Contents
Introduction vii
Part 1. Elliptic Problems
Chapter 1. An Introductory Problem 3
§1.1. Introduction and heuristic considerations 3
§1.2. A one-phase singular perturbation problem 6
§1.3. The free boundary condition 17
Chapter 2. Viscosity Solutions and Their Asymptotic Developments 25
§2.1. The notion of viscosity solution 25
§2.2. Asymptotic developments 27
§2.3. Comparison principles 30
Chapter 3. The Regularity of the Free Boundary 35
§3.1. Weak results 35
§3.2. Weak results for one-phase problems 36
§3.3. Strong results 40
Chapter 4. Lipschitz Free Boundaries Are
C1'7
43
§4.1. The main theorem. Heuristic considerations and strategy 43
§4.2. Interior improvement of the Lipschitz constant 47
§4.3. A Harnack principle. Improved interior gain 51
§4.4. A continuous family of i?-subsolutions 53
§4.5. Free boundary improvement. Basic iteration 62
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