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.

Introduction

A Background in Conformal Geometry

The Foundations of Quasiconformal Mappings

Complex Potentials

The Measurable Riemann Mapping Theorem: The Existence Theory of Quasiconformal Mapping

Parameterizing General Linear Elliptic Systems

The Concept of Ellipticity

Solving General Nonlinear First-Order Elliptic Systems

Nonlinear Riemann Mapping Theorems

Conformal Deformations and Beltrami Systems

Quasilinear Cauchy Problem

Holomorphic Motions

Higher Intergrability

Theory of Beltrami Operators

Schauder Estimates for Beltrami Operators

Applications to Partial Differential Equations

PDEs Not of Divergence Type:Pucci's Conjecture

Quasiconformal Methods in Impedance Tomography: Calderon's Problem

Integral Estimates for the Jacobian

Solving the Beltrami Equation: Degenerate Elliptic Case

Aspects of the Calculus of Variations

Appendix: Elements of Sobolev Theory and Function Spaces

Basic Notation

Bibliography

Index