A Geometric Approach to Free Boundary Problems (Graduate Studies in Mathematics) (Graduate Studies in Mathematics, 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. "Free boundary (or moving boundary or phase transition) problems surface 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 we present a series of ideas, methods, and techniques for treating the most basic issues of such a problem. In particular, we 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. We hope that the tools and ideas presented here will serve as a basis for the study of more complex phenomena and problems."--BOOK JACKET

free Or Moving Boundary Problems Are Common In May Areas Of Mathematics And Science, Including Shape Optimization, Phase Transitions, Fluid Dynamics, Probability And Statistics. This Text Covers Such Topics In Free Boundary Problems As Elliptic Problems (such As Viscosity Solutions And They Asymptotic Developments), Evolution Problems (such As Lipschitz Free Boundaries), Main Tools Such As The Boundary Behavior Of Harmonic Functions And Caloric Functions And Monotonicity Formulas And Their Applications. Annotation ©2005 Book News, Inc., Portland, Or