Skip to main content
Log in

The Dirichlet-to-Neumann Operator on Exterior Domains

  • Published:
Potential Analysis Aims and scope Submit manuscript

Abstract

We define two versions of the Dirichlet-to-Neumann operator on exterior domains and study convergence properties when the domain is truncated.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Alpay, D., Behrndt, J.: Generalized Q-functions and Dirichlet-to-Neumann maps for elliptic differential operators. J. Funct. Anal. 257, 1666–1694 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Arendt, W., Batty, C.J.K.: Domination and ergodicity for positive semigroups. Proc. Amer. Math. Soc. 114, 743–747 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  3. Arendt, W., Daners, D.: The Dirichlet problem by variational methods. Bull. London Math. Soc. 40, 51–56 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Arendt, W., Elst, A.F.M. ter: The Dirichlet-to-Neumann operator on rough domains. J. Diff. Eq. 251, 2100–2124 (2011)

    Article  MATH  Google Scholar 

  5. Arendt, W., Elst, A.F.M. ter: From forms to semigroups. In: Arendt, W., Ball, J.A., Behrndt, J., Förster, K.-H., Mehrmann, V., Trunk, C. (eds.) Spectral Theory, Mathematical System Theory, Evolution Equations, Differential and Difference Equations, pp 47–70. Birkäuser, Basel (2012)

  6. Sectorial forms and degenerate differential operators. J. Operator Theory 67, 33–72 (2012)

  7. Arendt, W., Elst, A. F. M. ter, Kennedy, J. B., Sauter, M.: The Dirichlet-to-Neumann operator via hidden compactness. J. Funct. Anal. 266, 1757–1786 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. Arendt, W., Mazzeo, R.: Friedlander’s eigenvalue inequalities and the Dirichlet-to-Neumann semigroup. Commun. Pure Appl. Anal. 11, 2201–2212 (2012)

    Article  MathSciNet  Google Scholar 

  9. Arendt, W. Heat kernels, 2006, Internet Seminar. http://tulka.mathematik.uni-ulm.de/2005/lectures/internetseminar.pdf

  10. Behrndt, J., Elst, A. F. M. ter: Dirichlet-to-Neumann maps on bounded Lipschitz domains, 2014. arXiv:1403.3167

  11. Behrndt, J., Langer, M.: Elliptic operators, Dirichlet-to-Neumann maps and quasi boundary triples. In: Operator methods for boundary value problems, London Math. Soc. Lecture Note Ser. 404, 121–160. Cambridge Univ. Press, Cambridge (2012)

  12. Brown, B.M., Marletta, M.: Spectral inclusion and spectral exactness for PDEs on exterior domains. IMA J. Numer. Anal. 24, 21–43 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  13. Behrndt, J., Rohleder, J.: An inverse problem of Calderón type with partial data. Comm. Partial Differential Equations 37, 1141–1159 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Brézis, H.: Analyse fonctionnelle, théorie et applications, Collection Mathématiques appliquées pour la maîtrise, Masson, Paris etc. (1983)

  15. Dautray, R., Lions, J.L.: Mathematical analysis and numerical methods for science and technology, vol 1. Springer, Berlin (1990)

    Book  Google Scholar 

  16. Gesztesy, F., Mitrea, M.: Generalized Robin boundary conditions, Robin-to-Dirichlet maps, and Krein-type resolvent formulas for Schrödinger operators on bounded Lipschitz domains. In: Perspectives in partial differential equations, harmonic analysis and applications, Proc. Sympos. Pure Math. 79 105–173, Amer. Math. Soc., Providence, RI (2008)

  17. Gesztesy, F., Mitrea, M., Zinchenko, M.: Variations on a theme of Jost and Pais. J. Funct. Anal. 253, 399–448 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  18. Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order, 2nd edn. Springer, Berlin (1983)

    Book  MATH  Google Scholar 

  19. Helms, L.L.: Potential theory, 2nd edn. Springer, London (2014)

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  21. Lu, G., Ou, B.: A Poincaré inequality on n and its application to potential fluid flows in space. Comm. Appl. Nonlinear Anal. 12, 1–24 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  22. Marletta, M.: Eigenvalue problems on exterior domains and Dirichlet to Neumann maps. J. Comput. Appl. Math. 171, 367–391 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  23. Malý, J., Ziemer, W.P.: Fine regularity of solutions of elliptic partial differential equations, Math. Surveys and Monographs 51, Amer. Math. Soc., Providence, RI (1997)

  24. Maz’ja, V.G.: Sobolev spaces. Springer Series in Soviet Mathematics. Springer, Berlin (1985)

    Google Scholar 

  25. Nagel, R. (ed.): One-parameter semigroups of positive operators, Lecture Notes in Mathematics, vol. 1184. Springer, Berlin (1986)

  26. Nittka, R.: Regularity of solutions of linear second order elliptic and parabolic boundary value problems on Lipschitz domains. J. Differential Equations 251, 860–880 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  27. Ouhabaz, E.-M.: Invariance of closed convex sets and domination criteria for semigroups. Potential Anal. 5, 611–625 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  28. Ouhabaz, E.-M.: Analysis of heat equations on domains, vol. 31 of London Mathematical Society Monographs Series. Princeton University Press, Princeton (2005)

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to A. F. M. ter Elst.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Arendt, W., ter Elst, A.F.M. The Dirichlet-to-Neumann Operator on Exterior Domains. Potential Anal 43, 313–340 (2015). https://doi.org/10.1007/s11118-015-9473-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11118-015-9473-6

Keywords

Mathematics Subject Classification (2010)

Navigation