Index ACF monotonicity formula, 32, 34, 36, 44, 70, 106, 108, 143, 157, 163, 179 ACF estimate, 39 case of equality, 39, 70, 157 for harmonic functions, 32 Almgren’s frequency formula, 173, 175 for harmonic functions, 174 for Problem S, 175 Almgren’s frequency function, 174 Almgren’s monotonicity formula, see Almgren’s frequency formula almost monotonicity formula, 39, 41 Alt-Caffarelli-Friedman monotonicity formula, see ACF monotonicity formula American options, perpetual, 170 Andersson-Weiss counterexample, 49 balanced energy, 77, 134 blowdowns, see shrinkdowns blowups, 3, 4, 66 at a point, 66 classification energetic, Problem A, 78 Problems A,B,C, 72 Problem S, 177 continuous dependence classical obstacle problem, singular points, 140 homogeneity, 76 obstacle-type problems, strong, 69 Problem S, 176 Problem S, κ-homogeneous, 189 uniqueness classical obstacle problem, singular points, 139 contact points, 162 Problem A, B, variable centers, 92 Problem A, regular points, 85 Problem B, regular points, 89 Problem S, singular points, 190 uniqueness of type, 72 Problems A, B, C, 73 with variable centers (over a sequence), 66, 92 boundary Harnack principle, 120, 123, 186 boundary thin obstacle problem, 169 branch points, see Problem C Brownian motion, 14 C1 touch, 159, 161 Caffarelli-Jerison-Kenig estimate, see CJK estimate Calder´ on-Zygmund estimates, 17, 21 Cauchy-Kovalevskaya theorem, 10, 25, 188, 200 characteristic constant, 35, 38 CJK almost monotonicity formula, see CJK estimate CJK estimate, 41, 44, 48 classical obstacle problem, 1, 7, 8 C1,1 regularity, 31 W 2,p regularity, 18 blowups continuous dependence, singular points, 140 uniqueness, singular points, 139 global solutions, 100 convexity, 100 optimal regularity, 29 singular set structure, 141 217

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