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G-operators and \(\Gamma\)-convergence. (English) Zbl 0568.35025
Proc. Int. Congr. Math., Warszawa 1983, Vol. 2, 1175-1191 (1984).
[For the entire collection see Zbl 0553.00001.]
The author presents the general theory of \(\Gamma\)-convergence. This theory represents a substantial generalization of earlier work on quadratic functionals related with limit problems for second order elliptic partial differential equations. In general it is observed that pointwise convergence does not guarantee the convergence of the minimum values of functionals, even for real smooth functions. A type of convergence which ensures, under suitable hypotheses, the convergence of the minimum values is just \(\Gamma\)-convergence. The definition of \(\Gamma\)-convergence, the principal properties and further generalizations are given. Moreover the general results and calculus of variations are considered. Illustrating direct methods in \(\Gamma\)- convergence the author recalls the compactness theorem with respect to the \(\Gamma\)-convergence of classes of integral functionals. But examples are provided in which the \(\Gamma\)-limit of a sequence of functionals exists, but has a form very different from the approximating functionals. In the last section he mentions the applications of \(\Gamma\)-convergence to optimal control problems, differential equations, stationary points, Pareto minima and random homogenization.
Reviewer: M.Codegone

MSC:
35J25 Boundary value problems for second-order elliptic equations
35B40 Asymptotic behavior of solutions to PDEs
49J27 Existence theories for problems in abstract spaces