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:
[1]

L. Caffarelli, Certain multiple valued harmonic functions,, \emph{Proc. Amer. Math. Soc.}, 54 (1976), 90.

[2]

L. Caffarelli, On the Hölder continuity of multiple valued harmonic functions,, \emph{Indiana Univ. Math. J.}, 25 (1976), 79.

[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,, \emph{Comm. Anal. Geom.}, 14 (2006), 411.

[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.

[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.

[7]

G. C. Evans, A necessary and sufficient condition of Wiener,, \emph{Amer. Math. Monthly}, 54 (1947), 151.

[8]

G. C. Evans, Surfaces of minimal capacity,, \emph{Proc. Nat. Acad. Sci. U. S. A.}, 26 (1940), 489.

[9]

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

[10]

W. J. Firey, Shapes of worn stones,, \emph{Mathematika}, 21 (1974), 1.

[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.

[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.

[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.

[14]

G. Levi, Generalization of a spatial angle theorem,, (Russian) \emph{Translated from the English by Ju. V. Egorov, 26 (1971), 199.

[15]

G. M. Lieberman, Second Order Parabolic Differential Equations,, World Scientific Publishing Co., (1996).

[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.

[17]

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

[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.

[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.

[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.

[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.

[22]

Y. Zhan, Viscosity Solutions of Nonlinear Degenerate Parabolic Equations and Several Applications,, Ph.D thesis, (2000).

show all references

References:
[1]

L. Caffarelli, Certain multiple valued harmonic functions,, \emph{Proc. Amer. Math. Soc.}, 54 (1976), 90.

[2]

L. Caffarelli, On the Hölder continuity of multiple valued harmonic functions,, \emph{Indiana Univ. Math. J.}, 25 (1976), 79.

[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,, \emph{Comm. Anal. Geom.}, 14 (2006), 411.

[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.

[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.

[7]

G. C. Evans, A necessary and sufficient condition of Wiener,, \emph{Amer. Math. Monthly}, 54 (1947), 151.

[8]

G. C. Evans, Surfaces of minimal capacity,, \emph{Proc. Nat. Acad. Sci. U. S. A.}, 26 (1940), 489.

[9]

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

[10]

W. J. Firey, Shapes of worn stones,, \emph{Mathematika}, 21 (1974), 1.

[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.

[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.

[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.

[14]

G. Levi, Generalization of a spatial angle theorem,, (Russian) \emph{Translated from the English by Ju. V. Egorov, 26 (1971), 199.

[15]

G. M. Lieberman, Second Order Parabolic Differential Equations,, World Scientific Publishing Co., (1996).

[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.

[17]

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

[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.

[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.

[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.

[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.

[22]

Y. Zhan, Viscosity Solutions of Nonlinear Degenerate Parabolic Equations and Several Applications,, Ph.D thesis, (2000).

[1]

Alessio Figalli, Young-Heon Kim. Partial regularity of Brenier solutions of the Monge-Ampère equation. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 559-565. doi: 10.3934/dcds.2010.28.559

[2]

Shouchuan Hu, Haiyan Wang. Convex solutions of boundary value problem arising from Monge-Ampère equations. Discrete & Continuous Dynamical Systems - A, 2006, 16 (3) : 705-720. doi: 10.3934/dcds.2006.16.705

[3]

Jiakun Liu, Neil S. Trudinger. On Pogorelov estimates for Monge-Ampère type equations. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 1121-1135. doi: 10.3934/dcds.2010.28.1121

[4]

Barbara Brandolini, Carlo Nitsch, Cristina Trombetti. Shape optimization for Monge-Ampère equations via domain derivative. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 825-831. doi: 10.3934/dcdss.2011.4.825

[5]

Qi-Rui Li, Xu-Jia Wang. Regularity of the homogeneous Monge-Ampère equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 6069-6084. doi: 10.3934/dcds.2015.35.6069

[6]

Jingang Xiong, Jiguang Bao. The obstacle problem for Monge-Ampère type equations in non-convex domains. Communications on Pure & Applied Analysis, 2011, 10 (1) : 59-68. doi: 10.3934/cpaa.2011.10.59

[7]

Cristian Enache. Maximum and minimum principles for a class of Monge-Ampère equations in the plane, with applications to surfaces of constant Gauss curvature. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1347-1359. doi: 10.3934/cpaa.2014.13.1347

[8]

Bo Guan, Qun Li. A Monge-Ampère type fully nonlinear equation on Hermitian manifolds. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1991-1999. doi: 10.3934/dcdsb.2012.17.1991

[9]

Haitao Yang, Yibin Chang. On the blow-up boundary solutions of the Monge -Ampére equation with singular weights. Communications on Pure & Applied Analysis, 2012, 11 (2) : 697-708. doi: 10.3934/cpaa.2012.11.697

[10]

Adam M. Oberman. Wide stencil finite difference schemes for the elliptic Monge-Ampère equation and functions of the eigenvalues of the Hessian. Discrete & Continuous Dynamical Systems - B, 2008, 10 (1) : 221-238. doi: 10.3934/dcdsb.2008.10.221

[11]

Diego Maldonado. On interior \begin{document} $C^2$ \end{document}-estimates for the Monge-Ampère equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1427-1440. doi: 10.3934/dcds.2018058

[12]

Siegfried Carl, Christoph Tietz. Quasilinear elliptic equations with measures and multi-valued lower order terms. Discrete & Continuous Dynamical Systems - S, 2018, 11 (2) : 193-212. doi: 10.3934/dcdss.2018012

[13]

Yejuan Wang, Peter E. Kloeden. The uniform attractor of a multi-valued process generated by reaction-diffusion delay equations on an unbounded domain. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4343-4370. doi: 10.3934/dcds.2014.34.4343

[14]

Francesca Da Lio. Remarks on the strong maximum principle for viscosity solutions to fully nonlinear parabolic equations. Communications on Pure & Applied Analysis, 2004, 3 (3) : 395-415. doi: 10.3934/cpaa.2004.3.395

[15]

Gregorio Díaz, Jesús Ildefonso Díaz. On the free boundary associated with the stationary Monge--Ampère operator on the set of non strictly convex functions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1447-1468. doi: 10.3934/dcds.2015.35.1447

[16]

Maria Michaela Porzio. Existence of solutions for some "noncoercive" parabolic equations . Discrete & Continuous Dynamical Systems - A, 1999, 5 (3) : 553-568. doi: 10.3934/dcds.1999.5.553

[17]

Mariane Bourgoing. Viscosity solutions of fully nonlinear second order parabolic equations with $L^1$ dependence in time and Neumann boundary conditions. Existence and applications to the level-set approach. Discrete & Continuous Dynamical Systems - A, 2008, 21 (4) : 1047-1069. doi: 10.3934/dcds.2008.21.1047

[18]

Mariane Bourgoing. Viscosity solutions of fully nonlinear second order parabolic equations with $L^1$ dependence in time and Neumann boundary conditions. Discrete & Continuous Dynamical Systems - A, 2008, 21 (3) : 763-800. doi: 10.3934/dcds.2008.21.763

[19]

Thierry Cazenave, Flávio Dickstein, Fred B. Weissler. Multi-scale multi-profile global solutions of parabolic equations in $\mathbb{R}^N $. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 449-472. doi: 10.3934/dcdss.2012.5.449

[20]

Ting Li. Pullback attractors for asymptotically upper semicompact non-autonomous multi-valued semiflows. Communications on Pure & Applied Analysis, 2007, 6 (1) : 279-285. doi: 10.3934/cpaa.2007.6.279

2016 Impact Factor: 0.801

Metrics

  • PDF downloads (0)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]