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Edge quantisation of elliptic operators

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Abstract

The ellipticity of operators on a manifold with edge is defined as the bijectivity of the components of a principal symbolic hierarchy \({\sigma=(\sigma_\psi,\sigma_\wedge)}\) , where the second component takes values in operators on the infinite model cone of the local wedges. In the general understanding of edge problems there are two basic aspects: Quantisation of edge-degenerate operators in weighted Sobolev spaces, and verifying the ellipticity of the principal edge symbol \({\sigma_\wedge}\) which includes the (in general not explicitly known) number of additional conditions of trace and potential type on the edge. We focus here on these questions and give explicit answers for a wide class of elliptic operators that are connected with the ellipticity of edge boundary value problems and reductions to the boundary. In particular, we study the edge quantisation and ellipticity for Dirichlet–Neumann operators with respect to interfaces of some codimension on a boundary. We show analogues of the Agranovich–Dynin formula for edge boundary value problems.

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References

  1. Atiyah, M.F., Bott, R.: The index problem for manifolds with boundary. In: Coll. Differential Analysis, Tata Institute Bombay, pp. 175–186. Oxford University Press, Oxford (1964)

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

    Article  MathSciNet  Google Scholar 

  3. Dines N., Harutjunjan G., Schulze B.W.: Mixed boundary value problems and parametrices in the edge calculus. Bull. Sci. Math. 131, 325–360 (2007)

    Article  MathSciNet  Google Scholar 

  4. Dines N., Schulze B.W.: Mellin-edge-representations of elliptic operators. Math. Meth. Appl. Sci. 28(18), 2133–2172 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  5. Egorov, Ju.V., Schulze, B.W.: Pseudo-differential operators, singularities, applications. In: Operator Theory: Advances and Applications, vol. 93. Birkhäuser, Basel (1997)

  6. Eskin, G.I.: Boundary value problems for elliptic pseudodifferential equations. Transl. Math. Monogr., vol. 52. American Mathematics Society, Providence (1980)

  7. Gil J.B., Krainer T., Mendoza G.: Geometry and spectra of closed extensions of elliptic cone operators. Can. J. Math. 59(4), 742–794 (2007)

    MathSciNet  Google Scholar 

  8. Gil J.B., Krainer T., Mendoza G.: Resolvents of elliptic cone operators. J. Funct. Anal. 241, 1–55 (2006)

    Article  MATH  MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  11. Grubb G.: Functional Calculus of Pseudo-Differential Boundary Problems, 2nd edn. Birkhäuser, Boston (1996)

    Google Scholar 

  12. Kapanadze, D., Schulze, B.W.: Crack theory and edge singularities. In: Mathematics and its Applications, vol. 561. Kluwer, Dordrecht (2003)

  13. Kondratyev V.A.: Boundary problems for elliptic equations in domains with conical or angular points. Trans. Mosc. Math. Soc. 16, 227–313 (1967)

    Google Scholar 

  14. Kondratyev V.A., Oleynik O.A.: Boundary problems for partial differential equations on non-smooth domains. Usp. Mat. Nauk. 38(2), 3–76 (1983)

    Google Scholar 

  15. Krainer T.: On the inverse of parabolic boundary value problems for large times. Jpn. J. Math. 30(1), 91–163 (2004)

    MATH  MathSciNet  Google Scholar 

  16. Krainer T., Schulze B.W.: Long-time asymptotics with geometric singularities in the spatial variables. Contemp. Math. 364, 103–126 (2004)

    MathSciNet  Google Scholar 

  17. Lauter, R., Nistor, V.: Analysis of geometric operators on open manifolds: a groupoid approach. In: Landsman, N., Pflaum, M., Schlichenmaier, M. (eds.) Quantization of Singular Symplectic Quotients: Progress in Mathematics, vol. 198, pp. 181–229. Birkhäuser, Basel (2001)

  18. Liu X., Schulze B.W.: Boundary value problems in edge representation. Math. Nachr. 280(5-6), 581–621 (2007)

    Article  MathSciNet  Google Scholar 

  19. Loya, P.: Index theory of Dirac operators on manifolds with corners up to codimension two. In: Gil, J., Krainer, T., Witt, I. (eds.) Advances in Partial Differential Equations (Aspects of Boundary Problems in Analysis and Geometry), Oper. Theory Adv. Appl., pp. 131–169. Birkhäuser, Basel (2004)

  20. Nazaikinskij, V., Savin, A., Schulze, B.W., Sternin, B.Ju.: Elliptic theory on manifolds with nonisolated singularities: II. Products in elliptic theory on manifolds with edges. Preprint 2002/15, Institut für Mathematik, Potsdam (2002)

  21. Nazaikinskij, V., Savin, A., Schulze, B.W., Sternin, B.Ju.: On the homotopy clssification of elliptic operators on manifolds with edges. Preprint 2004/16, Institut für Mathematik, Potsdam (2004)

  22. Nistor V.: Higher index theorems and the boundary map in cyclic homology. Documenta 2, 263–295 (1997)

    MATH  MathSciNet  Google Scholar 

  23. Plamenevskij, B.A.: Algebras of Pseudo-Differential Operators. Nauka, Moscow (1986)

    Google Scholar 

  24. Rempel S., Schulze B.W.: Index Theory of Elliptic Boundary Problems. Akademie, Berlin (1982)

    MATH  Google Scholar 

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

  26. Schrohe, E., Schulze, B.W.: Boundary value problems in Boutet de Monvel’s calculus for manifolds with conical singularities II. In: Advances in Partial Differential Equations (Boundary Value Problems, Schrödinger Operators, Deformation Quantization), pp. 70–205. Akademie, Berlin (1995)

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

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

  29. Schulze, B.W.: Operators with symbol hierarchies and iterated asymptotics. Publ. RIMS, Kyoto Univ., vol. 38(4), pp. 735–802 (2004)

  30. Schulze, B.W.: Toeplitz operators, and ellipticity of boundary value problems with global projection conditions. In: Gil, J., Krainer, T., Witt, I. (eds.) Advances in Partial Differential Equations (Aspects of Boundary Problems in Analysis and Geometry), Oper. Theory Adv. Appl., pp. 342–429. Birkhäuser, Basel (2004)

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

    Google Scholar 

  32. 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 

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Authors and Affiliations

  1. Institute of Mathematics, Georg-August-University, Göttingen, 37073, Göttingen, Germany

    Nicoleta Dines

  2. School of Mathematics and Statistics, Wuhan University, 430072, Wuhan, People’s Republic of China

    Xiaochun Liu

  3. Institute of Mathematics, University of Potsdam, 14469, Potsdam, Germany

    Bert-Wolfgang Schulze

Authors
  1. Nicoleta Dines
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  2. Xiaochun Liu
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  3. Bert-Wolfgang Schulze
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Corresponding author

Correspondence to Xiaochun Liu.

Additional information

Communicated by A. Jüngel.

Nicoleta Dines and Bert-Wolfgang Schulze were supported by Chinese-German Cooperation Program “Partial Differential Equations”, NNSF of China and DFG of Germany. Xiaochun Liu was supported by NNSF of China through Grant No. 10501034, and Chinese-German Cooperation Program “Partial Differential Equations”, NNSF of China and DFG of Germany.

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Dines, N., Liu, X. & Schulze, BW. Edge quantisation of elliptic operators. Monatsh Math 156, 233–274 (2009). https://doi.org/10.1007/s00605-008-0058-y

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