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Nonsmooth pseudodifferential boundary value problems on manifolds

  • Helmut AbelsEmail author
  • Carolina Neira Jiménez
Article
  • 14 Downloads

Abstract

We study pseudodifferential boundary value problems in the context of the Boutet de Monvel calculus or Green operators, with nonsmooth coefficients on smooth compact manifolds with boundary. In order to have a definition that is independent of the choice of (smooth) coordinates, we prove that nonsmooth Green operators are invariant under smooth coordinate transformations.

Keywords

Pseudodifferential boundary value problems Nonsmooth pseudodifferential operators Coordinate changes 

Mathematics Subject Classification

35 S 15 35 J 55 

Notes

Acknowledgements

The second author gratefully acknowledges support through DFG, GRK 1692 “Curvature, Cycles and Cohomology”, GRK 1463 “Analysis, Geometry and String Theory”, and Grant HERMES Code 43443 Departamento de Matemáticas, Universidad Nacional de Colombia Sede Bogotá, during parts of the work.

References

  1. 1.
    Abels, H.: Pseudodifferential boundary value problems with non-smooth coefficients. Comm. Part. Diff. Eq. 30, 1463–1503 (2005)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Abels, H.: Reduced and generalized Stokes resolvent equations in asymptotically flat layers, part II: \(H_{\infty }\)-calculus. J. Math. Fluid. Mech. 7, 223–260 (2005)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Abels, H.: Pseudodifferential and Singular Integral Operators. An Introduction with Applications. De Gruyter, Berlin (2011)CrossRefGoogle Scholar
  4. 4.
    Abels, H., Grubb, G., Wood, I.G.: Extension theory and Kreĭ n-type resolvent formulas for nonsmooth boundary value problems. J. Funct. Anal. 266(7), 4037–4100 (2014)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Abels, H., Pfeuffer, C.: Spectral invariance of non-smooth pseudodifferential operators. Integral Eq. Oper. Theory 86(1), 41–70 (2016)CrossRefGoogle Scholar
  6. 6.
    Abels, H., Pfeuffer, C.: Characterization of non-smooth pseudodifferential operators. J. Fourier Anal. Appl. 24(2), 371–415 (2017)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Abels, H., Pfeuffer, C.: Fredholm Property of Non-Smooth Pseudodifferential Operators (2018). arXiv:1806.01113
  8. 8.
    Abels, H., Terasawa, Y.: On Stokes operators with variable viscosity in bounded and unbounded domains. Math. Ann. 344(2), 381–429 (2009)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Barraza-Martínez, B., Denk, R., Hernández-Monzón, J.: Pseudodifferential operators with non-regular operator-valued symbols. Manuscr. Math. 144(3–4), 349–372 (2014)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Boutet de Monvel, L.: Boundary problems for pseudo-differential operators. Acta Math. 126, 11–51 (1971)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Grubb, G.: Functional Calculus of Pseudodifferential Boundary Problems, 2nd edn. Birkhäuser, Basel (1996)CrossRefGoogle Scholar
  12. 12.
    Grubb, G.: Nonhomogeneous Dirichlet Navier–Stokes problems in low regularity \({L^{p}}\)-Sobolev spaces. J. Math. Fluid Mech. 3(1), 57–81 (2001)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Grubb, G.: Spectral asymptotics for nonsmooth singular Green operators. Commun. Partial Differ. Eq. 39(3), 530–573 (2014). With an appendix by H. AbelsMathSciNetCrossRefGoogle Scholar
  14. 14.
    Grubb, G., Schrohe, E.: Trace expansions and the noncommutative residue for manifolds with boundary. J. Reine Angew. Math. 536, 167–207 (2001)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Grubb, G., Solonnikov, V.A.: Boundary value problems for the nonstationary Navier–Stokes equations treated by pseudo-differential methods. Math. Scand. 69, 217–290 (1991)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Köppl, D.: Pseudodifferential operatos with non–smooth coefficients on manifolds. Diploma thesis (in German), Universität Regensburg (2011)Google Scholar
  17. 17.
    Kumano-Go, H.: Pseudo-Differential Operators. MIT Press, Cambridge (1974)zbMATHGoogle Scholar
  18. 18.
    Kumano-Go, H., Nagase, M.: Pseudo-differential operators with non-regular symbols and applications. Funkcial Ekvac. 21, 151–192 (1978)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Marschall, J.: Nonregular pseudo-differential operators. Z. Anal. Anwendungen 15(1), 109–148 (1996)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Pfeuffer, C.: Characterization of non-smooth pseudodifferential operators. Ph.D. thesis, Universität Regensburg, (2015). http://epub.uni-regensburg.de/31776/ [23.06.2015] (2015)
  21. 21.
    Prüss, J., Simonett, G.: Moving Interfaces and Quasilinear Parabolic Evolution Equations. Monographs in Mathematics, vol. 105. Birkhäuser, Basel (2016)zbMATHGoogle Scholar
  22. 22.
    Rempel, S., Schulze, B.-W.: Index Theory of Elliptic Boundary Problems. Akademie Verlag, Berlin (1982)zbMATHGoogle Scholar
  23. 23.
    Taylor, M.E.: Pseudodifferential Operators and Nonlinear PDE. Birkhäuser, Basel (1991)CrossRefGoogle Scholar
  24. 24.
    Taylor, M.E.: Tools for PDE. Mathematical Surveys and Monographs. AMS, Providence (2000)Google Scholar
  25. 25.
    Witt, I.: A calculus for classical pseudo-differential operators with non-smooth symbols. Math. Nachr. 194, 239–284 (1998)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Wloka, J.T., Rowley, B., Lawruk, B.: Boundary Value Problems for Elliptic Systems. Cambridge University Press, Cambridge (1995)CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Fakultät für MathematikUniversität RegensburgRegensburgGermany
  2. 2.Departamento de MatemáticasUniversidad Nacional de ColombiaBogotáColombia

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