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Analytic dependence of riemann mappings for bounded domains and minimal surfaces
Author(s) -
Wu Sijue
Publication year - 1993
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.3160461002
Subject(s) - mathematics , bounded function , differentiable function , pure mathematics , open set , jordan curve theorem , mathematical analysis , riemann surface , euclidean space , space (punctuation) , function (biology) , euclidean geometry , geometry , computer science , evolutionary biology , biology , operating system
In this paper, we prove that the functional which takes a closed Lavrentiev curve to the corresponding Riemann mapping is locally Lipl on the set Ω of all closed Lavrentiev curves. This set is a subset of BMO(T). It is, however, not open in BMO(T). We also prove that the previous functional is analytic for certain classes of closed Lavrentiev curves, including the class of curves which have some symmetry with respect to the unit circle. These classes of curves are submanifolds of BMO(T). Finally, we consider the functional which takes a Lavrentiev curve (closed or not) in n ‐dimensional Euclidean space to the corresponding minimal surface, and we study the differentiability and analyticity of this functional on certain function spaces. © 1993 John Wiley & Sons, Inc.