$C^{1,1}$ regularity for an obstacle problem of Hessian equations on Riemannian manifolds
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Abstract:
In this paper, we study an obstacle problem for a class of fully nonlinear equations on Riemannian manifolds. Using some new ideas, the $C^{1,1}$ regularity for the greatest viscosity solution is established under essentially optimal structure conditions.References
- Gejun Bao, Weisong Dong, and Heming Jiao, Regularity for an obstacle problem of Hessian equations on Riemannian manifolds, J. Differential Equations 258 (2015), no. 3, 696–716. MR 3279350, DOI 10.1016/j.jde.2014.10.001
- Luis A. Caffarelli and Robert J. McCann, Free boundaries in optimal transport and Monge-Ampère obstacle problems, Ann. of Math. (2) 171 (2010), no. 2, 673–730. MR 2630054, DOI 10.4007/annals.2010.171.673
- 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
- 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
- Claus Gerhardt, Hypersurfaces of prescribed mean curvature over obstacles, Math. Z. 133 (1973), 169–185. MR 324528, DOI 10.1007/BF01237902
- Bo Guan, The Dirichlet problem for Hessian equations on Riemannian manifolds, Calc. Var. Partial Differential Equations 8 (1999), no. 1, 45–69. MR 1666866, DOI 10.1007/s005260050116
- Bo Guan, Second-order estimates and regularity for fully nonlinear elliptic equations on Riemannian manifolds, Duke Math. J. 163 (2014), no. 8, 1491–1524. MR 3284698, DOI 10.1215/00127094-2713591
- B. Guan, The Dirichlet problem for fully nonlinear elliptic equations on Riemannian manifolds, arXiv:1403.2133.
- Bo Guan and Heming Jiao, Second order estimates for Hessian type fully nonlinear elliptic equations on Riemannian manifolds, Calc. Var. Partial Differential Equations 54 (2015), no. 3, 2693–2712. MR 3412389, DOI 10.1007/s00526-015-0880-8
- Bo Guan and Heming Jiao, The Dirichlet problem for Hessian type elliptic equations on Riemannian manifolds, Discrete Contin. Dyn. Syst. 36 (2016), no. 2, 701–714. MR 3392900, DOI 10.3934/dcds.2016.36.701
- Bo Guan, Shujun Shi, and Zhenan Sui, On estimates for fully nonlinear parabolic equations on Riemannian manifolds, Anal. PDE 8 (2015), no. 5, 1145–1164. MR 3393676, DOI 10.2140/apde.2015.8.1145
- Heming Jiao and Yong Wang, The obstacle problem for Hessian equations on Riemannian manifolds, Nonlinear Anal. 95 (2014), 543–552. MR 3130542, DOI 10.1016/j.na.2013.10.004
- David Kinderlehrer, How a minimal surface leaves an obstacle, Acta Math. 130 (1973), 221–242. MR 419997, DOI 10.1007/BF02392266
- Ki-ahm Lee, The obstacle problem for Monge-Ampére equation, Comm. Partial Differential Equations 26 (2001), no. 1-2, 33–42. MR 1842427, DOI 10.1081/PDE-100002244
- Yan Yan Li, Some existence results for fully nonlinear elliptic equations of Monge-Ampère type, Comm. Pure Appl. Math. 43 (1990), no. 2, 233–271. MR 1038143, DOI 10.1002/cpa.3160430204
- Jiakun Liu and Bin Zhou, An obstacle problem for a class of Monge-Ampère type functionals, J. Differential Equations 254 (2013), no. 3, 1306–1325. MR 2997372, DOI 10.1016/j.jde.2012.10.017
- Ovidiu Savin, The obstacle problem for Monge Ampere equation, Calc. Var. Partial Differential Equations 22 (2005), no. 3, 303–320. MR 2118901, DOI 10.1007/s00526-004-0275-8
- 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
- John Urbas, Hessian equations on compact Riemannian manifolds, Nonlinear problems in mathematical physics and related topics, II, Int. Math. Ser. (N. Y.), vol. 2, Kluwer/Plenum, New York, 2002, pp. 367–377. MR 1972006, DOI 10.1007/978-1-4615-0701-7_{2}0
- Jingang Xiong and Jiguang Bao, The obstacle problem for Monge-Ampère type equations in non-convex domains, Commun. Pure Appl. Anal. 10 (2011), no. 1, 59–68. MR 2746527, DOI 10.3934/cpaa.2011.10.59
Additional Information
- Heming Jiao
- Affiliation: Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, People’s Republic of China
- MR Author ID: 1044324
- ORCID: 0000-0002-6595-8303
- Email: jiao@hit.edu.cn
- Received by editor(s): April 3, 2015
- Received by editor(s) in revised form: September 30, 2015
- Published electronically: February 2, 2016
- Additional Notes: This work was supported by the Fundamental Research Funds for the Central Universities and Program for Innovation Research of Science in Harbin Institute of Technology, No. 61509066.
- Communicated by: Guofang Wei
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 3441-3453
- MSC (2010): Primary 35B45, 35B65, 58J32
- DOI: https://doi.org/10.1090/proc/12988
- MathSciNet review: 3503712