American Institute of Mathematical Sciences

Advanced
May  2014, 13(3): 1061-1074. doi: 10.3934/cpaa.2014.13.1061

Multi-valued solutions to a class of parabolic Monge-Ampère equations

Limei Dai 1,

1. 

School of Mathematics and information Science, Weifang University, Shandong Weifang, 261061, China

Received  March 2013 Revised  September 2013 Published  December 2013

In this paper, we investigate the multi-valued solutions of a class of parabolic Monge-Ampère equation $-u_{t}\det(D^{2}u)=f$. Using the Perron method, we obtain the existence of finitely valued and infinitely valued solutions to the parabolic Monge-Ampère equations. We generalize the results of elliptic Monge-Ampère equations and Hessian equations.
Keywords: Perron method, multi-valued solutions, Parabolic Monge-Ampère equations, viscosity solutions, existence..
Mathematics Subject Classification: Primary: 35K96, 35D4.
Citation: Limei Dai. Multi-valued solutions to a class of parabolic Monge-Ampère equations. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1061-1074. doi: 10.3934/cpaa.2014.13.1061
References:
[1]

L. Caffarelli, Certain multiple valued harmonic functions, Proc. Amer. Math. Soc., 54 (1976), 90-92.  Google Scholar

[2]

L. Caffarelli, On the Hölder continuity of multiple valued harmonic functions, Indiana Univ. Math. J., 25 (1976), 79-84.  Google Scholar

[3]

L. Caffarelli, Monge-Ampère equation, Div-Curl theorems in Lagrangian coordinates, Compression and Rotation,, lecture notes., ().   Google Scholar

[4]

L. Caffarelli and Y. Y. Li, Some multi-valued solutions to Monge-Ampère equations, Comm. Anal. Geom., 14 (2006), 411-441.  Google Scholar

[5]

L. M. Dai and J. G. Bao, Multi-valued solutions to Hessian equations, Nonlinear Differential Equations Appl., 18 (2011), 447-457. doi: 10.1007/s00030-011-0103-8.  Google Scholar

[6]

L. M. Dai, Multi-valued solutions to Hessian quotient equations, Commun. Math. Sci., 10 (2012), 717-733. doi: 10.4310/CMS.2012.v10.n2.a14.  Google Scholar

[7]

G. C. Evans, A necessary and sufficient condition of Wiener, Amer. Math. Monthly, 54 (1947), 151-155.  Google Scholar

[8]

G. C. Evans, Surfaces of minimal capacity, Proc. Nat. Acad. Sci. U. S. A., 26 (1940), 489-491.  Google Scholar

[9]

G. C. Evans, Lectures on multiple valued harmonic functions in space, Univ. California Publ. Math. (N.S.), 1 (1951), 281-340.  Google Scholar

[10]

W. J. Firey, Shapes of worn stones, Mathematika, 21 (1974), 1-11.  Google Scholar

[11]

C. E. Gutiérrez and Q. B. Huang, $W^{2,p}$ estimates for the parabolic Monge-Ampère equation, Arch. Ration. Mech. Anal., 159 (2001), 137-177. doi: 10.1007/s002050100151.  Google Scholar

[12]

C. E. Gutiérrez and Q. B. 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.  Google Scholar

[13]

C. E. Gutiérrez and Q. B. Huang, Geometric properties of the sections of solutions to the Monge-Ampère equation, Trans. Amer. Math. Soc., 352 (2000), 4381-4396. doi: 10.1090/S0002-9947-00-02491-0.  Google Scholar

[14]

G. Levi, Generalization of a spatial angle theorem, (Russian) Translated from the English by Ju. V. Egorov, Uspekhi Mat. Nauk, 26 (1971), 199-204.  Google Scholar

[15]

G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996.  Google Scholar

[16]

N. V. Krylov, Sequences of convex functions, and estimates of the maximum of the solution of a parabolic equation, (Russian) Sibirsk. Mat. Ž, 17 (1976), 290-303.  Google Scholar

[17]

K. Tso, Deforming a hypersurface by its Gauss-Kronecker curvature, Comm. Pure Appl. Math., 38 (1985), 867-882. doi: 10.1002/cpa.3160380615.  Google Scholar

[18]

K. Tso, On an Aleksandrov-Bakelman type maximum principle for second-order parabolic equations, Comm. Partial Differential Equations, 10 (1985), 543-553. doi: 10.1080/03605308508820388.  Google Scholar

[19]

R. H. Wang and G. L. Wang, On the existence, uniqueness and regularity of viscosity solution for the first initial boundary value problem to parabolic Monge-Ampère equations, Northeast Math. J., 8 (1992), 417-446.  Google Scholar

[20]

R. H. Wang and G. L. Wang, The geometric measure theoretical characterization of viscosity solutions to parabolic Monge-Ampère type equation, J. Partial Differential Equations, 6 (1993), 237-254.  Google Scholar

[21]

J. G. Xiong and J. G. Bao, On Jörgens, Calabi, and Pogorelov type theorem and isolated singularities of parabolic Monge-Ampère equations, J. Differential Equations, 250 (2011), 367-385. doi: 10.1016/j.jde.2010.08.024.  Google Scholar

[22]

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

show all references

References:
[1]

L. Caffarelli, Certain multiple valued harmonic functions, Proc. Amer. Math. Soc., 54 (1976), 90-92.  Google Scholar

[2]

L. Caffarelli, On the Hölder continuity of multiple valued harmonic functions, Indiana Univ. Math. J., 25 (1976), 79-84.  Google Scholar

[3]

L. Caffarelli, Monge-Ampère equation, Div-Curl theorems in Lagrangian coordinates, Compression and Rotation,, lecture notes., ().   Google Scholar

[4]

L. Caffarelli and Y. Y. Li, Some multi-valued solutions to Monge-Ampère equations, Comm. Anal. Geom., 14 (2006), 411-441.  Google Scholar

[5]

L. M. Dai and J. G. Bao, Multi-valued solutions to Hessian equations, Nonlinear Differential Equations Appl., 18 (2011), 447-457. doi: 10.1007/s00030-011-0103-8.  Google Scholar

[6]

L. M. Dai, Multi-valued solutions to Hessian quotient equations, Commun. Math. Sci., 10 (2012), 717-733. doi: 10.4310/CMS.2012.v10.n2.a14.  Google Scholar

[7]

G. C. Evans, A necessary and sufficient condition of Wiener, Amer. Math. Monthly, 54 (1947), 151-155.  Google Scholar

[8]

G. C. Evans, Surfaces of minimal capacity, Proc. Nat. Acad. Sci. U. S. A., 26 (1940), 489-491.  Google Scholar

[9]

G. C. Evans, Lectures on multiple valued harmonic functions in space, Univ. California Publ. Math. (N.S.), 1 (1951), 281-340.  Google Scholar

[10]

W. J. Firey, Shapes of worn stones, Mathematika, 21 (1974), 1-11.  Google Scholar

[11]

C. E. Gutiérrez and Q. B. Huang, $W^{2,p}$ estimates for the parabolic Monge-Ampère equation, Arch. Ration. Mech. Anal., 159 (2001), 137-177. doi: 10.1007/s002050100151.  Google Scholar

[12]

C. E. Gutiérrez and Q. B. 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.  Google Scholar

[13]

C. E. Gutiérrez and Q. B. Huang, Geometric properties of the sections of solutions to the Monge-Ampère equation, Trans. Amer. Math. Soc., 352 (2000), 4381-4396. doi: 10.1090/S0002-9947-00-02491-0.  Google Scholar

[14]

G. Levi, Generalization of a spatial angle theorem, (Russian) Translated from the English by Ju. V. Egorov, Uspekhi Mat. Nauk, 26 (1971), 199-204.  Google Scholar

[15]

G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996.  Google Scholar

[16]

N. V. Krylov, Sequences of convex functions, and estimates of the maximum of the solution of a parabolic equation, (Russian) Sibirsk. Mat. Ž, 17 (1976), 290-303.  Google Scholar

[17]

K. Tso, Deforming a hypersurface by its Gauss-Kronecker curvature, Comm. Pure Appl. Math., 38 (1985), 867-882. doi: 10.1002/cpa.3160380615.  Google Scholar

[18]

K. Tso, On an Aleksandrov-Bakelman type maximum principle for second-order parabolic equations, Comm. Partial Differential Equations, 10 (1985), 543-553. doi: 10.1080/03605308508820388.  Google Scholar

[19]

R. H. Wang and G. L. Wang, On the existence, uniqueness and regularity of viscosity solution for the first initial boundary value problem to parabolic Monge-Ampère equations, Northeast Math. J., 8 (1992), 417-446.  Google Scholar

[20]

R. H. Wang and G. L. Wang, The geometric measure theoretical characterization of viscosity solutions to parabolic Monge-Ampère type equation, J. Partial Differential Equations, 6 (1993), 237-254.  Google Scholar

[21]

J. G. Xiong and J. G. Bao, On Jörgens, Calabi, and Pogorelov type theorem and isolated singularities of parabolic Monge-Ampère equations, J. Differential Equations, 250 (2011), 367-385. doi: 10.1016/j.jde.2010.08.024.  Google Scholar

[22]

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

[1]

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
[2]

Fan Cui, Huaiyu Jian. Symmetry of solutions to a class of Monge-Ampère equations. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1247-1259. doi: 10.3934/cpaa.2019060
[3]

Yahui Niu. Monotonicity of solutions for a class of nonlocal Monge-Ampère problem. Communications on Pure & Applied Analysis, 2020, 19 (11) : 5269-5283. doi: 10.3934/cpaa.2020237
[4]

Luca Codenotti, Marta Lewicka. Visualization of the convex integration solutions to the Monge-Ampère equation. Evolution Equations & Control Theory, 2019, 8 (2) : 273-300. doi: 10.3934/eect.2019015
[5]

Alessio Figalli, Young-Heon Kim. Partial regularity of Brenier solutions of the Monge-Ampère equation. Discrete & Continuous Dynamical Systems, 2010, 28 (2) : 559-565. doi: 10.3934/dcds.2010.28.559
[6]

Shouchuan Hu, Haiyan Wang. Convex solutions of boundary value problem arising from Monge-Ampère equations. Discrete & Continuous Dynamical Systems, 2006, 16 (3) : 705-720. doi: 10.3934/dcds.2006.16.705
[7]

Ziwei Zhou, Jiguang Bao, Bo Wang. A Liouville theorem of parabolic Monge-AmpÈre equations in half-space. Discrete & Continuous Dynamical Systems, 2021, 41 (4) : 1561-1578. doi: 10.3934/dcds.2020331
[8]

Barbara Brandolini, Carlo Nitsch, Cristina Trombetti. Shape optimization for Monge-Ampère equations via domain derivative. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 825-831. doi: 10.3934/dcdss.2011.4.825
[9]

Limei Dai, Hongyu Li. Entire subsolutions of Monge-Ampère type equations. Communications on Pure & Applied Analysis, 2020, 19 (1) : 19-30. doi: 10.3934/cpaa.2020002
[10]

Jiakun Liu, Neil S. Trudinger. On Pogorelov estimates for Monge-Ampère type equations. Discrete & Continuous Dynamical Systems, 2010, 28 (3) : 1121-1135. doi: 10.3934/dcds.2010.28.1121
[11]

Juhua Shi, Feida Jiang. The degenerate Monge-Ampère equations with the Neumann condition. Communications on Pure & Applied Analysis, 2021, 20 (2) : 915-931. doi: 10.3934/cpaa.2020297
[12]

Qi-Rui Li, Xu-Jia Wang. Regularity of the homogeneous Monge-Ampère equation. Discrete & Continuous Dynamical Systems, 2015, 35 (12) : 6069-6084. doi: 10.3934/dcds.2015.35.6069
[13]

Mengni Li. Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions. Communications on Pure & Applied Analysis, 2021, 20 (1) : 301-317. doi: 10.3934/cpaa.2020267
[14]

Jingang Xiong, Jiguang Bao. The obstacle problem for Monge-Ampère type equations in non-convex domains. Communications on Pure & Applied Analysis, 2011, 10 (1) : 59-68. doi: 10.3934/cpaa.2011.10.59
[15]

Cristian Enache. Maximum and minimum principles for a class of Monge-Ampère equations in the plane, with applications to surfaces of constant Gauss curvature. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1347-1359. doi: 10.3934/cpaa.2014.13.1347
[16]

Bo Guan, Qun Li. A Monge-Ampère type fully nonlinear equation on Hermitian manifolds. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1991-1999. doi: 10.3934/dcdsb.2012.17.1991
[17]

Haitao Yang, Yibin Chang. On the blow-up boundary solutions of the Monge -Ampére equation with singular weights. Communications on Pure & Applied Analysis, 2012, 11 (2) : 697-708. doi: 10.3934/cpaa.2012.11.697
[18]

Adam M. Oberman. Wide stencil finite difference schemes for the elliptic Monge-Ampère equation and functions of the eigenvalues of the Hessian. Discrete & Continuous Dynamical Systems - B, 2008, 10 (1) : 221-238. doi: 10.3934/dcdsb.2008.10.221
[19]

Zhijun Zhang. Optimal global asymptotic behavior of the solution to a singular monge-ampère equation. Communications on Pure & Applied Analysis, 2020, 19 (2) : 1129-1145. doi: 10.3934/cpaa.2020053
[20]

Diego Maldonado. On interior $C^2$-estimates for the Monge-Ampère equation. Discrete & Continuous Dynamical Systems, 2018, 38 (3) : 1427-1440. doi: 10.3934/dcds.2018058

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (46)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences