Description: 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.
Review Quotes: "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
"This book provides an excellent account of the theory of BV functions (apparently for the first time in book form), and a nice introduction to geometric measure theory, as well as a rigorous survey of results for 'free discontinuity' problems modeled to the Mumford-Shah problem."--The Bulletin of the London Mathematical Society"The most important contribution of this book and what makes it unique is that it offers a comprehensive, unified, and self-contained treatment of contemporary BV theory that nicely complements the existing literature. ... The book also contains new contributions, such as the formula of push-forward in BV, and some proofs of known results have been greatly simplified. ... This is a book with staying power, an invaluable source for experts in the calculus of variations, geometric measure theory, and partial differential equations, and a graduate textbook for advanced Ph.D. students ..."--Society for Industrial and Applied Mathematics