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Hardcover ISBN:  9780821837849 
Product Code:  GSM/68 
List Price:  $99.00 
MAA Member Price:  $89.10 
AMS Member Price:  $79.20 
eBook ISBN:  9781470421090 
Product Code:  GSM/68.E 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $68.00 
Hardcover ISBN:  9780821837849 
eBook ISBN:  9781470421090 
Product Code:  GSM/68.B 
List Price:  $184.00 $141.50 
MAA Member Price:  $165.60 $127.35 
AMS Member Price:  $147.20 $113.20 

Book DetailsGraduate Studies in MathematicsVolume: 68; 2005; 270 ppMSC: Primary 35
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.
ReadershipGraduate students and research mathematicians interested in partial differential equations.

Table of Contents

Part 1. Elliptic problems

Chapter 1. An introductory problem

Chapter 2. Viscosity solutions and their asymptotic developments

Chapter 3. The regularity of the free boundary

Chapter 4. Lipschitz free boundaries are $C^{1,\gamma }$

Chapter 5. Flat free boundaries are Lipschitz

Chapter 6. Existence theory

Part 2. Evolution problems

Chapter 7. Parabolic free boundary problems

Chapter 8. Lipschitz free boundaries: Weak results

Chapter 9. Lipschitz free boundaries: Strong results

Chapter 10. Flat free boundaries are smooth

Part 3. Complementary chapters: Main tools

Chapter 11. Boundary behavior of harmonic functions

Chapter 12. Monotonicity formulas and applications

Chapter 13. Boundary behavior of caloric functions


Additional Material

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 problembased 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 wellwritten 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 wellwritten and the style is clear. It is suitable for graduate students and researchers interested in partial differential equations.
Zentralblatt MATH


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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.
Graduate students and research mathematicians interested in partial differential equations.

Part 1. Elliptic problems

Chapter 1. An introductory problem

Chapter 2. Viscosity solutions and their asymptotic developments

Chapter 3. The regularity of the free boundary

Chapter 4. Lipschitz free boundaries are $C^{1,\gamma }$

Chapter 5. Flat free boundaries are Lipschitz

Chapter 6. Existence theory

Part 2. Evolution problems

Chapter 7. Parabolic free boundary problems

Chapter 8. Lipschitz free boundaries: Weak results

Chapter 9. Lipschitz free boundaries: Strong results

Chapter 10. Flat free boundaries are smooth

Part 3. Complementary chapters: Main tools

Chapter 11. Boundary behavior of harmonic functions

Chapter 12. Monotonicity formulas and applications

Chapter 13. Boundary behavior of caloric functions

The book will be a great resource, especially for scientists with an application in mind who want to find out what a free boundary problembased 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 wellwritten 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 wellwritten and the style is clear. It is suitable for graduate students and researchers interested in partial differential equations.
Zentralblatt MATH