Astala / Iwaniec / Martin Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane

Course Book
ISBN: 978-1-4008-3011-4
Verlag: De Gruyter
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 696 Seiten

Reihe: 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.

Autoren/Hrsg.

Astala, Kari

Iwaniec, Tadeusz

Martin, Gaven

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Differentialrechnungen und -gleichungen
• Mathematik | Informatik Mathematik Mathematische Analysis Funktionentheorie, Komplexe Analysis
• Mathematik | Informatik Mathematik Mathematische Analysis Funktionalanalysis

Inhaltsverzeichnis

FrontMatter, pg. i
Contents, pg. vii
Preface, pg. xv
Chapter 1. Introduction, pg. 1
Chapter 2. A Background In Conformal Geometry, pg. 12
Chapter 3. The Foundations Of Quasiconformal Mappings, pg. 48
Chapter 4. Complex Potentials, pg. 92
Chapter 5. The Measurable Riemann Mapping Theorem: The Existence Theory Of Quasiconformal Mappings, pg. 161
Chapter 6. Parameterizing General Linear Elliptic Systems, pg. 195
Chapter 7. The Concept Of Ellipticity, pg. 210
Chapter 8. Solving General Nonlinear First-Order Elliptic Systems, pg. 235
Chapter 9. Nonlinear Riemann Mapping Theorems, pg. 259
Chapter 10. Conformal Deformations And Beltrami Systems, pg. 275
Chapter 11. A Quasilinear Cauchy Problem, pg. 289
Chapter 12. Holomorphic Motions, pg. 293
Chapter 13. Higher Integrability, pg. 316
Chapter 14. Lp-Theory Of Beltrami Operators, pg. 362
Chapter 15. Schauder Estimates For Beltrami Operators, pg. 389
Chapter 16. Applications To Partial Differential Equations, pg. 403
Chapter 17. PDEs Not Of Divergence Type: Pucci’S Conjecture, pg. 472
Chapter 18. Quasiconformal Methods In Impedance Tomography: Calderón’s Problem, pg. 490
Chapter 19. Integral Estimates For The Jacobian, pg. 514
Chapter 20. Solving The Beltrami Equation: Degenerate Elliptic Case, pg. 527
Chapter 21. Aspects Of The Calculus Of Variations, pg. 586
Appendix: Elements Of Sobolev Theory And Function Spaces, pg. 624
Basic Notation, pg. 643
Bibliography, pg. 647
Index, pg. 671

Über Autor(innen)

Kari Astala is the Finnish Academy Professor of Mathematics at the University of Helsinki. Tadeusz Iwaniec is the John Raymond French Distinguished Professor of Mathematics at Syracuse University. Gaven Martin is the Distinguished Professor of Mathematics at Massey University.