On the Problem of the Averaging of Solutions to the Laplace Operator in Domains with Narrow Slanting Channels

We consider the problem of the averaging of solutions to the Laplace operator in domains with narrow slanting channels of length O(ε^q), q = const > 0, and diameter a _ε = o(ε ^q), where ε is a small parameter. The number of channels is N _ε = O (ε ¹⁻ⁿ ), where n is the dimension of the space. We study the asymptotic behavior of solutions, obtain the limit problem, and estimate the closeness of the initial and limit problem.__________Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 10, Suzdal Conference-4, 2003.