Functions of Bounded Variation and Free Discontinuity Problems

Hardcover | May 15, 2000

byLuigi Ambrosio, Nicola Fusco, Diego Pallara

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This book deals with a class of mathematical problems which involve the minimization of the sum of a volume and a surface energy and have lately been refered to as 'free discontinuity problems'. The aim of this book is twofold: The first three chapters present all the basic prerequisites forthe treatment of free discontinuity and other variational problems in a systematic, general, and self-contained way. In the later chapters, the reader is introduced to the theory of free discontinuity problems, to the space of special functions of bounded variation, and is presented with a detailedanalysis of the Mumford-Shah image segmentation problem. Existence, regularity and qualitative properties of solutions are explained and a survey is given on the current knowledge of this challenging mathematical problem. Free discontinuity problems reveal a wide range of applications. The theoryembodies classical problems, e.g. related to phase transitions, or fracture and plasticity in continuum mechanics, as well as more recent ones like edge detection in image analysis. This book provides the reader with a solid introduction to the field, written by principle contributors to the theory.The first half of the book contains a comprehensive and updated treatment of the theory of Functions of Bounded Variation and of the mathematical prerequisites of that theory, that is Abstract Measure Theory and Geometric Measure Theory.

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This book deals with a class of mathematical problems which involve the minimization of the sum of a volume and a surface energy and have lately been refered to as 'free discontinuity problems'. The aim of this book is twofold: The first three chapters present all the basic prerequisites forthe treatment of free discontinuity and other...

Luigi Ambrosio is at Scuola Normale Superiore, Pisa. Nicola Fusco is in the Department of Mathematics, Universita di Firenze.

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Format:HardcoverPublished:May 15, 2000Publisher:Oxford University PressLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:0198502451

ISBN - 13:9780198502456

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Table of Contents

Measure TheoryBasic Geometric Measure TheoryFunctions of bounded variationSpecial functions of bounded variationSemicontinuity in BVThe Mumford-Shah functionalMinimisers of free continuity problemsRegularity of the free discontinuity setReferencesIndex

Editorial Reviews

' a systematic and self-contained approach to this ongoing research topic with also incorporates an elementary introduction to geometric measure theory' Aslib