Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane

Kari Astala, Tadeusz Iwaniec, Gaven Martin

• Publisher: Princeton University Press
• Year: 2008
• Print ISBN:
  9780691137773
• eBook ISBN:
  9781400830114
• Topic: Mathematics
• Subtopic: Complex Analysis
• Index: Mathematics

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.

Table of contents

• Contents
• Preface
• 1 Introduction
• 2 A Background in Conformal Geometry
• 3 The Foundations of Quasiconformal Mappings
• 4 Complex Potentials
• 5 The Measurable Riemann Mapping Theorem: The Existence Theory of Quasiconformal Mappings
• 6 Parameterizing General Linear Elliptic Systems
• 7 The Concept of Ellipticity
• 8 Solving General Nonlinear First-Order Elliptic Systems
• 9 Nonlinear Riemann Mapping Theorems
• 10 Conformal Deformations and Beltrami Systems
• 11 A Quasilinear Cauchy Problem
• 12 Holomorphic Motions
• 13 Higher Integrability
• 14 L[sup(p)]-Theory of Beltrami Operators
• 15 Schauder Estimates for Beltrami Operators
• 16 Applications to Partial Diffierential Equations
• 17 PDEs Not of Divergence Type: Pucci's Conjecture
• 18 Quasiconformal Methods in Impedance Tomography: Calderón's Problem
• 19 Integral Estimates for the Jacobian
• 20 Solving the Beltrami Equation: Degenerate Elliptic Case
• 21 Aspects of the Calculus of Variations
• Appendix: Elements of Sobolev Theory and Function Spaces
• Basic Notation
• Bibliography
• Index