Skip to main content

Singular perturbation methods in a one-dimensional free boundary problem

Part of the Lecture Notes in Mathematics book series (LNM,volume 711)

Abstract

The optimal cost function associated to a stopping time problem for a dynamical system perturbed by an additive noise term with small positive coefficient ɛ satisfies a singularly perturbed variational inequality with an obstacle. Characteristic for this type of variational inequalities is the occurrence of a free boundary. Here we shall study the behaviour as ɛ → 0 of the solution and the free boundary of the variational inequality induced by a one-dimensional randomly perturbed differential equation. Our results are derived by standard techniques in the theory of asymptotic expansions and the maximum principle.

Keywords

  • Variational Inequality
  • Maximum Principle
  • Free Boundary
  • Singular Perturbation
  • Differential Inequality

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   34.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   46.00
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bensoussan, A. and J.L. Lions, Problèmes de temps d'arrèt optimal et inéquations variationnelles paraboliques, Applicable Analysis 3 (1973), 267–295.

    CrossRef  MathSciNet  MATH  Google Scholar 

  2. Bensoussan, A. and J.L. Lions, Problèmes de temps d'arrèt optimal et de perturbations singulières dans les inéquations variationnelles variationelles, Lecture Notes in Economics and Mathematical systems 107, Springer-Verlag, Berlin, 1975.

    MATH  Google Scholar 

  3. Brézis, H. and G. Stampacchia, Sur la régularité de la solution d'inéquations elliptiques, Bull. Soc. Math. France 96 (1968), 153–180.

    MathSciNet  MATH  Google Scholar 

  4. Dorr, F.W., Parter, S.V. and L.F. Shampine, Applications of the maximum principle to singular perturbation problems, SIAM Review 15 (1973), 43–88.

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. Eckhaus, W., Matched Asymptotic Expansions and Singular Perturbations, Mathematics Studies 6, North-Holland, Amsterdam, 1973.

    MATH  Google Scholar 

  6. Eckhaus, W., Formal approximations and singular perturbations, SIAM Review 19 (1977), 593–633.

    CrossRef  MathSciNet  MATH  Google Scholar 

  7. Eckhaus, W., Asymptotic Analysis of Singular Perturbation Problems, North-Holland, Amsterdam, to appear in 1979.

    MATH  Google Scholar 

  8. Eckhaus, W. and E.M. de Jager, Asymptotic solutions of singular perturbation problems for linear differential equations of elliptic type, Arch. Rational Mech. Anal. 23 (1966), 26–86.

    CrossRef  MathSciNet  MATH  Google Scholar 

  9. Eckhaus, W. and H.J.K. Moet, Asymptotic solutions in free boundary problems of singularly perturbed elliptic variational inequalities, in: W. Eckhaus and E.M. de Jager, ed., Differential Equations and Apllications, Mathematics Studies 31, North-Holland, Amsterdam, 1978.

    Google Scholar 

  10. Freidlin, M.I., Markov processes and differential equations, in: R.V. Gamkrelidze, ed., Progress in Mathematics, Vol. 3, Plenum Press, New York, 1969.

    Google Scholar 

  11. Friedman, A., Stochastic Differential Equations and Applications, I & II, Academic Press, 1975/1976.

    Google Scholar 

  12. Lions, J.L., Perturbations Singulières dans les Problèmes aux Limites et en Contrôle Optimal, Lecture Notes in Mathematics 323, Springer-Verlag, Berlin, 1973.

    MATH  Google Scholar 

  13. Lions, J.L., Partial differential inequalities, Russ. Math. Surveys 27 (1972), 91–159.

    CrossRef  MATH  Google Scholar 

  14. Lions, J.L. and G. Stampacchia, Variational inequalities, Comm. Pure Appl. Math. 20 (1967), 493–519.

    CrossRef  MathSciNet  MATH  Google Scholar 

  15. Moet, H.J.K., Asymptotic analysis of singularly perturbed variational inequalities, to appear.

    Google Scholar 

  16. Mosco, U., Introduction to variational and quasi-variational inequalities, in: Abdul Salam, ed., Control Theory and Topics in Functional Analysis, Vol. III, International Atomic Energy Agency, Vienna, 1976.

    Google Scholar 

  17. Protter, M.H. and H.F. Weinberger, Maximum Principles in Differential Equations, Prentice-Hall, New Jersey, 1967.

    MATH  Google Scholar 

  18. van Harten, A., Singularly Perturbed Nonlinear 2nd Order Elliptic Boundary Value Problems, Utrecht, 1975 (thesis).

    Google Scholar 

  19. Wong, E., Stochastic Processes in Information and Dynamical Systems, McGraw-Hill, New York, 1971.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1979 Springer-Verlag

About this paper

Cite this paper

Moet, H.J.K. (1979). Singular perturbation methods in a one-dimensional free boundary problem. In: Verhulst, F. (eds) Asymptotic Analysis. Lecture Notes in Mathematics, vol 711. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0062947

Download citation

  • DOI: https://doi.org/10.1007/BFb0062947

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-09245-2

  • Online ISBN: 978-3-540-35332-4

  • eBook Packages: Springer Book Archive