Functions of bounded variation and free discontinuity problems.

*(English)*Zbl 0957.49001
Oxford Mathematical Monographs. Oxford: Clarendon Press. xviii, 434 p. (2000).

The book is divided into two parts, the first one dealing with the theory of functions of bounded variation, the second part gives insight in a class of variational problems which involve the minimization of the sum of a volume and a surface energy (“free discontinuity problems”) and which have been object of intensive research of the three authors over the last years. Both parts are linked through the fact that free discontinuity problems require a deep analysis of the fine properties of BV-functions which is not presented in the monographs of, e.g., V. G. Maz’ja [“Sobolev spaces” (1985; Zbl 0692.46023)], A. I. Vol’pert and S. I. Hudjaev [“Analysis in classes of discontinuous functions and equations of mathematical physics” (1985; Zbl 0564.46025)], H. Federer [“Geometric measure theory” (1969; Zbl 0176.00801)], E. Giusti [“Minimal surfaces and functions of bounded variation” (1984; Zbl 0545.49018)], U. Massari and M. Miranda [“Minimal surfaces of codimension one” (1984; Zbl 0565.49030)], and M. Giaquinta, M. Modica and J. Souček [“Cartesian currents in the calculus of variations. I: Cartesian currents” (1998; Zbl 0914.49001), “II: Variational integrals” (1998; Zbl 0914.49002)].

In order to provide a selfcontained exposition the authors present the necessary tools from measure theory in two introductory chapters: Chapter 1 includes the basic notions of positive, real and vector measures; the reader will find useful facts about the spaces \(L^p\), moreover, weak convergence in spaces of measures, the concept of outer measures and Carathéodory’s construction are discussed. Chapter 2 is entitled “Basic geometric measure theory” presenting the concepts of Hausdorff measures, rectifiable sets, approximate tangent spaces, etc., moreover, the reader will find information on Young measures and on functionals defined on measures. Chapter 3 then is entirely devoted to the study of BV-functions and sets of finite perimeter. Besides of “standard results” the authors discuss in great detail the decomposition of the distributional derivative into an absolutely continuous part, a jump part and a Cantor part and prove several properties of the different parts of the derivative. This analysis serves as a basis for Chapter 4 where the space SBV of special BV-functions is introduced. SBV-functions were introduced by E. De Giorgi and L. Ambrosio [Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 82, No. 2, 199-210 (1988; Zbl 0715.49014)] as good candidates for variational problems involving volume and surface energies. By definition, a function is of class SBV if the Cantor part of the distributional derivative vanishes. The first part of Chapter 4 investigates the properties of such functions as well as of the space SBV, for example, closure and compactness theorems are established. In the second part of Chapter 4 a list of prominent free discontinuity problems is given including sets with prescribed mean curvature, optimal partitions, the Mumford-Shah image segmentation problem and structured deformations. Here the reader can learn about the history of these problems, moreover, a suitable weak formulation is given. Chapter 5 is a survey on the semicontinuity properties of volume and surface energies including on the one hand classical results as lower semicontinuity theorems for functionals like \(\int_\Omega f(\cdot,u,\nabla u) dx\) on Sobolev spaces, and on the other hand very recent results on lower-semicontinuous functionals on SBV. The last three chapters of the book are mainly devoted to the study of the Mumford-Shah functional, and it is for the first time that the collection of deep results in this field including the contributions of the three authors is presented in form of a monograph. Chapter 6 gives an introduction into the state of the art, i.e., the reader will find a complete survey of those results concerning, for example, the regularity of minimizing pairs for the Mumford-Shah functional. In order to give a reasonable limitation on the size of the book, the authors decided just to comment on the material contained in Chapter 6 and not to state proofs in detail. Chapter 7 starts with the result that the classical and the weak formulation of the Mumford-Shah problem are equivalent in any space dimension, in the second part of Chapter 7 the first variation is discussed, and, assuming some initial smoothness higher regularity follows from these Euler equations. Finally, Chapter 8 is devoted to the regularity of the free discontinuity set; the main result is a partial regularity theorem for the discontinuity set of quasi minimizers of functionals like \(F(u, \Omega)= \int_\Omega |\nabla u|^2 dx+{\mathcal H}^{n- 1}(S_u\cap \Omega)\), \(S_u\) denoting the approximate discontinuity set of \(u\).

The book can be highly recommended not only to experts in variational calculus: the clear and attractive style makes the material understandable also for graduate students who want to get familiar with the ideas which are behind and around problems like image segmentation. On the other hand, the systematic approach towards the theory of BV-functions presented in the first half of the monograph, puts the book on top of the list of standard references for spaces of BV-type. It should also be noted that a list of nearly 280 references is given and that each chapter is followed by a collection of “exercises” which sometimes are not easy to solve.

In order to provide a selfcontained exposition the authors present the necessary tools from measure theory in two introductory chapters: Chapter 1 includes the basic notions of positive, real and vector measures; the reader will find useful facts about the spaces \(L^p\), moreover, weak convergence in spaces of measures, the concept of outer measures and Carathéodory’s construction are discussed. Chapter 2 is entitled “Basic geometric measure theory” presenting the concepts of Hausdorff measures, rectifiable sets, approximate tangent spaces, etc., moreover, the reader will find information on Young measures and on functionals defined on measures. Chapter 3 then is entirely devoted to the study of BV-functions and sets of finite perimeter. Besides of “standard results” the authors discuss in great detail the decomposition of the distributional derivative into an absolutely continuous part, a jump part and a Cantor part and prove several properties of the different parts of the derivative. This analysis serves as a basis for Chapter 4 where the space SBV of special BV-functions is introduced. SBV-functions were introduced by E. De Giorgi and L. Ambrosio [Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 82, No. 2, 199-210 (1988; Zbl 0715.49014)] as good candidates for variational problems involving volume and surface energies. By definition, a function is of class SBV if the Cantor part of the distributional derivative vanishes. The first part of Chapter 4 investigates the properties of such functions as well as of the space SBV, for example, closure and compactness theorems are established. In the second part of Chapter 4 a list of prominent free discontinuity problems is given including sets with prescribed mean curvature, optimal partitions, the Mumford-Shah image segmentation problem and structured deformations. Here the reader can learn about the history of these problems, moreover, a suitable weak formulation is given. Chapter 5 is a survey on the semicontinuity properties of volume and surface energies including on the one hand classical results as lower semicontinuity theorems for functionals like \(\int_\Omega f(\cdot,u,\nabla u) dx\) on Sobolev spaces, and on the other hand very recent results on lower-semicontinuous functionals on SBV. The last three chapters of the book are mainly devoted to the study of the Mumford-Shah functional, and it is for the first time that the collection of deep results in this field including the contributions of the three authors is presented in form of a monograph. Chapter 6 gives an introduction into the state of the art, i.e., the reader will find a complete survey of those results concerning, for example, the regularity of minimizing pairs for the Mumford-Shah functional. In order to give a reasonable limitation on the size of the book, the authors decided just to comment on the material contained in Chapter 6 and not to state proofs in detail. Chapter 7 starts with the result that the classical and the weak formulation of the Mumford-Shah problem are equivalent in any space dimension, in the second part of Chapter 7 the first variation is discussed, and, assuming some initial smoothness higher regularity follows from these Euler equations. Finally, Chapter 8 is devoted to the regularity of the free discontinuity set; the main result is a partial regularity theorem for the discontinuity set of quasi minimizers of functionals like \(F(u, \Omega)= \int_\Omega |\nabla u|^2 dx+{\mathcal H}^{n- 1}(S_u\cap \Omega)\), \(S_u\) denoting the approximate discontinuity set of \(u\).

The book can be highly recommended not only to experts in variational calculus: the clear and attractive style makes the material understandable also for graduate students who want to get familiar with the ideas which are behind and around problems like image segmentation. On the other hand, the systematic approach towards the theory of BV-functions presented in the first half of the monograph, puts the book on top of the list of standard references for spaces of BV-type. It should also be noted that a list of nearly 280 references is given and that each chapter is followed by a collection of “exercises” which sometimes are not easy to solve.

Reviewer: M.Fuchs (Saarbrücken)

##### MSC:

49-02 | Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control |

26-02 | Research exposition (monographs, survey articles) pertaining to real functions |

49J10 | Existence theories for free problems in two or more independent variables |

49Q20 | Variational problems in a geometric measure-theoretic setting |

49J45 | Methods involving semicontinuity and convergence; relaxation |

##### MathOverflow Questions:

Is a function \(u\in \mathrm{SBV}(\Omega)\) with these additional properties essentially bounded?Is the intersection of two Caccioppoli (i.e. finite perimeter) sets Caccioppoli?