Attouch, Hedy; Buttazzo, Giuseppe; Michaille, Gérard Variational analysis in Sobolev and BV spaces. Applications to PDEs and optimization. 2nd revised ed. (English) Zbl 1311.49001 MOS-SIAM Series on Optimization 17. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM); Philadelphia, PA: Mathematical Optimization Society (ISBN 978-1-611973-47-1/hbk; 978-1-61197-348-8/ebook). xii, 793 p. (2014). It is an extended second edition of this well–written and self–contained book. The book is an excellent source of information for anyone interested in variational analysis, optimization, and PDEs. It is divided into two parts. The first one – Basic Variational Principles and the second one – Advanced Variational Analysis. The first part in chapters 1-9 covers the following subjects: weak solution methods, abstract variational principles, complements on measure theory, Sobolev spaces, classical examples, the finite element method, spectral analysis, convex duality and optimization. The second part in chapters 10–17 presents the following subjects: BV and SBV spaces, relaxation in Sobolev, BV, and Young measures spaces, \(\Gamma\)-convergence and applications, integral functionals of the calculus of variations, applications in mechanics and computer vision, variational problems with a lack of coercivity, shape optimization problems and gradient flows.The book is addressed to PhD students, researchers and practicioners who want to study the field of variational analysis in a systematic way. Summing up, everybody who is interested in variational methods, optimization, and PDEs should have this valuable book in his/her library. Reviewer: Wiesław Kotarski (Sosnowiec) Cited in 1 ReviewCited in 158 Documents MSC: 49-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to calculus of variations and optimal control 49K20 Optimality conditions for problems involving partial differential equations 49J45 Methods involving semicontinuity and convergence; relaxation 49J52 Nonsmooth analysis 49Q10 Optimization of shapes other than minimal surfaces 49Q20 Variational problems in a geometric measure-theoretic setting 90C48 Programming in abstract spaces Keywords:variational analysis; Sobolev spaces; BV spaces; PDEs; optimization; \(\Gamma\)-convergence; relaxation; Young measures; shape optimization Citations:Zbl 1095.49001 PDFBibTeX XMLCite \textit{H. Attouch} et al., Variational analysis in Sobolev and BV spaces. Applications to PDEs and optimization. 2nd revised ed. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM); Philadelphia, PA: Mathematical Optimization Society (2014; Zbl 1311.49001) Full Text: DOI