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Semicontinuity, relaxation and integral representation in the calculus of variations. (English) Zbl 0669.49005
Pitman Research Notes in Mathematics Series, 207. Harlow: Longman Scientific & Technical; New York: John Wiley & Sons. 222 p. £16.00 (1989).
The main subject of these research notes is related to the direct method in the calculus of variations to prove the existence of minimizers for various types of integral functionals. The standard approach is to establish coerciveness and lower semicontinuity with respect to an appropriate topology and to show in this way that limit points of minimizing sequences are minimizers for the given integral functional. This usually works well if semicontinuity is related to some qualitative properties of the integrand like convexity or quasiconvexity but may fail in the absence of such properties. In this latter case a convenient tool is to study minimizers of relaxed functionals being defined as the lower semicontinuous envelope of the given functional. For obvious reasons it is desirable to have an integral representation of the relaxed functional which, however, is not always easy to obtain.
The presented material is subdivided into five chapters. In Chapter 1 an introduction is given to the direct method and the idea of relaxation in an abstract framework. Then, Chapters 2-4 specialize on lower semicontinuity, relaxation and integral representation for functionals defined on \(L^ p\) spaces (Chapter 2), the space of measures (Chapter 3) and Sobolev spaces (Chapter 4). Finally, Chapter 5 deals with relaxation in optimal control. Again, the basic results are first derived in an abstract setting and then applied to problems with ordinary and elliptic state equations.
The book is nicely organized and written and gives an excellent survey on the recent developments in this area of the calculus of variations.
Reviewer: R.H.W.Hoppe

49J45 Methods involving semicontinuity and convergence; relaxation
49-02 Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control