Advertisement

Potential Analysis

, Volume 38, Issue 4, pp 1123–1171 | Cite as

Nonlinear Neumann–Transmission Problems for Stokes and Brinkman Equations on Euclidean Lipschitz Domains

  • Mirela Kohr
  • Massimo Lanza de Cristoforis
  • Wolfgang L. Wendland
Article

Abstract

The purpose of this paper is to use a layer potential analysis and the Leray–Schauder degree theory to show an existence result for a nonlinear Neumann–transmission problem corresponding to the Stokes and Brinkman operators on Euclidean Lipschitz domains with boundary data in L p spaces, Sobolev spaces, and also in Besov spaces.

Keywords

Stokes and Brinkman operators Lipschitz domain Nonlinear boundary value problem Layer potential operators 

Mathematics Subject Classifications (2010)

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 Press, New York (1975)MATHGoogle Scholar
  2. 2.
    Amann, H.: Compact embeddings of vector-valued Sobolev and Besov spaces. Glas. Mat. 35, 161–177 (2000)MathSciNetMATHGoogle Scholar
  3. 3.
    Ammari, H., Kang, H., Lee, H.: Layer Potential Techniques in Spectral Analysis. American Mathematical Society, Providence (2009)MATHGoogle Scholar
  4. 4.
    Begehr, H., Hile, G.N.: Nonlinear Riemann boundary value problems for a nonlinear elliptic system in the plane. Math. Z. 179, 241–261 (1982)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Begehr, H., Hsiao, G.C.: Nonlinear boundary value problems for a class of elliptic systems. In: Complexe Analysis und ihre Anwendung auf partielle Differentialgleichungen, pp. 90–102. Martin–Luther–Unniversität, Halle-Wittenberg (1980)Google Scholar
  6. 6.
    Begehr, H., Hsiao, G.C.: Nonlinear boundary value problems of Riemann–Hilbert type. Contemp. Math. 11, 139–153 (1982)MATHCrossRefGoogle Scholar
  7. 7.
    Burenkov, V.I., Lanza de Cristoforis, M.: Spectral stability of the Robin Laplacian. Proc. Steklov Inst. Math. 260, 68–89 (2008)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Carleman, T.: Über eine nichtlineare Randwertaufgabe bei der Gleichung Δu = 0. Math. Z. 9, 35–43 (1921)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Chang, T.K.: Boundary integral operators over Lipschitz surfaces for a Stokes equation in \({\mathbb R}^n\). Potential Anal. 29, 105–117 (2008)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Chkadua, O., Mikhailov, S.E., Natroshvili, D.: Localized direct segregated boundary-domain integral equations for variable coefficient transmission problems with interface crack. Mem. Differ. Equ. Math. Phys. 52, 17–64 (2011)MathSciNetMATHGoogle Scholar
  11. 11.
    Costabel, M.: Boundary integral operators on Lipschitz domains: elementary results. SIAM J. Math. Anal. 19, 613–626 (1988)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Costabel, M.: Some historical remarks on the positivity of boundary integral operators. In: Schanz, M., Steinbach, O. (eds.) Boundary Element Analysis, pp. 1–27. Springer, Berlin (2007)CrossRefGoogle Scholar
  13. 13.
    Cwikel, M.: Real and complex interpolation and extrapolation of compact operators. Duke Math. J. 65, 333–343 (1992)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Dahlberg, B.E.J., Kenig, C.: Hardy spaces and the Neumann problem in L p for Laplace’s equation in Lipschitz domains. Ann. Math. 125, 437–465 (1987)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Dahlberg, B.E.J., Kenig, C., Verchota, G.C.: Boundary value problems for the systems of elastostatics in Lipschitz domains. Duke Math. J. 57, 795–818 (1988)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Dalla Riva, M., Lanza de Cristoforis, M.: Hypersingularly perturbed loads for a nonlinear traction boundary value problem. A functional analytic approach. Eurasian Math. J. 1, 31–58 (2010)MathSciNetMATHGoogle Scholar
  17. 17.
    Dindos̆, M.: Hardy spaces and potential theory on C 1 domains in Riemannian manifolds. Mem. Am. Math. Soc. 191, 894 (2008)MathSciNetGoogle Scholar
  18. 18.
    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)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Efendiev, M.A., Schmitz, H., Wendland, W.: On some nonlinear potential problems. Electr. J. Differ. Equ. 1999, 1–17 (1999)MathSciNetGoogle Scholar
  20. 20.
    Escauriaza, L., Mitrea, M.: Transmission problems and spectral theory for singular integral operators on Lipschitz domains. J. Funct. Anal. 216, 141–171 (2004)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Fabes, E., Jodeit, M., Rivère, N.: Potential techniques for boundary value problems on C 1-domains. Acta Math. 141, 165–186 (1978)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Fabes, E., Kenig, C., Verchota, G.: The Dirichlet problem for the Stokes system on Lipschitz domains. Duke Math. J. 57, 769–793 (1988)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    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)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Folland, G.B.: Real analysis. In: Modern Techniques and their Applications, 2nd edn. Wiley, New York (1999)Google Scholar
  25. 25.
    Gatica, G.N., Meddahi, S.: A dual-dual mixed formulation for nonlinear exterior transmission problems. Math. Comput. 70, 1461–1480 (2000)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Geng, J., Shen, Z.: The Neumann problem and Helmholtz decomposition in convex domains. J. Funct. Anal. 259, 2147–2164 (2010)MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Gesztesy, F., Mitrea, M.: Robin-to-Robin maps and Krein-Type resolvent formulas for Schrödinger operators on bounded Lipschitz domains. In: Modern Analysis and Applications: Mark Krein Centenary Conference. Differential Operators and Mechanics, Book Series: Operator Theory Advances and Applications, vol. 191, pp. 81–113 (2009)Google Scholar
  28. 28.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of 2nd Order. Springer, Berlin (2001)Google Scholar
  29. 29.
    Hofmann, S., Mitrea, M., Taylor, M.: Singular integrals and elliptic boundary problems on regular Semmes–Kenig–Toro domains. Int. Math. Res. Not. 14, 2567–2865 (2010)MathSciNetGoogle Scholar
  30. 30.
    Hsiao, G.C., Wendland, W.L.: A finite element method for an integral equation of the first kind. J. Math. Anal. Appl. 58, 449–481 (1977)MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Hsiao, G.C., Wendland, W.L.: Boundary Integral Equations: Variational Methods. Springer, Heidelberg (2008)MATHCrossRefGoogle Scholar
  32. 32.
    Jerison, D.S., Kenig, C.E.: The Neumann problem on Lipschitz domains. Bull. Am. Math. Soc. 4, 203–207 (1981)MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Jerison, D.S., Kenig, C.E.: Boundary behavior of harmonic functions in nontangentially accesible domains. Adv. Math. 46, 80–147 (1982)MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    Jerison, D.S., Kenig, C.: The inhomogeneous Dirichlet problem in Lipschitz domains. J. Funct. Anal. 130, 161–219 (1995)MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    Kalton, N.J., Mayboroda, S., Mitrea, M.: Interpolation of Hardy–Sobolev–Besov–Triebel–Lizorkin spaces and applications to problems in partial differential equations. Contemp. Math. 445, 121–177 (2007)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Kalton, N., Mitrea, M.: Stability of Fredholm properties on interpolation scales of quasi-Banach spaces and applications. Trans. Am. Math. Soc. 350, 3837–2901 (1998)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Kenig, C.E.: Harmonic analysis techniques for 2nd order elliptic boundary value problems. In: AMS CBMS vol. 83 (1994)Google Scholar
  38. 38.
    Kim, A.S., Shen, Z.: The Neumann problem in L p on Lipschitz and convex domains. J. Funct. Anal. 255, 1817–1830 (2010)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Klingelhöfer, K.: Über nichtlineare Randwertaufgaben der Potentialtheorie. Mitt. Math. Semin. Giessen Heft. 76, 1–70 (1967)Google Scholar
  40. 40.
    Klingelhöfer, K.: Modified Hammerstein integral equations and nonlinear harmonic boundary value problems. J. Math. Anal. Appl. 28, 77–87 (1969)MathSciNetMATHCrossRefGoogle Scholar
  41. 41.
    Klingelhöfer, K.: Nonlinear harmonic boundary value problems. I. Arch. Ration. Mech. Anal. 31, 364–371 (1968)/(1969)MATHCrossRefGoogle Scholar
  42. 42.
    Klingelhöfer, K.: Nonlinear harmonic boundary value problems. II. Modified Hammerstein integral equations. J. Math. Anal. Appl. 25, 592–606 (1969)MathSciNetMATHCrossRefGoogle Scholar
  43. 43.
    Kohr, M.: The interior Neumann problem for the Stokes resolvent system in a bounded domain in \({\mathbb R}^n\). Arch. Mech. 59, 1–22 (2007)MathSciNetGoogle Scholar
  44. 44.
    Kohr, M., Pintea, C., Wendland, W.L.: Stokes-Brinkman transmission problems on Lipschitz and C 1 domains in Riemannian manifolds. Commun. Pure Appl. Anal. 9, 493–537 (2010)MathSciNetMATHCrossRefGoogle Scholar
  45. 45.
    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)MathSciNetMATHCrossRefGoogle Scholar
  46. 46.
    Kohr, M., Pintea, C., Wendland, W.L.: Dirichlet-transmission problems for general Brinkman operators on Lipschitz and C 1 domains in Riemannian manifolds. Discrete Continuous Dyn. Syst. B. 15, 999–1018 (2011)MathSciNetMATHCrossRefGoogle Scholar
  47. 47.
    Kohr M., Pintea C., Wendland W.L.: Layer potential analysis for pseudodifferential matrix operators in Lipschitz domains on compact Riemannian manifolds: applications to pseudodifferential Brinkman operators. Int. Math. Res. Not. (2012). doi: 10.1093/imrn/RNS158 Google Scholar
  48. 48.
    Kohr, M., Pop, I.: Viscous Incompressible Flow for Low Reynolds Numbers. WIT Press, Southampton, UK (2004)MATHGoogle Scholar
  49. 49.
    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)MathSciNetMATHCrossRefGoogle Scholar
  50. 50.
    Kohr, M., Wendland, W.L.: Boundary integral equations for a three-dimensional Brinkman flow problem. Math. Nachr. 282, 1–29 (2009)MathSciNetCrossRefGoogle Scholar
  51. 51.
    Lamberti, P.D., Lanza de Cristoforis, M.: A global Lipschitz continuity result for a domain dependent Neumann eigenvalue problem for the Laplace operator. J. Differ. Equ. 216, 109–133 (2005)MathSciNetMATHCrossRefGoogle Scholar
  52. 52.
    Lamberti, P.D., Lanza de Cristoforis, M.: Persistence of eigenvalues and multiplicity in the Neumann problem for the Laplace operator on nonsmooth domains. Rend. Circ. Mat. Palermo, Series II, Suppl. 76, 413–427 (2005)MathSciNetGoogle Scholar
  53. 53.
    Lanza de Cristoforis, M.: Singular perturbation problems in potential theory and applications. In: Complex Analysis and Potential Theory, pp. 131–139. World Sci. Publ., NJ (2007)Google Scholar
  54. 54.
    Lanzani, L., Shen Z.: On the Robin boundary condition for Laplace’s equation in Lipschitz domains. Commun. Partial Differ. Equ. 29, 91–109 (2004)MathSciNetMATHCrossRefGoogle Scholar
  55. 55.
    Mayboroda, S., Mitrea, M.: Sharp estimates for Green potentials on non-smooth domains. Math. Res. Lett. 11, 481–492 (2004)MathSciNetMATHGoogle Scholar
  56. 56.
    Maz’ya, V.G.: Sobolev Spaces. Springer, Berlin (1985)Google Scholar
  57. 57.
    Maz’ya, V., Mitrea, M., Shaposhnikova, T.: The inhomogeneous Dirichlet problem for the Stokes system in Lipschitz domains with unit normal close to VMO. Funct. Anal. Appl. 43, 217–235 (2009)MathSciNetCrossRefGoogle Scholar
  58. 58.
    McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge Univ. Press, Cambridge (2000)MATHGoogle Scholar
  59. 59.
    Medková, D.: Convergence of the Neumann series in BEM for the Neumann problem of the Stokes system. Acta Appl. Math. 116, 281–304 (2011)MathSciNetMATHCrossRefGoogle Scholar
  60. 60.
    Medkov\(\acute{a}\), D.: Integral representation of a solution of the Neumann problem for the Stokes system. Numer. Algorithms 54, 459–484 (2010)MathSciNetMATHCrossRefGoogle Scholar
  61. 61.
    Mikhailov, S.E.: Traces, extensions and co-normal derivatives for elliptic systems on Lipschitz domains. J. Math. Anal. Appl. 378, 324–342 (2011)MathSciNetMATHCrossRefGoogle Scholar
  62. 62.
    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
  63. 63.
    Mitrea, D., Mitrea, M., Taylor, M.: Layer potentials, the Hodge Laplacian and global boundary problems in non-smooth Riemannian manifolds. Mem. Am. Math. Soc. 150, 713 (2001)MathSciNetGoogle Scholar
  64. 64.
    Mitrea, M., Monniaux, S.: On the analyticity of the semigroup generated by the Stokes operator with Neumann-type boundary conditions on Lipschitz subdomains of Riemannian manifolds. Trans. Am. Math. Soc. 361, 3125–3157 (2009)MathSciNetMATHCrossRefGoogle Scholar
  65. 65.
    Mitrea, M., Monniaux, S., Wright, M.: The Stokes operator with Neumann boundary conditions in Lipschitz domains. J. Math. Sci. (N.Y.) 176(3), 409–457 (2011)MathSciNetCrossRefGoogle Scholar
  66. 66.
    Mitrea, M., Taylor, M.: Boundary layer methods for Lipschitz domains in Riemannian manifolds. J. Funct. Anal. 163, 181–251 (1999)MathSciNetMATHCrossRefGoogle Scholar
  67. 67.
    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)MathSciNetMATHCrossRefGoogle Scholar
  68. 68.
    Mitrea, M., Taylor, M.: Navier–Stokes equations on Lipschitz domains in Riemannian manifolds. Math. Ann. 321, 955–987 (2001)MathSciNetMATHCrossRefGoogle Scholar
  69. 69.
    Mitrea, M., Taylor, M.: Sobolev and Besov space estimates for solutions to 2nd order PDE on Lipschitz domains in manifolds with Dini or Hölder continuous metric tensors. Commun. Partial Differ. Equ. 30, 1–37 (2005)MathSciNetMATHCrossRefGoogle Scholar
  70. 70.
    Mitrea, M., Wright, M.: Boundary value problems for the Stokes system in arbitrary Lipschitz domains. Astérisque 344, viii+241 pp (2012)MathSciNetGoogle Scholar
  71. 71.
    Nakamori, K., Suyama, Y.: On a nonlinear boundary problem for the equations Δu = 0 and Δu = f(x,y) (Esperanto). Mem. Fac. Sci. Kyǔsyǔ Univ. A. 5, 99–106 (1950)MathSciNetGoogle Scholar
  72. 72.
    Power, H., Wrobel, L.C.: Boundary Integral Methods in Fluid Mechanics. WIT Press: Computational Mechanics Publ. Southampton (1995)Google Scholar
  73. 73.
    Reidinger, B., Steinbach, O.: A symmetric boundary element method for the Stokes problem in multiple connected domains. Math. Methods Appl. Sci. 26, 77–93 (2003)MathSciNetMATHCrossRefGoogle Scholar
  74. 74.
    Russo, R.: On Stokes’ problem. In: Rannacher, R., Sequeira, A. (eds.) Advances in Mathematical Fluid Mechanics, pp. 473–511. Springer, Berlin (2010)CrossRefGoogle Scholar
  75. 75.
    Shen, Z.: The L p boundary value problems on Lipschitz domains. Adv. Math. 216, 212–254 (2007)MathSciNetMATHCrossRefGoogle Scholar
  76. 76.
    Starita, G., Tartaglione, A.: On the traction problem for the Stokes system. Math. Models Methods Appl. Sci. 12, 813–834 (2002)MathSciNetMATHCrossRefGoogle Scholar
  77. 77.
    Steinbach, O.: A note on the ellipticity of the single layer potential in two-dimensional elastostatics. J. Math. Anal. Appl. 294, 1–6 (2004)MathSciNetMATHCrossRefGoogle Scholar
  78. 78.
    Steinbach, O., Wendland, W.L.: On C. Neumann’s method for 2nd-order elliptic systems in domains with non-smooth boundaries. J. Math. Anal. Appl. 262, 733–748 (2001)MathSciNetMATHCrossRefGoogle Scholar
  79. 79.
    Toft, J.: Continuity properties for modulation spaces, with applications to pseudo-differential calculus-I. J. Funct. Anal. 207, 399–429 (2004)MathSciNetMATHCrossRefGoogle Scholar
  80. 80.
    Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. North-Holland Publ. Co. Amsterdam (1978)Google Scholar
  81. 81.
    Varnhorn, W.: The Stokes Equations. Akademie Verlag, Berlin (1994)MATHGoogle Scholar
  82. 82.
    Verchota, G.C.: Layer potentials and boundary value problems for Laplace’s equation in Lipschitz domains. J. Funct. Anal. 59, 572–611 (1984)MathSciNetMATHCrossRefGoogle Scholar
  83. 83.
    Wegert, E., Khimshiashvili, G., Spitkovsky, I.: Nonlinear transmission problems. International symposium on differential equations and mathematical physics (Tbilisi, 1997). Mem. Differ. Equ. Math. Phys. 12, 223–230 (1997)MathSciNetMATHGoogle Scholar
  84. 84.
    Wong, M.W.: An Introduction to Pseudo-Differential Operators. World Sci., Singapore (1991)MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  • Mirela Kohr
    • 1
  • Massimo Lanza de Cristoforis
    • 2
  • Wolfgang L. Wendland
    • 3
  1. 1.Faculty of Mathematics and Computer ScienceBabeş-Bolyai UniversityCluj-NapocaRomania
  2. 2.Dipartimento di MatematicaUniversità di PadovaPadovaItaly
  3. 3.Institut für Angewandte Analysis und Numerische SimulationUniversität StuttgartStuttgartGermany

Personalised recommendations