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This monograph is dedicated to new findings in the theory of conformal maps of nonsmooth surfaces in Euclidian space. Although a multidimensional and explanatory theory of conformal maps between plain domains does exist, the theory of conformal maps of nonsmooth surfaces has not yet been introduced to public.
Conformal maps play a significant role in surface triangulation questions, grid generation on the surfaces, qualitative research of harmonious functions on the surfaces and etc.
Generally, the author restricts his research to locally Lipschitz surfaces. It seems that this class of surfaces is necessary in generalization by minimum and is rather sufficient for applications.
The following problems are touched upon in this book: the existence and uniqueness of maps, the boundary behavior of maps and prime ends of surfaces, theorems of the Ahlfors and Warschawski type, applications in a problem of the gas dynamics equation, and qualitative questions of the theory of minimal surfaces.
It should be mentioned that a class of problems presented in the monograph does not require an application of a complex variable. Thus, there could be another title courageously offered for this book such as Conformal Map without Complex Variable. Nevertheless, in such places where the application of complex variables is justified by convenience, the language of complex variables should be used. The author attempts to present the material in a simplified manner, hence, it could be comprehensible for young mathematicians who are in the beginning of their scientific career.
This book formulates a number of unsolved problems for starters.
The author hopes that the content of his book will help a reader to apply the conformal map method in practice.
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ISBN13 (TP): 978-1-4363-3692-5
ISBN13 (HB): 978-1-4363-3693-2
Library of Congress Number: 2008903546