Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane (PMS-48) (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 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. 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 Differential Equations Chapter 17. PDEs Not Of Divergence Type: Pucci’S Conjecture Chapter 18. Quasiconformal Methods In Impedance Tomography: Calderón’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