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October  2020, 19(10): 4921-4936. doi: 10.3934/cpaa.2020218

## Existence and uniqueness of viscosity solutions to the exterior problem of a parabolic Monge-Ampère equation

 School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, China

* Corresponding author

Received  December 2019 Revised  May 2020 Published  July 2020

Fund Project: The authors are supported in part by NSFC (11631002 and 11871102)

In this paper, we use the Perron method to prove the existence and uniqueness of the exterior problem for a kind of parabolic Monge-Ampère equation $-u_t+\log\det D^2u = f(x)$ with prescribed asymptotic behavior at infinity, where $f$ is asymptotically close to a radial function at infinity. We generalize the results of both the elliptic exterior problems and the parabolic interior problems for the Monge-Ampère equations.

Citation: Shuyu Gong, Ziwei Zhou, Jiguang Bao. Existence and uniqueness of viscosity solutions to the exterior problem of a parabolic Monge-Ampère equation. Communications on Pure & Applied Analysis, 2020, 19 (10) : 4921-4936. doi: 10.3934/cpaa.2020218
##### References:
 [1] J. Bao, H. Li and L. Zhang, Monge-Ampère equation on exterior domains, Calc. Var. Partial Differ. Equ., 52 (2015), 39-63.  doi: 10.1007/s00526-013-0704-7.  Google Scholar [2] J. Bao, J. Xiong and Z. Zhou, Existence of entire solutions of Monge-Ampère equations with prescribed asymptotic behaviors, Calc. Var. Partial Differ. Equ., 58 (2019), Art. 193, 12 pp. doi: 10.1007/s00526-019-1639-4.  Google Scholar [3] L. Caffarelli, Interior $W^{2, p}$ estimates for solutions of the Monge-Ampère equation, Ann. of Math., (2) 131 (1990), no. 1, 135-150.  Google Scholar [4] L. Caffarelli, L. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations. I. Monge-Ampère equation, Commun. Pure Appl. Math., 37 (1984), 369-402.  doi: 10.1002/cpa.3160370306.  Google Scholar [5] L. Caffarelli and Y. Li, An extension to a theorem of J$\ddot{o}$rgens, Calabi, and Pogorelov, Commun. Pure Appl. Math., 56 (2003), 549–583. doi: 10.1002/cpa.10067.  Google Scholar [6] K. Chou and X. Wang, A logarithmic Gauss curvature flow and the Minkowski problem, Ann. Inst. Henri Poincare Anal. Non Lineaire, 17 (2001), 733-751.  doi: 10.1016/S0294-1449(00)00053-6.  Google Scholar [7] 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.  Google Scholar [8] L. Dai, Exterior problems for a parabolic Monge-Ampère equation, Nonlinear Anal. Theory Meth. Appl., 100 (2014), 99-110.  doi: 10.1016/j.na.2014.01.011.  Google Scholar [9] L. Ferrer, A. Martínez and F. Mil$\rm\acute{a}$n, An extension of a theorem by K. J$\ddot{o}$rgens and a maximum principle at infinity for parabolic affine spheres, Math. Z., 230 (1999), 471-486.  doi: 10.1007/PL00004700.  Google Scholar [10] L. Ferrer, A. Martínez and F. Mil$\rm\acute{a}$n, The space of parabolic affine spheres with fixed compact boundary, Monatsh. Math., 130 (2000), 19-27.  doi: 10.1007/s006050050084.  Google Scholar [11] H. Ishii and P.L. Lions, Viscosity solutions of fully nonlinear second-order elliptic partial differential equations, J. Partial Differ. Equ., 83 (1990), 26-78.  Google Scholar [12] N. M. Ivochkina and O. A. Ladyzhenskaya, On parabolic equations generated by symmetric functions of the principal curvatures of the evolving surfacing, or of the eigenvalues of Hessian, part Ⅰ: parabolic Monge-Ampère equations, St. Petersburg Math. J., 6 (1994), 527-594.   Google Scholar [13] T. Jin and J. Xiong, Solutions of some Monge-Ampère equations with isolated and line singularities, Adv. Math, 289 (2016), 114-141.  doi: 10.1016/j.aim.2015.11.029.  Google Scholar [14] Y. Li and S. Lu, Existence and nonexistence to exterior Dirichlet problem for Monge Ampère equation, Calc. Var. Partial Differ. Equ., (2018), 57–161. doi: 10.1007/s00526-018-1428-5.  Google Scholar [15] G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific, 1996. doi: 10.1142/3302.  Google Scholar [16] J. Rauch and B. A. Taylor, The Dirichlet problem for the multidimensional Monge-Ampère equation, Rocky Mountain J. Math., 7 (1977), 345-364. doi: 10.1216/rmj-1977-7-2-345.  Google Scholar [17] R. Wang and G. Wang, On existence, uniqueness and regularity of viscosity solutions for the first initial boundary value problems to parabolic Monge-Ampère equation, Northeast. Math. J., 8 (1992), 417-446.   Google Scholar [18] 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.   Google Scholar [19] R. Wang and G. Wang, On another kind of parabolic Monge-Ampère equation: the existence, uniqueness and regularity of the viscosity solution, Northeast. Math. J., 10 (1994), 434-454.  doi: 10.13447/j.1674-5647.1994.04.002.  Google Scholar [20] Y. Zhan, Viscosity solutions of nonlinear degenerate parabolic equations and several applications, Ph.D thesis, University of Toronto (Canada), ProQuest LLC, Ann Arbor, MI, 2000.  Google Scholar

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##### References:
 [1] J. Bao, H. Li and L. Zhang, Monge-Ampère equation on exterior domains, Calc. Var. Partial Differ. Equ., 52 (2015), 39-63.  doi: 10.1007/s00526-013-0704-7.  Google Scholar [2] J. Bao, J. Xiong and Z. Zhou, Existence of entire solutions of Monge-Ampère equations with prescribed asymptotic behaviors, Calc. Var. Partial Differ. Equ., 58 (2019), Art. 193, 12 pp. doi: 10.1007/s00526-019-1639-4.  Google Scholar [3] L. Caffarelli, Interior $W^{2, p}$ estimates for solutions of the Monge-Ampère equation, Ann. of Math., (2) 131 (1990), no. 1, 135-150.  Google Scholar [4] L. Caffarelli, L. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations. I. Monge-Ampère equation, Commun. Pure Appl. Math., 37 (1984), 369-402.  doi: 10.1002/cpa.3160370306.  Google Scholar [5] L. Caffarelli and Y. Li, An extension to a theorem of J$\ddot{o}$rgens, Calabi, and Pogorelov, Commun. Pure Appl. Math., 56 (2003), 549–583. doi: 10.1002/cpa.10067.  Google Scholar [6] K. Chou and X. Wang, A logarithmic Gauss curvature flow and the Minkowski problem, Ann. Inst. Henri Poincare Anal. Non Lineaire, 17 (2001), 733-751.  doi: 10.1016/S0294-1449(00)00053-6.  Google Scholar [7] 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.  Google Scholar [8] L. Dai, Exterior problems for a parabolic Monge-Ampère equation, Nonlinear Anal. Theory Meth. Appl., 100 (2014), 99-110.  doi: 10.1016/j.na.2014.01.011.  Google Scholar [9] L. Ferrer, A. Martínez and F. Mil$\rm\acute{a}$n, An extension of a theorem by K. J$\ddot{o}$rgens and a maximum principle at infinity for parabolic affine spheres, Math. Z., 230 (1999), 471-486.  doi: 10.1007/PL00004700.  Google Scholar [10] L. Ferrer, A. Martínez and F. Mil$\rm\acute{a}$n, The space of parabolic affine spheres with fixed compact boundary, Monatsh. Math., 130 (2000), 19-27.  doi: 10.1007/s006050050084.  Google Scholar [11] H. Ishii and P.L. Lions, Viscosity solutions of fully nonlinear second-order elliptic partial differential equations, J. Partial Differ. Equ., 83 (1990), 26-78.  Google Scholar [12] N. M. Ivochkina and O. A. Ladyzhenskaya, On parabolic equations generated by symmetric functions of the principal curvatures of the evolving surfacing, or of the eigenvalues of Hessian, part Ⅰ: parabolic Monge-Ampère equations, St. Petersburg Math. J., 6 (1994), 527-594.   Google Scholar [13] T. Jin and J. Xiong, Solutions of some Monge-Ampère equations with isolated and line singularities, Adv. Math, 289 (2016), 114-141.  doi: 10.1016/j.aim.2015.11.029.  Google Scholar [14] Y. Li and S. Lu, Existence and nonexistence to exterior Dirichlet problem for Monge Ampère equation, Calc. Var. Partial Differ. Equ., (2018), 57–161. doi: 10.1007/s00526-018-1428-5.  Google Scholar [15] G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific, 1996. doi: 10.1142/3302.  Google Scholar [16] J. Rauch and B. A. Taylor, The Dirichlet problem for the multidimensional Monge-Ampère equation, Rocky Mountain J. Math., 7 (1977), 345-364. doi: 10.1216/rmj-1977-7-2-345.  Google Scholar [17] R. Wang and G. Wang, On existence, uniqueness and regularity of viscosity solutions for the first initial boundary value problems to parabolic Monge-Ampère equation, Northeast. Math. J., 8 (1992), 417-446.   Google Scholar [18] 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.   Google Scholar [19] R. Wang and G. Wang, On another kind of parabolic Monge-Ampère equation: the existence, uniqueness and regularity of the viscosity solution, Northeast. Math. J., 10 (1994), 434-454.  doi: 10.13447/j.1674-5647.1994.04.002.  Google Scholar [20] Y. Zhan, Viscosity solutions of nonlinear degenerate parabolic equations and several applications, Ph.D thesis, University of Toronto (Canada), ProQuest LLC, Ann Arbor, MI, 2000.  Google Scholar
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