**Graduate Studies in Mathematics**

Volume: 68;
2005;
270 pp;
Hardcover

MSC: Primary 35;

Print ISBN: 978-0-8218-3784-9

Product Code: GSM/68

List Price: $57.00

Individual Member Price: $45.60

**Electronic ISBN: 978-1-4704-2109-0
Product Code: GSM/68.E**

List Price: $57.00

Individual Member Price: $45.60

#### Supplemental Materials

# A Geometric Approach to Free Boundary Problems

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*Luis Caffarelli; Sandro Salsa*

Free or moving boundary problems appear in many areas of analysis, geometry, and applied mathematics. A typical example is the evolving interphase between a solid and liquid phase: if we know the initial configuration well enough, we should be able to reconstruct its evolution, in particular, the evolution of the interphase.

In this book, the authors present a series of ideas, methods, and techniques for treating the most basic issues of such a problem. In particular, they describe the very fundamental tools of geometry and real analysis that make this possible: properties of harmonic and caloric measures in Lipschitz domains, a relation between parallel surfaces and elliptic equations, monotonicity formulas and rigidity, etc. The tools and ideas presented here will serve as a basis for the study of more complex phenomena and problems.

This book is useful for supplementary reading or will be a fine independent study text. It is suitable for graduate students and researchers interested in partial differential equations.

Also available from the AMS by Luis
Caffarelli is *Fully
Nonlinear Elliptic Equations*, as Volume 43 in the AMS series, Colloquium Publications.

#### Table of Contents

# Table of Contents

## A Geometric Approach to Free Boundary Problems

- Cover Cover11 free
- Title i2 free
- Copyright ii3 free
- Contents iii4 free
- Introduction vii8 free
- Part 1. Elliptic Problems 112 free
- Chapter 1. An Introductory Problem 314
- Chapter 2. Viscosity Solutions and Their Asymptotic Developments 2536
- Chapter 3. The Regularity of the Free Boundary 3546
- Chapter 4. Lipschitz Free Boundaries Are C[sup(1)],γ 4354
- Chapter 5. Flat Free Boundaries Are Lipschitz 6576
- Chapter 6. Existence Theory 8798
- §6.1. Introduction 8798
- §6.2. u[sup(+)] is locally Lipschitz 90101
- §6.3. u is Lipschitz 91102
- §6.4. u[sup(+)] is nondegenerate 95106
- §6.5. u is a viscosity supersolution 96107
- §6.6. u is a viscosity subsolution 99110
- §6.7. Measure-theoretic properties of F(u) 101112
- §6.8. Asymptotic developments 103114
- §6.9. Regularity and compactness 106117

- Part 2. Evolution Problems 109120
- Chapter 7. Parabolic Free Boundary Problems 111122
- Chapter 8. Lipschitz Free Boundaries: Weak Results 121132
- Chapter 9. Lipschitz Free Boundaries: Strong Results 131142
- §9.1. Nondegenerate problems: main result and strategy 131142
- §9.2. Interior gain in space (parabolic homogeneity) 135146
- §9.3. Common gain 138149
- §9.4. Interior gain in space (hyperbolic homogeneity) 141152
- §9.5. Interior gain in time 143154
- §9.6. A continuous family of subcaloric functions 149160
- §9.7. Free boundary improvement. Propagation lemma 153164
- §9.8. Regularization of the free boundary in space 157168
- §9.9. Free boundary regularity in space and time 160171

- Chapter 10. Flat Free Boundaries Are Smooth 165176
- §10.1. Main result and strategy 165176
- §10.2. Interior enlargement of the monotonicity cone 168179
- §10.3. Control of u[sub(v)] at a "contact point" 172183
- §10.4. A continuous family of perturbations 174185
- §10.5. Improvement of ε-monotonicity 177188
- §10.6. Propagation of cone enlargement to the free boundary 180191
- §10.7. Proof of the main theorem 183194
- §10.8. Finite time regularization 185196

- Part 3. Complementary Chapters: Main Tools 189200
- Bibliography 265276
- Index 269280 free
- Back Cover Back Cover1282

#### Readership

Graduate students and research mathematicians interested in partial differential equations.

#### Reviews

The book will be a great resource, especially for scientists with an application in mind who want to find out what a free boundary problem-based approach can offer them. ... The book is written by two of the most renowned specialists in the study of free boundary problems, with deep contributions in this field. ... For anyone who later will do research on free boundary problems, this is probably the best introduction ever written. But the potential audience of this volume is much wider; his approach is just right for a book at the introductory level. The result is not only a comprehensive overview of the area itself, but also a very informative and inspiring monograph. Overall, this is a fine text for a graduate or postgraduate course in free boundary problems and a valuable reference that should be on the shelves of researchers and those teaching applied partial differential equations.

-- Vicentiu Radulescu, MAA Reviews

In this very interesting and well-written book, the authors present many techniques and ideas to investigate free boundary problems (hereafter, denoted FBP) of both elliptic and parabolic type.

-- Mathematical Reviews

The tools and ideas presented in this book will serve as a basis for the study of more complex phenomena and problems. The book is well-written and the style is clear. It is suitable for graduate students and researchers interested in partial differential equations.

-- Zentralblatt MATH