American Institute of Mathematical Sciences

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