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Singular sets of minimizers for the Mumford-Shah functional. (English) Zbl 1086.49030

Progress in Mathematics 233. Basel: Birkhäuser (ISBN 3-7643-7182-X/hbk). xiv, 581 p. (2005).
This monograph is the Ferran Sunyer i Balaguer 2004 prize winner. The prize is awarded yearly by the Fundació Ferran Sunyer i Balaguer and the Institute d’Etudis Catalans for a mathematical monograph of expository nature.
In the author’s intentions, the book wants to provide a wide survey on the machineries recently developed in the framework of the study of the Mumford-Shah functional, and on the progresses in the subject, in the hope that this can lead to some achievements in the analysis of the Mumford-Shah conjecture, or even in some other contexts. Recall that the Mumford-Shah conjecture says that, in dimension 2, the singular set of a reduced minimizer of the Mumford-Shah functional is locally a \(C^1\)-curve, except at a finite number of points.
The book follows an approach to the problem different from the bounded variation one, and tries to be as much as possible accessible and self-contained. It consists of nine chapters.
The first one provides a general presentation of the Mumford-Shah functional and its origin in the framework of image segmentation. Then, the Mumford-Shah conjecture and some known results on it are described. After a review on simple facts about Sobolev spaces, the first regularity results for minimizers and quasiminimizers in dimension 2 are presented with proofs very close to the original ones. In particular, the Ahlfors regularity results, the Carleson measure estimates, the projection and concentration properties, and the local uniform rectifiability property are discussed. Then, the Dal Maso-Morel-Solimini concentration property is used in order to prove that limits of minimizers or of almost-minimizers are still minimizers or almost-minimizers. The almost everywhere \(C^1\)-regularity result for almost-minimizers in dimension 2 is provided in Chapter 5 by means of a decay estimate for the normalized energy coming from a variant of a Bonnet’s monotonicity argument. The properties of global minimizers in the plane are described in Chapter 6 by using the above concentration property, the monotonicity argument, and a variant of the Léger’s formula. Some additional regularity results for almost-minimizers in a domain are then deduced, and it is checked that the Mumford-Shah conjecture follows from its counterpart for global minimizers in the plane. In Chapters 8 and 9 a quick description of the situation in higher dimension is provided, and the boundary regularity results for singular sets are obtained. In dimension 2 a good description of such set near the boundary as a finite union of \(C^1\) curves that meet the boundary orthogonally is obtained as well. The book ends with a section of questions and open problems.

MSC:

49Q20 Variational problems in a geometric measure-theoretic setting
58E12 Variational problems concerning minimal surfaces (problems in two independent variables)
94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
68U10 Computing methodologies for image processing
49-02 Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control
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