zbMATH — the first resource for mathematics

Calculus of variations and homogenization. (English) Zbl 0939.35019
Cherkaev, Andrej (ed.) et al., Topics in the mathematical modelling of composite materials. Boston, MA: Birkhäuser. Prog. Nonlinear Differ. Equ. Appl. 31, 139-173 (1997).
This paper is a very interesting synthesis of results obtained by the authors from 1970. Many of them are well known, because they were presented in conferences and seminars, but they have not been published in this form until now. Proofs are not included. First part concerns the concept of generalized solutions and relaxation procedures applied to the derivation of the necessary optimality conditions in Pontryagin’s principle. Rest of the paper deals with homogenization techniques (\(H\)-convergence of operators). An important well described application is to determine the properties of heterogeneous materials made of some basic materials. Another interesting and difficult application is related with an optimal design problem.
For the entire collection see [Zbl 0870.00018].

35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
49J45 Methods involving semicontinuity and convergence; relaxation
74Q05 Homogenization in equilibrium problems of solid mechanics
49M20 Numerical methods of relaxation type