Homogenization of a boundary‐value problem with a nonlinear boundary condition in a thick junction of type 3:2:1
Taras Mel’nyk · Mathematical Methods in the Applied Sciences · 2007
Abstract We consider a boundary‐value problem for the Poisson equation in a thick junction Ωε, which is the union of a domain Ω0and a large number of ε‐periodically situated thin curvilinear cylinders. The following nonlinear Robin boundary condition ∂νuε+ εκ(uε)=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 spaceH1(Ωε) is proved. Copyright © 2007 John Wiley & Sons, Ltd.