eBook ISBN:  9781470404994 
Product Code:  MEMO/191/893.E 
List Price:  $68.00 
MAA Member Price:  $61.20 
AMS Member Price:  $40.80 
eBook ISBN:  9781470404994 
Product Code:  MEMO/191/893.E 
List Price:  $68.00 
MAA Member Price:  $61.20 
AMS Member Price:  $40.80 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 191; 2008; 92 ppMSC: Primary 35
The “measurable Riemann Mapping Theorem” (or the existence theorem for quasiconformal mappings) has found a central rôle in a diverse variety of areas such as holomorphic dynamics, Teichmüller theory, low dimensional topology and geometry, and the planar theory of PDEs. Anticipating the needs of future researchers, the authors give an account of the “state of the art” as it pertains to this theorem, that is, to the existence and uniqueness theory of the planar Beltrami equation, and various properties of the solutions to this equation. The classical theory concerns itself with the uniformly elliptic case (quasiconformal mappings). Here the authors develop the theory in the more general framework of mappings of finite distortion and the associated degenerate elliptic equations.
The authors recount aspects of this classical theory for the uninitiated, and then develop the more general theory. Much of this is either new at the time of writing, or provides a new approach and new insights into the theory. Indeed, it is the substantial recent advances in nonlinear harmonic analysis, Sobolev theory and geometric function theory that motivated their approach here. The concept of a principal solution and its fundamental role in understanding the natural domain of definition of a given Beltrami operator is emphasized in their investigations. The authors believe their results shed considerable new light on the theory of planar quasiconformal mappings and have the potential for wide applications, some of which they discuss.

Table of Contents

Chapters

1. Introduction

2. Quasiconformal mappings

3. Partial differential equations

4. Mappings of finite distortion

5. Hardy Spaces and BMO

6. The principal solution

7. Solutions for integrable distortion

8. Some technical results


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The “measurable Riemann Mapping Theorem” (or the existence theorem for quasiconformal mappings) has found a central rôle in a diverse variety of areas such as holomorphic dynamics, Teichmüller theory, low dimensional topology and geometry, and the planar theory of PDEs. Anticipating the needs of future researchers, the authors give an account of the “state of the art” as it pertains to this theorem, that is, to the existence and uniqueness theory of the planar Beltrami equation, and various properties of the solutions to this equation. The classical theory concerns itself with the uniformly elliptic case (quasiconformal mappings). Here the authors develop the theory in the more general framework of mappings of finite distortion and the associated degenerate elliptic equations.
The authors recount aspects of this classical theory for the uninitiated, and then develop the more general theory. Much of this is either new at the time of writing, or provides a new approach and new insights into the theory. Indeed, it is the substantial recent advances in nonlinear harmonic analysis, Sobolev theory and geometric function theory that motivated their approach here. The concept of a principal solution and its fundamental role in understanding the natural domain of definition of a given Beltrami operator is emphasized in their investigations. The authors believe their results shed considerable new light on the theory of planar quasiconformal mappings and have the potential for wide applications, some of which they discuss.

Chapters

1. Introduction

2. Quasiconformal mappings

3. Partial differential equations

4. Mappings of finite distortion

5. Hardy Spaces and BMO

6. The principal solution

7. Solutions for integrable distortion

8. Some technical results