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

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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

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

Received March 2013 Revised September 2013 Published December 2013

Abstract
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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 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.

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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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