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

Abstract

We consider a boundary-value problem for the Poisson equation in a thick junction ε , which is the union of a domain 0 and a large number of ε-periodically situated thin curvilinear cylinders. The following nonlinear Robin boundary condition * u ε +ε (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 H 1 ( ε ) is proved.

Homogenization of quasilinear optimal control problems involving a thick multilevel junction of type 3 : 2 : 1

“…4δ , a, b, δ > 0) and (1.2), we deduce from (1.1) the following estimates Similarly as in [19] we get from (0.3) the following inequalities…”

“…At first glance it may seem that there is no difference between the boundary conditions A similar phenomenon is observed in [19] for a boundary-value problem in a thick one-level junction.…”

“…Therefore boundary-value problems in thick junctions of different types are very extensively investigated at present (see [1][2][3]6,8,10], [19,20,24,25] and references therein). The aim of these researches is to develop rigorous methods to study the asymptotic behavior of solutions when the number of the attached thin domains of a thick junction infinitely increases and their thickness vanishes.…”

“…Using the approach of paper [19], we can state that for fixed values of the parameters ε, α ∈ R, β ∈ R, and for every function θ ε ∈ K…”

“…Mathematical justification of this fact is presented in [19] with due regard for the very small chemical activity between the surface of a thick absorber and water.…”