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Variational analysis in Sobolev and BV spaces. Applications to PDEs and optimization. (English) Zbl 1095.49001

MPS/SIAM Series on Optimization 6. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM). Philadelphia, PA: MPS, Mathematical Programming Society (ISBN 0-89871-600-4/pbk; 978-0-89871-878-2/ebook). xii, 634 p. (2006).
This volume is on various functional analytic and function space developments that are relevant in some areas of modern variational calculus (“modern” meaning, roughly, the last quarter century). Along the way, the authors develop essentially from scratch all necessary “classical” tools (e.g., Sobolev spaces, Hausdorff measures, the functional analytic setup for linear partial differential operators, variational theory of differential operators and the attendant finite element method, Young measures) and cover such classical applications as the Courant-Fisher minimax and maximin characterization of eigenvalues of selfadjoint operators. Among more recent (but already classical) topics are nonsmooth analysis, convex analysis, and Ekeland’s \(\epsilon\)-variational principle, whose influence in the development of calculus of variations and infinite dimensional control theory has been (and continues to be) enormous.
In the last chapters, the reader is introduced to numerous areas that are presently in a state of evolution and where research proceeds at a fast pace. Among these areas are the spaces \(BV(\Omega)\) and \(SBV(\Omega)\) of multivariable functions of bounded variation, which complement and complete Sobolev spaces, and existence theorems via relaxation (Young measures appear in new guises here). Several applications are also covered or sketched (fracture mechanics, phase transitions). Material is included that has not previously seen print in book form.
This book should provide (nearly) self contained reading to anybody having a solid grounding on Lebesgue integration, linear functional analysis and some partial differential equations. Classroom uses suggest themselves, for instance in graduate courses in applied functional analysis (where the choice of topics would admit many combinations) and also in courses that aim for a more specific approach to active fields of research in variational calculus.

MSC:

49-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to calculus of variations and optimal control
35A15 Variational methods applied to PDEs
35D05 Existence of generalized solutions of PDE (MSC2000)
26B30 Absolutely continuous real functions of several variables, functions of bounded variation
49K20 Optimality conditions for problems involving partial differential equations
49J52 Nonsmooth analysis
49J45 Methods involving semicontinuity and convergence; relaxation
49Q20 Variational problems in a geometric measure-theoretic setting
35-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to partial differential equations
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