Journal d’Analyse Mathematique

, Volume 76, Issue 1, pp 67–92 | Cite as

On solutions of the Beltrami equation

  • Melkana A. Brakalova
  • James A. Jenkins


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.


Compact Subset Conformal Mapping Accumulation Point Connected Domain Quasiconformal Mapping 
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Copyright information

© Hebrew University of Jerusalem 1998

Authors and Affiliations

  1. 1.Department of MathematicsThe Hotchkiss SchoolLakevilleUSA
  2. 2.Department of MathematicsWashington UniversitySt. LouisUSA

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