Introduction
Part 1. Elliptic Problems
Chapter 1. An Introductory Problem
1.1. Introduction and heuristic considerations
1.2. A one-phase singular perturbation problemg
1.3. The free boundary condition
Chapter 2. Viscosity Solutions and Their Asymptotic Developments
2.1. The notion of viscosity solution
2.2. Asymptotic developments
2.3. Comparison principles
Chapter 3. The Regularity of the Free Boundary
3.1. Weak results
3.2. Weak results for one- phase problemsg
3.3. Strong results
Chapter 4. Lipschitz Free Boundaries Are CLr
4.1. The main theorem. Heuristic considerations and strategy
4.2. Interior improvement of the Lipschitz constant
4.3. A Harnack principle. Improved interior gainZO
4.4. A continuous family of R-subsolutions
4.5. Free boundary improvement. Basic iteration
Chapter 5.Heuristic considederations
5.1. Heuristic considerations
5.2. An auxiliary family of functions
5.3. Level surfaces of normal perturbations of e-monotone functions
5.4. A continuou ataeJiic s
5.5. Proofof Theorem 5:l
5.6. A degenerate case
Chapter 6. Existence Theory
6.1. Introduction
6.2. μ+ is loally Lipschitz poidos
6.3. μ is Lipschitz
6.4. μ+ is nondegenerave nmaldors oiaqlla
6.5. μ is a viscosity supersolution
6.6. μ is a viscosity subsolution
6.7. Measuretheoreti properties of F(u)
6.8. Asymptotic developments eadq-odo,
6.9. Regularity and compactness moovuabauod erd edT
Part 2. Evolution Problems ndF ban anoituloB vieooiY
Chapter 7. Parabolic Free Boundary Problems aiv tó noion od F
7.1. Introduction
7.2. A class of free boundary problems and their viscosity solutions
7.3. Asymptotic behavior and free boundary relationerT8
7.4. R-subsolutions and a comparison principl
Chapter 8. Lipschitz Free Boundaries: Weak Results
8.1. Lipschitz continuity of viscosity solutions
8.2. Asymptotic behavior and free boundary relation
8.3. Counterexamples
Chapter 9.Lipschitz Free Boundaries: Strong Results
9.1. Nondegenerate problems: main result and strategy
9.2. Interior gain in space (parabolic homogeneity)uim
