Asymptotic approximation of the solution to the Robin problem in a thick multi-structure

We consider a mixed boundary value problem for the Poisson equation in a thick multistructure Ωɛ, which is the union of a domain Ω0 and a large number N of ɛ-periodically situated thin rings with variable thickness of order ɛ = O(N-1): The Robin conditions are given on the lateral boundaries of the thin rings. The leading terms of the asymptotic expansion for the solution are constructed and the corresponding estimates in the Sobolev space H1(Ωɛ) are proved (as ɛ-> 0) with minimal conditions for the right-hand side.