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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 535629, 8 pages
http://dx.doi.org/10.1155/2013/535629
Research Article

The Initial and Neumann Boundary Value Problem for a Class Parabolic Monge-Ampère Equation

1School of Mathematics, Physics and Biological Engineering, Inner Mongolia University of Science and Technology, Baotou 014010, China
2Department of Applied Mathematics, University of Waterloo, Waterloo, ON, Canada N2L 3G1

Received 26 March 2013; Accepted 11 June 2013

Academic Editor: Sergey Piskarev

Copyright © 2013 Juan Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. J.-F. Cossette and P. K. Smolarkiewicz, “A Monge-Ampère enhancement for semi-Lagrangian methods,” Computers & Fluids, vol. 46, pp. 180–185, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  2. Z. Liu and Y. He, “Solving the elliptic Monge-Ampère equation by Kansa's method,” Engineering Analysis with Boundary Elements, vol. 37, no. 1, pp. 84–88, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  3. J. X. Hong, “The global smooth solutions of Cauchy problems for hyperbolic equation of Monge-Ampère type,” Nonlinear Analysis: Theory, Methods & Applications, vol. 24, no. 12, pp. 1649–1663, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. E. J. Dean and R. Glowinski, “Numerical methods for fully nonlinear elliptic equations of the Monge-Ampère type,” Computer Methods in Applied Mechanics and Engineering, vol. 195, no. 13-16, pp. 1344–1386, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. C. E. Gutiérrez, The Monge-Ampère Equation, Birkhäuser, Basel, Switzerland, 2011.
  6. O. C. Schnürer and K. Smoczyk, “Neumann and second boundary value problems for Hessian and Gauss curvature flows,” Annales de l'Institut Henri Poincaré. Analyse Non Linéaire, vol. 20, no. 6, pp. 1043–1073, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. P.-L. Lions, N. S. Trudinger, and J. I. E. Urbas, “The Neumann problem for equations of Monge-Ampère type,” Communications on Pure and Applied Mathematics, vol. 39, no. 4, pp. 539–563, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. J. Urbas, “The second boundary value problem for a class of Hessian equations,” Communications in Partial Differential Equations, vol. 26, no. 5-6, pp. 859–882, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. J. Urbas, “Oblique boundary value problems for equations of Monge-Ampère type,” Calculus of Variations and Partial Differential Equations, vol. 7, no. 1, pp. 19–39, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. J. Urbas, “On the second boundary value problem for equations of Monge-Ampère type,” Journal für die Reine und Angewandte Mathematik, vol. 487, pp. 115–124, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. W. S. Zhou and S. Z. Lian, “The third initial-boundary value problem for an equation of parabolic Monge-Ampère type,” Journal of Jilin University, no. 1, pp. 23–30, 2001. View at Zentralblatt MATH · View at MathSciNet
  12. J. C. Kuang, Applied Inequalities, Shandong Science and Technology Press, 3rd edition, 2004.
  13. L. C. Evans, Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, USA, 1998. View at MathSciNet
  14. Y. Z. Chen, “Some methods of Krylov for the a priori estimation of solutions of fully nonlinear equations,” Advances in Mathematics, vol. 15, no. 1, pp. 63–101, 1986. View at MathSciNet
  15. G. C. Dong, “Initial and nonlinear oblique boundary value problems for fully nonlinear parabolic equations,” Journal of Partial Differential Equations. Series A, vol. 1, no. 2, pp. 12–42, 1988. View at Zentralblatt MATH · View at MathSciNet
  16. O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural’zeva, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, 1995.
  17. C. Gerhardt, “Hypersurfaces of prescribed curvature in Lorentzian manifolds,” Indiana University Mathematics Journal, vol. 49, no. 3, pp. 1125–1153, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet