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The Journal of Geometric Analysis

, Volume 29, Issue 1, pp 656–706 | Cite as

Elliptic Complexes on Manifolds with Boundary

  • B.-W. Schulze
  • J. SeilerEmail author
Article
  • 46 Downloads

Abstract

We show that elliptic complexes of (pseudo) differential operators on smooth compact manifolds with boundary can always be complemented to a Fredholm problem by boundary conditions involving global pseudodifferential projections on the boundary (similarly as the spectral boundary conditions of Atiyah, Patodi, and Singer for a single operator). We prove that boundary conditions without projections can be chosen if, and only if, the topological Atiyah–Bott obstruction vanishes. These results make use of a Fredholm theory for complexes of operators in algebras of generalized pseudodifferential operators of Toeplitz type which we also develop in the present paper.

Keywords

Elliptic complexes Manifolds with boundary Atiyah–Bott obstruction Toeplitz-type pseudodifferential operators 

Mathematics Subject Classification

Primary 58J10 47L15 Secondary 35S15 58J40 

References

  1. 1.
    Ambrozie, C.-G., Vasilescu, F.-H.: Banach Space Complexes. Mathematics and Its Applications, vol. 334. Kluwer Academic Publishers Group, Dordrecht (1995)CrossRefzbMATHGoogle Scholar
  2. 2.
    Atiyah, M.F., Bott, R.: A Lefschetz fixed point formula for elliptic complexes I. Ann. Math. 86(2), 374–407 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Atiyah, M.F., Bott, R.: The index problem for manifolds with boundary. Differ. Anal. 1, 175–186 (1964)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Atiyah, M.F., Patodi, V., Singer, I.M.: Spectral asymmetry and Riemannian geometry I. Math. Proc. Camb. Philos. Soc. 77, 43–69 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Atiyah, M.F., Patodi, V., Singer, I.M.: Spectral asymmetry and Riemannian geometry II. Math. Proc. Camb. Philos. Soc. 78, 405–432 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Atiyah, M.F., Patodi, V., Singer, I.M.: Spectral asymmetry and Riemannian geometry III. Math. Proc. Camb. Philos. Soc. 79, 315–330 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Boutet de Monvel, L.: Boundary value problems for pseudo-differential operators. Acta Math. 126(1–2), 11–51 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Brüning, J., Lesch, M.: Hilbert complexes. J. Funct. Anal. 108(1), 88–132 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Dynin, A.: Elliptic boundary problems for pseudo-differential complexes. Funct. Anal. Appl. 6(1), 75–76 (1972)CrossRefGoogle Scholar
  10. 10.
    Grubb, G.: Pseudo-differential boundary problems in \(L_p\) spaces. Commun. Partial Differ. Equ. 15(3), 289–340 (1990)CrossRefzbMATHGoogle Scholar
  11. 11.
    Grubb, G.: Functional Calculus of Pseudodifferential Boundary Problems, 2nd edn. Birkhäuser, Boston (1996)CrossRefzbMATHGoogle Scholar
  12. 12.
    Krupchyk, K., Tarkhanov, N., Tuomela, J.: Elliptic quasicomplexes in Boutet de Monvel algebra. J. Funct. Anal. 247(1), 202–230 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Melrose, R.B.: The Atiyah–Patodi–Singer Index Theorem. AK Peters, Wellesley (1993)CrossRefzbMATHGoogle Scholar
  14. 14.
    Nazaikinskii, V.E., Schulze, B.-W., Sternin, BYu., Shatalov, V.E.: Spectral boundary value problems and elliptic equations on manifolds with singularities. Differ. Uravn. 34(5), 695–708, 720 (1998). Translation. Differ. Equ. 34(5), 696–710 (1998)Google Scholar
  15. 15.
    Pillat, U., Schulze, B.-W.: Elliptische Randwertprobleme für Komplexe von Pseudodifferentialoperatoren. Math. Nachr. 94, 173–210 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Rempel, S., Schulze, B.-W.: Index Theory of Elliptic Boundary Problems. Akademie, Oxford (1982)zbMATHGoogle Scholar
  17. 17.
    Schrohe, E.: A short introduction to Boutet de Monvel’s calculus. In: Gil, J.B., et al. (eds.) Approaches to Singular Analysis, pp. 1–29. Birkhäuser, Boston (2001)Google Scholar
  18. 18.
    Schulze, B.-W.: Pseudo-differential Operators on Manifolds with Singularities. Studies in Mathematics and Its Applications, vol. 24. North-Holland Publishing Co., Amsterdam (1991)Google Scholar
  19. 19.
    Schulze, B.-W.: An algebra of boundary value problems not requiring Shapiro–Lopatinskij conditions. J. Funct. Anal. 179(2), 374–408 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Schulze, B.-W.: On a paper of Krupchyk, Tarkhanov, and Toumela. J. Funct. Anal. 256, 1665–1667 (2008)CrossRefzbMATHGoogle Scholar
  21. 21.
    Schulze, B.-W., Seiler, J.: Pseudodifferential boundary value problems with global projection conditions. J. Funct. Anal. 206(2), 449–498 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Schulze, B.-W., Seiler, J.: Edge operators with conditions of Toeplitz type. J. Inst. Math. Jussieu 5(1), 101–123 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Schulze, B.-W., Sternin, B.Yu., Shatalov, V.E.: On general boundary value problems for elliptic equations. Sb. Math. 189(10), 1573–1586 (1998)Google Scholar
  24. 24.
    Seeley, R.T.: Singular integrals and boundary value problems. Am. J. Math. 88(4), 781–809 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Segal, G.: Equivariant \(K\)-theory. Inst. Hautes Études Sci. Publ. Math. 34, 129–151 (1968)CrossRefzbMATHGoogle Scholar
  26. 26.
    Segal, G.: Fredholm complexes. Q. J. Math. Oxford Ser. (2) 21, 385–402 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Seiler, J.: Ellipticity in pseudodifferential algebras of Toeplitz type. J. Funct. Anal. 263(5), 1408–1434 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Spanier, E.H.: Algebraic Topology. McGraw-Hill Book Company, New York (1966)zbMATHGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2018

Authors and Affiliations

  1. 1.Universität Potsdam, Institut für MathematikPotsdamGermany
  2. 2.Dipartimento di MatematicaUniversità di TorinoTurinItaly

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