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A Geometric Approach to Free Boundary Problems
Luis Caffarelli, University of Texas, Austin, TX, and Sandro Salsa, Politecnico di Milano, Italy
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2005; 270 pp; hardcover
Volume: 68
ISBN-10: 0-8218-3784-2
ISBN-13: 978-0-8218-3784-9
List Price: US$54 Member Price: US$43.20
Order Code: GSM/68

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.

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."

"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

Elliptic problems
• An introductory problem
• Viscosity solutions and their asymptotic developments
• The regularity of the free boundary
• Lipschitz free boundaries are $$C^{1,\gamma}$$
• Flat free boundaries are Lipschitz
• Existence theory
Evolution problems
• Parabolic free boundary problems
• Lipschitz free boundaries: Weak results
• Lipschitz free boundaries: Strong results
• Flat free boundaries are smooth
Complementary chapters: Main tools
• Boundary behavior of harmonic functions
• Monotonicity formulas and applications
• Boundary behavior of caloric functions
• Bibliography
• Index