Applied Mathematics and Optimization

, Volume 23, Issue 1, pp 1–15

Remarks on elliptic singular perturbation problems

  • Hitoshi Ishii
  • Shigeaki Koike
Article

Abstract

We show the effectiveness of viscosity-solution methods in asymptotic problems for second-order elliptic partial differential equations (PDEs) with a small parameter. Our stress here is on the point that the methods, based on stability results [3], [16], apply without hard PDE calculations. We treat two examples from [11] and [23]. Moreover, we generalize the results to those for Hamilton—Jacobi—Bellman equations with a small parameter.

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References

  1. 1.
    M. Bardi, An asymptotic formula for the Green's function of an elliptic operator, Ann. Scuola Norm. Sup. Pisa, to appear.Google Scholar
  2. 2.
    M. Bardi, Work in preparation.Google Scholar
  3. 3.
    G. Barles and B. Perthame, Discontinuous solutions of deterministic optimal stopping time problems, Math. Modeling Numer. Anal. 21 (1987), 557–579.Google Scholar
  4. 4.
    M. G. Crandall, L. C. Evans, and P.-L. Lions, Some properties of viscosity solutions of Hamilton—Jacobi equations, Trans. Amer. Math. Soc. 282 (1984), 487–502.Google Scholar
  5. 5.
    M. G. Crandall and P.-L. Lions, Viscosity solutions of Hamilton—Jacobi equations, Trans. Amer. Math. Soc. 277 (1983), 1–42.Google Scholar
  6. 6.
    M. G. Crandall and P.-L. Lions, Two approximations of solutions of Hamilton—Jacobi equations, Math. Comp. 43 (1984), 1–19.Google Scholar
  7. 7.
    P. Dupuis and H. Kushner, Minimizing escape probabilities; a large deviations approach (preprint).Google Scholar
  8. 8.
    L. C. Evans, Classical solutions of the Hamilton—Jacobi—Bellman equation for uniformly elliptic operators, Trans. Amer. Math. Soc. 275 (1983), 245–255.Google Scholar
  9. 9.
    L. C. Evans and H. Ishii, A PDE approach to some asymptotic problems concerning random differential equations with small noise intensities, Ann. Inst. H. Poincare Anal. Non Linéaire 2 (1985), 1–20.Google Scholar
  10. 10.
    L. C. Evans and P. E. Souganidis, Differential games and representation formulas for solutions of Hamilton—Jacobi—Isaacs equations, Indiana Univ. Math. J. 33 (1984), 773–797.Google Scholar
  11. 11.
    W. H. Fleming, Exit probabilities and stochastic optimal control, Appl. Math. Optim. 4 (1978), 329–346.Google Scholar
  12. 12.
    W. H. Fleming and P. E. Souganidis, A PDE approach to asymptotic estimates for optimal exit probabilities, Ann. Scuola Norm. Sup. Pisa 13 (1986), 171–192.Google Scholar
  13. 13.
    W. H. Fleming and C.-P. Tsai, Optimal exit probabilities and differential games, Appl., Math. Optim. 7 (1981), 253–282.Google Scholar
  14. 14.
    M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems, Springer-Verlag, New York, 1984.Google Scholar
  15. 15.
    H. Ishii, A Simple, direct proof of uniqueness for solutions of the Hamilton—Jacobi equations of eikonal type, Proc. Amer. Math. Soc. 100 (1987), 247–251.Google Scholar
  16. 16.
    H. Ishii, A boundary value problem of the Dirichlet type for Hamilton—Jacobi equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 16 (1989), 105–135.Google Scholar
  17. 17.
    S. Kamin, Exponential descent of solutions of elliptic singular perturbation problems, Comm. Partial Differential Equations 9 (1984), 197–213.Google Scholar
  18. 18.
    S. Koike, An asymptotic formula for solutions of Hamilton—Jacobi—Bellman equations, Nonlinear Anal. TMA 11 (1987), 429–436.Google Scholar
  19. 19.
    P.-L. Lions, Generalized Solutions of Hamilton—Jacobi Equations. Pitman, Boston, 1982.Google Scholar
  20. 20.
    P.-L. Lions, Optimal control of diffusion processes and Hamilton—Jacobi—Bellman equations; Part II, viscosity solutions and uniqueness, Comm. Partial Differential Equations 8 (1983), 1229–1276.Google Scholar
  21. 21.
    N. S. Trudinger, Fully nonlinear, uniformly elliptic equations under natural structure conditions, Trans. Amer. Math. Soc. 278 (1983), 751–769.Google Scholar
  22. 22.
    I. I. Tsitovich, On the time of first exit from a domain, Theory Probab. Appl. 23 (1978), 117–129.Google Scholar
  23. 23.
    S. R. S. Varadhan, On the behavior of the fundamental solution of the heat equation with variable coefficients, Comm. Pure Appl. Math. 20 (1967), 431–455.Google Scholar
  24. 24.
    A. D. Vent-tsel’ and M. I. Freidlin, Some problems concerning stability under small random perturbations, Theory Probab. Appl. 17 (1972), 269–283.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1991

Authors and Affiliations

  • Hitoshi Ishii
    • 1
  • Shigeaki Koike
    • 2
  1. 1.Department of MathematicsChuo UniversityTokyoJapan
  2. 2.Department of MathematicsTokyo Metropolitan UniversityTokyoJapan

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