Functions of Bounded Variation and Free Discontinuity Problems [Hardcover]

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 referred to as 'free discontinuity problems'. The aim of this book is twofold: The first three chapters present all the basic prerequisites for the 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 detailed analysis 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. The theory embodies 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.

Basic terminology and notation
1. Measure theory
2. Basic geometric measure theory
3. Functions of bounded variation
4. Special functions of bounded variation
5. Semicontinuity inBV
6. The Mumford-Shah functional
7. Minimisers of free continuity problems
8. Regularity of the free discontinuity set
References
Index

The book has two distinct parts. The first part begins with some topics from abstract measure theory, continues with an introduction to basic geometric measure theory, and concludes with BV functions and the theory of sets of finite perimeter. It is assumed that the reader has a basic knowledge of measure and integration theory. The second part of the book is oriented towards the study of specific variational problems. -- --Mathematical Reviews

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