Correctors in the homogenization of elasticity problems on thin structures.

*(English. Russian original)*Zbl 1272.74532
Dokl. Math. 71, No. 2, 177-182 (2005); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 401, No. 1, 21-26 (2005).

Summary: A homogenization principle for elasticity problems considered on periodic grids of critical thickness was obtained in V.V. Zhikov and S.E. Pastukhova [Sb. Math. 194, No. 5, 697–732 (2003); translation from Mat. Sb. 194, No. 5, 61–96 (2003; Zbl 1077.35023)]. This result relied heavily on two-scale convergence in a variable space. In this paper, we hardly use two-scale convergence but rather return to the classical method of asymptotic expansions and prove a corrector theorem that refines the result loc. cited above. Asymptotic expansions are constructed from solutions to auxiliary problems considered on a one-periodic grid whose thickness \(h\) tends to zero. The solutions are elements of the vector space \(L^2\) with a variable measure \(\mu^h\), and the substantiation of the asymptotic expansion is based heavily on the convergence properties in this space. The proof of the corrector theorem is derived by analyzing these properties and passing to the limit in particular problems related to the expansion.