Developments in Mathematics

Transkript

1 Developments in Mathematics VOLUME 26 Series Editors: Krishnaswami Alladi, University of Florida Hershel M. Farkas, Hebrew University of Jerusalem Robert Guralnick, University of Southern California For further volumes:

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3 Vladimir Gutlyanskii Vladimir Ryazanov Uri Srebro Eduard Yakubov The Beltrami Equation A Geometric Approach 123

4 Vladimir Gutlyanskii Department of the Partial Differential Equations Institute of Applied Mathematics and Mechanics National Academy of Sciences of Ukraine Donetsk, Ukraine Uri Srebro Faculty of Mathematics Technion-Israel Institute of Technology Haifa, Israel Vladimir Ryazanov Department of the Function Theory Institute of Applied Mathematics and Mechanics National Academy of Sciences of Ukraine Donetsk, Ukraine Eduard Yakubov Faculty of Sciences H.I.T. Holon Institute of Technology Holon, Israel ISSN ISBN e-isbn DOI / Springer New York Dordrecht Heidelberg London Library of Congress Control Number: Mathematics Subject Classification (2010): 30C62, 30H35, 34M03, 35A01, 35J46, 35J56, 35J67, 35J70, 35Q15, 35Q35, 37F30, 57M10 c Springer Science+Business Media, LLC 2012 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (

5 Dedicated to Professor Bogdan Bojarski who is an important contributor to the Beltrami equation theory

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7 Preface The book is a summation of many years work on the study of general Beltrami equations with singularities. This is not only a summary of our own long-term collaboration but also with that of many other authors in the field. We show that our geometric approach based on the modulus and capacity developed by us makes it possible to derive the main known existence theorems, including sophisticated and more general existence theorems that have been recently established. The Beltrami equation plays a significant role in geometry, analysis, and physics, and, in particular, in the theory of quasiconformal mappings and their generalizations, Kleinian groups, and Teichmüller spaces. There has been renewed interest and activity in these areas and, in particular, in the study of degenerate and alternating Beltrami equations since the early 1990s. In this monograph, we restrict ourselves to the study of very basic properties of solutions in the degenerate and in the alternating cases like existence, uniqueness, distortion, boundary behavior, and mapping problems that can be derived by extremal length methods. The monograph can serve as a textbook for a one- or two-semester graduate course. Donetsk, Ukraine Donetsk, Ukraine Haifa, Israel Holon, Israel Vladimir Gutlyanskii Vladimir Ryazanov Uri Srebro Eduard Yakubov vii

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9 Contents 1 Introduction The Beltrami Equation Historical Remarks Applications of Beltrami Equations Classification of Beltrami Equations ACL Solutions Ellipticity of the Beltrami Equation Preliminaries BMO Functions in C BMO Functions in C Removability of Isolated Singularities of BMO Functions BMO Functions, qc Mappings, qc Arc, and Symmetric Extensions FMO Functions Examples of Functions ϕ FMO \ BMO loc On Sobolev s Classes Modulus and Capacity Convergence Theorems for Homeomorphisms Ring Q-Homeomorphisms at Inner Points On Some Equivalent Integral Conditions One More Integral Condition On Weakly Flat and Strongly Accessible Boundaries The Classical Beltrami Equation μ < Quasiconformal Mappings The Main Problems Integrability The Classical Existence and Uniqueness Theorem ix

10 x Contents 3.5 Methods of Proof of Uniqueness and Existence Uniqueness Existence Smoothness of the Solutions Analytic Dependence on Parameters The Degenerate Case Examples Example One Example Two The Singular Set Auxiliary Results Case (i): The Singular Set E is Specified and E D Existence and Uniqueness Boundary Behavior Proof of Theorem Proof of Theorem Examples Case (ii): The Singular Set E is Specified and E D Case (iii): The Singular Set E Is Not Specified Pesin Miklyukov and Suvorov Lehto Brakalova and Jenkins Iwaniec and Martin Gutlyanskii, Martio, Sugawa, and Vuorinen David Tukia Ryazanov, Srebro and Yakubov Modulus Inequalities BMO- and FMO-Quasiconformal Mappings Introduction Inclusions, Integrability, and Group Properties Distortion Lemmas One Existence Theorem Uniqueness and Approximation Good Approximation Other Properties The Main Lemma on FMO Estimate of Distortion Further Existence Theorems RingQ-Homeomorphisms at Boundary Points Introduction Examples and Properties

11 Contents xi 6.3 The Completeness of Ring Homeomorphisms One Integral Inequality Distortion Estimates On Removability of Isolated Singularities On Extending Inverse Mappings to the Boundary On Extending Direct Mappings to the Boundary Consequences for Quasiextremal Distance Domains On Singular Null Sets for Extremal Distances Strong Ring Solutions of Beltrami Equations Introduction General Existence Lemma and Corollaries Extension of Lehto s Existence Theorem Sufficient Integral Conditions for Solvability Necessary Integral Conditions for Solvability Representation, Factorization, and Uniqueness Theorems Historical Comments and Final Remarks On the Dirichlet Problem for Beltrami Equations Introduction Characterization of Ring Q-Homeomorphisms at the Boundary The Regular Homeomorphisms On Extension of Regular Homeomorphisms to the Boundary Existence Theorems for the Dirichlet Problem Examples of the Dirichlet Problem On Integral Conditions in the Dirichlet Problem On the Beltrami Equations with Two Characteristics Introduction One Convergence Theorem The Main Lemma Some Existence Theorems Consequences for the Reduced Beltrami Equations Existence Theorems with Integral Conditions Alternating Beltrami Equation Introduction The Geometric Configuration and the Conditions on μ A Symmetric Form of the Alternating Beltrami Equation Branched Folded Maps Definition of BF-Maps Discrete Maps, Canonical Maps, and Classification of Critical Points Branch Points, Power Maps, and Winding Maps

12 xii Contents Folding Maps (p,q)-cusp Maps and (p,q)-cusp Points Umbrella and Simple Umbrella Maps Branched Folded Maps BF-Maps Which Are Locally Different from the Canonical Maps Discrete Maps Which Are Not BF-Maps The Degree of a BF-Map and Prime BF-Maps Alternating Beltrami Equations and FQR-Maps Proper Folding, Cusp, and Umbrella Solutions Example Definition Existence of Local Folding Solutions Assumptions on the Equation and Geometric Configuration Uniformization and Folds Example A Archives of the Existence Theorems A.1 The Existence Theorems of Brakalova Jenkins A.1.1 Introduction A.1.2 Statements of the Main Results A.1.3 Construction of the Function f(z) A.1.4 f(z) Is a Homeomorphism A.1.5 Differentiability Properties of f(z) A.1.6 The Limit Function Satisfies the Beltrami Equation A.1.7 The Inverse Function g(w) of f(z) A.1.8 Proof of the Main Results A.1.9 Examples and Comparisons A.1.10 Formulations of New Existence Theorems A.1.11 Auxiliary Results and an Equivalent Statement A.1.12 Construction of a Solution A.1.13 The Solution Is a Homeomorphism A.1.14 Differentiability Properties of the Solution A.1.15 The Inverse Mapping A.1.16 The Case of the Whole Plane A.2 The Existence Theorems of Gutlyanski Martio Sugawa Vuorinen A.2.1 Introduction A.2.2 Sequences of Self-Homeomorphisms A.2.3 Existence Theorems A.2.4 Uniqueness A.2.5 Dominating Factors and Modulus Bounds

13 Contents xiii A.3 The Existence and Uniqueness Theorems of Martio Miklyukov A.3.1 Introduction and Main Results A.3.2 Proof of Theorem A A.3.3 Proof of Theorem A A.4 The Existence Theorems of Miklyukov Suvorov A.4.1 Notations A.4.2 The main result A.4.3 The main lemma A.4.4 Proof of Theorem A A.4.5 One more theorem A.4.6 The final conclusions A.5 One Example of Iwaniec Martin References Index

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