# American Institute of Mathematical Sciences

October  2022, 21(10): 3407-3420. doi: 10.3934/cpaa.2022106

## On the exterior problem for parabolic k-Hessian equations

 1 School of Statistics, University of International Business and Economics, Beijing 100029, China 2 School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, China

* Corresponding author

Received  January 2022 Revised  May 2022 Published  October 2022 Early access  June 2022

Fund Project: The first author is supported by "the Fundamental Research Funds for the Central Universities" in UIBE (21QD20). The second author is supported in part by the NSFC (11871102)

We use Perron method to prove the existence of ancient solutions of the exterior problem for parabolic k-Hessian equations $-u_tS_k(D^2u) = 1$ with prescribed asymptotic behavior at infinity.

Citation: Ziwei Zhou, Jiguang Bao. On the exterior problem for parabolic k-Hessian equations. Communications on Pure and Applied Analysis, 2022, 21 (10) : 3407-3420. doi: 10.3934/cpaa.2022106
##### References:
 [1] J. Bao, J. Chen, B. Guan and M. Ji, Liouville property and regularity of a Hessian quotient equation, Amer. J. Math., 125 (2003), 301-316. [2] J. Bao, H. Li and Y. Li, On the exterior Dirichlet problem for Hessian equations, Transactions of the American Mathematical Society, 366 (2014), 6183-6200.  doi: 10.1090/S0002-9947-2014-05867-4. [3] M. G. Crandall, H. Ishii and P. L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67.  doi: 10.1090/S0273-0979-1992-00266-5. [4] L. Dai, Exterior problems of parabolic Monge-Ampère equations for n = 2, Comput. Math. Appl., 67 (2014), 1497-1506.  doi: 10.1016/j.camwa.2014.02.009. [5] L. Dai, Exterior problems for a parabolic Monge-Ampère equation, Nonlinear Anal., 100 (2014), 99-110.  doi: 10.1016/j.na.2014.01.011. [6] L. Dai, Exterior problems for more general parabolic Monge-Ampère equation in more general domain, J. Math. Anal. Appl., 427 (2015), 1190-1204.  doi: 10.1016/j.jmaa.2015.02.087. [7] L. Dai, Exterior problems for parabolic Hessian equations, (Chinese), Adv. Math. (China), 45 (2016), 561-571. [8] C. E. Gutiérrez and Q. Huang, A generalization of a theorem by Calabi to the parabolic Monge-Ampère equation, Indiana Univ. Math. J., 47 (1998), 1459-1480.  doi: 10.1512/iumj.1998.47.1563. [9] R. Jensen, The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations, Arch. Rational Mech. Anal., 101 (1988), 1-27.  doi: 10.1007/BF00281780. [10] G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific. 1996. doi: 10.1142/3302. [11] S. Nakamori and K. Takimoto, A Bernstein type theorem for parabolic k-Hessian equations, Nonlinear Anal., 117 (2015), 211-220.  doi: 10.1016/j.na.2015.01.010. [12] R. Wang and G. Wang, The geometric measure theoretical characterization of viscosity solutions to parabolic Monge-Ampère type equation, J. Partial Differ. Equ., 6 (1993), 237-254. [13] J. Xiong and J. Bao, On Jögens, Calabi, and Pogorelov type theorem and isolated singularities of parabolic Monge-Ampère equations, J. Differ. Equ., 250 (2011), 367-385.  doi: 10.1016/j.jde.2010.08.024. [14] Y. Zhan, Viscosity Solutions of Nonlinear Degenerate Parabolic Equations and Several Applications, Ph.D thesis, University of Toronto in Canada, 2000.

show all references

##### References:
 [1] J. Bao, J. Chen, B. Guan and M. Ji, Liouville property and regularity of a Hessian quotient equation, Amer. J. Math., 125 (2003), 301-316. [2] J. Bao, H. Li and Y. Li, On the exterior Dirichlet problem for Hessian equations, Transactions of the American Mathematical Society, 366 (2014), 6183-6200.  doi: 10.1090/S0002-9947-2014-05867-4. [3] M. G. Crandall, H. Ishii and P. L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67.  doi: 10.1090/S0273-0979-1992-00266-5. [4] L. Dai, Exterior problems of parabolic Monge-Ampère equations for n = 2, Comput. Math. Appl., 67 (2014), 1497-1506.  doi: 10.1016/j.camwa.2014.02.009. [5] L. Dai, Exterior problems for a parabolic Monge-Ampère equation, Nonlinear Anal., 100 (2014), 99-110.  doi: 10.1016/j.na.2014.01.011. [6] L. Dai, Exterior problems for more general parabolic Monge-Ampère equation in more general domain, J. Math. Anal. Appl., 427 (2015), 1190-1204.  doi: 10.1016/j.jmaa.2015.02.087. [7] L. Dai, Exterior problems for parabolic Hessian equations, (Chinese), Adv. Math. (China), 45 (2016), 561-571. [8] C. E. Gutiérrez and Q. Huang, A generalization of a theorem by Calabi to the parabolic Monge-Ampère equation, Indiana Univ. Math. J., 47 (1998), 1459-1480.  doi: 10.1512/iumj.1998.47.1563. [9] R. Jensen, The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations, Arch. Rational Mech. Anal., 101 (1988), 1-27.  doi: 10.1007/BF00281780. [10] G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific. 1996. doi: 10.1142/3302. [11] S. Nakamori and K. Takimoto, A Bernstein type theorem for parabolic k-Hessian equations, Nonlinear Anal., 117 (2015), 211-220.  doi: 10.1016/j.na.2015.01.010. [12] R. Wang and G. Wang, The geometric measure theoretical characterization of viscosity solutions to parabolic Monge-Ampère type equation, J. Partial Differ. Equ., 6 (1993), 237-254. [13] J. Xiong and J. Bao, On Jögens, Calabi, and Pogorelov type theorem and isolated singularities of parabolic Monge-Ampère equations, J. Differ. Equ., 250 (2011), 367-385.  doi: 10.1016/j.jde.2010.08.024. [14] Y. Zhan, Viscosity Solutions of Nonlinear Degenerate Parabolic Equations and Several Applications, Ph.D thesis, University of Toronto in Canada, 2000.
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