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**Parametrized measures and variational principles.**
*(English)*
Zbl 0879.49017

Progress in Nonlinear Differential Equations and their Applications. 30. Basel: Birkhäuser. xi, 212 p. (1997).

The purpose of this book is to give a different and new approach to some questions of the calculus of variations, based on the notion of parametric measure or Young measure. Given a integral functional, the argument of Direct Methods is based on the semicontinuity of this functional, which is related with the convexity property on the integrand, and on convergence of minimizing sequence in the same weak topology. In many cases the lack of convexity leads to oscillating minimizing sequences whose weak limit is not a minimizer. The Young measures permit to understand and also predict the highly oscillatory nature of the minimizing sequences and determine the character of the semicontinuity of the functionals. In this direction, the book is very interesting and selfcontaining.

The plan of the book is the following. In Ch. 1, the basic concept of the direct methods and parametric measures are introduced. Ch. 2 is devoted to consider some important variational problems, optimal control and optimal design problems, with the purpose to give an idea of the usefulness of parametric measures. Moreover, the large time behaviour of complicated turbulence system is considered. In Ch. 3 the weak lower semicontinuity under convexity is studied and some existence results are given. Moreover, the polyconvexity and applications to nonlinear elasticity are also studied. In Ch. 4 the nonconvex problems and the relaxation theory are analyzed. Moreover, some variational problems, where parametric measures are involved, are considered. In Ch. 5 phase transitions and microstructure are introduced, and some examples from continuum mechanics and micromagnetics are considered. Ch. 6 and Ch. 7 are devoted to a systematic study of the parametric measures and to introduce the basic tools to deal with these families of probability measures. In particular, the case in which parametric measures can be generated by some sequences of functions is studied. Ch. 8 is devoted to the study of parametric measures associated with sequences of gradients, which are very interesting in variational principles, since in this case the integrand depends on the gradients. In Ch. 9 the relationship between quasi-convexity and rank-one convexity of vectorial integral functionals is considered. Moreover, the structure of laminates, which gives an explicit example of application of gradient parametrized measures, is analyzed. Finally, Ch. 10 is devoted to the analysis of divergence free parametrized measures.

The plan of the book is the following. In Ch. 1, the basic concept of the direct methods and parametric measures are introduced. Ch. 2 is devoted to consider some important variational problems, optimal control and optimal design problems, with the purpose to give an idea of the usefulness of parametric measures. Moreover, the large time behaviour of complicated turbulence system is considered. In Ch. 3 the weak lower semicontinuity under convexity is studied and some existence results are given. Moreover, the polyconvexity and applications to nonlinear elasticity are also studied. In Ch. 4 the nonconvex problems and the relaxation theory are analyzed. Moreover, some variational problems, where parametric measures are involved, are considered. In Ch. 5 phase transitions and microstructure are introduced, and some examples from continuum mechanics and micromagnetics are considered. Ch. 6 and Ch. 7 are devoted to a systematic study of the parametric measures and to introduce the basic tools to deal with these families of probability measures. In particular, the case in which parametric measures can be generated by some sequences of functions is studied. Ch. 8 is devoted to the study of parametric measures associated with sequences of gradients, which are very interesting in variational principles, since in this case the integrand depends on the gradients. In Ch. 9 the relationship between quasi-convexity and rank-one convexity of vectorial integral functionals is considered. Moreover, the structure of laminates, which gives an explicit example of application of gradient parametrized measures, is analyzed. Finally, Ch. 10 is devoted to the analysis of divergence free parametrized measures.

Reviewer: E.Mascolo (Firenze)

### MSC:

49J45 | Methods involving semicontinuity and convergence; relaxation |

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

49J40 | Variational inequalities |

49M20 | Numerical methods of relaxation type |