# zbMATH — the first resource for mathematics

On solutions of the Beltrami equation. (English) Zbl 0921.30015
The result of G. David [Ann. Acad., Sci. Fenn., Ser. A I, Math. 13, 25-70 (1988; Zbl 0619.30024)] concerning plane BMO – quasiconformal mappings has attracted a lot of research. In general $$\| \mu_f\|_\infty=1$$ for such a mapping $$f$$. The authors introduce two integral conditions for $$\mu_f$$ and prove the existence and uniqueness result for $$f$$ under these conditions; these provide an extension of the result of G. David. The proofs are based on the approximation method for $$f$$ where the cut-off approach for $$\mu_f$$ is used. This was already employed by G. David but the estimates are more elaborate in this case.

##### MSC:
 30C62 Quasiconformal mappings in the complex plane
##### Keywords:
Beltrami equation; BMO; quasiconformal mappings
Full Text:
##### References:
 [1] L. Ahlfors,Lectures on Quasiconformal Mappings, Van Nostrand, London, 1966. · Zbl 0138.06002 [2] G. David,Solutions de l’équation de Beltrami avec ||{$$\text{M}$$}||= 1, Ann. Acad. Sci. Fenn. Ser. A I Math.13 (1988), 25–70. · Zbl 0619.30024 [3] O. Lehto and K. Virtanen,Quasiconformal Mappings in the Plane, Springer-Verlag, Berlin, 1973. · Zbl 0267.30016 [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. [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. [6] E. Reich and H. Walczak,On the behavior of quasiconformal mappings at a point, Trans. Amer. Math. Soc.117 (1965), 338–351. · Zbl 0178.08301 [7] W. Ziemer,Weakly Dijferentiable Functions, Springer-Verlag, Berlin, 1989.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.