On the averaging of a boundary value problem for the biharmonic equation in a domain containing thin short channels. (Q1432704)

We study the asymptotic behavior of solutions of a boundary value problem for the biharmonic equation in the union of domains joined by thin channels of width \(a_{\varepsilon}\) and length \(\varepsilon^q, q>0\), under the condition that the distance beween adjoining channels is equal to \(\varepsilon\), where \(\varepsilon\) is a small parameter, \(a_{\varepsilon}\to 0\) as \(\varepsilon \to 0\), and the number of these channels is equal to \(N(\varepsilon)\), with \(N(\varepsilon)\simeq d\varepsilon^{-1}\to \infty\) when \(\varepsilon \to 0\). \textit{V. A. Marchenko} and \textit{E. Ya. Khruslov} [Boundary value problems in domains with a fine-grained boundary (Russian), Izdat. ``Naukova Dumka'', Kiev (1974; Zbl 0289.35002)] and \textit{T. A. Shaposhnikova} [Sb. Math. 192, No. 10, 1553--1585 (2001); translation from Mat. Sb. 192, No. 10, 131--160 (2001; Zbl 1041.35014)] considered a similar problem for the Poisson equation.