Skip to main content
SpringerLink
Log in

The Edge Algebra Structure of Boundary Value Problems

  • Published:
Annals of Global Analysis and Geometry Aims and scope Submit manuscript

Abstract

Boundary value problems for pseudodifferential operators (with orwithout the transmission property) are characterised as a substructureof the edge pseudodifferential calculus with constant discreteasymptotics. The boundary in this case is the edge and the inner normalthe model cone of local wedges. Elliptic boundary value problems fornoninteger powers of the Laplace symbol belong to the examples as wellas problems for the identity operator in the interior with a prescribednumber of trace and potential conditions. Transmission operators arecharacterised as smoothing Mellin and Green operators with meromorphicsymbols.

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. Agranovich, M. S. and Vishik, M. I.: Elliptic problems with parameter and parabolic problems of general type, Uspekhi Mat. Nauk 19(3) (1964), 53-161.

    Google Scholar 

  2. Atiyah, M. F. and Bott, R.: The index problem for manifolds with boundary, in: Coll. Differential Analysis, Tata Institute Bombay, Oxford University Press, Oxford, 1964, pp. 175-186.

    Google Scholar 

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

    Google Scholar 

  4. Brüning, J. and Lesch, M.: On the eta-invariant of certain nonlocal boundary value problems, Duke Math. J. 96(2) (1999), 425-468.

    Google Scholar 

  5. Dorschfeldt, Ch.: Algebras of Pseudo-Differential Operators near Edge and Corner Singularities, Math. Res. 102, Akademie Verlag, Berlin, 1998.

    Google Scholar 

  6. Egorov, Ju. V. and Schulze, B.-W.: Pseudo-Differential Operators, Singularities, Applications, Oper. Theor.: Adv. Appl. 93, Birkhäuser Verlag, Basel, 1997.

    Google Scholar 

  7. Eskin, G. I.: Boundary Value Problems for Elliptic Pseudodifferential Equations, Math. Monogr. 52, Amer. Math. Soc., Providence, RI, 1980 (Transl. of Nauka, Moscow, 1973).

    Google Scholar 

  8. Gil, J. B., Schulze, B.-W. and Seiler, J.: Holomorphic operator-valued symbols for edgedegenerate pseudo-differential operators, in: M. Demuth and B.-W. Schulze, (eds), Differential Equations, Asymptotic Analysis, and Mathematical Physics, Math. Res. 100, Akademie Verlag, Berlin, 1997, pp. 113-137.

    Google Scholar 

  9. Gil, J. B., Schulze, B.-Wand Seiler, J.: Cone pseudodifferential operators in the edge symbolic calculus, Osaka J. Math. 37 (2000), 219-258.

    Google Scholar 

  10. Gohberg, I. and Krupnik, N.: The algebra generated by the one-dimensional singular integral operators with piecewise continuous coefficients, Funktsional. Anal. i Prilozen. 4(3) (1970), 26-36.

    Google Scholar 

  11. Grubb, G.: Singular Green operators and their spectral asymptotics, Duke Math. J. 51 (1984), 477-528.

    Google Scholar 

  12. Grubb, G.: Pseudo-differential boundary value problems in Lp spaces, Comm. Partial Differential Equations 15 (1990), 289-340.

    Google Scholar 

  13. Grubb, G.: Functional Calculus of Pseudo-Differential Boundary Problems, second edition, Birkhäuser Verlag, Boston, MA, 1996.

    Google Scholar 

  14. Kapanadze, D. and Schulze, B.-W.: Crack theory and edge singularities, Preprint No. 2001/05-09, Institute for Mathematics, Potsdam, 2001.

    Google Scholar 

  15. Komech, A. I.: Elliptic boundary problems for pseudo-differential operators on manifolds with conical points, Mat. Sb. 86(2) (1971), 268-298.

    Google Scholar 

  16. Kondratyev, V. A.: Boundary value problems for elliptic equations in domains with conical points, Trudy Mosk. Mat. Obshch. 16 (1967), 209-292.

    Google Scholar 

  17. Lesch, M.: Differential Operators of Fuchs Type, Conical Singularities, and Asymptotic Methods, Teubner-Texte Math., B. G. Teubner, Stuttgart, 1997.

    Google Scholar 

  18. Myshkis, P. A.: On an algebra generated by two-sided pseudodifferential operators on a manifold, Uspekhi Mat. Nauk. 31(4) (1976), 269-270 [in Russian].

    Google Scholar 

  19. Plamenevskij, B. A.: On the boundedness of singular integrals in spaces with weight, Mat. Sb. 76(4) (1968), 573-592.

    Google Scholar 

  20. Rempel, S. and Schulze, B.-W.: Index Theory of Elliptic Boundary Problems, Akademie Verlag, Berlin, 1982.

    Google Scholar 

  21. Rempel, S. and Schulze, B.-W.: Parametrices and boundary symbolic calculus for elliptic boundary problems without transmission property, Math. Nachr. 105 (1982), 45-149.

    Google Scholar 

  22. Rempel, S. and Schulze, B.-W.: Complete Mellin and Green symbolic calculus in spaces with conormal asymptotics, Ann. Global Anal. Geom. 4(2), 1986, 137-224.

    Google Scholar 

  23. Schrohe, E. and Schulze, B.-W.: Boundary value problems in Boutet de Monvel's calculus for manifolds with conical singularities I, in: M. Demuth, E. Schrohe and B.-W. Schulze (eds), Advances in Partial Differential Equations (Pseudo-Differential Calculus and Mathematical Physics), Akademie Verlag, Berlin, 1994, pp. 97-209.

    Google Scholar 

  24. Schrohe, E. and Schulze, B.-W.: Boundary value problems in Boutet de Monvel's calculus for manifolds with conical singularities II, in: M. Demuth, E. Schrohe and B.-W. Schulze (eds), Advances in Partial Differential Equations (Boundary Value Problems, Schrödinger Operators, Deformation Quantization), Akademie Verlag, Berlin, 1995, pp. 70-205.

    Google Scholar 

  25. Schulze, B.-W.: Mellin representations of pseudo-differential operators on manifolds with corners, Ann. Glob. Anal. Geom. 8(3) (1990), 261-297.

    Google Scholar 

  26. Schulze, B.-W., Pseudo-differential Operators on Manifolds with Singularities, North-Holland, Amsterdam, 1991.

    Google Scholar 

  27. Schulze, B.-W.: The Mellin pseudo-differential calculus on manifolds with corners, in: B.-W. Schulze and H. Triebel (eds), Analysis in Domains and on Manifolds with Singularities, Breitenbrunn, 1990, Teubner-Texte Math. 131, Teubner, Leipzig, 1992, pp. 208-289.

    Google Scholar 

  28. Schulze, B.-W.: Pseudo-Differential Boundary Value Problems, Conical Singularities, and Asymptotics, Akademie Verlag, Berlin, 1994.

    Google Scholar 

  29. Schulze, B.-W.: Boundary Value Problems and Singular Pseudo-Differential Operators, Wiley, Chichester, 1998.

    Google Scholar 

  30. Schulze, B.-W.: An algebra of boundary value problems not requiring Shapiro-Lopatinskij conditions, J. Funct. Anal. 179 (2001), 374-408.

    Google Scholar 

  31. Schulze, B.-W.: Operator algebras with symbol hierarchies on manifolds with singularities, in: J. Gil, D. Grieser and M. Lesch (eds), Advances in Partial Differential Equations (Approaches to Singular Analysis, Oper. Theor: Adv. Appl., Birkhäuser Verlag, Basel, 2001, pp. 167-207.

    Google Scholar 

  32. Schulze, B.-W. and Seiler, J.: Pseudodifferential boundary value problems with global projection conditions, Preprint No. 2002/04, Institute for Mathematics, Potsdam, 2002.

    Google Scholar 

  33. Seiler, J.: Mellin and Green pseudodifferential operators associated with non-compact edges, Integral Equations Oper. Theor. 31 (1998), 214-245.

    Google Scholar 

  34. Seiler, J.: Pseudodifferential calculus on manifolds with non-compact edges, Ph.D. Thesis, University of Potsdam, 1998.

  35. Seiler, J.: The cone algebra and a kernel characterization of Green operators, in: J. Gil, D. Grieser and M. Lesch (eds), Advances in Partial Differential Equations (Approaches to Singular Analysis), Oper. Theor: Adv. Appl., Birkhäuser Verlag, Basel, 2001, pp. 1-29.

    Google Scholar 

  36. Vishik, M. I. and Eskin, G. I.: Convolution equations in a bounded region, Uspekhi Mat. Nauk 20(3) (1965), 89-152.

    Google Scholar 

  37. Vishik, M. I. and Eskin, G. I.: Convolution equations in bounded domains in spaces with weighted norms, Mat. Sb. 69(1) (1966), 65-110.

    Google Scholar 

  38. Widom, H.: Asymptotic Expansions for Pseudo-Differential Operators on Bounded Domains, Lecture Notes Math. 1152, Springer-Verlag, Berlin, 1985.

    Google Scholar 

  39. Witt, I.: A calculus for classical pseudo-differential operators with non-smooth symbols, Math. Nachr. 194 (1998), 239-284.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Mathematik, Universität Potsdam, Postfach 601553, 14415, Potsdam, Germany

    B. W. Schulze & J. Seiler

Authors
  1. B. W. Schulze
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. Seiler
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Schulze, B.W., Seiler, J. The Edge Algebra Structure of Boundary Value Problems. Annals of Global Analysis and Geometry 22, 197–265 (2002). https://doi.org/10.1023/A:1019939316595

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1019939316595

Search

Navigation