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

References:

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[2]	L. Caffarelli, On the Hölder continuity of multiple valued harmonic functions,, \emph{Indiana Univ. Math. J.}, 25 (1976), 79. 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,, \emph{Comm. Anal. Geom.}, 14 (2006), 411. Google Scholar
[5]	L. M. Dai and J. G. Bao, Multi-valued solutions to Hessian equations,, \emph{Nonlinear Differential Equations Appl.}, 18 (2011), 447. doi: 10.1007/s00030-011-0103-8. Google Scholar
[6]	L. M. Dai, Multi-valued solutions to Hessian quotient equations,, \emph{Commun. Math. Sci.}, 10 (2012), 717. doi: 10.4310/CMS.2012.v10.n2.a14. Google Scholar
[7]	G. C. Evans, A necessary and sufficient condition of Wiener,, \emph{Amer. Math. Monthly}, 54 (1947), 151. Google Scholar
[8]	G. C. Evans, Surfaces of minimal capacity,, \emph{Proc. Nat. Acad. Sci. U. S. A.}, 26 (1940), 489. Google Scholar
[9]	G. C. Evans, Lectures on multiple valued harmonic functions in space,, \emph{Univ. California Publ. Math. (N.S.)}, 1 (1951), 281. Google Scholar
[10]	W. J. Firey, Shapes of worn stones,, \emph{Mathematika}, 21 (1974), 1. Google Scholar
[11]	C. E. Gutiérrez and Q. B. Huang, $W^{2,p}$ estimates for the parabolic Monge-Ampère equation,, \emph{Arch. Ration. Mech. Anal.}, 159 (2001), 137. 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,, \emph{Indiana Univ. Math. J.}, 47 (1998), 1459. 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,, \emph{Trans. Amer. Math. Soc.}, 352 (2000), 4381. doi: 10.1090/S0002-9947-00-02491-0. Google Scholar
[14]	G. Levi, Generalization of a spatial angle theorem,, (Russian) \emph{Translated from the English by Ju. V. Egorov, 26 (1971), 199. Google Scholar
[15]	G. M. Lieberman, Second Order Parabolic Differential Equations,, World Scientific Publishing Co., (1996). Google Scholar
[16]	N. V. Krylov, Sequences of convex functions, and estimates of the maximum of the solution of a parabolic equation,, (Russian) \emph{Sibirsk. Mat. $\breveZ$}, 17 (1976), 290. Google Scholar
[17]	K. Tso, Deforming a hypersurface by its Gauss-Kronecker curvature,, \emph{Comm. Pure Appl. Math.}, 38 (1985), 867. doi: 10.1002/cpa.3160380615. Google Scholar
[18]	K. Tso, On an Aleksandrov-Bakelman type maximum principle for second-order parabolic equations,, \emph{Comm. Partial Differential Equations}, 10 (1985), 543. 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,, \emph{Northeast Math. J.}, 8 (1992), 417. Google Scholar
[20]	R. H. Wang and G. L. Wang, The geometric measure theoretical characterization of viscosity solutions to parabolic Monge-Ampère type equation,, \emph{J. Partial Differential Equations}, 6 (1993), 237. 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,, \emph{J. Differential Equations}, 250 (2011), 367. 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, (2000). Google Scholar

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References:

[1]	L. Caffarelli, Certain multiple valued harmonic functions,, \emph{Proc. Amer. Math. Soc.}, 54 (1976), 90. Google Scholar
[2]	L. Caffarelli, On the Hölder continuity of multiple valued harmonic functions,, \emph{Indiana Univ. Math. J.}, 25 (1976), 79. 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,, \emph{Comm. Anal. Geom.}, 14 (2006), 411. Google Scholar
[5]	L. M. Dai and J. G. Bao, Multi-valued solutions to Hessian equations,, \emph{Nonlinear Differential Equations Appl.}, 18 (2011), 447. doi: 10.1007/s00030-011-0103-8. Google Scholar
[6]	L. M. Dai, Multi-valued solutions to Hessian quotient equations,, \emph{Commun. Math. Sci.}, 10 (2012), 717. doi: 10.4310/CMS.2012.v10.n2.a14. Google Scholar
[7]	G. C. Evans, A necessary and sufficient condition of Wiener,, \emph{Amer. Math. Monthly}, 54 (1947), 151. Google Scholar
[8]	G. C. Evans, Surfaces of minimal capacity,, \emph{Proc. Nat. Acad. Sci. U. S. A.}, 26 (1940), 489. Google Scholar
[9]	G. C. Evans, Lectures on multiple valued harmonic functions in space,, \emph{Univ. California Publ. Math. (N.S.)}, 1 (1951), 281. Google Scholar
[10]	W. J. Firey, Shapes of worn stones,, \emph{Mathematika}, 21 (1974), 1. Google Scholar
[11]	C. E. Gutiérrez and Q. B. Huang, $W^{2,p}$ estimates for the parabolic Monge-Ampère equation,, \emph{Arch. Ration. Mech. Anal.}, 159 (2001), 137. 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,, \emph{Indiana Univ. Math. J.}, 47 (1998), 1459. 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,, \emph{Trans. Amer. Math. Soc.}, 352 (2000), 4381. doi: 10.1090/S0002-9947-00-02491-0. Google Scholar
[14]	G. Levi, Generalization of a spatial angle theorem,, (Russian) \emph{Translated from the English by Ju. V. Egorov, 26 (1971), 199. Google Scholar
[15]	G. M. Lieberman, Second Order Parabolic Differential Equations,, World Scientific Publishing Co., (1996). Google Scholar
[16]	N. V. Krylov, Sequences of convex functions, and estimates of the maximum of the solution of a parabolic equation,, (Russian) \emph{Sibirsk. Mat. $\breveZ$}, 17 (1976), 290. Google Scholar
[17]	K. Tso, Deforming a hypersurface by its Gauss-Kronecker curvature,, \emph{Comm. Pure Appl. Math.}, 38 (1985), 867. doi: 10.1002/cpa.3160380615. Google Scholar
[18]	K. Tso, On an Aleksandrov-Bakelman type maximum principle for second-order parabolic equations,, \emph{Comm. Partial Differential Equations}, 10 (1985), 543. 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,, \emph{Northeast Math. J.}, 8 (1992), 417. Google Scholar
[20]	R. H. Wang and G. L. Wang, The geometric measure theoretical characterization of viscosity solutions to parabolic Monge-Ampère type equation,, \emph{J. Partial Differential Equations}, 6 (1993), 237. 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,, \emph{J. Differential Equations}, 250 (2011), 367. 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, (2000). Google Scholar

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