Acknowledgments ............................................... vii
Introduction ................................................... ix
Chapter 1 Results .............................................. 1
1.1 Partial regularity ......................................... 1
1.2 Singular sets estimates .................................... 3
1.3 Extended Calderon-Zygmund theory ........................... 5
1.4 Outline of the paper ....................................... 9
Chapter 2 Basic material, assumptions ......................... 11
2.1 Notation, parabolic cylinders ............................. 11
2.2 Basic assumptions, especially for partial regularity ...... 12
2.3 General technical results ................................. 14
2.4 Compactness in parabolic spaces ........................... 15
2.5 Function spaces, preliminaries ............................ 16
2.6 Parabolic Hausdorff dimension ............................. 17
Chapter 3 The A-caloric approximation lemma ................... 25
3.1 A-caloric maps and approximation .......................... 25
Chapter 4 Partial regularity .................................. 35
4.1 Caccioppoli's inequality for parabolic systems with
p-growth .................................................. 35
4.2 Linearization via A-caloric approximation ................. 41
4.3 A decay estimate .......................................... 44
4.4 Iteration, and description of regular points .............. 48
4.5 Regular points ............................................ 54
Chapter 5 Some basic regularity results and a priori
estimates ........................................... 61
5.1 Estimates for differentiable systems ...................... 61
5.2 Reverse Hölder inequality on intrinsic cylinders .......... 71
5.3 Self-improving nature of the higher integrability ......... 72
5.4 Improved reversed Hölder inequality on intrinsic
cylinders ................................................. 76
Chapter 6 Dimension estimates ................................. 79
6.1 Smoothing the vector field x → α(x, •) .................... 79
6.2 Basic comparison estimate ................................. 81
6.3 Fractional estimates ...................................... 83
6.4 Proof of Theorem 1.3 ...................................... 86
6.5 Lowering the regularity of t → α(x, t, ω) ................. 87
Chapter 7 Hölder continuity of u .............................. 89
Chapter 8 Non-linear Calderón-Zygmund theory .................. 91
8.1 Proof of Theorem 1.6: set up .............................. 91
8.2 Reverse Hölder inequalities and intrinsic geometry ........ 91
8.3 Approximation in the case of VMO-coefficients ............. 94
8.4 Approximation for continuous vector fields ................ 97
8.5 Proof of the a priori estimate ............................ 98
8.6 Exit times ................................................ 98
8.7 Construction of comparison maps .......................... 100
8.8 Estimates on cylinders ................................... 103
8.9 Estimates for super-level sets ........................... 105
8.10 Estimate 1.20 and proof of Theorem 1.6 concluded ......... 107
8.11 Proof of Theorem 1.5 ..................................... 110
8.12 Proof of Theorems 1.7 and 1.9 ............................ 112
8.13 Interpolative nature of estimate 1.20 .................... 113
Bibliography .................................................. 115
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