Boundary Value Problems with the Transmission Property

  • B.-W. Schulze
Part of the Operator Theory: Advances and Applications book series (OT, volume 205)


We give a survey on the calculus of (pseudo-differential) boundary value problems with the transmision property at the boundary, and ellipticity in the Shapiro—Lopatinskij sense. A part from the original results of the work of Boutet de Monvel we present an approach based on the ideas of the edge calculus. In a final section we introduce symbols with the anti-transmission property.


Pseudo-differential boundary value problems transmission and anti-transmission property boundary symbolic calculus Shapiro-Lopatinskij ellipticity parametrices 

Mathematics Subject Classification (2000)

35J40 58J32 58J40 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  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.zbMATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    M. F. Atiyah and R. Bott, The index problem for manifolds with boundary, in Differential Analysis, Oxford University Press, 1964, 175–186.Google Scholar
  3. [3]
    M. F. Atiyah, V. Patodi and I. M. Singer, Spectral asymmetry and Riemannian geometry I, II, III, Math. Proc. Cambridge Phil. Soc. 77, 78, 79 (1975, 1976, 1976), 43–69, 405–432, 315–330.zbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    L. Boutet de Monvel, Boundary problems for pseudo-differential operators, Acta Math. 126 (1971), 11–51.zbMATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    L. Boutet de Monvel, Comportement d’un opérateur pseudo-différentiel sur une variété à bord, J. Anal. Math. 17 (1966), 241–304.CrossRefMathSciNetGoogle Scholar
  6. [6]
    Ju. V. Egorov and B.-W. Schulze, Pseudo-Differential Operators, Singularities, Applications, Operator Theory: Advances and Applications, 93, Birkhäuser, 1997.Google Scholar
  7. [7]
    G. I. Eskin, Boundary Value Problems for Elliptic Pseudodifferential Equations, Math. Monographs 52, American Mathematical Society, 1980.Google Scholar
  8. [8]
    G. Grubb, Functional Calculus of Pseudo-Differential Boundary Problems, Birkhäuser, 1996.Google Scholar
  9. [9]
    G. Grubb, Pseudo-differential boundary value problems in L p Comm. Partial Differential Equations 15 (1990), 289–340.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    G. Harutyunyan and B.-W. Schulze, Elliptic Mixed, Transmission and Singular Crack Problems, European Mathematical Society, 2008.Google Scholar
  11. [11]
    K. Jänich, Vektorraumbündel und der Raum der Fredholm-Operatoren, Math. Ann. 161 (1965), 129–142.zbMATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    D. Kapanadze and B.-W. Schulze, Crack Theory and Edge Singularities, Kluwer Academic Publishers, 2003.Google Scholar
  13. [13]
    J.-L. Lions and E. Magenes, Problèmes aux Limites non Homogènes et Applications 1, Dunod, Paris, 1968.zbMATHGoogle Scholar
  14. [14]
    P. A. Myshkis, On an algebra generated by two-sided pseudodifferential operators on a manifold, Uspechi Mat. Nauk. 31(4) (1976), 269–270.zbMATHGoogle Scholar
  15. [15]
    S. Rempel and B.-W. Schulze, Index Theory of Elliptic Boundary Problems, Akademie-Verlag, 1982.Google Scholar
  16. [16]
    S. Rempel and B.-W. Schulze Parametrices and boundary symbolic calculus for elliptic boundary problems without transmission property, Math. Nachr. 105 (1982), 45–149.zbMATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    B.-W. Schulze, Topologies and invertibility in operator spaces with symbolic structures, in Problems and Methods in Mathematical Physics, Teubner Texte zur Mathematik 111 (1989), 257–270.MathSciNetGoogle Scholar
  18. [18]
    B.-W. Schulze, Pseudo-differential operators on manifolds with edges, in Symp. Partial Differential Equations, Holzhau 1988, Teubner-Texte, zur Mathematik 112 (1989), 259–287.MathSciNetGoogle Scholar
  19. [19]
    B.-W. Schulze, The Mellin pseudo-differential calculus on manifolds with corners, in Symp. Analysis in Domains and on Manifolds with Singularities, Breitenbrunn 1990, Teubner-Texte zur Mathematik 131 (1990), 208–289.MathSciNetGoogle Scholar
  20. [20]
    B.-W. Schulze, Pseudo-Differential Operators on Manifolds with Singularities, North-Holland, 1991.Google Scholar
  21. [21]
    B.-W. Schulze, Pseudo-Differential Boundary Value Problems, Conical Singularities, and Asymptotics, Akademie Verlag, 1994.Google Scholar
  22. [22]
    B.-W. Schulze, Boundary Value Problems and Singular Pseudo-Differential Operators, Wiley, 1998.Google Scholar
  23. [23]
    B.-W. Schulze, Operator algebras with symbol hierarchies on manifolds with singularities, in Advances in Partial Differential Equations (Approaches to Singular Analysis), Editors: J. Gil, D. Grieser and M. Lesch, Operator Theory: Advances and Applications Birkhäuser, 2001, 167–207.Google Scholar
  24. [24]
    B.-W. Schulze, An algebra of boundary value problems not requiring Shapiro-Lopatinskij conditions, J. Funct. Anal. 179 (2001), 374–408.zbMATHCrossRefMathSciNetGoogle Scholar
  25. [25]
    B.-W. Schulze and J. Seiler, The edge algebra structure of boundary value problems, Ann. Global Anal. Geom. 22 (2002), 197–265.zbMATHCrossRefMathSciNetGoogle Scholar
  26. [26]
    B.-W. Schulze and J. Seiler, Pseudodifferential boundary value problems with global projection conditions, J. Funct. Anal. 206(2) (2004), 449–498.zbMATHCrossRefMathSciNetGoogle Scholar
  27. [27]
    B.-W. Schulze and J. Seiler, Edge operators with conditions of Toeplitz type, J. Inst. Math. Jussieu 5(1) (2006), 101–123.zbMATHCrossRefMathSciNetGoogle Scholar
  28. [28]
    J. Seiler, Continuity of edge and corner pseudo-differential operators, Math. Nachr. 205 (1999), 163–182.zbMATHMathSciNetGoogle Scholar
  29. [29]
    M.I. Vishik and G. I. Eskin, Convolution equations in bounded domains in spaces with weighted norms, Mat. Sb. 69,(1) (1966), 65–110.MathSciNetGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  • B.-W. Schulze
    • 1
  1. 1.Institut für MathematikUniversität PotsdamPotsdamGermany

Personalised recommendations