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The Initial and Neumann Boundary Value Problem for A Class Parabolic Monge–Ampère Equation

  • Juan Wang
  • Huizhao Liu
  • Jinlin Yang
Conference paper
Part of the Lecture Notes in Electrical Engineering book series (LNEE, volume 113)

Abstract

Monge–Ampère equation is a typical fully nonlinear non-uniformly equation. The study of MA is motivated by the following two problems: Minkowski problem and Weyl problem. We consider the existence and uniqueness of a classical solution to the initial and Neumann boundary value problem for a class nonlinear parabolic equation of Monge–Ampère type. We show that such a solution exists for all times and is unique.

Keywords

Parabolic Monge–Ampère equation Neumann Boundary value 

References

  1. 1.
    Schnurer OC, Smoczyk K (2003) Neumann and second boundary value problems for Hessian and Gauss Curvature flows. Ann I H Poincar′e-AN 20(6):1043–1073MathSciNetGoogle Scholar
  2. 2.
    Lions PL, Trudinger NS, Urbas JIE (1986) The Neumann problem for equations of Monge–Amp`ere type. Comm Pure Appl Math 39:539–563CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Lions PL, Sznitman AS (1984) Stochastic differential equations with reflecting boundary conditions. Commun Pur Appl Math 37(4):511–537CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Alvino A, Trombetti G, Diaz JI, Lions PL (1996) Elliptic equations and Steiner symmetrization. Commun Pur Appl Math 49(3):217–236CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Chen YZ (1986) Krylov’s a priori estimates methods on fully nonlinear equation. Adv Math(China) 15(1):63–101MATHGoogle Scholar
  6. 6.
    Dong GC (1998) Initial and nonlinear oblique boundary value problem for fully nonlinear partial equations. J PDE 2 Series A 1:12–42Google Scholar
  7. 7.
    Dong GC (1994) Elliptic and parabolic equations partial differential equations in China mathematics and its applications. Kluwer, Dordrecht, pp 30–41Google Scholar
  8. 8.
    Dong GC, Mu CL (1998) The measurable viscosity solutions for fully nonlinear elliptic equations. Nonlinear Anal 33(4):401–412CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Ladyzenskaja A, Solonnikov VA, Ural’zeva NN (1967) Linear and quasilinear equations of parabolic type. (Russian) Translated from the Russian by S.Smith.Translations of mathematical monographs. vol 23 American Mathematical Society, Providence, R.I. xi + 648Google Scholar
  10. 10.
    Temam R (1982) Behaviour at time t = 0 of the solutions of semilinear evolution equations. J Differ Equ 43(1):73–92CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Thomee V, Wahlbin L (1974) Convergence rates of parabolic difference schemes for non-smooth data. Math Comp 28:1–13CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Bahri A, Lions PL (1988) Morse index of some min–max critical points I Application to multiplicity results. Commun Pur Appl Math 41(8):1027–1037CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.School of Mathematics,Physics and Biological EngineeringInner Mongolia University of Science and TechnologyBaotouChina
  2. 2.Mathematical Institute HebeiUniversity of TechnologyTianjinChina

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