One of the most topical models studied in the modern calculus of variations is the Mumford-Shah functional. It is motivated by the question of best approximation by piecewise smooth functions. Similar problems are studied in mathematical theory of image processing. A competitor for the Mumford-Shah functional is a function u of two variables, which is smooth except for a singular set, where it can jump. The Mumford-Shah functional is a sum of two terms: The first term is the ordinary Dirichlet integral over the regular part of the domain of u. The second term is the length (precisely, one-dimensional Hausdorff measure) of the singular part of the domain. Any minimizer of the Mumford-Shah functional must be a harmonic function in the regular part. The cracktip is a canonical example of a harmonic function in the complement of the half-line y=0, x<0. In polar coordinates, u(r cos t, r sin t) = const r 1/2 sin (t/2). In his paper from 1991, E. De Giorgi raised the conjecture that the cracktip could be a global minimizer of the Mumford-Shah functional. Although the conjecture was supported by experiments, the question of a rigorous proof became a famous problem. Now, the solution of the problem is presented by Alexis Bonnet and Guy David. The fact that they need the extent of a monograph to describe the proof certifies that the problem was really considerably deep. The method of the proof exploits a careful analysis of the harmonic conjugate to the competitor and its level set. Blow up techniques and monotonicity of the energy functional are also used. In the existence part, weak compactness properties in SBV are bypassed and all is done in the framework of strong minimizers and competitors with closed singular sets. The presentation is well organized so that the reader can recognize where to learn the strategy of the proof and where to look for particular technical details. The aim of the monograph is to give an evidence that the problem is solved. Certainly, the book is a valuable source of inspiration for researchers, which try to attack problems of a similar nature.

Reviewer:

jama