Skip to main content
SpringerLink
Log in

Dirichlet-to-Neumann Operator and Zaremba Problem

  • Chapter
  • First Online:
Advances in Microlocal and Time-Frequency Analysis

Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

Abstract

Starting with the Zaremba problem for the Laplacian, a boundary value problem with jumping conditions from Dirichlet to Neumann data or also with discontinuous Dirichlet- or Neumann data, a reduction to the boundary in terms of Boutet de Monvel’s calculus gives rise to an interface problem which can be interpreted as a boundary value problem on the Neumann side for the Dirichlet-to-Neumann operator. This is a first order elliptic classical pseudo-differential operator on the boundary without the transmission property at the interface. A specific choice of edge quantization admits an interpretation within the edge calculus, and we apply the formalism of the edge algebra together with interface conditions.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. S. Behm, Pseudo-Differential Operators with Parameters on Manifolds with Edges, Ph-D thesis, Potsdam, 1995.

    Google Scholar 

  2. L. Boutet de Monvel, Boundary problems for pseudo-differential operators, Acta Math. 126 (1971), 11–51.

    Article  MathSciNet  Google Scholar 

  3. D.-C. Chang, N. Habal, and B.-W. Schulze, The edge algebra structure of the Zaremba problem, NCTS Preprints in Mathematics 2013-6-002, Taiwan, 2013. J. Pseudo-Differ. Oper. Appl. 5 (2014), 69–155, DOI: https://doi.org/10.1007/s11868-013-0088-7

  4. N. Dines, X. Liu, and B.-W. Schulze, Edge quantisation of elliptic operators, Monatshefte für Math. 156 (2009), 233–274.

    Article  MathSciNet  Google Scholar 

  5. G.I. Eskin, Boundary value problems for elliptic pseudodifferential equations, Transl. of Nauka, Moskva, 1973, Math. Monographs, Amer. Math. Soc. 52, Providence, Rhode Island 1980.

    Google Scholar 

  6. J.B. Gil, B.-W. Schulze, and J. Seiler, Holomorphic operator-valued symbols for edge-degenerate pseudo-differential operators, in “Differential Equations, Asymptotic Analysis and Mathematical Physics”, eds. M. Demuth et al. Mathematical Research, Vol. 100, Akademie Verlag (1997), pp. 113–137.

    Google Scholar 

  7. J.B. Gil, B.-W. Schulze, and J. Seiler, Cone pseudodifferential operators in the edge symbolic calculus, Osaka J. Math. 37 (2000), 221–260.

    MathSciNet  MATH  Google Scholar 

  8. G. Grubb, Functional calculus of pseudo-differential boundary problems, Second Edition, Birkhäuser Verlag, Boston, 1996.

    Book  Google Scholar 

  9. N. Habal, Operators on singular manifolds, Ph.D. thesis, University of Potsdam, 2013.

    Google Scholar 

  10. G. Harutyunyan and B.-W. Schulze, Elliptic mixed, transmission and singular crack problems, European Mathematical Soc., Zürich, 2008.

    Google Scholar 

  11. L. Hörmander, Pseudo-differential operators and non-elliptic boundary problems, Ann. of Math. 83, 1 (1966), 129–200.

    Article  MathSciNet  Google Scholar 

  12. S. Khalil and B.-W. Schulze, Boundary Problems on a Manifold with Edge, Asian-European Journal of Mathematics, AEJM, Vol. 10, No.2 (2017) 1750087 World Scientific Publ. Company DOI: https://doi.org/10.1142/S1793557117500875

  13. S. Khalil and B.-W. Schulze, Calculus on a manifold with edge and boundary, CAOT 13 (2019), 2627–2670, DOI: https://doi.org/10.1007/s11785-018-0800-y.

  14. S. Khalil and B.-W. Schulze, Boundary value problems in Boutet de Monvel’s calculus on manifolds with edge, Proc. Conf. “ Mathematics, Informatics in Natural Sciences and Engineering”, AMINSE, December 6–9, 2017, Tbilisi, Georgia, (eds. G. Jaiani, D. Natroshvili), Springer.

    Google Scholar 

  15. T. Krainer, The calculus of Volterra Mellin pseudo-differential operators with operator-valued symbols, Oper. Theory Adv. Appl. 138, Adv. in Partial Differential Equations “Parabolicity, Volterra Calculus, and Conical Singularities” (S. Albeverio, M. Demuth, E. Schrohe, and B.-W. Schulze, eds.), Birkhäuser Verlag, Basel, 2002, pp. 47–91.

    Google Scholar 

  16. X. Liu and B.-W. Schulze, Boundary Value Problems with Global Projection Conditions, Operator Theory, Advances and Applications, Vol. 265, Springer/Birkhäuser, Basel. 2018.

    Google Scholar 

  17. S. Rempel and B.-W. Schulze, Index theory of elliptic boundary problems, Akademie-Verlag, Berlin, 1982; North Oxford Acad. Publ. Comp., Oxford, 1985. (Transl. to Russian: Mir, Moscow, 1986).

    Google Scholar 

  18. S. Rempel and B.-W. Schulze, Asymptotics for elliptic mixed boundary problems (pseudo-differential and Mellin operators in spaces with conormal singularity), Math. Res. 50, Akademie-Verlag, Berlin, 1989.

    Google Scholar 

  19. E. Schrohe and B.-W. Schulze, Boundary value problems in Boutet de Monvel’s calculus for manifolds with conical singularities I, Adv. in Partial Differential Equations “Pseudo-Differential Calculus and Mathematical Physics”, Akademie Verlag, Berlin, 1994, pp. 97–209.

    Google Scholar 

  20. E. Schrohe and B.-W. Schulze, Boundary value problems in Boutet de Monvel’s calculus for manifolds with conical singularities II, Adv. in Partial Differential Equations “Boundary Value Problems, Schrödinger Operators, Deformation Quantization”, Akademie Verlag, Berlin, 1995, pp. 70–205.

    Google Scholar 

  21. E. Schrohe and B.-W. Schulze, A symbol algebra for pseudo-differential boundary value problems on manifolds with edges, in “Differential Equations, Asymptotic Analysis and Mathematical Physics”, eds. M. Demuth et al. Mathematical Research, Vol. 100, Akademie Verlag (1997), pp. 292–324.

    Google Scholar 

  22. E. Schrohe and B.-W. Schulze, Edge-degenerate boundary value problems on cones, Proc. “Evolution Equations and their Applications in Physical and Life Sciences”, Bad Herrenalb, Karlsruhe, 2000.

    Google Scholar 

  23. B.-W. Schulze, Topologies and invertibility in operator spaces with symbolic structures, Teubner-Texte zur Mathematik 111, “Problems and Methods in Mathematical Physics”, BSB Teubner, Leipzig, 1989, pp. 257–270.

    Google Scholar 

  24. B.-W. Schulze, Pseudo-differential operators on manifolds with edges, Teubner-Texte zur Mathematik 112, Symp. “Partial Differential Equations, Holzhau 1988”, BSB Teubner, Leipzig, 1989, pp. 259–287.

    Google Scholar 

  25. B.-W. Schulze, Pseudo-differential operators on manifolds with singularities, North-Holland, Amsterdam, 1991.

    MATH  Google Scholar 

  26. B.-W. Schulze, Pseudo-differential boundary value problems, conical singularities, and asymptotics, Akademie Verlag, Berlin, 1994.

    MATH  Google Scholar 

  27. B.-W. Schulze, Boundary value problems and singular pseudo-differential operators, J. Wiley, Chichester, 1998.

    MATH  Google Scholar 

  28. B.-W. Schulze and J. Seiler, Pseudodifferential boundary value problems with global projection conditions, J. Funct. Anal. 206, 2 (2004), 449–498.

    Article  MathSciNet  Google Scholar 

  29. B.-W. Schulze and J. Seiler, The edge algebra structure of boundary value problems, Ann. Glob. Anal. Geom. 22 (2002), 197–265.

    Article  MathSciNet  Google Scholar 

  30. B.-W. Schulze and J. Seiler, Truncation quantization in the edge calculus, (in preparation).

    Google Scholar 

  31. B.-W. Schulze and G. Wildenhain, Methoden der Potentialtheorie für Elliptische Differentialgleichungen beliebiger Ordnung, Akademie-Verlag, Berlin; Birkhäuser Verlag, Basel, 1977.

    Book  Google Scholar 

  32. J. Seiler, Continuity of edge and corner pseudo-differential operators, Math. Nachr. 205 (1999), 163–182.

    Article  MathSciNet  Google Scholar 

  33. J. Seiler, Pseudodifferential calculus on manifolds with non-compact edges, Ph.D. thesis, University of Potsdam, 1997.

    Google Scholar 

  34. S. Zaremba, Sur in probleme mixte relatif a l’équation de Laplace, Bull. de l’Académie des Sciences de Cracovie, Classe des Sciences Mathématiques et Naturelles, Series A, 1910, pp. 313–344.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Mathematik, Universität Potsdam, Potsdam, Germany

    B.-W. Schulze

Authors
  1. B.-W. Schulze
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to B.-W. Schulze .

Editor information

Editors and Affiliations

  1. Dipartimento di Matematica “G. Peano”, Università degli Studi di Torino, Torino, Italy

    Paolo Boggiatto

  2. Dipartimento di Matematica “G. Peano”, Università degli Studi di Torino, Torino, Italy

    Marco Cappiello

  3. Dipartimento di Matematica “G. Peano”, Università degli Studi di Torino, Torino, Italy

    Elena Cordero

  4. Dipartimento di Matematica “G. Peano”, Università degli Studi di Torino, Torino, Italy

    Sandro Coriasco

  5. Dipartimento di Matematica “G. Peano”, Università degli Studi di Torino, Torino, Italy

    Gianluca Garello

  6. Dipartimento di Matematica “G. Peano”, Università degli Studi di Torino, Torino, Italy

    Alessandro Oliaro

  7. Dipartimento di Matematica "G. Peano", Università degli Studi di Torino, Torino, Italy

    Jörg Seiler

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Schulze, BW. (2020). Dirichlet-to-Neumann Operator and Zaremba Problem. In: Boggiatto, P., et al. Advances in Microlocal and Time-Frequency Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-36138-9_24

Download citation

Publish with us

Policies and ethics

Search

Navigation