Skip to main content
SpringerLink
Log in

On Elliptic Non-divergence Operators with Measurable Coefficients

  • Chapter
Nonlinear Elliptic and Parabolic Problems

Part of the book series: Progress in Nonlinear Differential Equations and Their Applications ((PNLDE,volume 64))

Abstract

We study properties of the coefficient matrices of non-divergence operators on ℝn aiming at sectoriality and R-sectoriality of these operators. In particular, we present results on approximation, scaling, and the behavior in the L p -scale.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. H. Amann, M. Hieber, G. Simonett, Bounded H∞ calculus for elliptic operators, Differential Integral Equations 7, 613–653 (1994).

    MathSciNet  Google Scholar 

  2. P. Auscher, Ph. Tchamitchian, Square root problem for divergence operators and related topics, Asteérisque 249, Paris, Société Mathématique de France (1998).

    Google Scholar 

  3. S. Blunck, P.C. Kunstmann, Weighted norm estimates and maximal regularity, Adv. Differ. Equ. 7, 1513–1532 (2002).

    MathSciNet  Google Scholar 

  4. E.B. Davies, Uniformly elliptic operators with measurable coefficients, J. Funct. Anal. 132, 141–169 (1995).

    Article  MATH  MathSciNet  Google Scholar 

  5. R. Denk, M. Hieber, J. Prüss, R-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Mem. Am. Math. Soc. 788 (2003).

    Google Scholar 

  6. R. Denk, G. Dore, M. Hieber, J. Prüss, A. Venni, New thoughts on old results of R. T. Seeley, Math. Ann. 328, 545–583 (2004).

    Article  MathSciNet  Google Scholar 

  7. X.T. Duong, G. Simonett, H∞-calculus for elliptic operators with non-smooth coefficients, Differ. Integral Equ. 10, 201–217 (1997).

    MathSciNet  Google Scholar 

  8. X.T. Duong, Li Xin Yan, Bounded holomorphic functional calculus for nondivergence form differential operators, Differ. Integral Equ. 15, 709–730 (2002).

    Google Scholar 

  9. R. Haller, H. Heck, M. Hieber, Muckenhoupt weights and maximal LP-regularity, Arch. Math. 81, 422–430 (2003).

    Article  MathSciNet  Google Scholar 

  10. H. Heck, M. Hieber, Maximal LP-regularity for elliptic operators with VMO-coefficients, J. Evol. Equ. 3, 332-359 (2003).

    Google Scholar 

  11. N.J. Kalton, P.C. Kunstmann, L. Weis, Perturbation and interpolation theorems for the H∞-calculus with applications to differential operators, to appear in Math. Ann.

    Google Scholar 

  12. N.J. Kalton, M. Mitrea, Stability results on interpolation scales of quasi-Banach spaces and applications, Trans. Am. Math. Soc. 350, 3903–3922 (1998).

    Article  MathSciNet  Google Scholar 

  13. P.C. Kunstmann, Maximal LP-regularity for second-order elliptic operators with uniformly continuous coefficients on domains, in Iannelli, Mimmo (ed.) et al., Evolution equations: applications to physics, industry, life sciences and economics, Basel, Birkhäuser. Prog. Nonlinear Differ. Equ. Appl. 55, 293–305 (2003).

    Google Scholar 

  14. P.C. Kunstmann, L. Weis, Maximal L P -regularity for parabolic equations, Fourier multiplier theorems and H∞-functional calculus, in M. Iannelli, R. Nagel, S. Piazzera (eds.), Functional Analytic Methods for Evolution Equations, Springer Lecture Notes Math. 1855, 65–311 (2004).

    Google Scholar 

  15. A. Maugeri, D.K. Palagachev, L.G. Softova, Elliptic and Parabolic Equations with Discontinuous Coefficients, Mathematical Research Vol. 109, Wiley-VCH, Berlin, 2000.

    Google Scholar 

  16. J. Prüss, H. Sohr, Imaginary powers of elliptic second-order differential operators in LP-spaces, Hiroshima Math. J. 23, 161–192 (1993).

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematisches Institut I, Universität Karlsruhe, Englerstr. 2, D-76128, Karlsruhe, Germany

    Peer Christian Kunstmann

Authors
  1. Peer Christian Kunstmann
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Universitá Pierre et Marie Curie, Paris

    Haim Brezis

  2. Rutgers University, New Brunswick, N.J.

    Haim Brezis

  3. Angewandte Mathematik, Universität Zurich, Winterthurerstr. 190, 8057, Zürich, Switzerland

    Michel Chipot

  4. Institute of Applied Mathematics, University of Hannover, Welfengarten 1, 30167, Hannover, Germany

    Joachim Escher

Rights and permissions

Reprints and permissions

Copyright information

© 2005 Birkhäuser Verlag Basel/Switzerland

About this chapter

Cite this chapter

Kunstmann, P.C. (2005). On Elliptic Non-divergence Operators with Measurable Coefficients. In: Brezis, H., Chipot, M., Escher, J. (eds) Nonlinear Elliptic and Parabolic Problems. Progress in Nonlinear Differential Equations and Their Applications, vol 64. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7385-7_15

Download citation

Publish with us

Policies and ethics

Search

Navigation