Surface Evolution Equations: A Level Set Approach (Monographs in Mathematics)

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9783764324308: Surface Evolution Equations: A Level Set Approach (Monographs in Mathematics)

This book is intended to be a self-contained introduction to analytic foundations of a level set method for various surface evolution equations including curvature ?ow equations. These equations are important in various ?elds including material sciences, image processing and di?erential geometry. The goal of this book is to introduce a generalized notion of solutions allowing singularities and solve the initial-value problem globally-in-time in a generalized sense. Various equivalent de?nitions of solutions are studied. Several new results on equivalence are also presented. Wepresentherearathercompleteintroductiontothetheoryofviscosityso- tionswhichis a keytoolforthe levelsetmethod. Alsoa self-containedexplanation isgivenforgeneralsurfaceevolutionequationsofthe secondorder.Althoughmost ofthe resultsin this book aremoreor lessknown, they arescatteredinseveralr- erences, sometimes without proof. This book presents these results in a synthetic way with full proofs. However, the references are not exhaustive at all. The book is suitable for applied researchers who would like to know the detail of the theory as well as its ?avour.No familiarity with di?erential geometry and the theory of viscosity solutions is required. The prerequisites are calculus, linear algebra and some familiarity with semicontinuous functions. This book is also suitable for upper level under graduate students who are interested in the ?eld

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This book presents a self-contained introduction to the analytic foundation of a level set approach for various surface evolution equations including curvature flow equations. These equations are important in many applications, such as material sciences, image processing and differential geometry. The goal is to introduce a generalized notion of solutions allowing singularities, and to solve the initial-value problem globally-in-time in a generalized sense. Various equivalent definitions of solutions are studied. Several new results on equivalence are also presented. Moreover, structures of level set equations are studied in detail. Further, a rather complete introduction to the theory of viscosity solutions is contained, which is a key tool for the level set approach.
Although most of the results in this book are more or less known, they are scattered in several references, sometimes without proofs. This book presents these results in a synthetic way with full proofs.
The intended audience are graduate students and researchers in various disciplines who would like to know the applicability and detail of the theory as well as its flavour. No familiarity with differential geometry or the theory of viscosity solutions is required. Only prerequisites are calculus, linear algebra and some basic knowledge about semicontinuous functions.

Review:

In view of its detailed and thorough presentation this book will be a valuable source for everyone interested in the level set approach to surface evolution equations.

Zentralblatt MATH

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