Chapter 1. Introduction ......................................... 1
Chapter 2. Quasiconformal Mappings .............................. 5
2.1. Analytic Definition of Quasiconformality ................. 5
2.2. The Beltrami Equation .................................... 6
2.3. Radial Stretchings ....................................... 7
2.4. Classical Regularity Theory .............................. 8
Chapter 3. Partial Differential Equations ...................... 11
3.1. The Transformation Formula .............................. 11
3.2. A Fundamental Example ................................... 12
3.3. The Construction ........................................ 13
3.4. Cavitation and Riemann Surfaces ......................... 15
Chapter 4. Mappings of Finite Distortion ....................... 17
4.1. Orlicz-Sobolev Spaces ................................... 18
4.2. Monotonicity ............................................ 21
4.3. A Class of Orlicz Functions ............................. 22
4.4. The Monotonicity Theorem ................................ 23
4.5. Modulus of Continuity ................................... 24
Chapter 5. Hardy Spaces and BMO ................................ 27
5.1. Moilifiers .............................................. 27
5.2. Hardy-Orlicz Spaces ..................................... 28
5.3. BMO ..................................................... 29
5.4. L log L-Integrability ................................... 30
5.5. Liouville Type Theorems ................................. 30
Chapter 6. The Principal Solution .............................. 33
6.1. Solutions ............................................... 33
6.2. Uniqueness of Principal Solutions ....................... 34
6.3. Stoilow Factorization ................................... 35
Chapter 7. Solutions for Integrable Distortion ................. 39
7.1. Distortion in the Exponential Class ..................... 41
7.2. An Example .............................................. 42
7.3. Results ................................................. 43
7.4. Distortion in the Subexponential Class .................. 45
7.5. An Example .............................................. 45
7.6. Further Generalities .................................... 47
7.7. Existence Theory ........................................ 48
7.8. Global Solutions ........................................ 60
7.9. Holomorphic Dependence .................................. 64
7.10. Examples and Non-Uniqueness ............................. 67
7.11. Equations in the Plane .................................. 73
7.12. Compactness ............................................. 77
7.13. Removable Singularities ................................. 79
7.14. Final Comments .......................................... 80
Chapter 8. Some Technical Results .............................. 81
8.1. The Divergence Condition ................................ 81
8.2. Integration by Parts .................................... 84
8.3. Higher Integrability .................................... 86
Bibliography ................................................... 89
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