Homogenization of the mixed boundary value problem for a~formally self-adjoint system in a~periodically perforated domain

G. Cardone ; A. Corbo Esposito ; S. A. Nazarov

Algebra i analiz, Tome 21 (2009) no. 4, pp. 126-173.

Résumé

A generalized Gårding-Korn inequality is established in a domain $\Omega(h)\subset{\mathbb{R}}^n$ with a small, of size $O(h)$, periodic perforation, without any restrictions on the shape of the periodicity cell, except for the usual assumptions that the boundary is Lipschitzian, which ensures the Korn inequality in a general domain. Homogenization is performed for a formally selfadjoint elliptic system of second order differential equations with the Dirichlet or Neumann conditions on the outer or inner parts of the boundary, respectively; the data of the problem are assumed to satisfy assumptions of two types: additional smoothness is required from the dependence on either the “slow” variables $x$, or the “fast” variables $y=h^{-1}x$. It is checked that the exponent $\delta\in(0,1/2]$ in the accuracy $O(h^\delta)$ $O(h^\delta)$ of homogenization depends on the smoothness properties of the problem data.

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