Homogenization of a boundary-value problem with a nonlinear boundary condition in a thick junction of type 3:2:1

We consider a boundary-value problem for the Poisson equation in a thick junction Ω_ε, which is the union of a domain Ω₀ and a large number of ε-periodically situated thin curvilinear cylinders. The following nonlinear Robin boundary condition ∂_νu_ε + εκ(u_ε)=0 is given on the lateral surfaces of the thin cylinders. The asymptotic analysis of this problem is performed as ε → 0, i.e. when the number of the thin cylinders infinitely increases and their thickness tends to zero. We prove the convergence theorem and show that the nonlinear Robin boundary condition is transformed (as ε → 0) in the blow-up term of the corresponding ordinary differential equation in the region that is filled up by the thin cylinders in the limit passage. The convergence of the energy integral is proved as well. Using the method of matched asymptotic expansions, the approximation for the solution is constructed and the corresponding asymptotic error estimate in the Sobolev space H¹(Ω_ε) is proved. Copyright © 2007 John Wiley & Sons, Ltd.