Skip to main content
SpringerLink
Log in

Continuous and variable branching asymptotics

  • Published:
Journal of Pseudo-Differential Operators and Applications Aims and scope Submit manuscript

Abstract

The regularity of solutions to elliptic equations on a manifold with singularities, say, an edge, can be formulated in terms of asymptotics in the distance variable \(r>0\) to the singularity. In simplest form such asymptotics turn to a meromorphic behaviour under applying the Mellin transform on the half-axis. Poles, multiplicity, and Laurent coefficients form a system of asymptotic data which depend on the specific operator. Moreover, these data may depend on the variable \(y\) along the edge. We then have \(y\)-dependent families of meromorphic functions with variable poles, jumping multiplicities and a discontinuous dependence of Laurent coefficients on \(y.\) We study here basic phenomena connected with such variable branching asymptotics, formulated in terms of variable continuous asymptotics with a \(y\)-wise discrete behaviour.

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. Bennish, J.: Asymptotics for elliptic boundary value problems for systems of pseudo-differential equations. J. Math. Anal. Appl. 179, 417–445 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bennish, J.: Variable discrete asymptotics of solutions to elliptic boundary-value problems, Math. Res., vol. 100. In: Differential Equations, Asymptotic Analysis, and Mathematical Physics. pp. 26–31, Akademie Verlag, Berlin (1997)

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

    Article  MATH  MathSciNet  Google Scholar 

  4. Chang, D.-C., Habal, N., Schulze, B.-W.: 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 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  5. Dorschfeldt, C.: Algebras of Pseudo-Differential Operators Near Edge and Corner Singularities, Math. Res. vol. 102. Akademie Verlag, Berlin (1998)

    Google Scholar 

  6. Egorov, J.V., Schulze, B.-W.: Pseudo-Differential Operators, Singularities, Applications, Oper. Theory Adv. Appl. Vol. 93. Birkhäuser Verlag, Basel (1997)

    Book  Google Scholar 

  7. Eskin, G.I.: Boundary value problems for elliptic pseudodifferential equations, Translation of Nauka, Moskva, 1973, Math. Monographs, vol. 52, Amer. Math. Soc., Providence, Rhode Island (1980)

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

    MATH  MathSciNet  Google Scholar 

  9. Gohberg, I.C., Sigal, E.I.: An operator generalization of the logarithmic residue theorem and the theorem of Rouché. Math. USSR Sbornik 13(4), 603–625 (1971)

    Article  Google Scholar 

  10. Gramsch, B.: Meromorphie in der Theorie von Fredholmoperatoren mit Anwendungen auf elliptische Differentialgleichungen. Math. Ann. 188, 97–112 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  11. Hirschmann, T.: Functional analysis in cone and edge Sobolev spaces. Ann. Global Anal. Geom. 8(2), 167–192 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  12. Hörmander, L.: The Analysis of Linear Partial Differential Operators, vol. 1. Springer, New York (1983)

    Google Scholar 

  13. Kapanadze, D., Schulze, B.-W.: Crack Theory and Edge singularities. Kluwer Academic Publ, Dordrecht (2003)

    Book  MATH  Google Scholar 

  14. Kapanadze, D., Schulze, B.-W., Seiler, J.: Operators with singular trace conditions on a manifold with edges. Integr. Equ. Oper. Theory 61, 241–279 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  15. Kumano-go, H.: Pseudo-differential operators. The MIT Press, Cambridge (1981)

    Google Scholar 

  16. Lyu, X.: Asymptotics in weighted corner spaces. Asian Euro. J. Math., 7(3), 1450050-1-1450050-36 (2014)

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

    Article  MATH  MathSciNet  Google Scholar 

  18. Rempel, S., Schulze, B.-W.: Branching of asymptotics for elliptic operators on manifolds with edges. In: Proceedings of Partial Differential Equations. Banach Center Publ. 19, PWN Polish Scientific Publisher, Warsaw (1984)

  19. Rempel, S., Schulze, B.-W.: Mellin symbolic calculus and asymptotics for boundary value problems, Seminar Analysis 1984/1985. Karl-Weierstrass Institut, Berlin (1985)

    Google Scholar 

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

  21. Schäfer, H.H.: Topological Vector Spaces. Graduate Texts in Mathematics. Springer, New York, Heidelberg, Berlin (1986)

  22. Schapira, P.: Théorie des Hyperfonctions, vol. 126 of Springer Lecture Notes in Math. Springer, Berlin, Heidelberg (1970)

    Google Scholar 

  23. Schrohe, E., Schulze, B.-W.: A symbol algebra for pseudodifferential boundary value problems on manifolds with edges. In: Math Research, vol. 100, Differential Equations, Asymptotic Analysis, and Mathematical Physics. pp. 292–324, Akademie Verlag, Berlin (1997)

  24. Schrohe, E., Schulze, B.-W.: Edge-degenerate boundary value problems on cones. In: Proceedings of Evolution Equations and their Applications in Physical and Life Sciences. Bad Herrenalb, Karlsruhe (2000)

  25. Schulze, B.-W.: Regularity with continuous and branching asymptotics for elliptic operators on manifolds with edges. Integr. Equs. Oper. Theory 11, 557–602 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  26. Schulze, B.-W.: Corner Mellin operators and reduction of orders with parameters. Ann. Sci. Norm. Sup. Pisa Cl. Sci. 16(1), 1–81 (1989)

    MATH  MathSciNet  Google Scholar 

  27. Schulze, B.-W.: Pseudo-differential operators on manifolds with edges, Teubner-Texte zur Mathematik, Symposium on Partial Differential Equations. Holzhau 1988, 112, Leipzig, pp. 259–287 (1989)

  28. Schulze, B.-W.: Pseudo-differential operators on manifolds with singularities. North-Holland, Amsterdam (1991)

    MATH  Google Scholar 

  29. Schulze, B.-W.: The variable discrete asymptotics in pseudo-differential boundary value problems I, Advances in Partial Differential Equations. Pseudo-Differential Calculus and Mathematical Physics. Akademie Verlag, Berlin, pp. 9–96 (1994)

  30. Schulze, B.-W.: The variable discrete asymptotics in pseudo-differential boundary value problems II, Advance in Partial Differential Equations Boundary Value Problems, Schrödinger Operators, Deformation Quantization. pp. 9–69, Akademie Verlag, Berlin (1995)

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

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  33. Schulze, B.-W., Tepoyan, L.: The Singular Functions of Branching Edge Asymptotics, Operator Theory: Advances and Applications, vol. 231, pp. 27–53. Springer, Basel (2013)

  34. Schulze, B.-W., Volpato, A.: Branching asymptotics on manifolds with edge. J. Pseudo Differ. Oper. Appl. 1, 433–493 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  35. Schulze, B.-W., Wong, M.W.: Mellin and Green operators of the corner calculus. J. Pseudo Differ. Oper. Appl. 2(4), 467–507 (2011)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  37. Seiler, J.: The cone algebra and a kernel characterization of Green operators. In: Gil, J., Grieser, D., Lesch, M. (eds) Oper. Theory Adv. Appl. 125 Adv. in Partial Differential Equations, Approaches to Singular Analysis. pp. 1–29, Birkhäuser, Basel (2001)

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

    Google Scholar 

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

    MathSciNet  Google Scholar 

  40. Witt, I.: On the factorization of meromorphic Mellin symbols. In: Albeverio, S., Demuth, M., Schrohe, E., Schulze, B.-W. (eds) Oper. Theory: Adv. Appl. 138, Advances in Partial Differential Equations Parabolicity, Volterra Calculus, and Conical Singularites. pp 279–306, Birkhäuser Verlag, Basel (2002)

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, University of Potsdam, Am Neuen Palais 10, 14469, Potsdam, Germany

    M. Hedayat Mahmoudi & B.-W. Schulze

  2. Department of Mathematics and Mechanics, Yerevan State University, A. Manoogian str. 1, 0025, Yerevan, Armenia

    L. Tepoyan

Authors
  1. M. Hedayat Mahmoudi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. B.-W. Schulze
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. L. Tepoyan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Hedayat Mahmoudi.

Additional information

Research supported by DAAD.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hedayat Mahmoudi, M., Schulze, BW. & Tepoyan, L. Continuous and variable branching asymptotics. J. Pseudo-Differ. Oper. Appl. 6, 69–112 (2015). https://doi.org/10.1007/s11868-015-0110-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11868-015-0110-3

Keywords

Mathematics Subject Classification

Search

Navigation