Elliptic partial differential equations and quasiconformal mappings in the plane (princeton mathematical series)

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 compellin...

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Bibliographic Details
Main Authors:	Astala, Kari, Iwaniec, Tadeusz, Martin, Gaven
Format:	eBook Book
Language:	English
Published:	Princeton Princeton University Press 2009
Edition:	1
Series:	Princeton Mathematical Series
Subjects:	Adjoint equation Analytic function Analytic proof Beltrami equation Boundary value problem Bounded mean oscillation Calculus of variations Cauchy–Riemann equations Central limit theorem Characterization (mathematics) Complex analysis Complex plane Conformal geometry Conjugate variables Continuous function (set theory) Degeneracy (mathematics) Differential equation Differential equations, Elliptic Dirichlet integral Dirichlet problem Distribution (mathematics) Elliptic operator Elliptic partial differential equation Equation Equations of motion Euler–Lagrange equation Explicit formulae (L-function) Factorization Fourier transform Fubini's theorem Geometric function theory Geometry Harmonic conjugate Harmonic function Hilbert transform Holomorphic function Homeomorphism Hyperbolic geometry Julia set Lagrangian (field theory) Laplace's equation Limit (mathematics) Linear differential equation Linear equation Linear fractional transformation Locally integrable function Lusin's theorem Mathematical optimization MATHEMATICS MATHEMATICS / Complex Analysis MATHEMATICS / Functional Analysis MATHEMATICS / Differential Equations / General Mirror symmetry (string theory) Moduli space Monodromy theorem Montel's theorem Operator (physics) Operator theory Partial differential equation Quadratic function Quasiconformal mapping Quasiconformal mappings Quasiconvex function Renormalization Riemann sphere Riemann surface Riemannian geometry Riesz–Thorin theorem Sign (mathematics) Sobolev space Square-integrable function Theorem Two-dimensional space Upper half-plane Variable (mathematics) Weyl's lemma (Laplace equation) Hyperbolic geometry Linear differential equation Monodromy theorem Laplace's equation Mirror symmetry (string theory) Partial differential equation Square-integrable function Hilbert transform Analytic proof Riesz–Thorin theorem Lagrangian (field theory) Distribution (mathematics) Degeneracy (mathematics) Characterization (mathematics) Two-dimensional space Julia set Explicit formulae (L-function) Complex plane Adjoint equation Linear fractional transformation Holomorphic function Fubini's theorem Riemann surface Conjugate variables Conformal geometry Harmonic function Boundary value problem Homeomorphism Lusin's theorem Mathematical optimization Beltrami equation Locally integrable function Quadratic function Bounded mean oscillation Continuous function (set theory) Sign (mathematics) Sobolev space Differential equation Complex analysis Harmonic conjugate Operator (physics) Dirichlet integral Fourier transform Cauchy–Riemann equations Operator theory Riemannian geometry Weyl's lemma (Laplace equation) Euler–Lagrange equation Linear equation Montel's theorem Moduli space Dirichlet problem Theorem Quasiconformal mapping Calculus of variations Upper half-plane Equation Analytic function Quasiconvex function Equations of motion Factorization Riemann sphere Geometric function theory Geometry Variable (mathematics) Central limit theorem Elliptic operator Renormalization Elliptic partial differential equation Limit (mathematics)
ISBN:	1400830117, 0691137773, 9780691137773, 9781400830114
Online Access:	Get full text
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Table of Contents:

Elliptic partial differential equations and quasiconformal mappings in the plane (princeton mathematical series) -- Contents -- Preface -- Chapter 1: Introduction -- Chapter 2: A Background in Conformal Geometry -- Chapter 3: The Foundations of Quasiconformal Mappings -- Chapter 4: Complex Potentials -- Chapter 5: The Measurable Riemann Mapping Theorem: The Existence Theory of Quasiconformal Mappings -- Chapter 6: Parameterizing General Linear Elliptic Systems -- Chapter 7: The Concept of Ellipticity -- Chapter 8: Solving General Nonlinear First-Order Elliptic Systems -- Chapter 9: Nonlinear Riemann Mapping Theorems -- Chapter 10: Conformal Deformations and Beltrami Systems -- Chapter 11: A Quasilinear Cauchy Problem -- Chapter 12: Holomorphic Motions -- Chapter 13: Higher Integrability -- Chapter 14: Lp-Theory of Beltrami Operators -- Chapter 15: Schauder Estimates for Beltrami Operators -- Chapter 16: Applications to Partial Diffierential Equations -- Chapter 17: PDEs Not of Divergence Type: Pucci's Conjecture -- Chapter 18: Quasiconformal Methods in Impedance Tomography: Calderon's Problem -- Chapter 19: Integral Estimates for the Jacobian -- Chapter 20: Solving the Beltrami Equation: Degenerate Elliptic Case -- Chapter 21: Aspects of the Calculus of Variations -- Appendix: Elements of Sobolev Theory and Function Spaces -- Basic Notation -- Bibliography -- Index
Front Matter Table of Contents Preface Chapter 1: Introduction Chapter 2: A Background in Conformal Geometry Chapter 3: The Foundations of Quasiconformal Mappings Chapter 4: Complex Potentials Chapter 5: The Measurable Riemann Mapping Theorem: Chapter 6: Parameterizing General Linear Elliptic Systems Chapter 7: The Concept of Ellipticity Chapter 8: Solving General Nonlinear First-Order Elliptic Systems Chapter 9: Nonlinear Riemann Mapping Theorems Chapter 10: Conformal Deformations and Beltrami Systems Chapter 11: A Quasilinear Cauchy Problem Chapter 12: Holomorphic Motions Chapter 13: Higher Integrability Chapter 14: ${L^p}$-Theory of Beltrami Operators Chapter 15: Schauder Estimates for Beltrami Operators Chapter 16: Applications to Partial Differential Equations Chapter 17: PDEs Not of Divergence Type: Chapter 18: Quasiconformal Methods in Impedance Tomography: Chapter 19: Integral Estimates for the Jacobian Chapter 20: Solving the Beltrami Equation: Chapter 21: Aspects of the Calculus of Variations Appendix: Basic Notation Bibliography Index
Intro -- Contents -- Preface -- 1 Introduction -- 1.1 Calculus of Variations, PDEs and Quasiconformal Mappings -- 1.2 Degeneracy -- 1.3 Holomorphic Dynamical Systems -- 1.4 Elliptic Operators and the Beurling Transform -- 2 A Background in Conformal Geometry -- 2.1 Matrix Fields and Conformal Structures -- 2.2 The Hyperbolic Metric -- 2.3 The Space S(2) -- 2.4 The Linear Distortion -- 2.5 Quasiconformal Mappings -- 2.6 Radial Stretchings -- 2.7 Hausdorff Dimension -- 2.8 Degree and Jacobian -- 2.9 A Background in Complex Analysis -- 2.9.1 Analysis with Complex Notation -- 2.9.2 Riemann Mapping Theorem and Uniformization -- 2.9.3 Schwarz-Pick Lemma of Ahlfors -- 2.9.4 Normal Families and Montel's Theorem -- 2.9.5 Hurwitz's Theorem -- 2.9.6 Bloch's Theorem -- 2.9.7 The Argument Principle -- 2.10 Distortion by Conformal Mapping -- 2.10.1 The Area Formula -- 2.10.2 Koebe 1/4-Theorem and Distortion Theorem -- 3 The Foundations of Quasiconformal Mappings -- 3.1 Basic Properties -- 3.2 Quasisymmetry -- 3.3 The Gehring-Lehto Theorem -- 3.3.1 The Differentiability of Open Mappings -- 3.4 Quasisymmetric Maps Are Quasiconformal -- 3.5 Global Quasiconformal Maps Are Quasisymmetric -- 3.6 Quasiconformality and Quasisymmetry: Local Equivalence -- 3.7 Lusin's Condition N and Positivity of the Jacobian -- 3.8 Change of Variables -- 3.9 Quasisymmetry and Equicontinuity -- 3.10 Hölder Regularity -- 3.11 Quasisymmetry and &amp -- #948 -- -Monotone Mappings -- 4 Complex Potentials -- 4.1 The Fourier Transform -- 4.1.1 The Fourier Transform in L[sup(1)] and L[sup(2)] -- 4.1.2 Fourier Transform on Measures -- 4.1.3 Multipliers -- 4.1.4 The Hecke Identities -- 4.2 The Complex Riesz Transforms R[sup(k)] -- 4.2.1 Potentials Associated with R[sup(k)] -- 4.3 Quantitative Analysis of Complex Potentials -- 4.3.1 The Logarithmic Potential -- 4.3.2 The Cauchy Transform
7 The Concept of Ellipticity -- 7.1 The Algebraic Concept of Ellipticity -- 7.2 Some Examples of First-Order Equations -- 7.3 General Elliptic First-Order Operators in Two Variables -- 7.3.1 Complexification -- 7.3.2 Homotopy Classification -- 7.3.3 Classification -- n = 1 -- 7.4 Partial Differential Operators with Measurable Coefficients -- 7.5 Quasilinear Operators -- 7.6 Lusin Measurability -- 7.7 Fully Nonlinear Equations -- 7.8 Second-Order Elliptic Systems -- 7.8.1 Measurable Coefficients -- 8 Solving General Nonlinear First-Order Elliptic Systems -- 8.1 Equations Without Principal Solutions -- 8.2 Existence of Solutions -- 8.3 Proof of Theorem 8.2.1 -- 8.3.1 Step 1: H Continuous, Supported on an Annulus -- 8.3.2 Step 2: Good Smoothing of H -- 8.3.3 Step 3: Lusin-Egoroff Convergence -- 8.3.4 Step 4: Passing to the Limit -- 8.4 Equations with Infinitely Many Principal Solutions -- 8.5 Liouville Theorems -- 8.6 Uniqueness -- 8.6.1 Uniqueness for Normalized Solutions -- 8.7 Lipschitz H(z, w, &amp -- #950 -- ) -- 9 Nonlinear Riemann Mapping Theorems -- 9.1 Ellipticity and Change of Variables -- 9.2 The Nonlinear Mapping Theorem: Simply Connected Domains -- 9.2.1 Existence -- 9.2.2 Uniqueness -- 9.3 Mappings onto Multiply Connected Schottky Domains -- 9.3.1 Some Preliminaries -- 9.3.2 Proof of the Mapping Theorem 9.3.4 -- 10 Conformal Deformations and Beltrami Systems -- 10.1 Quasilinearity of the Beltrami System -- 10.1.1 The Complex Equation -- 10.2 Conformal Equivalence of Riemannian Structures -- 10.3 Group Properties of Solutions -- 10.3.1 Semigroups -- 10.3.2 Sullivan-Tukia Theorem -- 10.3.3 Ellipticity Constants -- 11 A Quasilinear Cauchy Problem -- 11.1 The Nonlinear [Omitted]-Equation -- 11.2 A Fixed-Point Theorem -- 11.3 Existence and Uniqueness -- 12 Holomorphic Motions -- 12.1 The &amp -- #955 -- -Lemma -- 12.2 Two Compelling Examples
16 Applications to Partial Diffierential Equations -- 16.1 The Hodge * Method -- 16.1.1 Equations of Divergence Type: The A-Harmonic Operator -- 16.1.2 The Natural Domain of Definition -- 16.1.3 The A-Harmonic Conjugate Function -- 16.1.4 Regularity of Solutions -- 16.1.5 General Linear Divergence Equations -- 16.1.6 A-Harmonic Fields -- 16.2 Topological Properties of Solutions -- 16.3 The Hodographic Method -- 16.3.1 The Continuity Equation -- 16.3.2 The p-Harmonic Operator div|[Omitted]|[sup(p-2)][Omitted] -- 16.3.3 Second-Order Derivatives -- 16.3.4 The Complex Gradient -- 16.3.5 Hodograph Transform for the p-Laplacian -- 16.3.6 Sharp Hölder Regularity for p-Harmonic Functions -- 16.3.7 Removing the Rough Regularity in the Gradient -- 16.4 The Nonlinear A-Harmonic Equation -- 16.4.1 &amp -- #648 -- -Monotonicity of the Structural Field -- 16.4.2 The Dirichlet Problem -- 16.4.3 Quasiregular Gradient Fields and C[sup(1,&amp -- #945 -- )]-Regularity -- 16.5 Boundary Value Problems -- 16.5.1 A Nonlinear Riemann-Hilbert Problem -- 16.6 G-Compactness of Beltrami Diffierential Operators -- 16.6.1 G-Convergence of the Operators [Omitted] - &amp -- #956 -- [sub(j)][Omitted][sub(z)] -- 16.6.2 G-Limits and the Weak*-Topology -- 16.6.3 The Jump from [Omitted][sub(2)] -V[Omitted][sub(z)] to [Omitted][sub(z)] - &amp -- #956 -- [Omitted][sub(z)] -- 16.6.4 The Adjacent Operator's Two Primary Solutions -- 16.6.5 The Independence of [Omitted][sub(z)](z) and [Omitted][sub(z)](z) -- 16.6.6 Linear Families of Quasiregular Mappings -- 16.6.7 G-Compactness for Beltrami Operators -- 17 PDEs Not of Divergence Type: Pucci's Conjecture -- 17.1 Reduction to a First-Order System -- 17.2 Second-Order Caccioppoli Estimates -- 17.3 The Maximum Principle and Pucci's Conjecture -- 17.4 Interior Regularity -- 17.5 Equations with Lower-Order Terms -- 17.5.1 The Dirichlet Problem
12.2.1 Limit Sets of Kleinian Groups -- 12.2.2 Julia Sets of Rational Maps -- 12.3 The Extended &amp -- #955 -- -Lemma -- 12.3.1 Holomorphic Motions and the Cauchy Problem -- 12.3.2 Holomorphic Axiom of Choice -- 12.4 Distortion of Dimension in Holomorphic Motions -- 12.5 Embedding Quasiconformal Mappings in Holomorphic Flows -- 12.6 Distortion Theorems -- 12.7 Deformations of Quasiconformal Mappings -- 13 Higher Integrability -- 13.1 Distortion of Area -- 13.1.1 Initial Bounds for Distortion of Area -- 13.1.2 Weighted Area Distortion -- 13.1.3 An Example -- 13.1.4 General Area Estimates -- 13.2 Higher Integrability -- 13.2.1 Integrability at the Borderline -- 13.2.2 Distortion of Hausdorff Dimension -- 13.3 The Dimension of Quasicircles -- 13.3.1 Symmetrization of Beltrami Coefficients -- 13.3.2 Distortion of Dimension -- 13.4 Quasiconformal Mappings and BMO -- 13.4.1 Quasiconformal Jacobians and A[sub(p)]-Weights -- 13.5 Painlevé's Theorem: Removable Singularities -- 13.5.1 Distortion of Hausdorff Measure -- 13.6 Examples of Nonremovable Sets -- 14 L[sup(p)]-Theory of Beltrami Operators -- 14.1 Spectral Bounds and Linear Beltrami Operators -- 14.2 Invertibility of the Beltrami Operators -- 14.2.1 Proof of Invertibility -- Theorem 14.0.4 -- 14.3 Determining the Critical Interval -- 14.4 Injectivity in the Borderline Cases -- 14.4.1 Failure of Factorization in W[sup(1,q)] -- 14.4.2 Injectivity and Liouville-Type Theorems -- 14.5 Beltrami Operators -- Coefficients in VMO -- 14.6 Bounds for the Beurling Transform -- 15 Schauder Estimates for Beltrami Operators -- 15.1 Examples -- 15.2 The Beltrami Equation with Constant Coefficients -- 15.3 A Partition of Unity -- 15.4 An Interpolation -- 15.5 Hölder Regularity for Variable Coefficients -- 15.6 Hölder-Caccioppoli Estimates -- 15.7 Quasilinear Equations
4.4 Maximal Functions and Interpolation -- 4.4.1 Interpolation -- 4.4.2 Maximal Functions -- 4.5 Weak-Type Estimates and L[sup(p)]-Bounds -- 4.5.1 Weak-Type Estimates for Complex Riesz Transforms -- 4.5.2 Estimates for the Beurling Transform S -- 4.5.3 Weighted L[sup(p)]-Theory for S -- 4.6 BMO and the Beurling Transform -- 4.6.1 Global John-Nirenberg Inequalities -- 4.6.2 Norm Bounds in BMO -- 4.6.3 Orthogonality Properties of S -- 4.6.4 Proof of the Pointwise Estimates -- 4.6.5 Commutators -- 4.6.6 The Beurling Transform of Characteristic Functions -- 4.7 Hölder Estimates -- 4.7.1 Hölder Bounds for the Beurling Transform -- 4.7.2 The Inhomogeneous Cauchy-Riemann Equation -- 4.8 Beurling Transforms for Boundary Value Problems -- 4.8.1 The Beurling Transform on Domains -- 4.8.2 L[sup(p)]-Theory -- 4.8.3 Complex Potentials for the Dirichlet Problem -- 4.9 Complex Potentials in Multiply Connected Domains -- 5 The Measurable Riemann Mapping Theorem: The Existence Theory of Quasiconformal Mappings -- 5.1 The Basic Beltrami Equation -- 5.2 Quasiconformal Mappings with Smooth Beltrami Coefficient -- 5.3 The Measurable Riemann Mapping Theorem -- 5.4 L[sup(p)]-Estimates and the Critical Interval -- 5.4.1 The Caccioppoli Inequalities -- 5.4.2 Weakly Quasiregular Mappings -- 5.5 Stoilow Factorization -- 5.6 Factoring with Small Distortion -- 5.7 Analytic Dependence on Parameters -- 5.8 Extension of Quasisymmetric Mappings of the Real Line -- 5.8.1 The Douady-Earle Extension -- 5.8.2 The Beurling-Ahlfors Extension -- 5.9 Reflection -- 5.10 Conformal Welding -- 6 Parameterizing General Linear Elliptic Systems -- 6.1 Stoilow Factorization for General Elliptic Systems -- 6.2 Linear Families of Quasiconformal Mappings -- 6.3 The Reduced Beltrami Equation -- 6.4 Homeomorphic Solutions to Reduced Equations -- 6.4.1 Fabes-Stroock Theorem
17.6 Pucci's Example
Appendix: Elements Of Sobolev Theory And Function Spaces
Chapter 14. Lp-Theory Of Beltrami Operators
Chapter 16. Applications To Partial Differential Equations
Index
Chapter 4. Complex Potentials
Chapter 11. A Quasilinear Cauchy Problem
Chapter 2. A Background In Conformal Geometry
Chapter 19. Integral Estimates For The Jacobian
Chapter 13. Higher Integrability
Chapter 1. Introduction
Preface
Chapter 21. Aspects Of The Calculus Of Variations
Chapter 18. Quasiconformal Methods In Impedance Tomography: Calderón’s Problem
Chapter 20. Solving The Beltrami Equation: Degenerate Elliptic Case
Chapter 7. The Concept Of Ellipticity
Chapter 3. The Foundations Of Quasiconformal Mappings
Chapter 12. Holomorphic Motions
Basic Notation
Chapter 6. Parameterizing General Linear Elliptic Systems
Chapter 17. PDEs Not Of Divergence Type: Pucci’S Conjecture
Chapter 15. Schauder Estimates For Beltrami Operators
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Contents
Chapter 10. Conformal Deformations And Beltrami Systems
Frontmatter --
Chapter 8. Solving General Nonlinear First-Order Elliptic Systems
Chapter 5. The Measurable Riemann Mapping Theorem: The Existence Theory Of Quasiconformal Mappings
Chapter 9. Nonlinear Riemann Mapping Theorems
Bibliography