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Remarks on the point interaction approximation

  • R. Figari
  • G. Papanicolaou
  • J. Rubinstein
Part of the The IMA Volumes in Mathematics and Its Applications book series (IMA, volume 9)

Abstract

The point interaction approximation is a way to study boundary value problems in regions with many small inclusions. For example, heat conduction in a material with many small holes that absorb heat, fluid flow in a region with many small obstacles, etc. The main idea is to replace the inclusions along with the boundary conditions on them by an inhomogeneous term in the differential equation which is then to hold evrywhere. As the name Point Interaction suggests, the effect of the inclusions is localized so the approximation is valid for small volume fractions. In this paper we shall consider the heat conduction problem as follows.

Keywords

Continuum Limit Heat Conduction Problem Sphere Center Center Configuration Bubbly Liquid 
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References

  1. [1]
    E. I. Khruslov and V. A. Marchenko, Boundary value problems in regions with fine-grained boundaries, Naukova Dumka, Kiev, 1974.Google Scholar
  2. [2]
    M. Kac, Probabilistic methods in some problems of scattering theory, Rocky Mountain J. Math. 4, 1974, 511–538.MATHGoogle Scholar
  3. [3]
    J. Rauch and M. Taylor, Potential and scattering theory on wildly perturbed domains, J. Funct. Anal. 18, 1975, 27–59MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    G. Papanicolaou and S.R.S. Varadhan, Diffusion in regions with many small holes. In Stochastic Differential Systems (ed. B. Grigeliouis). Lecture Notes in Control and Information Theory 25, 190–206, Springer.Google Scholar
  5. [5]
    S. Ozawa., On an elaboration of M. Kac’s Theorem concerning eigenvalues of the Laplacian in a region with randomly distributed small obstacles, Comm. Math. Phys. 91, 1983, 473–487.MATHCrossRefGoogle Scholar
  6. [6]
    R. Figari, E. Orlandi and J. Teta, The Laplacian in regions with many small obstacles: fluctuations around the limit operator 41, 1985, 465–488.MathSciNetMATHGoogle Scholar
  7. [7]
    L. L. Foldy, The multiple scattering of waves, Phys. Rev. 67, 1945, 107–119.MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    R. Caflisch, M. Miksis, G. Papanicolaou and L. Ting, Effective equations for wave propagation in bubbly liquids, J. Fluid Mech. 153, 1985, 259–273 and also 160, 1–14.Google Scholar
  9. [9]
    J. Rubinstein, NYU Dissertation, 1985.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1987

Authors and Affiliations

  • R. Figari
    • 1
  • G. Papanicolaou
    • 2
  • J. Rubinstein
    • 3
  1. 1.Department of Theoretical PhysicsUniversity of NaplesNaplesItaly
  2. 2.Courant InstituteNew York UniversityNew YorkUSA
  3. 3.Department of MathematicsStanford UniversityStanfordUSA

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