Unbounded functionals in the calculus of variations. Representations, relaxation, and homogenization. (English) Zbl 1002.49018

Carbone, Luciano; De Arcangelis, Riccardo

Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics 125. Boca Raton, FL: Chapman & Hall/CRC (ISBN 1-58488-235-2/hbk). xiii, 394 p. (2002).

The book under review covers, under different points of view, the study of a class of functionals of the calculus of variations of the form \[ F(\Omega, u)= \int_\Omega f(x,\nabla u) dx\tag{1} \] that do not verify the standard growth assumptions: this is why they are called “unbounded functionals”. Indeed, in several examples coming for instance from the applications to elasto-plastic torsion problems, electrostatic screening, rubber-like nonlinear elastomers, the integrand \(f(x,z)\) above is convex and lower-semicontinuous in the \(z\) variable, but it may take the value \(+\infty\) somewhere. On the other hand, coercivity assumptions like \[ f(x,z)\geq \alpha|z|^p\qquad(\text{with }\alpha>0\text{ and }p>1)\tag{2} \] are not always available, which makes the setting of Sobolev spaces \(W^{1,p}(\Omega)\) too small to work with.
In the book, the extension of a functional like (1) to the space \(\text{BV}(\Omega)\) of functions with bounded variation is considered, by taking the largest lower-semicontinuous functional (for the weak\(^*\)-BV convergence) which is less than or equal to \(\int_\Omega f(x,\nabla u) dx\) for every smooth function \(u\).
When a functional of type (1) does not satisfy the standard \(p\)-growth assumptions and also the coercivity condition (2) fails, the variational tools as relaxation, integral representation, homogenization, become more delicate and require a deep analysis that in the book is performed under some suitable assumptions.
The authors made an effort to make the volume as much as possible self-contained; the Chapters 1 to 5 are indeed devoted to recall the main notions of convex analysis, measure theory, \(\Gamma\)-convergence, function spaces, and other tools that are used in the subsequent chapters. In Chapter 6 some of the most classical results of relaxation, integral representation, homogenization, for functionals with standard growth, are recalled, to introduce the reader into a field that is treated in a less standard framework later. Some physical models which require the use of unbounded functionals are also presented. Chapters 7 to 13 deal with the case of unbounded functionals, and most of the results of Chapter 6 are extended to this less regular framework. Finally, Chapter 14 deals with some examples where explicit computations can be made.

Reviewer: Giuseppe Buttazzo (Pisa)

MSC:

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

Keywords:

calculus of variations; lower-semicontinuous functional; relaxation; integral representation; homogenization; \(\Gamma\)-convergence; unbounded functionals