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1 COPYRIGHT NOTICE: Kari Astala, Tadeusz Iwaniec & Gaven Martin: Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane is published by Princeton University Press and copyrighted, 2009, by Princeton University Press. All rights reserved. No part of this book may be reproduced in any form by any electronic or mechanical means (including photocopying, recording, or information storage and retrieval) without permission in writing from the publisher, except for reading and browsing via the World Wide Web. Users are not permitted to mount this file on any network servers. Follow links Class Use and other Permissions. For more information, send to:

2 Contents Preface xv 1 Introduction Calculus of Variations, PDEs and Quasiconformal Mappings Degeneracy Holomorphic Dynamical Systems Elliptic Operators and the Beurling Transform A Background in Conformal Geometry Matrix Fields and Conformal Structures The Hyperbolic Metric The Space S(2) The Linear Distortion Quasiconformal Mappings Radial Stretchings Hausdorff Dimension Degree and Jacobian A Background in Complex Analysis Analysis with Complex Notation Riemann Mapping Theorem and Uniformization Schwarz-Pick Lemma of Ahlfors Normal Families and Montel s Theorem Hurwitz s Theorem Bloch s Theorem The Argument Principle Distortion by Conformal Mapping The Area Formula Koebe 1 4-Theorem and Distortion Theorem The Foundations of Quasiconformal Mappings Basic Properties Quasisymmetry The Gehring-Lehto Theorem The Differentiability of Open Mappings vii

3 viii 3.4 Quasisymmetric Maps Are Quasiconformal Global Quasiconformal Maps Are Quasisymmetric Quasiconformality and Quasisymmetry: Local Equivalence Lusin s Condition N and Positivity of the Jacobian Change of Variables Quasisymmetry and Equicontinuity Hölder Regularity Quasisymmetry and δ-monotone Mappings Complex Potentials The Fourier Transform The Fourier Transform in L 1 and L Fourier Transform on Measures Multipliers The Hecke Identities The Complex Riesz Transforms R k Potentials Associated with R k Quantitative Analysis of Complex Potentials The Logarithmic Potential The Cauchy Transform Maximal Functions and Interpolation Interpolation Maximal Functions Weak-Type Estimates and L p -Bounds Weak-Type Estimates for Complex Riesz Transforms Estimates for the Beurling Transform S Weighted L p -Theory for S BMO and the Beurling Transform Global John-Nirenberg Inequalities Norm Bounds in BMO Orthogonality Properties of S Proof of the Pointwise Estimates Commutators The Beurling Transform of Characteristic Functions Hölder Estimates Hölder Bounds for the Beurling Transform The Inhomogeneous Cauchy-Riemann Equation Beurling Transforms for Boundary Value Problems The Beurling Transform on Domains L p -Theory Complex Potentials for the Dirichlet Problem Complex Potentials in Multiply Connected Domains

4 ix 5 The Measurable Riemann Mapping Theorem: The Existence Theory of Quasiconformal Mappings The Basic Beltrami Equation Quasiconformal Mappings with Smooth Beltrami Coefficient The Measurable Riemann Mapping Theorem L p -Estimates and the Critical Interval The Caccioppoli Inequalities Weakly Quasiregular Mappings Stoilow Factorization Factoring with Small Distortion Analytic Dependence on Parameters Extension of Quasisymmetric Mappings of the Real Line The Douady-Earle Extension The Beurling-Ahlfors Extension Reflection Conformal Welding Parameterizing General Linear Elliptic Systems Stoilow Factorization for General Elliptic Systems Linear Families of Quasiconformal Mappings The Reduced Beltrami Equation Homeomorphic Solutions to Reduced Equations Fabes-Stroock Theorem The Concept of Ellipticity The Algebraic Concept of Ellipticity Some Examples of First-Order Equations General Elliptic First-Order Operators in Two Variables Complexification Homotopy Classification Classification; n = Partial Differential Operators with Measurable Coefficients Quasilinear Operators Lusin Measurability Fully Nonlinear Equations Second-Order Elliptic Systems Measurable Coefficients Solving General Nonlinear First-Order Elliptic Systems Equations Without Principal Solutions Existence of Solutions Proof of Theorem Step 1: H Continuous, Supported on an Annulus Step 2: Good Smoothing of H Step 3: Lusin-Egoroff Convergence Step 4: Passing to the Limit

5 x 8.4 Equations with Infinitely Many Principal Solutions Liouville Theorems Uniqueness Uniqueness for Normalized Solutions Lipschitz H(z, w, ζ) Nonlinear Riemann Mapping Theorems Ellipticity and Change of Variables The Nonlinear Mapping Theorem: Simply Connected Domains Existence Uniqueness Mappings onto Multiply Connected Schottky Domains Some Preliminaries Proof of the Mapping Theorem Conformal Deformations and Beltrami Systems Quasilinearity of the Beltrami System The Complex Equation Conformal Equivalence of Riemannian Structures Group Properties of Solutions Semigroups Sullivan-Tukia Theorem Ellipticity Constants A Quasilinear Cauchy Problem The Nonlinear -Equation A Fixed-Point Theorem Existence and Uniqueness Holomorphic Motions The λ-lemma Two Compelling Examples Limit Sets of Kleinian Groups Julia Sets of Rational Maps The Extended λ-lemma Holomorphic Motions and the Cauchy Problem Holomorphic Axiom of Choice Distortion of Dimension in Holomorphic Motions Embedding Quasiconformal Mappings in Holomorphic Flows Distortion Theorems Deformations of Quasiconformal Mappings Higher Integrability Distortion of Area Initial Bounds for Distortion of Area Weighted Area Distortion

6 xi An Example General Area Estimates Higher Integrability Integrability at the Borderline Distortion of Hausdorff Dimension The Dimension of Quasicircles Symmetrization of Beltrami Coefficients Distortion of Dimension Quasiconformal Mappings and BMO Quasiconformal Jacobians and A p -Weights Painlevé s Theorem: Removable Singularities Distortion of Hausdorff Measure Examples of Nonremovable Sets L p -Theory of Beltrami Operators Spectral Bounds and Linear Beltrami Operators Invertibility of the Beltrami Operators Proof of Invertibility; Theorem Determining the Critical Interval Injectivity in the Borderline Cases Failure of Factorization in W 1,q Injectivity and Liouville-Type Theorems Beltrami Operators; Coefficients in V MO Bounds for the Beurling Transform Schauder Estimates for Beltrami Operators Examples The Beltrami Equation with Constant Coefficients A Partition of Unity An Interpolation Hölder Regularity for Variable Coefficients Hölder-Caccioppoli Estimates Quasilinear Equations Applications to Partial Differential Equations The Hodge Method Equations of Divergence Type: The A-Harmonic Operator The Natural Domain of Definition The A-Harmonic Conjugate Function Regularity of Solutions General Linear Divergence Equations A-Harmonic Fields Topological Properties of Solutions The Hodographic Method The Continuity Equation

7 xii The p-harmonic Operator div p Second-Order Derivatives The Complex Gradient Hodograph Transform for the p-laplacian Sharp Hölder Regularity for p-harmonic Functions Removing the Rough Regularity in the Gradient The Nonlinear A-Harmonic Equation δ-monotonicity of the Structural Field The Dirichlet Problem Quasiregular Gradient Fields and C 1,α -Regularity Boundary Value Problems A Nonlinear Riemann-Hilbert Problem G-Compactness of Beltrami Differential Operators G-Convergence of the Operators z µ j z G-Limits and the Weak*-Topology The Jump from z ν z to z µ z The Adjacent Operator s Two Primary Solutions The Independence of Φ z (z) and Ψ z (z) Linear Families of Quasiregular Mappings G-Compactness for Beltrami Operators PDEs Not of Divergence Type: Pucci s Conjecture Reduction to a First-Order System Second-Order Caccioppoli Estimates The Maximum Principle and Pucci s Conjecture Interior Regularity Equations with Lower-Order Terms The Dirichlet Problem Pucci s Example Quasiconformal Methods in Impedance Tomography: Calderón s Problem Complex Geometric Optics Solutions The Hilbert Transform H σ Dependence on Parameters Nonlinear Fourier Transform Argument Principle Subexponential Growth The Solution to Calderón s Problem Integral Estimates for the Jacobian The Fundamental Inequality for the Jacobian Rank-One Convexity and Quasiconvexity Burkholder s Theorem L 1 -Integrability of the Jacobian

8 xiii 20 Solving the Beltrami Equation: Degenerate Elliptic Case Mappings of Finite Distortion; Continuity Topological Monotonicity Proof of Continuity in W 1, Integrable Distortion; W 1,2 -Solutions and Their Properties A Critical Example Distortion in the Exponential Class Example: Regularity in Exponential Distortion Beltrami Operators for Degenerate Equations Decay of the Neumann Series Existence Above the Critical Exponent Exponential Distortion: Existence of Solutions Optimal Regularity Uniqueness of Principal Solutions Stoilow Factorization Failure of Factorization in W 1,q When q < Optimal Orlicz Conditions for the Distortion Function Global Solutions Solutions on C Solutions on Ĉ A Liouville Theorem Applications to Degenerate PDEs Lehto s Condition Aspects of the Calculus of Variations Minimizing Mean Distortion Formulation of the General Problem The L 1 -Grötzsch Problem Sublinear Growth: Failure of Minimization Inverses of Homeomorphisms of Integrable Distortion The Traces of Mappings with Integrable Distortion Variational Equations The Lagrange-Euler Equations Equations for the Inverse Map Mean Distortion, Annuli and the Nitsche Conjecture Polar Coordinates Free Lagrangians Lower Bounds by Free Lagrangians Weighted Mean Distortion Minimizers within the Nitsche Range Beyond the Nitsche Bound The Minimizing Sequence and Its BV-limit Correction Lemma

9 xiv Appendix: Elements of Sobolev Theory and Function Spaces 624 A.1 Schwartz Distributions A.2 Definitions of Sobolev Spaces A.3 Mollification A.4 Pointwise Coincidence of Sobolev Functions A.5 Alternate Characterizations A.6 Embedding Theorems A.7 Duals and Compact Embeddings A.8 Hardy Spaces and BMO A.9 Reverse Hölder Inequalities A.10 Variations of Sobolev Mappings Basic Notation Bibliography Index