Contents Preface ix Introduction 1 Suggestions for reading/teaching 5 Chapter 1. Model problems 7 §1.1. Catalog of problems 7 §1.2. Model Problems A, B, C 15 §1.3. W 2,p regularity of solutions 17 Notes 22 Exercises 24 Chapter 2. Optimal regularity of solutions 29 §2.1. Optimal regularity in the classical obstacle problem 29 §2.2. ACF monotonicity formula and generalizations 32 §2.3. Optimal regularity in obstacle-type problems 42 §2.4. Optimal regularity up to the fixed boundary 45 §2.5. A counterexample 49 Notes 51 Exercises 53 Chapter 3. Preliminary analysis of the free boundary 57 §3.1. Nondegeneracy 57 §3.2. Lebesgue and Hausdorff measures of the free boundary 61 §3.3. Classes of solutions, rescalings, and blowups 65 v
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