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Book review

Author(s) OLEINIK, O.A.
Title Mathematical Problems in Elasticity and Homogenization
Publisher North-Holland
Year of publication 1992
Reviewed by L. Trabucho

The main purpose of this book is to study homogenization problems in the framework of linear elastostatics.

The first chapter contains most of the functional analysis tools needed in the book and which are relevant to elasticity problems. These include function spaces; Korn's type inequalities for different domains including the case where the domain depends on a small parameter, which goes to zero; existence results in linear elasticity for the Dirichlet, Neumann and mixed boundary value problems both in classical and perforated domains.

After an analysis of Saint Venant's principle for periodic solutions the chapter ends up with a study of necessary and sufficient conditions for the strong G-convergence of elasticity type operators.

Chapter II deals with the homogenization of the system of linear elasticity both for composite and perforated materials. This includes the analysis of different types of boundary conditions on the holes.

Next, several periodic cases are studied using asymptotic expansion techniques and the results are generalized to the case of almost periodic coefficients. The corresponding results in terms of G-convergence are also presented.

The last chapter deals with spectral problems. After reviewing some theorems from functional analysis, the authors study the homogenization of eigenvalues and eigenfunctions associated with different types of boundary value problems for strongly nonhomogeneous elastic bodies. They apply these results to the Dirichlet problem for second order elliptic equations in perforated domains, and also to mixed boundary value problems for second order elliptic equations in domains with rapidly oscillating boundary.

The book ends up with some results, presented briefly, for the limit behaviour of eigenvalues in domains with different types of cavities; homogenization of eigenvalues for ordinary differential operators; an asymptotic expansion analysis for the Sturm-Liouville problem for equations with rapidly oscillating coefficients together with some additional results about G-convergence.

In summary a very usefull and complete book in homogenization applied to linear elasticity problems, which also contains some results that were available up to now only in research papers and some of them not easily obtainable.