An introduction to \(\Gamma\)-convergence.

*(English)*Zbl 0816.49001
Progress in Nonlinear Differential Equations and their Applications. 8. Basel: Birkhäuser. xiv, 340 p. (1993).

This really well-written book presents a notion of “variational convergence”, called \(\Gamma\)-convergence (originated by E. De Giorgi and his group of collaborators), which proved to be very useful in the study of variational problems characterized by severe perturbations of the data. It has been applied with great success to the study of phase transitions, singular perturbations, boundary value and control problems in widely perturbed domains and various homogenization problems in mathematical theory of composite materials.

The first part of the book contains the abstract theory of \(\Gamma\)- convergence for functionals defined on arbitrary topological spaces, while the second one is devoted to the study of the asymptotic behaviour of integral functionals of the calculus of variations, with applications to homogenization problems.

All topics presented in the book are treated in great detail and precision. We mention here only some of them: the topology of \(\Gamma\)- convergence, representation theorems for local functionals, relations between \(\Gamma\)-convergence of quadratic functionals and G-convergence of the Green operators associated to them…Also many interesting examples and counterexamples are given.

A large (almost complete to 1991) bibliography and a superbe guide to the literature related to the subjects treated in the book provide an invaluable source of information not only for newcomers in the field.

The first part of the book contains the abstract theory of \(\Gamma\)- convergence for functionals defined on arbitrary topological spaces, while the second one is devoted to the study of the asymptotic behaviour of integral functionals of the calculus of variations, with applications to homogenization problems.

All topics presented in the book are treated in great detail and precision. We mention here only some of them: the topology of \(\Gamma\)- convergence, representation theorems for local functionals, relations between \(\Gamma\)-convergence of quadratic functionals and G-convergence of the Green operators associated to them…Also many interesting examples and counterexamples are given.

A large (almost complete to 1991) bibliography and a superbe guide to the literature related to the subjects treated in the book provide an invaluable source of information not only for newcomers in the field.

Reviewer: Z.Denkowski (Kraków)

##### MSC:

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

49J45 | Methods involving semicontinuity and convergence; relaxation |

49J40 | Variational inequalities |

35B27 | Homogenization in context of PDEs; PDEs in media with periodic structure |

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |