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Distortion of quasiconformal mappings with identity boundary values
Author(s) -
Vuorinen Matti,
Zhang Xiaohui
Publication year - 2014
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jdu043
Subject(s) - mathematics , quasiconformal mapping , pointwise , bounded function , domain (mathematical analysis) , boundary (topology) , pure mathematics , mathematical analysis , upper and lower bounds , regular polygon , distortion (music) , geometry , physics , amplifier , optoelectronics , cmos
Teichmüller's classical mapping problem for plane domains concerns finding a lower bound for the maximal dilatation of a quasiconformal homeomorphism which holds the boundary pointwise fixed, maps the domain onto itself and maps a given point of the domain to another given point of the domain. For a domain D ⊂ R n , n ⩾ 2 , we consider the class of all K ‐quasiconformal maps of D onto itself with identity boundary values and Teichmüller's problem in this context. Given a map f of this class and a point x ∈ D , we show that the maximal dilatation of f has a lower bound in terms of the distance of x and f ( x ) . We improve recent results for the unit ball and consider this problem in other more general domains. For instance, convex domains, bounded domains and domains with uniformly perfect boundaries are studied.