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

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

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.
Citation: Limei Dai. Multi-valued solutions to a class of parabolic Monge-Ampère equations. Communications on Pure and 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.

[2]

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

[3]

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

[4]

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

[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.

[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.

[7]

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

[8]

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

[9]

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

[10]

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

[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.

[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.

[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.

[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.

[15]

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

[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.

[17]

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

[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.

[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.

[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.

[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.

[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.

show all references

References:
[1]

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

[2]

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

[3]

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

[4]

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

[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.

[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.

[7]

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

[8]

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

[9]

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

[10]

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

[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.

[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.

[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.

[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.

[15]

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

[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.

[17]

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

[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.

[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.

[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.

[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.

[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.

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