Potential Analysis

, Volume 32, Issue 3, pp 229–273 | Cite as

Brinkman-type Operators on Riemannian Manifolds: Transmission Problems in Lipschitz and C 1 Domains

  • Mirela Kohr
  • Cornel Pintea
  • Wolfgang L. Wendland
Article

Abstract

In this paper we use the method of boundary integral equations to treat some transmission problems for Brinkman-type operators on Lipschitz and C 1 domains in Riemannian manifolds.

Keywords

Brinkman operator Stokes operator Lipschitz and C1 domains Riemannian manifold Transmission problem Layer potentials Sobolev spaces 

Mathematics Subject Classifications (2000)

35J25 42B20 46E35 76D 76M 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Adams, R.A.: Sobolev Spaces. Academic, New York (1975)MATHGoogle Scholar
  2. 2.
    Brown, R., Mitrea, I., Mitrea, M., Wright, M.: Mixed boundary value problems for the Stokes system. Trans. Am. Math. Soc. (2009, in press)Google Scholar
  3. 3.
    Costabel, M.: Boundary integral operators on Lipschitz domains: elementary results. SIAM J. Math. Anal. 19, 613–626 (1988)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Dahlberg, B., Kenig, C., Verchota, C.: Boundary value problems for the system of elastostatics on Lipschitz domains. Duke Math. J. 57, 795–818 (1988)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Dindos̆, M.: Hardy spaces and potential theory on C 1 domains in Riemannian manifolds. Memoirs AMS. 191(894) (2008)Google Scholar
  6. 6.
    Dindos̆, M., Mitrea, M.: The stationary Navier-Stokes system in nonsmooth manifolds: the Poisson problem in Lipschitz and C 1 domains. Arch. Ration. Mech. Anal. 174, 1–47 (2004)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Ding, Z.: A proof of the trace theorem of Sobolev spaces on Lipschitz domains. Proc. Am. Math. Soc. 124, 591–600 (1996)MATHCrossRefGoogle Scholar
  8. 8.
    Escauriaza, L., Mitrea, M.: Transmission problems and spectral theory for singular integral operators on Lipschitz domains. J. Funct. Anal. 216, 141–171 (2004)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Fabes, E., Jodeit, M., Rivère, N.: Potential techniques for boundary value problems on C 1-domains. Acta Math. 141, 165–186 (1978)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Fabes, E., Mendez, O., Mitrea, M.: Boundary layers on Sobolev-Besov spaces and Poisson’s equation for the Laplacian in Lipschitz domains. J. Funct. Anal. 159, 323–368 (1998)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Fabes, E., Kenig, C., Verchota, G.: The Dirichlet problem for the Stokes system on Lipschitz domains. Duke Math. J. 57, 769–793 (1988)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Hirsch, M.W.: Differential Topology. Springer, Berlin (1976)MATHGoogle Scholar
  13. 13.
    Hsiao, G.C., Wendland, W.L.: Boundary Integral Equations. Springer, Heidelberg (2008)MATHCrossRefGoogle Scholar
  14. 14.
    Jerison, D., Kenig, C.: The inhomogeneous Dirichlet problem in Lipschitz domains. J. Funct. Anal. 130, 161–219 (1995)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Kohr, M.: The Dirichlet problems for the Stokes resolvent equations in bounded and exterior domains in ℝn. Math. Nachr. 280, 534–559 (2007)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Kohr, M., Pintea, C., Wendland, W.L.: Stokes-Brinkman transmission problems on Lipschitz and C 1 domains in Riemannian manifolds (2009, in press)Google Scholar
  17. 17.
    Kohr, M., Pop, I.: Viscous Incompressible Flow for Low Reynolds Numbers. WIT, Southampton (2004)MATHGoogle Scholar
  18. 18.
    Kohr, M., Raja Sekhar, G.P., Blake, J.R.: Green’s function of the Brinkman equation in a 2D anisotropic case. IMA J. Appl. Math. 73, 374–392 (2008)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Kohr, M., Raja Sekhar, G.P., Wendland, W.L.: Boundary integral method for Stokes flow past a porous body. Math. Methods Appl. Sci. 31, 1065–1097 (2008)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Kohr, M., Raja Sekhar, G.P., Wendland, W.L.: Boundary integral equations for a three-dimensional Stokes-Brinkman cell model. Math. Models Methods Appl. Sci. 18, 2055–2085 (2008)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Kohr, M., Wendland, W.L.: Boundary integral equations for a three-dimensional Brinkman flow problem. Math. Nachr. 282 (2009). doi: 10.1002/mana.200710797 MathSciNetGoogle Scholar
  22. 22.
    Kohr, M., Wendland, W.L., Raja Sekhar, G.P.: Boundary integral equations for two-dimensional low Reynolds number flow past a porous body. Math. Methods Appl. Sci. 32, 922–962 (2009)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Kress, R.: Linear Integral Equations. Springer, Berlin (1999)MATHGoogle Scholar
  24. 24.
    Lawson, H.B. Jr., Michelsohn, M.-L.: Spin Geometry. Princeton University Press, Princeton (1989)MATHGoogle Scholar
  25. 25.
    Mitrea, D., Mitrea, M., Qiang, S.: Variable coefficient transmission problems and singular integral operators on non-smooth manifolds. J. Integral Equ. Appl. 18, 361–397 (2006)MATHCrossRefGoogle Scholar
  26. 26.
    Mitrea, D., Mitrea, M., Taylor, M.: Layer potentials, the Hodge Laplacian and global boundary problems in non-smooth Riemannian manifolds. Memoirs AMS 150(713) (2001)Google Scholar
  27. 27.
    Mitrea, M., Monniaux, S.: On the analyticity of the semigroup generated by the Stokes operator with Neumann-type boundary conditions on Lipschitz subdomains of Riemannain manifolds. Trans. Am. Math. Soc. 361, 3125–3157 (2009)MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Mitrea, M., Taylor, M.: Boundary layer methods for Lipschitz domains in Riemannian manifolds. J. Funct. Anal. 163, 181–251 (1999)MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Mitrea, M., Taylor, M.: Potential theory on Lipschitz domains in Riemannian manifolds: Hölder continuous metric tensors. Commun. Partial Differ. Equ. 25, 1487–1536 (2000)MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Mitrea, M., Taylor, M.: Potential theory on Lipschitz domains in Riemannian manifolds: Sobolev-Besov space results and the Poisson problem. J. Funct. Anal. 176, 1–79 (2000)MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Mitrea, M., Taylor, M.: Potential theory on Lipschitz domains in Riemannian manifolds: Lp, Hardy and Hölder type results. Commun. Anal. Geom. 57, 369–421 (2001)MathSciNetGoogle Scholar
  32. 32.
    Mitrea, M., Taylor, M.: Potential theory on Lipschitz domains in Riemannian manifolds: the case of Dini metric tensors. Trans. AMS 355, 1961–1985 (2002)CrossRefMathSciNetGoogle Scholar
  33. 33.
    Mitrea, M., Taylor, M.: Sobolev and Besov space estimates for solutions to second order PDE on Lipschitz domains in manifolds with Dini or Hölder continuous metric tensors. Commun. Partial Differ. Equ. 30, 1–37 (2005)MATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Mitrea, M., Taylor, M.: Navier-Stokes equations on Lipschitz domains in Riemannian manifolds. Math. Anal. 321, 955–987 (2001)MATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Mitrea, M., Wright, M.: Boundary value problems for the Stokes system in arbitrary Lipschitz domains (2008, submitted)Google Scholar
  36. 36.
    Taylor, M.: Pseudodifferential Operators. Princeton University Press, Princeton (1981)MATHGoogle Scholar
  37. 37.
    Taylor, M.: Partial Differential Equations, vols. 1–3. Springer, New York (1996–2000)Google Scholar
  38. 38.
    Verchota, G.: Layer potentials and regularity for the Dirichlet problem for Laplace’s operator in Lipschitz domains. J. Funct. Anal. 59, 572–611 (1984)MATHCrossRefMathSciNetGoogle Scholar
  39. 39.
    Wloka, J.T., Rowley, B., Lawruk, B.: Boundary Value Problems for Elliptic Systems. Cambridge University Press, Cambridge (1995)MATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  • Mirela Kohr
    • 1
  • Cornel Pintea
    • 1
  • Wolfgang L. Wendland
    • 2
  1. 1.Faculty of Mathematics and Computer ScienceBabeş-Bolyai UniversityCluj-NapocaRomania
  2. 2.Institut für Angewandte Analysis und Numerische SimulationUniversität StuttgartStuttgartGermany

Personalised recommendations