自由或移动边界问题出现在分析、几何和应用数学的许多领域中。一个典型的例子是介于固相和液相之间不断演变的界面：如果我们对初始构形有足够的了解，便应该能够重新构造出它的演变过程，特别是界面的演变。
《自由边界问题的几何方法（英文版）》中，作者路易斯·卡法雷、桑德罗·萨尔萨提出了一系列处理这种问题中最基本情况的思想、方法和技术。特别地，他们描述了使构造成为可能的几何和实分析的极其基本的工具：在Lipschitz区域中的调和与热量测度的性质，平行曲面与椭圆方程之间的关系，单调性公式与刚性，等等。本书所给出的工具和思想可以作为研究更加复杂现象和问题的基础。
本书对于对偏微分方程感兴趣的研究生和研究者来说是一本有用的补充读物，也是一本很好的独立学习的教材。

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
9.3. Common gain uabauod
9.4. Interior gain in space (hyperbolic homogeneity)
9.5. Interior gain in time
9.6. A continuous family of subcaloric functions
9.7. Free boundary improvement. Propagation lemmag
9.8. Regularization of the free boundary in space
9.9. Free boundary regularity in space and time
Chapter 10. Flat Free Boundaries Are Smooth
10.1. Main result and strategy
10.2. Interior enlargement of the monotonicity coneg
10.3. Control of uv at a contact pointg
10.4. A continuous family of perturbationsg
10.5. Improvement of e-monotonicity
10.6. Propagation of cone enlargement to the free boundaryg
10.7. Proof of the main theoremj
10.8. Finite time regularization
Part 3. Complementary Chapters: Main Tools
Chapter 11. Boundary Behavior of Harmonic Functions
11.1. Harmonic functions in Lipschitz domainsf
11.2. Boundary Harnack principles
11.3. An excursion on harmonic measureg
11.4. Monotonicity properties
11.5. e-monotonicity and full monotonicity
11.6. Linear behavior at regular boundary points
Chapter 12. Monotonicity Formulas and Applications
12.1. A 2-dimensional formula
12.2. The n-dimensional formula
12.3. Consequences and applications
12.4. A parabolic monotonicity formula
12.5. A singular perturbation parabolic problem
Chapter 13 Boundary Behavior of Caloric Functionsg
13.1. Caloric functions in Lip(1, 1/2) domainsg
13.2. Caloric functions in Lipschitz domainsg
13.3. Asymptotic behavior near the zero setg
13.4. e-monotonicity and full monotonicity
13.5. An excursion on caloric measure
Bibliography
Index