## Averaging of functionals of the calculus of variations and elasticity theory.(English. Russian original)Zbl 0599.49031

Math. USSR, Izv. 29, No. 1, 33-66 (1987); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 50, No. 4, 675-710, 877 (1986).
The author begins his discussion with a comment that variational problems of mechanics appear to have increasingly more complex Lagrangians. Some change their properties in different regions, some are singular, some vary pointwise. In general, Lagrangians $$f(x, \xi)$$ are measurable functions of $$x$$ and convex in $$\xi$$ for a.e. $$x\in\mathbf R^n$$. $$\xi$$ could be a vector or a symmetric matrix $$\{\xi_{ij}\}$$, which is the case in classical elasticity. With each Lagrangian $$f(x,\xi)$$ is associated a dual Lagrangian $$f^*(x, \xi)=\sup_\eta \{\eta\cdot\xi-f(x,\eta)\}$$. This Young-type duality is used in a systematic manner to derive averaging results for nonlinear variational problems, particularly for the so-called regular Lagrangians obeying the inequalities $$c_1|\xi|^\alpha-c_0\leq f(x, \xi)\leq c_2|\xi|^\alpha+c_0$$, $$c_i>0$$, $$\alpha>1$$.
These averaging methods are applied to random Lagrangians, to problems in plasticity with singular Lagrangians, and to Lagrangians representing the behavior of composite materials, or materials with inclusions. This is a development that follows the ingenious applications of the Young-Fenchel duality developed by Soviet authors such as V. L. Berdichevskiĭ, who has written a book “Variational principles of continuum mechanics” (Russian), “Nauka”, Moscow (1983)] and a series of articles.
This approach competes with regularization ideas in deciding when averaging is legal, particularly in problems dealing with critical loads and the corresponding designs of elastic systems.

### MSC:

 49S05 Variational principles of physics 74S30 Other numerical methods in solid mechanics (MSC2010)
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