Lagrangian equations, Invent. Math. 150 (2002), no. 1, 117–125. MR 1930884, DOI 10.1007/s00222-002-0232-0
- 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
- Micah Warren and Yu Yuan, A Liouville type theorem for special Lagrangian equations with constraints, Comm. Partial Differential Equations 33 (2008), no. 4-6, 922–932. MR 2424382, DOI 10.1080/03605300801970986
- Dake Wang and Yu Yuan, Hessian estimates for special Lagrangian equations with critical and supercritical phases in general dimensions, Amer. J. Math. 136 (2014), no. 2, 481–499. MR 3188067, DOI 10.1353/ajm.2014.0009
- Yu Yuan, A Bernstein problem for special Lagrangian equations, Invent. Math. 150 (2002), no. 1, 117–125. MR 1930884, DOI 10.1007/s00222-002-0232-0
- Yu Yuan, Global solutions to special Lagrangian equations, Proc. Amer. Math. Soc. 134 (2006), no. 5, 1355–1358. MR 2199179, DOI 10.1090/S0002-9939-05-08081-0
Additional Information
- Zhisu Li
- Affiliation: Beijing International Center for Mathematical Research, Peking University, Beijing 100871, People’s Republic of China – and – School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, People’s Republic of China
- MR Author ID: 1204312
- Email: lizhisu@bicmr.pku.edu.cn; lizhisu@stu.xjtu.edu.cn
- Received by editor(s): April 23, 2017
- Received by editor(s) in revised form: March 15, 2018
- Published electronically: April 18, 2019
- Additional Notes: This research is partially supported by National Natural Science Foundation of China (Grant No. 11801015) and by China Postdoctoral Science Foundation (Grant No. 2018M631230)
- © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 372 (2019), 889-924
- MSC (2010): Primary 35J25, 35J60; Secondary 35D40, 35J15
- DOI: https://doi.org/10.1090/tran/7594
- MathSciNet review: 3968791