Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane
Author(s): Kari Astala (Author), Tadeusz Iwaniec (Author), Gaven Martin (Author)
- Publisher: Princeton University Press
- Publication Date: 29 Dec. 2008
- Language: English
- Print length: 696 pages
- ISBN-10: 0691137773
- ISBN-13: 9780691137773
Book Description
This book explores the most recent developments in the theory of planar quasiconformal mappings with a particular focus on the interactions with partial differential equations and nonlinear analysis. It gives a thorough and modern approach to the classical theory and presents important and compelling applications across a spectrum of mathematics: dynamical systems, singular integral operators, inverse problems, the geometry of mappings, and the calculus of variations. It also gives an account of recent advances in harmonic analysis and their applications in the geometric theory of mappings. The book explains that the existence, regularity, and singular set structures for second-order divergence-type equations--the most important class of PDEs in applications--are determined by the mathematics underpinning the geometry, structure, and dimension of fractal sets; moduli spaces of Riemann surfaces; and conformal dynamical systems. These topics are inextricably linked by the theory of quasiconformal mappings. Further, the interplay between them allows the authors to extend classical results to more general settings for wider applicability, providing new and often optimal answers to questions of existence, regularity, and geometric properties of solutions to nonlinear systems in both elliptic and degenerate elliptic settings.
Editorial Reviews
Review
"The nature of the writing is impressive, and any library owning this volume, and other volumes of he series, will be a rich library indeed. This book can work out well as a text for further study at higher graduate level and beyond. For many a mathematician, it works well as a collection of enjoyable chapters; and most importantly, it can comfortably serve well as a reference resource and study material. They will be grateful to the publishers and the authors, for the volume includes a wealth of interesting and useful information on many important topics in the subject. . . . In short, a scintillating volume, full of detailed and thought-provoking contributions. Readers who bring to this book a reasonably strong background of the topics treated in the volume and an open mind will be well rewarded."-- "Current Engineering Practice"
About the Author
Kari Astala is the Finnish Academy Professor of Mathematics at the University of Helsinki. Tadeusz Iwaniec is the John Raymond French Distinguished Professor of Mathematics at Syracuse University. Gaven Martin is the Distinguished Professor of Mathematics at Massey University.
Excerpt. © Reprinted by permission. All rights reserved.
Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane
By Kari Astala Tadeusz Iwaniec Gaven Martin
Princeton University Press
Copyright © 2009 Princeton University Press
All right reserved.
ISBN: 978-0-691-13777-3
Contents
Preface........................................................................................................xv1 Introduction................................................................................................11.1 Calculus of Variations, PDEs and Quasiconformal Mappings..................................................21.2 Degeneracy................................................................................................61.3 Holomorphic Dynamical Systems.............................................................................81.4 Elliptic Operators and the Beurling Transform.............................................................92 A Background in Conformal Geometry..........................................................................122.1 Matrix Fields and Conformal Structures....................................................................122.2 The Hyperbolic Metric.....................................................................................152.3 The Space S(2)............................................................................................172.4 The Linear Distortion.....................................................................................212.5 Quasiconformal Mappings...................................................................................222.6 Radial Stretchings........................................................................................282.7 Hausdorff Dimension.......................................................................................302.8 Degree and Jacobian.......................................................................................322.9 A Background in Complex Analysis..........................................................................342.10 Distortion by Conformal Mapping...........................................................................413 The Foundations of Quasiconformal Mappings..................................................................483.1 Basic Properties..........................................................................................483.2 Quasisymmetry.............................................................................................493.3 The Gehring-Lehto Theorem.................................................................................513.4 Quasisymmetric Maps Are Quasiconformal....................................................................583.5 Global Quasiconformal Maps Are Quasisymmetric.............................................................643.6 Quasiconformality and Quasisymmetry: Local Equivalence....................................................703.7 Lusin's Condition N and Positivity of the Jacobian........................................................723.8 Change of Variables.......................................................................................763.9 Quasisymmetry and Equicontinuity..........................................................................783.10 Hlder Regularity.........................................................................................803.11 Quasisymmetry and -Monotone Mappings......................................................................834 Complex Potentials..........................................................................................924.1 The Fourier Transform.....................................................................................984.2 The Complex Riesz Transforms [R.sup.k]....................................................................1024.3 Quantitative Analysis of Complex Potentials...............................................................1044.4 Maximal Functions and Interpolation.......................................................................1174.5 Weak-Type Estimates and [L.sup.p]-Bounds..................................................................1244.6 BMO and the Beurling Transform............................................................................1314.7 Hlder Estimates..........................................................................................1474.8 Beurling Transforms for Boundary Value Problems...........................................................1504.9 Complex Potentials in Multiply Connected Domains..........................................................1585 The Measurable Riemann Mapping Theorem: The Existence Theory of Quasiconformal Mappings.....................1615.1 The Basic Beltrami Equation...............................................................................1635.2 Quasiconformal Mappings with Smooth Beltrami Coefficient..................................................1655.3 The Measurable Riemann Mapping Theorem....................................................................1685.4 [L.sup.p]-Estimates and the Critical Interval.............................................................1725.5 Stoilow Factorization.....................................................................................1785.6 Factoring with Small Distortion...........................................................................1845.7 Analytic Dependence on Parameters.........................................................................1855.8 Extension of Quasisymmetric Mappings of the Real Line.....................................................1895.9 Reflection................................................................................................1925.10 Conformal Welding.........................................................................................1936 Parameterizing General Linear Elliptic Systems..............................................................1956.1 Stoilow Factorization for General Elliptic Systems........................................................1966.2 Linear Families of Quasiconformal Mappings................................................................1986.3 The Reduced Beltrami Equation.............................................................................2026.4 Homeomorphic Solutions to Reduced Equations...............................................................2047 The Concept of Ellipticity..................................................................................2107.1 The Algebraic Concept of Ellipticity......................................................................2117.2 Some Examples of First-Order Equations....................................................................2137.3 General Elliptic First-Order Operators in Two Variables...................................................2147.4 Partial Differential Operators with Measurable Coefficients...............................................2217.5 Quasilinear Operators.....................................................................................2227.6 Lusin Measurability.......................................................................................2237.7 Fully Nonlinear Equations.................................................................................2267.8 Second-Order Elliptic Systems.............................................................................2318 Solving General Nonlinear First-Order Elliptic Systems......................................................2358.1 Equations Without Principal...
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