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Summary: This article covers recent developments in the analysis of microstructures that arise from solid-solid phase transition, involving methods and models from the variational calculus. The material is presented in seven sections, starting with a comprehensive presentation of basic problems related to the formation of microstructures. Section 2 contains examples known as $k$-gradient problems, and collects some results concerning approximate and exact solutions for $k\le 4$. In section 3 the reader becomes familiar with the notion of Young measures and with the way how the Young measures can be used to represent the limits of variational integrals. Section 4 is devoted to the study of gradient Young measures, i.e. Young measures which arise from sequences of gradients. Among other things, the author describes here the classification of gradient Young measures due to {\it D. Kinderlehrer} and {\it P. Pedregal} [J. Geom. Anal. 4, No. 1, 59--90 (1994;

Zbl 0808.46046)] which relies on the concept of quasiconvexity of variational integrals. Moreover, Šveràk’s counterexample is reviewed, and it is indicated how to obtain {\it J. Kristensen}’s result [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 16, No. 1, 1--13 (1999;

Zbl 0932.49015)] saying that the quasiconvexity is not a local condition. In Section 5 the problem of exact solutions is discussed, i.e. the problem of finding all Lipschitz maps $u$ satisfying $Du\in K$ a.e. for a given compact set $K$ in the space of matrices (for example, the author considers the case $K=\text{SO}(2)A\cup \text{SO}(2)B$ for special choices of $A$ and $B$). Section 6 briefly describes some reasonable penalty terms which can be added to the pure elastic energy in order to exclude an infinitesimally fine mixture of phases. The final section 7 contains alternative descriptions of microstructures, gives comments on the dynamics and computation of microstructures, and finishes with some solved and unsolved problems.
The material is presented with great care, showing the author’s ability to explain the main methods which have been developed in this rapidly growing area of applied mathematics. Clearly, most of the proofs have not been written down, the interested reader who wants to go into details will find an exhaustive list of more than 230 references together with some comments on the history. The article is addressed to graduate students and researchers in applied analysis, applied mathematics, mechanics, material science and engineering. For the entire collection see [

Zbl 0927.00029].

Reviewer: M. Fuchs (Saarbrücken)