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Author Astala, Kari
Title Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane (PMS-48)
Imprint Princeton : Princeton University Press, 2009
©2009
book jacket
Descript 1 online resource (696 pages)
text txt rdacontent
computer c rdamedia
online resource cr rdacarrier
Series Princeton Mathematical Ser
Princeton Mathematical Ser
Note 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 δ-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
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
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, ζ) -- 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 λ-Lemma -- 12.2 Two Compelling Examples
12.2.1 Limit Sets of Kleinian Groups -- 12.2.2 Julia Sets of Rational Maps -- 12.3 The Extended λ-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 -- 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 ʈ-Monotonicity of the Structural Field -- 16.4.2 The Dirichlet Problem -- 16.4.3 Quasiregular Gradient Fields and C[sup(1,α)]-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] - μ[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)] - μ[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 -- 17.6 Pucci's Example -- 18 Quasiconformal Methods in Impedance Tomography: Calderón's Problem
18.1 Complex Geometric Optics Solutions
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
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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2020. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries
Link Print version: Astala, Kari Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane (PMS-48) Princeton : Princeton University Press,c2009 9780691137773
Subject Differential equations, Elliptic.;Quasiconformal mappings
Electronic books
Alt Author Iwaniec, Tadeusz
Martin, Gaven
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