Advertisement

Journal of Evolution Equations

, Volume 18, Issue 3, pp 1395–1425 | Cite as

Bounded \(H_\infty \)-calculus for cone differential operators

  • E. SchroheEmail author
  • J. Seiler
Article
  • 49 Downloads

Abstract

We prove that parameter-elliptic extensions of cone differential operators have a bounded \(H_\infty \)-calculus. Applications concern the Laplacian and the porous medium equation on manifolds with warped conical singularities.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    P. Clément, S. Li. Abstract parabolic quasilinear equations and application to a groundwater flow problem. Adv. Math. Sci. Appl.  3, Special Issue (1993/94), 17–32.Google Scholar
  2. 2.
    P. Clément and J. Prüss. An operator-valued transference principle and maximal regularity on vector-valued \(L_p\)-spaces. In: G. Lumer and L. Weis (eds.), Proc. of the 6th. International Conference on Evolution Equations. Marcel Dekker (2001).Google Scholar
  3. 3.
    S. Coriasco, E. Schrohe and J. Seiler. Differential operators on conic manifolds: Maximal regularity and parabolic equations. Bull. Soc. Roy. Sci. Liège  70 (2001) 207–229.MathSciNetzbMATHGoogle Scholar
  4. 4.
    S. Coriasco, E. Schrohe, J. Seiler. Bounded imaginary powers of differential operators on manifolds with conical singularities. Math. Z.  244 (2003), no. 2, 235–269.MathSciNetCrossRefGoogle Scholar
  5. 5.
    S. Coriasco, E. Schrohe, J. Seiler. Bounded \(H_\infty \)-calculus for differential operators on conic manifolds with boundary. Comm. Partial Differential Equations  32 (2007), 229–255.MathSciNetCrossRefGoogle Scholar
  6. 6.
    S. Coriasco, E. Schrohe, J. Seiler. Realizations of differential operators on conic manifolds with boundary. Ann. Global Anal. Geom.  31 (2007), 223–285.MathSciNetCrossRefGoogle Scholar
  7. 7.
    R. Denk, M. Hieber, J. Prüss. \(\mathscr {R}\) -boundedness, Fourier multipliers and problems of elliptic and parabolic type . Mem. Amer. Math. Soc. 166  (2003).Google Scholar
  8. 8.
    J.B. Gil, G.A. Mendoza. Adjoints of elliptic cone operators. Amer. J. Math.  125 (2003), 357–408.MathSciNetCrossRefGoogle Scholar
  9. 9.
    J.B. Gil, T. Krainer, G.A. Mendoza. Resolvents of elliptic cone operators. J. Funct. Anal.  241 (2006), 1–55.MathSciNetCrossRefGoogle Scholar
  10. 10.
    J.B. Gil, Th. Krainer, G.A. Mendoza. Geometry and spectra of closed extensions of elliptic cone operators. Canad. J. Math.  59  (2007), 742–794.MathSciNetCrossRefGoogle Scholar
  11. 11.
    Th. Krainer. Parabolic pseudo-differential operators and long-time asymptotics of solutions. Ph.D.-thesis, University of Potsdam, 2000.Google Scholar
  12. 12.
    P. C. Kunstmann, L. Weis. Maximal \(L_p\)-regularity for parabolic equations, Fourier multiplier theorems and \(H^{\infty }\)-functional calculus. Functional Analytic Methods for Evolution Equations, Lecture Notes in Mathematics 1855, Springer Berlin Heidelberg (2004), 65–311.CrossRefGoogle Scholar
  13. 13.
    M. Lesch. Operators of Fuchs type, Conical Singularities, and Asymptotic Methods. Teubner-Texte Math. 136, Teubner-Verlag, 1997.Google Scholar
  14. 14.
    J.-L. Lions, E. Magenes, Problèmes aux limites non homogènes et applications. Vol. 1. Travaux et Recherches Mathématiques, No. 17 Dunod, Paris 1968.Google Scholar
  15. 15.
    N. Roidos. Local geometry effect on the solutions of evolution equations: The case of the Swift-Hohenberg equation on closed manifolds. Preprint, 2016 (arXiv:1612.08766).
  16. 16.
    N. Roidos, E. Schrohe. Bounded imaginary powers of cone differential operators on higher order Mellin-Sobolev spaces and applications to the Cahn-Hilliard equation. J. Differential Equations  257  (2014), 611–637.MathSciNetCrossRefGoogle Scholar
  17. 17.
    N. Roidos, E. Schrohe. Existence and maximal \(L^{p}\)-regularity of solutions for the porous medium equation on manifolds with conical singularities. Comm. Partial Differential Equations  41  (2016), 1441–1471.MathSciNetCrossRefGoogle Scholar
  18. 18.
    E. Schrohe, B.-W. Schulze. Boundary value problems in Boutet de Monvel’s calculus for manifolds with conical singularities II. In M. Demuth, E. Schrohe, B.-W. Schulze (eds.), Boundary Value Problems, Schrödinger Operators, Deformation Quantization, Math. Topics, Vol. 8: Advances in Part. Diff. Equ., Akademie-Verlag, 1995.Google Scholar
  19. 19.
    E. Schrohe, J. Seiler. Ellipticity and invertibility in the cone algebra on \(L_p\)-Sobolev spaces. Integral Equations Operator Theory  41  (2001), 93–114.MathSciNetCrossRefGoogle Scholar
  20. 20.
    E. Schrohe, J. Seiler. The resolvent of closed extensions of cone differential operators. Canad. J. Math.  57  (2005), 771–811.MathSciNetCrossRefGoogle Scholar
  21. 21.
    B.-W. Schulze. Pseudo-Differential Operators on Manifolds with Singularities. North-Holland, Amsterdam, 1991.zbMATHGoogle Scholar
  22. 22.
    B.-W. Schulze. Boundary value problems and singular pseudo-differential operators. Pure and Applied Mathematics (New York). John Wiley & Sons, Ltd., , 1998.Google Scholar
  23. 23.
    J. Seiler. Natural domains for edge degenerate differential operators. Preprint, 2010 (arXiv:1006.0039).
  24. 24.
    J. Seiler. The cone algebra and a kernel characterization of Green operators. In: J.B. Gil et al. (eds.), Approaches to Singular Analysis (Berlin, 1999), 1–29. Oper. Theory Adv. Appl. 125, Birkhäuser, 2001.Google Scholar
  25. 25.
    L. Weis. Operator-valued Fourier multiplier theorems and maximal \(L_{p}\)-regularity. Math. Ann.  319  (2001), 735–758.Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institut für AnalysisLeibniz Universität HannoverHannoverGermany
  2. 2.Dipartimento di MatematicaUniversità degli Studi di TorinoTurinItaly

Personalised recommendations