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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

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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

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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

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

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

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

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

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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

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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

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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

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