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Birkhäuser - Birkhäuser Mathematics | Singular Sets of Minimizers for the Mumford-Shah Functional

Singular Sets of Minimizers for the Mumford-Shah Functional

Ferran Sunyer i Balaguer Award winning monograph

Series: Progress in Mathematics, Vol. 233

David, Guy

2005, XIV, 581 p.

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Award-winning monograph of the Ferran Sunyer i Balaguer Prize 2004.

This book studies regularity properties of Mumford-Shah minimizers. The Mumford-Shah functional was introduced in the 1980s as a tool for automatic image segmentation, but its study gave rise to many interesting questions of analysis and geometric measure theory. The main object under scrutiny is a free boundary K where the minimizer may have jumps. The book presents an extensive description of the known regularity properties of the singular sets K, and the techniques to get them. Some time is spent on the C^1 regularity theorem (with an essentially unpublished proof in dimension 2), but a good part of the book is devoted to applications of A. Bonnet's monotonicity and blow-up techniques. In particular, global minimizers in the plane are studied in full detail.
The book is largely self-contained and should be accessible to graduate students in analysis.The core of the book is composed of regularity results that were proved in the last ten years and which are presented in a more detailed and unified way.

Content Level » Research

Keywords » Minimierungsprobleme - Mumford-Shah-Funktional - Randwertprobleme - Sobolev space - Variationsrechnung - partielle Differentialgleichungen

Related subjects » Birkhäuser Mathematics

Table of contents 

Presentation of the Mumford-Shah Functional.- Functions in the Sobolev Spaces W1,p.- Regularity Properties for Quasiminimizers.- Limits of Almost-Minimizers.- Pieces of C1 Curves for Almost-Minimizers.- Global Mumford-Shah Minimizers in the Plane.- Applications to Almost-Minimizers (n = 2).- Quasi- and Almost-Minimizers in Higher Dimensions.- Boundary Regularity.

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