Found Comput Math (2009) 9: 515–516 DOI 10.1007/s10208-008-9028-y

H. Attouch, G. Buttazzo, and G. Michaille. Variational Analysis in Sobolev and BV Spaces, in MPS–SIAM Series on Optimization SIAM and MPS, Philadelphia, 2006. ISBN: 0-89871-600-4 (pbk) D. Azé

Received: 6 February 2008 / Accepted: 11 February 2008 / Published online: 2 April 2008 © SFoCM 2008

The topic of this monograph ranges beyond its title. We are in presence of a volume which gives a large survey of calculus of variations in several variables with detours into partial differential equations of variational type, but also captures some parts of mathematical programming and convex analysis. It is destined for graduate and postgraduate students, but more advanced students in mathematics will also find useful information in these pages, along with other scientists interested in applied mathematics. A clear presentation of the historical Dirichlet problem in Chapter 2 kicks off the first part. The classical tools of variational analysis such as weak topologies and the introduction of Sobolev spaces are introduced in this chapter. The Lax–Milgram theorem and the direct method in the calculus of variations are developed in Chapter 3 along with an introduction to convex optimization. An original and intuitive proof is given for the Ekeland variational principle based on the convergence of a discrete dynamical system. Chapter 4 contains complements on measure theory: Hausdorff measures, duality and introduction to Young measures, which makes the link between classical and modern calculus of variations. Chapter 5 deals with a complete survey of Sobolev spaces with some useful complements on capacity and potential theory. The next chapter contains classical applications to boundary value problems of the Dirichlet and Neumann type, along with an introduction to the p-Laplacian. D. Azé () Institut de Mathématiques de Toulouse, UMR CNRS 5219, Université Paul Sabatier, 118 Route de Narbonne, 31062 Toulouse cedex 4, France e-mail: [email protected]

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Found Comput Math (2009) 9: 515–516

A glimpse is taken in Chapter 7 into numerical analysis of PDEs via the Galerkin procedure and the finite elements method. In Chapter 8, the spectral properties of the Laplacian are discussed. The first part of the book ends in Chapter 9 with a sober treatment of convex duality (with applications in calculus of variations along the lines of Ekeland–Temam), convex programming, and Karush–Kuhn–Tucker optimality conditions. All this classical stuff is clearly presented and pleasant to read. At this stage, the authors set the scene to the recent developments of variational analysis that occurred during the last 30 years. The modeling of some problems in physics or image processing needs new classes of functions whose derivatives in the distributional sense are no longer functions but measures. These new classes of functions take into account some discontinuity behavior of the phenomena under modeling. The second part starts with Chapter 10 devoted to the introduction of the space BV() of functions of bounded variations on an open subset  of Rd with Lipschitz boundary and of its basic properties: density of regular functions, traces on the boundary, Green formula, and compact embedding results in a suitable Lp (). The lower semicontinuity is one of the main ingredients of the so-called direct method in the calculus of variations. In this spirit, Chapter 11 deals with relaxation of integral functionals with domain W 1,p (, Rm ), that is, computation of the lower semicontinuous envelope of these functionals. The cases p > 1 and p = 1 lead to very different conclusions. In the first case, the domain of the relaxed functional is W 1,p (, Rm ) and the integrand associated to the relaxed functional is the quasi-convex (in the sense of Morrey) envelope of the integrand defining the functional. In the second case, the domain of the relaxed integrand is now BV(, Rm ) and the relaxed functional contains an extra term which is an integral functional of measures. This dense and technical chapter ends with analogous results in the space of Young measures. When applied to sequences of integral functionals, the methods used in Chapter 11 lead to the introduction of  convergence (or epi-convergence). This is done in Chapter 12 along with applications to homogenization of composite media, phase transition, and image segmentation via the Mumford–Shah model. Chapters 13 and 14 are devoted to the lower semicontinuity of the integral functional of the calculus of variations with applications to mechanics and to image segmentation. At last, the books ends with two chapters on noncoercive variational problems and an introduction to shape optimization. In conclusion, this monograph is well organized and the increasing complexity of its contents culminating in the last chapters is well managed. Many parts are really new and original. It will be useful not only for beginners in mathematical research (despite the absence of exercises), but also to experienced people.