Skip to main content
SpringerLink
Log in

On solutions of the Beltrami equation

  • Published:
Journal d’Analyse Mathematique Aims and scope

Abstract

In this paper we study the existence and uniqueness of solutions of the Beltrami equationf -z (z) =Μ(z)f z (z), whereΜ(z) is a measurable function defined almost everywhere in a plane domain ‡ with ‖ΜΜ∞ = 1-Here the partialsf z andf z of a complex valued functionf z exist almost everywhere. In case ‖Μ‖∞ ≤9 < 1, it is well-known that homeomorphic solutions of the Beltrami equation are quasiconformal mappings. In case ‖Μ‖∞= 1, much less is known. We give sufficient conditions onΜ(z) which imply the existence of a homeomorphic solution of the Beltrami equation, which isACL and whose partial derivativesf z andf z are locally inL q for anyq < 2. We also give uniqueness results. The conditions we consider improve already known results.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. L. Ahlfors,Lectures on Quasiconformal Mappings, Van Nostrand, London, 1966.

    MATH  Google Scholar 

  2. G. David,Solutions de l’équation de Beltrami avec ‖Μ‖∞ = 1, Ann. Acad. Sci. Fenn. Ser. A I Math.13 (1988), 25–70.

    MATH  MathSciNet  Google Scholar 

  3. O. Lehto and K. Virtanen,Quasiconformal Mappings in the Plane, Springer-Verlag, Berlin, 1973.

    MATH  Google Scholar 

  4. O. Lehto,Homeomorphisms with a given dilatation, Proc. 15th Scand. Congr. Oslo, 1968, Lecture Notes in Math. Vol. 118, Springer-Verlag, Berlin, 1970, pp. 53–73.

    Google Scholar 

  5. O. Lehto,Remarks on generalized Beltrami equations and conformai mappings, Proceedings of the Romanian-Finnish Seminar on Teichmüller Spaces and Quasiconformal Mappings, Brasov, Romania, 1969, Publishing House of the Academy of the Socialist Republic of Romania, Bucharest, 1971, pp. 203–214.

    Google Scholar 

  6. E. Reich and H. Walczak,On the behavior of quasiconformal mappings at a point, Trans. Amer. Math. Soc.117 (1965), 338–351.

    Article  MATH  MathSciNet  Google Scholar 

  7. W. Ziemer,Weakly Dijferentiable Functions, Springer-Verlag, Berlin, 1989.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, The Hotchkiss School, 06039, Lakeville, CT, USA

    Melkana A. Brakalova

  2. Department of Mathematics, Washington University, Campus Box 1146, 63130-4899, St. Louis, MO, USA

    James A. Jenkins

Authors
  1. Melkana A. Brakalova
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. James A. Jenkins
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Melkana A. Brakalova.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Brakalova, M.A., Jenkins, J.A. On solutions of the Beltrami equation. J. Anal. Math. 76, 67–92 (1998). https://doi.org/10.1007/BF02786930

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02786930

Keywords

Search

Navigation