Quasiconformal Mappings in the Plane: Parametrical Methods
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978
Julian t.awrynowicz in cooperation with
Jan Krzyz
Ouasiconformal Mappings in the Plane: Parametrical Methods
---------------Springer-Verlag Berlin Heidelberg New York Tokyo 1983
Authors
Julian -lawrynowicz Institute of Mathematics of the Polish Academy of Sciences, -!:..odz Branch Kilinskieqo 86, PL-90-012 ..J:.odz, Poland in cooperation with Jan Krzyz Institute of Mathematics of the Maria-Curie-Sklodowska University in Lublin Nowotki 10, PL-20-031 Lublin, Poland
AMS Subject Classifications (1980): 30 C 60 ISBN 3-540-11989-2 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-11989-2 Springer-Verlag New York Heidelberg Berlin Tokyo
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FOREWORD These lecture notes contain an exposition of analytic properties of quasiconformal mappings in the plane (Chapter I), a detailed and systematic study of the parametrical method with complete proofs (exploring these analytic properties and including results of the author), partly new and simplified (Chapter II), and a brief account of variational methods (Chapter III). In contrast to the books by Lehto and Virtanen [1, 2J and by Ahlfors [5J the present author starts in Chapter I with defining the class of quasiconformal mappings as the closure of the GrBtzsch class with respect to uniform convergence on compact subsets. Then an important part of this chapter is devoted to proving the fundamental theorem on existence and uniqueness of quasiconformal mappings with a preassigned complex dilatation, established in particular cases by Gauss [1J, Lichtenstein [1J and Lavrentieff [2J, and in the general case by Morrey [1J. The present author chooses the proof due to Bojarski [2J which is based a.o. on the fundamental results of Calderon and Zygmunt [2J connected with properties of the Hilbert transform and on reducing the problem in question to solving some linear integral equation. Parametrical and variational methods belong to the most powerful research tools for extremal problems in the complex analysis. The parametrical method for conformal mappings of the unit disc {z: Izi 0, where the infimum is taken over all sets E' with IE'I = O. Tlle expressions if and only if, almost every [where J, and with spect to are abbreviated by iff, and respectively, while gQ means guasiconformal[ityJ and ACL absolutely continuous on lines.
I. BASIC CONCEPTS ANTI THEOREMS IN THE ANALYTIC THEORY OF QUASICONFORMAL MAPPINGS 1. The class of regular quasiconformal mappings and its closure We begin with the notion of a
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