Calculus on a Manifold with Edge and Boundary

Article

Abstract

We study elements of the calculus of boundary value problems in a variant of Boutet de Monvel’s algebra (Acta Math 126:11–51, 1971) on a manifold N with edge and boundary. If the boundary is empty then the approach corresponds to Schulze (Symposium on partial differential equations (Holzhau, 1988), BSB Teubner, Leipzig, 1989) and other papers from the subsequent development. For non-trivial boundary we study Mellin-edge quantizations and compositions within the structure in terms a new Mellin-edge quantization, compared with a more traditional technique. Similar structures in the closed case have been studied in Gil et al. (Osaka J Math 37:221–260, 2000).

Keywords

Pseudo-differential operators Manifold with edge Boutet de Monvel’s algebra Mellin quantization 

References

  1. 1.
    Boutet de Monvel, L.: Boundary problems for pseudo-differential operators. Acta Math. 126, 11–51 (1971)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Chang, D.-C.: Corner spaces and Mellin quantization. JNCA 19(2), 179–185 (2018)Google Scholar
  3. 3.
    Chang, D.-C., Qian, T., Schulze, B.-W.: Corner boundary value problems. Complex Anal. Oper. Theory 9(5), 1157–1210 (2014).  https://doi.org/10.1007/s11785-014-0424-9 MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Chang, D.-C., Habal, N., Schulze, B.-W.: The edge algebra structure of the Zaremba problem. J. Pseudo-Differ. Oper. Appl. 5, 69–155 (2014).  https://doi.org/10.1007/s11868-013-0088-7 MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Dorschfeldt, C.: Algebras of Pseudo-Differential Operators Near Edge and Corner Singularities (Mathematical Research), vol. 102. Wiley-VCH, Berlin (1998)MATHGoogle Scholar
  6. 6.
    Egorov, J.V., Schulze, B.-W.: Pseudo-Differential Operators, Singularities, Applications, Operator Theory: Advances and Applications, vol. 93. Birkhäuser Verlag, Basel (1997)CrossRefMATHGoogle Scholar
  7. 7.
    Gil, J.B., Schulze, B.W., Seiler, J.: Holomorphic operator-valued symbols for edge-degenerate pseudo-differential operators, differential equations, asymptotic analysis and mathematical physics. In: Demuth, M., et al. (eds.) Mathematical Research, vol. 100, pp. 113–137. Akademic Verlag, Berlin (1997)Google Scholar
  8. 8.
    Gil, J.B., Schulze, B.-W., Seiler, J.: Cone pseudodifferential operators in the edge symbolic calculus. Osaka J. Math. 37, 221–260 (2000)MathSciNetMATHGoogle Scholar
  9. 9.
    Grubb, G.: Functional Calculus of Pseudo-Differential Boundary Problems, 2nd edn. Birkhäuser Verlag, Boston (1996)CrossRefMATHGoogle Scholar
  10. 10.
    Harutyunyan, G., Schulze, B.-W.: Elliptic Mixed, Transmission and Singular Crack Problems. European Mathematical Society, Zürich (2008)MATHGoogle Scholar
  11. 11.
    Hedayat Mahmoudi, M., Schulze, B.-W.: A new approach to the second order edge calculus. J. Pseudo Differ. Oper. Appl (2017).  https://doi.org/10.1007/s11868-017-0191-2 Google Scholar
  12. 12.
    Hedayat Mahmoudi, M., Schulze, B.-W., Tepoyan, L.: Continuous and variable branching asymptotics. J. Pseudo Differ. Oper. Appl. 6(1), 69–112 (2015)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Jarchow, H.: Locally Convex Spaces, Mathematische Leitfäden. B.G. Teubner, Stuttgart (1981)CrossRefMATHGoogle Scholar
  14. 14.
    Kapanadze, D., Schulze, B.-W.: Crack Theory and Edge Singularities. Kluwer Academic Publishers, Dordrecht (2003)CrossRefMATHGoogle Scholar
  15. 15.
    Khalil, S., Schulze, B.W.: Boundary problems on a manifold with edge. AEJM 1, 1750087 (2017).  https://doi.org/10.1142/S1793557117500875. (43 pages)MathSciNetMATHGoogle Scholar
  16. 16.
    Krainer, T.: Parabolic Pseudodifferential Operators and Long-Time Asymptotics of Solutions, Ph.D. thesis, University of Potsdam (2000)Google Scholar
  17. 17.
    Krainer, T.: The calculus of Volterra Mellin pseudo-differential operators with operator-valued symbols, operator theory: advances and applications. In: Albeverio, S., Demuth, M., Schrohe, E., Schulze, B.-W. (eds.) Advanced in Partial Differential Equations “Parabolicity, Volterra Calculus, and Conical Singularities”, vol. 138, pp. 47–91. Birkhäuser Verlag, Basel (2002)CrossRefGoogle Scholar
  18. 18.
    Rempel, S., Schulze, B.-W.: Index Theory of Elliptic Boundary Problems. Akademie-Verlag, Berlin (1982); North Oxford Academic Publishing Company, Oxford, 1985 (Transl. to Russian: Mir, Moscow, 1986)Google Scholar
  19. 19.
    Rempel, S., Schulze, B.-W.: Pseudo-Differential and Mellin Operators in Spaces with Conormal Singularity (Boundary Symbols) Report R-Math-01/84. Karl-Weierstrass Institut, Berlin (1984)MATHGoogle Scholar
  20. 20.
    Rempel, S., Schulze, B.-W.: Asymptotics for Elliptic Mixed Boundary Problems (Pseudo-Differential and Mellin Operators in Spaces with Conormal Singularity) Mathematics Research, vol. 50. Akademie-Verlag, Berlin (1989)MATHGoogle Scholar
  21. 21.
    Schrohe, E., Schulze, B.-W.: A symbol algebra for pseudodifferential boundary value problems on manifolds with edges. In: Differential Equations, Asymptotic Analysis, and Mathematical Physics of Mathematical Research. Akademie Verlag, Berlin, vol. 100, pp. 292–324 (1997)Google Scholar
  22. 22.
    Schrohe, E., Schulze, B.-W.: Boundary value problems in Boutet de Monvel’s calculus for manifolds with conical singularities II, Advanced. In: Partial Differential Equations “Boundary Value Problems, Schrödinger Operators, Deformation Quantization”. Akademie Verlag, Berlin, pp. 70–205 (1995)Google Scholar
  23. 23.
    Schrohe, E., Schulze, B.-W.: Operators in a pseudodifferential calculus for boundary value problems on manifolds with edges. Preprint MPI 96-74, Max-Planck-Institut, Bonn, 1996. In: Mennicken, R., Tretter , C. (eds.) IWOTA 95 Proceedings. Operator Theory: Advances and Applications, Differential and Integral Operators. Birkhäuser, Basel, vol. 102, pp. 257–285 (1997)Google Scholar
  24. 24.
    Schrohe, E.: Functional calculus and Fredholm criteria for boundary value problems on noncompact manifolds. In: Operator Theory, Advances and Applications. Proceeding Lambrecht December 1991, 255–269, Birkhäuser, Boston, Basel, vol. 57 (1992)Google Scholar
  25. 25.
    Schrohe, E., Schulze, B.-W.: Boundary Value Problems in Boutet de Monvel’s Calculus for Manifolds with conical Singularities I, Advanced in Partial Differential Equations “Pseudo-Differential Calculus and Mathematical Physics”, pp. 97–209. Akademie Verlag, Berlin (1994)MATHGoogle Scholar
  26. 26.
    Schulze, B.-W.: Pseudo-differential operators on manifolds with edges. Teubner-Texte zur Mathematik. In: Symposium on Partial Differential Equations (Holzhau, 1988). BSB Teubner, Leipzig, vol. 112, pp. 259–287 (1989)Google Scholar
  27. 27.
    Schulze, B.-W.: Topologies and invertibility in operator spaces with symbolic structures, Teubner-Texte zur Mathematik. In: Problems and Methods in Mathematical Physics, BSB Teubner, Leipzig, vol. 111, pp. 257–270 (1989)Google Scholar
  28. 28.
    Schulze, B.-W.: Pseudo-Differential Operators on Manifolds with Singularities. North-Holland, Amsterdam (1991)MATHGoogle Scholar
  29. 29.
    Schulze, B.-W.: Pseudo-Differential Boundary Value Problems, Conical Singularities, and Asymptotics. Akademie Verlag, Berlin (1994)MATHGoogle Scholar
  30. 30.
    Schulze, B.-W.: Boundary Value Problems and Singular Pseudo-Differential Operators. Wiley, Chichester (1998)MATHGoogle Scholar
  31. 31.
    Seiler, J.: Pseudodifferential Calculus on Manifolds with Non-compact Edges, Ph.D. thesis, University of Potsdam (1997)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of PotsdamPotsdamGermany

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