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The Iterative Structure of the Corner Calculus

  • B.-W. SchulzeEmail author
Chapter
Part of the Operator Theory: Advances and Applications book series (OT, volume 213)

Abstract

We give a brief survey on some new developments on elliptic operators on manifolds with regular geometric singularities. The material corresponds to an extended version of talks given by the author at the Conference “Elliptic and Hyperbolic Equations on Singular Spaces”, October 27–31, 2008, at the MSRI, University of California at Berkeley, see also [56], and at the 7th ISAAC Congress, July 13–18, 2009, at Imperial College London.

Keywords

Ellipticity on manifolds with singularities stratified spaces corner pseudo-differential operators principal symbolic hierarchies 

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References

  1. 1.
    S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I, Comm. Pure Appl. Math., 12 (1959), 623–727.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    M.S. Agranovich and M.I. Vishik, Elliptic problems with parameter and parabolic problems of general type, Uspekhi Mat. Nauk 19, 3 (1964), 53–161.zbMATHGoogle Scholar
  3. 3.
    M.F. Atiyah and R. Bott, The index problem for manifolds with boundary, Coll. Differential Analysis, Tata Institute Bombay, Oxford University Press, Oxford , 1964, pp. 175–186.Google Scholar
  4. 4.
    K. Bekka and D. Trotman, Metric properties of stratified sets, Manuscripta math. 111 (2003), 71–95.MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    L. Boutet de Monvel, Boundary problems for pseudo-differential operators, Acta Math. 126 (1971), 11–51.MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    D. Calvo, C.-I. Martin, and B.-W. Schulze, Symbolic structures on corner manifolds, RIMS Conf. dedicated to L. Boutet de Monvel on “Microlocal Analysis and Asymptotic Analysis”, Kyoto, August 2004, Keio University, Tokyo, 2005, pp. 22–35.Google Scholar
  7. 7.
    D. Calvo and B.-W. Schulze, Edge symbolic structure of second generation, Math. Nachr. 282 (2009), 348–367.MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    H.O. Cordes, A global parametrix for pseudo-differential operators over Rn, with applications, Reprint, SFB 72, Universit¨at Bonn, 1976.Google Scholar
  9. 9.
    S. Coriasco and B.-W. Schulze, Edge problems on configurations with model cones of different dimensions, Osaka J. Math. 43 (2006), 1–40.MathSciNetGoogle Scholar
  10. 10.
    N. Dines, Elliptic operators on corner manifolds, Ph.D. thesis, University of Potsdam, 2006.Google Scholar
  11. 11.
    N. Dines, X. Liu, and B.-W. Schulze, Edge quantisation of elliptic operators, Monatshefte f¨ur Math. 156 (2009), 233–274.Google Scholar
  12. 12.
    Ju.V. Egorov and B.-W. Schulze, Pseudo-differential operators, singularities, applications, Oper. Theory: Adv. Appl. 93, Birkh¨auser Verlag, Basel, 1997.Google Scholar
  13. 13.
    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
  14. 14.
    H.-J. Flad, R. Schneider, and B.-W. Schulze Asymptotic regularity of solutions of Hartree-Fock equations with Coulomb potential, Math. Meth. in the Appl. Sci. 31, 18 (2008), 2172–2201.zbMATHCrossRefGoogle Scholar
  15. 15.
    W. Fulton and R. MacPherson, Categorical framework for the study of singular spaces, Memoirs of the AMS 243 (1981).Google Scholar
  16. 16.
    J.B. Gil, Full asymptotic expansion of the heat trace for non-self-adjoint elliptic cone operators, Math. Nachr. 250 (2003), 25–57.MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    J.B. Gil and G. Mendoza, Adjoints of the elliptic cone operators, Amer. J. Math. 125,2 (2003), 357–408.MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    J.B. Gil, B.-W. Schulze, and J. Seiler, Cone pseudodifferential operators in the edge symbolic calculus, Osaka J. Math. 37 (2000), 219–258.MathSciNetGoogle Scholar
  19. 19.
    I.C. Gohberg and E.I. Sigal, An operator generalization of the logarithmic residue theorem and the theorem of Rouch´e, Math. USSR Sbornik 13, 4 (1971), 603–625.zbMATHCrossRefGoogle Scholar
  20. 20.
    I.C. Gohberg and N.G. Krupnik, The algebra generated by the one-dimensional singular integral operators with piecewise continuous coefficients, Funk. Anal. i Prilozen. 4, 3 (1970), 26–36.MathSciNetGoogle Scholar
  21. 21.
    G. Harutjunjan and B.-W. Schulze, The relative index for corner singularities, Integr. Equ. Oper. Theory 54, 3 (2006), 385–426.MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    G. Harutjunjan and B.-W. Schulze, The Zaremba problem with singular interfaces as a corner boundary value problem, Potential Analysis 25, 4 (2006), 327–369.MathSciNetCrossRefGoogle Scholar
  23. 23.
    G. Harutjunjan and B.-W. Schulze, Elliptic mixed, transmission and singular crack problems, European Mathematical Soc., Z¨urich, 2008.Google Scholar
  24. 24.
    T. Hirschmann, Functional analysis in cone and edge Sobolev spaces, Ann. Global Anal. Geom. 8, 2 (1990), 167–192.MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    L. H¨ormander, The analysis of linear partial differential operators, vol. 1 and 2, Springer-Verlag, New York, 1983.Google Scholar
  26. 26.
    P. Jeanquartier, Transformation de Mellin et d´eveloppements asymptotiques, Enseign. Math. (2) 25 (1979), 285–308.Google Scholar
  27. 27.
    D. Kapanadze and B.-W. Schulze, Crack theory and edge singularities, Kluwer Academic Publ., Dordrecht, 2003.Google Scholar
  28. 28.
    H.C. King and D. Trotman, Poincar´e-Hopf theorems on singular spaces, manuscript (2007).Google Scholar
  29. 29.
    V.A. Kondratyev, Boundary value problems for elliptic equations in domains with conical points, Trudy Mosk. Mat. Obshch. 16, (1967), 209–292.Google Scholar
  30. 30.
    T. Krainer, The calculus of Volterra Mellin pseudo-differential operators with operator-valued ymbols, Oper. Theory Adv. Appl. 138, Adv. in Partial DifferentialGoogle Scholar
  31. 31.
    Equations “Parabolicity, Volterra Calculus, and Conical Singularities” (Albeverio, S. and Demuth, M. and Schrohe, E. and Schulze, B.-W., eds.), Birkh¨auser Verlag, Basel, 2002, pp. 47–91.Google Scholar
  32. 32.
    T. Krainer, On the inverse of parabolic boundary value problems for large times, Japan. J. Math. 30, 1 (2004), 91–163.MathSciNetzbMATHGoogle Scholar
  33. 33.
    T. Krainer and B.-W. Schulze, Long-time asymptotics with geometric singularities in the spatial variables, Contemporary Mathematics 364 (2004), 103–126.MathSciNetGoogle Scholar
  34. 34.
    X. Liu and B.-W. Schulze, Ellipticity on manifolds with edges and boundary, Monatshefte f¨ur Mathematik 146, 4 (2005),295–331.Google Scholar
  35. 35.
    L. Maniccia and B.-W. Schulze, An algebra of meromorphic corner symbols, Bull. des Sciences Math. 127, 1 (2003), 55–99.MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    C.-I. Martin and B.-W. Schulze, Parameter-dependent edge operators, arXiv: 0908.2030.v1 [math.AP], 2009.Google Scholar
  37. 37.
    R. Mazzeo, Elliptic theory of differential edge operators I, Comm. Partial Differential Equations. 16 (1991), 1615–1664.MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    R.B. Melrose, Transformation of boundary problems, Acta Math. 147 (1981), 149– 236.MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    R.B. Melrose and G.A. Mendoza, Elliptic operators of totally characteristic type, Preprint MSRI 047 – 83, Math. Sci. Res. Institute, 1983.Google Scholar
  40. 40.
    C. Parenti, Operatori pseudo-differenziali in Rn e applicazioni, Annali Mat. Pura Appl. (4) 93 (1972), 359–389.Google Scholar
  41. 41.
    S. Rempel and B.-W. Schulze, Parametrices and boundary symbolic calculus for elliptic boundary problems without transmission property, Math. Nachr. 105 (1982), 45–149.MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    S. Rempel and B.-W. Schulze, Complete Mellin and Green symbolic calculus in spaces with conormal asymptotics, Ann. Global Anal. Geom. 4, 2 (1986), 137–224.MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    S. Rempel and B.-W. Schulze, Asymptotics for elliptic mixed boundary problems (pseudo-differential and Mellin operators in spaces with conormal singularity), Math. Res., vol.50, Akademie-Verlag, Berlin, 1989.Google Scholar
  44. 44.
    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
  45. 45.
    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, Schr¨odinger Operators, Deformation Quantization”, Akademie Verlag, Berlin, 1995, pp. 70-205.Google Scholar
  46. 46.
    B.-W. Schulze, Pseudo-differential operators on manifolds with edges, Symp. “Partial Differential Equations”, Holzhau 1988, Teubner-Texte zur Mathematik, vol. 112, Teubner, Leipzig, 1989, pp. 259–287.Google Scholar
  47. 47.
    B.-W. Schulze, Pseudo-differential operators on manifolds with singularities, North- Holland, Amsterdam, 1991.Google Scholar
  48. 48.
    B.-W. Schulze, The Mellin pseudo-differential calculus on manifolds with corners, Symp. “Analysis in Domains and on Manifolds with Singularities”, Breitenbrunn 1990, Teubner-Texte zur Mathematik, vol. 131, Teubner, Leipzig, 1992, pp. 208–289.Google Scholar
  49. 49.
    B.-W. Schulze, Pseudo-differential boundary value problems, conical singularities, and asymptotics, Akademie Verlag, Berlin, 1994.Google Scholar
  50. 50.
    B.-W. Schulze, 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, 1994, pp. 9–96.Google Scholar
  51. 51.
    B.-W. Schulze, The variable discrete asymptotics in pseudo-differential boundary value problems II, Advances in Partial Differential Equations (Boundary Value Problems, Schr¨odinger Operators, Deformation Quantization), Akademie Verlag, Berlin, 1995, pp. 9–69.Google Scholar
  52. 52.
    B.-W. Schulze, Boundary value problems and singular pseudo-differential operators, J. Wiley, Chichester, 1998.Google Scholar
  53. 53.
    B.-W. Schulze, An algebra of boundary value problems not requiring Shapiro- Lopatinskij conditions, J. Funct. Anal. 179 (2001), 374–408.MathSciNetzbMATHCrossRefGoogle Scholar
  54. 54.
    B.-W. Schulze, Operator algebras with symbol hierarchies on manifolds with singularities, Advances in Partial Differential Equations (Approaches to Singular Analysis) (J.Gil, D.Grieser, and Lesch M., eds.), Oper. Theory Adv. Appl., Birkh¨auser Verlag, Basel, 2001, pp. 167–207.Google Scholar
  55. 55.
    B.-W. Schulze, Operators with symbol hierarchies and iterated asymptotics, Publications of RIMS, Kyoto University 38, 4 (2002),735–802.MathSciNetzbMATHCrossRefGoogle Scholar
  56. 56.
    B.-W. Schulze, Toeplitz operators, and ellipticity of boundary value problems with global projection conditions., Oper. Theory: Adv. Appl. 151, Advances in Partial Differential Equations “Aspects of Boundary Problems in Analysis and Geometry” (J. Gil, T. Krainer, and I. Witt, eds.), Birkh¨auser Verlag, Basel, 2004, pp. 342–429.Google Scholar
  57. 57.
    B.-W. Schulze, The iterative structure of corner operators, arXiv: 0901.1967v1 [math.AP], 2009.Google Scholar
  58. 58.
    B.-W. Schulze, Operators on corner manifolds, (manuscript in progress).Google Scholar
  59. 59.
    B.-W. Schulze, Boundary value problems with the transmission property, Oper. Theory: Adv. Appl. 205, “Pseudo-Differential Operators: Complex Analysis and Partial Differential Equations”, Birkh¨auser Verlag, Basel, 2009, pp. 1–50.Google Scholar
  60. 60.
    B.-W. Schulze and J. Seiler, The edge algebra structure of boundary value problems, Annals of Global Analysis and Geometry 22 (2002), 197–265.MathSciNetzbMATHCrossRefGoogle Scholar
  61. 61.
    B.-W. Schulze and J. Seiler, Pseudodifferential boundary value problems with global projection conditions, J. Funct. Anal. 206, 2 (2004), 449–498.MathSciNetzbMATHCrossRefGoogle Scholar
  62. 62.
    B.-W. Schulze and J. Seiler, Edge operators with conditions of Toeplitz type, J. of the Inst. Math. Jussieu. 5, 1 (2006), 101–123.MathSciNetzbMATHCrossRefGoogle Scholar
  63. 63.
    B.-W. Schulze and A. Volpato, Variable discrete and continuous asymptotics, (manuscript in progress).Google Scholar
  64. 64.
    B.-W. Schulze and Y. Wei, Edge-boundary problems with singular trace conditions, Ann. Global Anal. Geom., to appear.Google Scholar
  65. 65.
    J. Seiler, Continuity of edge and corner pseudo-differential operators, Math. Nachr. 205 (1999), 163–182.MathSciNetzbMATHCrossRefGoogle Scholar
  66. 66.
    J. Seiler, The cone algebra and a kernel characterization of Green operators, Advances in Partial Differential Equations (Approaches to Singular Analysis) (J.Gil, D.Grieser, and Lesch M., eds.), Oper. Theory Adv. Appl., Birkh¨auser, Basel, 2001, pp. 1–29.Google Scholar
  67. 67.
    M.A. Shubin, Pseudodifferential operators in Rn, Dokl. Akad. Nauk SSSR 196 (1971), 316–319.MathSciNetGoogle Scholar
  68. 68.
    M.I. Vishik and G.I. Eskin, Convolution equations in a bounded region, UspekhiMat. Nauk 20, 3 (1965), 89–152.Google Scholar
  69. 69.
    S. Weinberger, The topological classification of stratified spaces, Chicago Lectures in Mathematics, Univ. of Chicago Press, Chicago, 1994.Google Scholar
  70. 70.
    I. Witt, On the factorization of meromorphic Mellin symbols, Advances in Partial Differential Equations (Parabolicity, Volterra Calculus, and Conical Singularities) (S.Albeverio, M.Demuth, E.Schrohe, and B.-W. Schulze, eds.), Oper. Theory Adv. Appl., vol. 138, Birkh¨auser Verlag, Basel, 2002, pp. 279–306.Google Scholar

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© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Institut für MathematikUniversität PotsdamPotsdamGermany

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