# American Institute of Mathematical Sciences

September  2016, 15(5): 1769-1780. doi: 10.3934/cpaa.2016013

## A priori estimates for the obstacle problem of Hessian type equations on Riemannian manifolds

 1 Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, China, China, China

Received  September 2015 Revised  March 2016 Published  July 2016

We are concerned with A priori estimates for the obstacle problem of a wide class of fully nonlinear equations on Riemannian manifolds. We use new techniques introduced by Bo Guan and derive new results for A priori second order estimates of its singular perturbation problem under fairly general conditions. By approximation, the existence of a $C^{1,1}$ viscosity solution is proved.
Citation: Weisong Dong, Tingting Wang, Gejun Bao. A priori estimates for the obstacle problem of Hessian type equations on Riemannian manifolds. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1769-1780. doi: 10.3934/cpaa.2016013
##### References:
 [1] B. Andrews, Contraction of convex hypersurfaces in Euclidean space, Calc. Var. Partial Differential Equations, 2 (1994), 151-171. doi: 10.1007/BF01191340. [2] G.-J. Bao, W.-S. Dong and H.-M. Jiao, Regularity for an obstacle problem of Hessian equations on Riemannian manifolds, J. Differential Equations, 258 (2015), 696-716. doi: 10.1016/j.jde.2014.10.001. [3] L. A. Caffarelli and R. McCann, Free boundaries in optimal transport and Monge-Ampère obstacle problems, Ann. of Math., 171 (2010), 673-730. doi: 10.4007/annals.2010.171.673. [4] L. A. 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), 261-301. doi: 10.1007/BF02392544. [5] M. Crandall, H. Ishii and P. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc., 27 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5. [6] L. C. Evans, Classical solutions of fully nonlinear, convex, second order elliptic equations, Comm. Pure Appl. Math., 35 (1982), 333-363. doi: 10.1002/cpa.3160350303. [7] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations Of Second Order, 2nd edition, Springer-Verlag, New York, 1983. doi: 10.1007/978-3-642-61798-0. [8] C. Gerhardt, Hypersurfaces of prescribed mean curvature over obstacles, Math. Z., 133 (1973), 169-185. [9] C. Gerhardt, Closed Weingarten hypersurfaces in Riemannian manifolds, J. Differential Geom., 43 (1996), 612-641. [10] B. Guan, The Dirichlet problem for Hessian equations on Riemannian manifolds, Calc. Var. Partial Differential Equations, 8 (1999), 45-69. doi: 10.1007/s005260050116. [11] B. Guan, Second order estimates and regularity for fully nonlinear elliptic equations on Riemannian manifolds, Duke Math. J., 163 (2014), 1491-1524. doi: 10.1215/00127094-2713591. [12] B. Guan, The Dirichlet problem for fully nonlinear ellipitc equations on Riemannian manifolds,, preprint, (). [13] B. Guan, S.-J. Shi and Z.-N. Sui, On estimates for fully nonlinear parabolic equations on Riemannian manifolds, Anal. PDE, 8 (2015), 1145-1164.. doi: 10.2140/apde.2015.8.1145. [14] B. Guan and H.-M. Jiao, The Dirichlet problem for Hessian type elliptic equations on Riemannian manifolds, Discrete Conti. Dyn. Syst., 36 (2016), 701-714. doi: 10.3934/dcds.2016.36.701. [15] E. Giusti, Superfici minime cartesiane con ostaeoli diseontinui, Arch. Ration. Mech. Anal., 35 (1969), 47-82. [16] H.-M. Jiao, $C^{1,1}$ regularity for an obstacle problem of Hessian equations on Riemannian manifolds, Proc. Amer. Math. Soc., 144 (2016), 3441-3453. [17] H.-M. Jiao and Y. Wang, The obstacle problem for Hessian equations on Riemannian manifolds, Nonlinear Anal., 95 (2014), 543-552. [18] N. V. Krylov, Boundedly nonhomogeneous elliptic and parabolic equations in a domain, Izvestia Math. Ser., 47 (1983), 75-108. [19] D. S. Kinderlehrer, Variational inequalities with lower dimensional obstacles, Israel J. Math., 10 (1971), 339-348. [20] D. S. Kinderlehrer, How a minimal surface leaves an obstacle, Acta Math., 130 (1973), 221-242. [21] K. Lee, The obstacle problem for Monge-Ampère equation, Comm. Partial Differential Equations, 26 (2001), 33-42. doi: 10.1081/PDE-100002244. [22] Y.-Y. Li, Degree theory for second order nonlinear elliptic operators and its applications, Comm. Partial Differential Equations, 14 (1989), 1541-1578. doi: 10.1080/03605308908820666. [23] J.-K. Liu and B. Zhou, An obstacle problem for a class of Monge-Ampère type functionals, J. Differential Equations, 254 (2013), 1306-1325. doi: 10.1016/j.jde.2012.10.017. [24] A. Oberman, The convex envelope is the solution of a nonlinear obstacle problem, Proc. Amer. Math. Soc., 135 (2007), 1689-1694. doi: 10.1090/S0002-9939-07-08887-9. [25] A. Oberman and L. Silvestre, The Dirichlet problem for the convex envelope, Trans. Amer. Math. Soc., 363 (2011), 5871-5886. doi: 10.1090/S0002-9947-2011-05240-2. [26] O. Savin, A free boundary problem with optimal transportation, Comm. Pure Appl. Math., 57 (2004), 126-140. doi: 10.1002/cpa.3041. [27] O. Savin, The obstacle problem for Monge Ampere equation, Calc. Var. Partial Differential Equations, 22 (2005), 303-320. doi: 10.1007/s00526-004-0275-8. [28] N. S. Trudinger, The Dirichlet problem for the prescribed curvature equations, Arch. Ration. Mech. Anal., 111 (1990), 153-179. doi: 10.1007/BF00375406. [29] J. Urbas, Hessian equations on compact Riemannian manifolds, in Nonlinear Problems in Mathematical Physics and Related Topics, II, Kluwer/Plenum, (2002), 367-377. doi: 10.1007/978-1-4615-0701-7_20. [30] J.-G. Xiong and J.-G. Bao, The obstacle problem for Monge-Ampère type equations in non-convex domains, Commun. Pure Appl. Anal., 10 (2011), 59-68. doi: 10.3934/cpaa.2011.10.59.

show all references

##### References:
 [1] B. Andrews, Contraction of convex hypersurfaces in Euclidean space, Calc. Var. Partial Differential Equations, 2 (1994), 151-171. doi: 10.1007/BF01191340. [2] G.-J. Bao, W.-S. Dong and H.-M. Jiao, Regularity for an obstacle problem of Hessian equations on Riemannian manifolds, J. Differential Equations, 258 (2015), 696-716. doi: 10.1016/j.jde.2014.10.001. [3] L. A. Caffarelli and R. McCann, Free boundaries in optimal transport and Monge-Ampère obstacle problems, Ann. of Math., 171 (2010), 673-730. doi: 10.4007/annals.2010.171.673. [4] L. A. 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), 261-301. doi: 10.1007/BF02392544. [5] M. Crandall, H. Ishii and P. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc., 27 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5. [6] L. C. Evans, Classical solutions of fully nonlinear, convex, second order elliptic equations, Comm. Pure Appl. Math., 35 (1982), 333-363. doi: 10.1002/cpa.3160350303. [7] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations Of Second Order, 2nd edition, Springer-Verlag, New York, 1983. doi: 10.1007/978-3-642-61798-0. [8] C. Gerhardt, Hypersurfaces of prescribed mean curvature over obstacles, Math. Z., 133 (1973), 169-185. [9] C. Gerhardt, Closed Weingarten hypersurfaces in Riemannian manifolds, J. Differential Geom., 43 (1996), 612-641. [10] B. Guan, The Dirichlet problem for Hessian equations on Riemannian manifolds, Calc. Var. Partial Differential Equations, 8 (1999), 45-69. doi: 10.1007/s005260050116. [11] B. Guan, Second order estimates and regularity for fully nonlinear elliptic equations on Riemannian manifolds, Duke Math. J., 163 (2014), 1491-1524. doi: 10.1215/00127094-2713591. [12] B. Guan, The Dirichlet problem for fully nonlinear ellipitc equations on Riemannian manifolds,, preprint, (). [13] B. Guan, S.-J. Shi and Z.-N. Sui, On estimates for fully nonlinear parabolic equations on Riemannian manifolds, Anal. PDE, 8 (2015), 1145-1164.. doi: 10.2140/apde.2015.8.1145. [14] B. Guan and H.-M. Jiao, The Dirichlet problem for Hessian type elliptic equations on Riemannian manifolds, Discrete Conti. Dyn. Syst., 36 (2016), 701-714. doi: 10.3934/dcds.2016.36.701. [15] E. Giusti, Superfici minime cartesiane con ostaeoli diseontinui, Arch. Ration. Mech. Anal., 35 (1969), 47-82. [16] H.-M. Jiao, $C^{1,1}$ regularity for an obstacle problem of Hessian equations on Riemannian manifolds, Proc. Amer. Math. Soc., 144 (2016), 3441-3453. [17] H.-M. Jiao and Y. Wang, The obstacle problem for Hessian equations on Riemannian manifolds, Nonlinear Anal., 95 (2014), 543-552. [18] N. V. Krylov, Boundedly nonhomogeneous elliptic and parabolic equations in a domain, Izvestia Math. Ser., 47 (1983), 75-108. [19] D. S. Kinderlehrer, Variational inequalities with lower dimensional obstacles, Israel J. Math., 10 (1971), 339-348. [20] D. S. Kinderlehrer, How a minimal surface leaves an obstacle, Acta Math., 130 (1973), 221-242. [21] K. Lee, The obstacle problem for Monge-Ampère equation, Comm. Partial Differential Equations, 26 (2001), 33-42. doi: 10.1081/PDE-100002244. [22] Y.-Y. Li, Degree theory for second order nonlinear elliptic operators and its applications, Comm. Partial Differential Equations, 14 (1989), 1541-1578. doi: 10.1080/03605308908820666. [23] J.-K. Liu and B. Zhou, An obstacle problem for a class of Monge-Ampère type functionals, J. Differential Equations, 254 (2013), 1306-1325. doi: 10.1016/j.jde.2012.10.017. [24] A. Oberman, The convex envelope is the solution of a nonlinear obstacle problem, Proc. Amer. Math. Soc., 135 (2007), 1689-1694. doi: 10.1090/S0002-9939-07-08887-9. [25] A. Oberman and L. Silvestre, The Dirichlet problem for the convex envelope, Trans. Amer. Math. Soc., 363 (2011), 5871-5886. doi: 10.1090/S0002-9947-2011-05240-2. [26] O. Savin, A free boundary problem with optimal transportation, Comm. Pure Appl. Math., 57 (2004), 126-140. doi: 10.1002/cpa.3041. [27] O. Savin, The obstacle problem for Monge Ampere equation, Calc. Var. Partial Differential Equations, 22 (2005), 303-320. doi: 10.1007/s00526-004-0275-8. [28] N. S. Trudinger, The Dirichlet problem for the prescribed curvature equations, Arch. Ration. Mech. Anal., 111 (1990), 153-179. doi: 10.1007/BF00375406. [29] J. Urbas, Hessian equations on compact Riemannian manifolds, in Nonlinear Problems in Mathematical Physics and Related Topics, II, Kluwer/Plenum, (2002), 367-377. doi: 10.1007/978-1-4615-0701-7_20. [30] J.-G. Xiong and J.-G. Bao, The obstacle problem for Monge-Ampère type equations in non-convex domains, Commun. Pure Appl. Anal., 10 (2011), 59-68. doi: 10.3934/cpaa.2011.10.59.
 [1] Bo Guan, Heming Jiao. The Dirichlet problem for Hessian type elliptic equations on Riemannian manifolds. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 701-714. doi: 10.3934/dcds.2016.36.701 [2] Xavier Cabré, Manel Sanchón, Joel Spruck. A priori estimates for semistable solutions of semilinear elliptic equations. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 601-609. doi: 10.3934/dcds.2016.36.601 [3] Weisong Dong, Chang Li. Second order estimates for complex Hessian equations on Hermitian manifolds. Discrete and Continuous Dynamical Systems, 2021, 41 (6) : 2619-2633. doi: 10.3934/dcds.2020377 [4] Chunhui Qiu, Rirong Yuan. On the Dirichlet problem for fully nonlinear elliptic equations on annuli of metric cones. Discrete and Continuous Dynamical Systems, 2017, 37 (11) : 5707-5730. doi: 10.3934/dcds.2017247 [5] Martino Bardi, Paola Mannucci. On the Dirichlet problem for non-totally degenerate fully nonlinear elliptic equations. Communications on Pure and Applied Analysis, 2006, 5 (4) : 709-731. doi: 10.3934/cpaa.2006.5.709 [6] D. Bartolucci, L. Orsina. Uniformly elliptic Liouville type equations: concentration compactness and a priori estimates. Communications on Pure and Applied Analysis, 2005, 4 (3) : 499-522. doi: 10.3934/cpaa.2005.4.499 [7] Italo Capuzzo Dolcetta, Antonio Vitolo. Glaeser's type gradient estimates for non-negative solutions of fully nonlinear elliptic equations. Discrete and Continuous Dynamical Systems, 2010, 28 (2) : 539-557. doi: 10.3934/dcds.2010.28.539 [8] Paola Mannucci. The Dirichlet problem for fully nonlinear elliptic equations non-degenerate in a fixed direction. Communications on Pure and Applied Analysis, 2014, 13 (1) : 119-133. doi: 10.3934/cpaa.2014.13.119 [9] Yu-Zhao Wang. $\mathcal{W}$-Entropy formulae and differential Harnack estimates for porous medium equations on Riemannian manifolds. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2441-2454. doi: 10.3934/cpaa.2018116 [10] J.I. Díaz, D. Gómez-Castro. Steiner symmetrization for concave semilinear elliptic and parabolic equations and the obstacle problem. Conference Publications, 2015, 2015 (special) : 379-386. doi: 10.3934/proc.2015.0379 [11] Marco Ghimenti, A. M. Micheletti. Non degeneracy for solutions of singularly perturbed nonlinear elliptic problems on symmetric Riemannian manifolds. Communications on Pure and Applied Analysis, 2013, 12 (2) : 679-693. doi: 10.3934/cpaa.2013.12.679 [12] Limei Dai. Entire solutions with asymptotic behavior of fully nonlinear uniformly elliptic equations. Communications on Pure and Applied Analysis, 2011, 10 (6) : 1707-1714. doi: 10.3934/cpaa.2011.10.1707 [13] Pierpaolo Soravia. Uniqueness results for fully nonlinear degenerate elliptic equations with discontinuous coefficients. Communications on Pure and Applied Analysis, 2006, 5 (1) : 213-240. doi: 10.3934/cpaa.2006.5.213 [14] Hongxia Zhang, Ying Wang. Liouville results for fully nonlinear integral elliptic equations in exterior domains. Communications on Pure and Applied Analysis, 2018, 17 (1) : 85-112. doi: 10.3934/cpaa.2018006 [15] Lidan Wang, Lihe Wang, Chunqin Zhou. Classification of positive solutions for fully nonlinear elliptic equations in unbounded cylinders. Communications on Pure and Applied Analysis, 2021, 20 (3) : 1241-1261. doi: 10.3934/cpaa.2021019 [16] Antonio Vitolo, Maria E. Amendola, Giulio Galise. On the uniqueness of blow-up solutions of fully nonlinear elliptic equations. Conference Publications, 2013, 2013 (special) : 771-780. doi: 10.3934/proc.2013.2013.771 [17] Agnid Banerjee. A note on the unique continuation property for fully nonlinear elliptic equations. Communications on Pure and Applied Analysis, 2015, 14 (2) : 623-626. doi: 10.3934/cpaa.2015.14.623 [18] Gabrielle Nornberg, Delia Schiera, Boyan Sirakov. A priori estimates and multiplicity for systems of elliptic PDE with natural gradient growth. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3857-3881. doi: 10.3934/dcds.2020128 [19] Jianguo Huang, Jun Zou. Uniform a priori estimates for elliptic and static Maxwell interface problems. Discrete and Continuous Dynamical Systems - B, 2007, 7 (1) : 145-170. doi: 10.3934/dcdsb.2007.7.145 [20] Laura Baldelli, Roberta Filippucci. A priori estimates for elliptic problems via Liouville type theorems. Discrete and Continuous Dynamical Systems - S, 2020, 13 (7) : 1883-1898. doi: 10.3934/dcdss.2020148

2020 Impact Factor: 1.916