Preface......Page 6
Contents......Page 9
List of Figures......Page 11
List of Tables......Page 12
Basic Notation and Conventions......Page 13
1 Introduction......Page 16
1.1.1 Motivation......Page 19
1.1.2 Main Results for No-Sign Parabolic Obstacle-Type Problems......Page 22
1.1.3 Historical Review......Page 23
1.1.4 Main Strategy......Page 24
1.2.1 Estimates for Solutions to the Elliptic Obstacle Problem......Page 27
1.2.1.2 Main Result......Page 28
1.2.1.4 Outline of the Proof......Page 29
1.2.2 Estimates for Solutions to the Two-Phase EllipticProblem......Page 30
1.2.2.1 Examples of Two-Phase Elliptic Problems......Page 31
1.2.2.3 Historical Review......Page 32
1.2.2.4 Outline of the Proof......Page 33
1.2.3.1 Example of the Two-Phase Parabolic Problem......Page 35
1.2.3.3 Historical Review......Page 36
1.2.4.1 Motivation......Page 37
1.2.4.4 Outline of the Proof......Page 38
1.3.2 Appendix B: Auxiliary Results......Page 40
1.5 Open Problems......Page 41
2.1 Statement of the Problem and Main Results......Page 43
2.2 Optimal Regularity of Solutions......Page 47
2.3.1 Nondegeneracy......Page 58
2.3.2 Measure of ฮ(u)......Page 61
2.3.3 Convergence......Page 62
2.3.4 Blow-Up and Blow-Down......Page 63
2.3.5 Balanced Energy......Page 65
2.4 Classification of the Nonnegative Global Solutions......Page 67
2.5 Geometric Classification of the Global Solutions with No Sign Restrictions......Page 71
2.6 Characterization of the Free Boundary Points Near ฮ ......Page 74
2.7 Regularity Properties of Solutions......Page 77
2.8 Regularity Properties of the Free Boundary......Page 84
3.1.1 Statement of the Problem and Main Results......Page 86
3.1.2 Estimates for Mixed Derivatives on the Boundary......Page 88
3.1.3 Estimates for Pure Second Derivatives......Page 93
3.2.1 Statement of the Problem and Main Result......Page 98
3.2.2 Estimates of the Tangential Gradient Near the Boundary......Page 99
3.2.3 Boundary Estimates of the Second Derivatives......Page 102
3.3.1 Statement of the Problem and Main Result......Page 105
3.3.2 Lipschitz Estimate of the Normal Derivative at the Boundary Points......Page 106
3.3.3 Boundary Estimates of the Second Derivatives......Page 108
3.4.1 Statement of the Problem and Main Result......Page 111
3.4.2 Estimate of the Time Derivative......Page 113
3.4.3 Estimates of the Second Derivatives......Page 115
A.1 C-monotonicity Formula......Page 120
A.2 ACF-monotonicity Formula......Page 126
A.3 W-monotonicity Formula......Page 128
B Auxiliary Results......Page 135
C Additional Facts......Page 143
References......Page 148
Index......Page 153