Skip to main content
SpringerLink
Log in

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

  • Published:
Potential Analysis Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adams, R.A.: Sobolev Spaces. Academic, New York (1975)

    MATH  Google Scholar 

  2. Brown, R., Mitrea, I., Mitrea, M., Wright, M.: Mixed boundary value problems for the Stokes system. Trans. Am. Math. Soc. (2009, in press)

  3. Costabel, M.: Boundary integral operators on Lipschitz domains: elementary results. SIAM J. Math. Anal. 19, 613–626 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  5. Dindos̆, M.: Hardy spaces and potential theory on C 1 domains in Riemannian manifolds. Memoirs AMS. 191(894) (2008)

  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)

    Article  MathSciNet  Google Scholar 

  7. Ding, Z.: A proof of the trace theorem of Sobolev spaces on Lipschitz domains. Proc. Am. Math. Soc. 124, 591–600 (1996)

    Article  MATH  Google Scholar 

  8. Escauriaza, L., Mitrea, M.: Transmission problems and spectral theory for singular integral operators on Lipschitz domains. J. Funct. Anal. 216, 141–171 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  9. Fabes, E., Jodeit, M., Rivère, N.: Potential techniques for boundary value problems on C 1-domains. Acta Math. 141, 165–186 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  11. Fabes, E., Kenig, C., Verchota, G.: The Dirichlet problem for the Stokes system on Lipschitz domains. Duke Math. J. 57, 769–793 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  12. Hirsch, M.W.: Differential Topology. Springer, Berlin (1976)

    MATH  Google Scholar 

  13. Hsiao, G.C., Wendland, W.L.: Boundary Integral Equations. Springer, Heidelberg (2008)

    Book  MATH  Google Scholar 

  14. Jerison, D., Kenig, C.: The inhomogeneous Dirichlet problem in Lipschitz domains. J. Funct. Anal. 130, 161–219 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  15. Kohr, M.: The Dirichlet problems for the Stokes resolvent equations in bounded and exterior domains in ℝn. Math. Nachr. 280, 534–559 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  16. Kohr, M., Pintea, C., Wendland, W.L.: Stokes-Brinkman transmission problems on Lipschitz and C 1 domains in Riemannian manifolds (2009, in press)

  17. Kohr, M., Pop, I.: Viscous Incompressible Flow for Low Reynolds Numbers. WIT, Southampton (2004)

    MATH  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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

    MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  23. Kress, R.: Linear Integral Equations. Springer, Berlin (1999)

    MATH  Google Scholar 

  24. Lawson, H.B. Jr., Michelsohn, M.-L.: Spin Geometry. Princeton University Press, Princeton (1989)

    MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

  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)

    Article  MATH  MathSciNet  Google Scholar 

  28. Mitrea, M., Taylor, M.: Boundary layer methods for Lipschitz domains in Riemannian manifolds. J. Funct. Anal. 163, 181–251 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  34. Mitrea, M., Taylor, M.: Navier-Stokes equations on Lipschitz domains in Riemannian manifolds. Math. Anal. 321, 955–987 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  35. Mitrea, M., Wright, M.: Boundary value problems for the Stokes system in arbitrary Lipschitz domains (2008, submitted)

  36. Taylor, M.: Pseudodifferential Operators. Princeton University Press, Princeton (1981)

    MATH  Google Scholar 

  37. Taylor, M.: Partial Differential Equations, vols. 1–3. Springer, New York (1996–2000)

    Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  39. Wloka, J.T., Rowley, B., Lawruk, B.: Boundary Value Problems for Elliptic Systems. Cambridge University Press, Cambridge (1995)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, Babeş-Bolyai University, 1 M. Kogălniceanu Str., 400084, Cluj-Napoca, Romania

    Mirela Kohr & Cornel Pintea

  2. Institut für Angewandte Analysis und Numerische Simulation, Universität Stuttgart, Pfaffenwaldring, 57, 70569, Stuttgart, Germany

    Wolfgang L. Wendland

Authors
  1. Mirela Kohr
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Cornel Pintea
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Wolfgang L. Wendland
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mirela Kohr.

Additional information

Mirela Kohr is supported by the Romanian Ministry of Education and Research, UEFISCSU Grants, PN-II-ID-525/2007 and 524/2007.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kohr, M., Pintea, C. & Wendland, W.L. Brinkman-type Operators on Riemannian Manifolds: Transmission Problems in Lipschitz and C 1 Domains. Potential Anal 32, 229–273 (2010). https://doi.org/10.1007/s11118-009-9151-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11118-009-9151-7

Keywords

Mathematics Subject Classifications (2000)

Search

Navigation