Applied Mathematics and Optimization

, Volume 35, Issue 3, pp 265–282 | Cite as

A complex parabolic type monge-ampère equation

  • J. Spiliotis


The complex parabolic type Monge-Ampère equation we are dealing with is of the form\((\partial u/\partial t){\text{ det[}}\partial ^2 u/\partial z_i {\text{ }}\partial \overline z _j ] = f\) inB × (0,T),u=ϕ on Γ, whereB is the unit ball in ℂ d ,d>1, and Γ is the parabolic boundary ofB × (0,T). Solutionu is proved unique in the class\(C(\bar B \times [0,T]) \cap W_{\infty ,loc}^{2,1} (B \times (0,T))\).

Key Words

Monge-Ampère Complex First-initial Boundary Parabolic Stochastic integral Control 

AMS Classification

93E20 35K60 60H30 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bedford, E., and Taylor, B. A., The Dirichlet problem for a complex Monge-Ampère equation. Invent. Math., 37 (1976).Google Scholar
  2. 2.
    Evans, L. C., A convergence theorem for solutions of nonlinear second order elliptic equations. Indiana Univ. Math. J., 27 (5) (1978).Google Scholar
  3. 3.
    Friedmann, A., Stochastic Differential Equations and Applications, Vol. 1. Academic Press, New York, 1975.Google Scholar
  4. 4.
    Gaveau, B., Methodes de contrôle optimal en analyse complexe I. J. Funct. Anal., 25 (1977).Google Scholar
  5. 5.
    Gaveau, B., Methodes de contrôle optimal en analyse complexe II. Bull. Sci. Math. 2e serie, 102 (1978).Google Scholar
  6. 6.
    Krylov, N. V., On Ito’s stochastic integral equations. Theory of Probability and Its Applications, 14 (1969).Google Scholar
  7. 7.
    Krylov, N. V., On uniqueness of the solution of Bellman’s equation. Math. USSR Izv., 5 (6) (1971).Google Scholar
  8. 8.
    Krylov, N. V., Sequences of convex functions and estimations of the maximum of the solutions of a parabolic equation. Sib. Mat. Zhiurnal, 17 (2) (1976). English translation.Google Scholar
  9. 9.
    Krylov, N. V., Controlled Diffusion Processes. Springer-Verlag, New York, 1980.Google Scholar
  10. 10.
    Krylov, N. V., On controlled diffusion processes with unbounded coefficients. Math. USSR Izv., 19 (1) (1982). English translation.Google Scholar
  11. 11.
    Krylov, N. V., Smoothness of the value function for a controlled diffusion process in a domain. Math. USSR Izv., 34 (1) (1990). English translation.Google Scholar
  12. 12.
    Lions, P. L., Control of Diffusion Processes in ℝN. Comm. Pure Appl. Math. XXXIV (1981).Google Scholar
  13. 13.
    Lions, P. L., Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations, Part 1. Comm. Partial Differential Equations, 8 (10) (1983).Google Scholar
  14. 14.
    Spiliotis, J., Certain results on a parabolic type Monge-ampère equation. J. Math. Anal. Appl., 163 (2) (1992).Google Scholar
  15. 15.
    Stroock, D. W., and Varadhan, S. R. S., Multidimensional Diffusion Processes. Springer-Verlag, Berlin, 1979.Google Scholar

Copyright information

© Springer-Verlag New York Inc 1997

Authors and Affiliations

  • J. Spiliotis
    • 1
  1. 1.Department of MathematicsNational Technical UniversityAthensGreece

Personalised recommendations