Applied Mathematics and Optimization

, Volume 28, Issue 1, pp 57–85 | Cite as

The dam problem with leaky boundary conditions

  • J. Carrillo
  • M. Chipot
Article

Abstract

The goal of this paper is to study the fluid flow through a two-dimensional porous medium when we impose a leaky boundary condition. We show in particular that the situation is quite different from the one with the usual Dirichlet boundary condition.

Key words

Free boundary problems Fluid flow Porous media 

AMS classification

35A05 35J25 35J85 76S05 

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References

  1. [A]
    H. W. Alt: Strömungen durch inhomogene poröse Medien mit freiem Rand, J. Angew. Math. 305 (1979), 89–115.Google Scholar
  2. [Ba1]
    C. Baiocchi: Su un problema di frontiera libera connesso a questioni di idraulica. Ann. Mat. Pura Appl. 92 (1972), 107–127.Google Scholar
  3. [Ba2]
    C. Baiocchi: Free boundary problems in the theory of fluid flow through porous media. Proceedings of the International Congress of Mathematicians, Vancouver (1974), pp. 237–243.Google Scholar
  4. [Ba3]
    C. Baiocchi: Free boundary problems in fluid flow through porous media and variational inequalities. In: Free Boundary Problems (Proceedings of a seminar held in Pavia, Sept.–Oct. 1979), Vol. 1. Rome, 1980, pp. 175–191.Google Scholar
  5. [BC]
    C. Baiocchi and A. Cappello: Disequazioni variazionali et quasivariazionali. Applicazioni a problemi di frontiera libera. Vols. 1 and 2. Pitagora Editrice, Bologna, 1978.Google Scholar
  6. [Be]
    J. Bear: Hydraulics of Groundwater. McGraw-Hill, New York, 1979.Google Scholar
  7. [BKS]
    H. Brezis, D. Kinderlehrer, and G. Stampacchia: Sur une nouvelle formulation du problème de l'écoulement à travers une digue. C. R. Acad. Sci. Paris Sér. A 287 (1978), 711–714.Google Scholar
  8. [BL]
    A. Bensoussan and J. L. Lions, Applications des Inéquations Variationelles en contrôle stochatique. Dunod, Paris, 1978.Google Scholar
  9. [Br]
    H. Brezis: Problèmes Unilatéraux. Math. Pures Appl. 51 (1972), 1–168.Google Scholar
  10. [C]
    M. Chipot: Variational Inequalities and Flow in Porous Media. Applied Mathematical Sciences, Vol. 52. Springer-Verlag, New York, 1984.Google Scholar
  11. [CC]
    J. Carrillo and M. Chipot: On the dam problem. J. Differential Equations 45 (1982), 234–271.Google Scholar
  12. [CM]
    M. Chipot and G. Michaille: Uniqueness results and montonicity properties for the solution of some variational inequalities. Ann. Scuola Norm. Sup. Pisa (4) 16(1) (1989), 137–166.Google Scholar
  13. [ET]
    I. Ekeland and R. Temam: Convex Analysis and Variational Problems. North-Holland, Amsterdam, 1976.Google Scholar
  14. [G]
    P. Grisvard: Elliptic Problems in Nonsmooth Domains. Pitman, London, 1985.Google Scholar
  15. [GT]
    D. Gilbarg and N. S. Trudinger: Elliptic Partial Differential Equations of Second Order. Springer-Verlag, New York, 1977.Google Scholar
  16. [KNS1]
    D. Kinderlehrer, L. Nirenberg, and J. Spruck: Regularity in elliptic free boundary problems, I. J. Analyse Math., 34 (1978), 86–119.Google Scholar
  17. [KNS2]
    D. Kinderlehrer, L. Nirenberg, and J. Spruck: Regularity in elliptic free boundary problems, II. Ann. Scuola Norm. Sup. Pisa 6 (1979), 637–683.Google Scholar
  18. [KS]
    D. Kinderlehrer and G. Stampacchia: An Introduction to Variational Inequalities. Academic Press, New York, 1980.Google Scholar
  19. [L]
    J. L. Lions: Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod-Gauthier-Villars, Paris, 1969.Google Scholar
  20. [N]
    J. Nečas. Introduction to the Theory of Nonlinear Elliptic Equations. Wiley Interscience, New York, 1986.Google Scholar
  21. [R]
    J. F. Rodrigues: On the dam problem with boundary leaky condition. Portugal. Math. 39 (1980), 399–411.Google Scholar
  22. [RT]
    P. A. Raviard and J. M. Thomas: Introduction à l'analyse numérique des équations aux dérivées partielles. Masson, Paris, 1983.Google Scholar
  23. [V]
    A. Visintin. Study of a free boundary filtration problem by a nonlinear variational equation. Boll. Un. Mat. Ital. 5 (1979), 212–237.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1993

Authors and Affiliations

  • J. Carrillo
    • 1
  • M. Chipot
    • 2
  1. 1.Departamento de Matematica AplicadaUniversidad Complutense de MadridMadridSpain
  2. 2.Département de MathématiquesUniversité de MetzMetz-Cedex 01France

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