On the rate of convergence for perforated plates with a small interior Dirichlet zone

Abstract

The aim of the paper is to compare the asymptotic behavior of solutions of two boundary value problems for an elliptic equation posed in a thin periodically perforated plate. In the first problem, we impose homogeneous Dirichlet boundary condition only at the exterior lateral boundary of the plate, while at the remaining part of the boundary Neumann condition is assigned. In the second problem, Dirichlet condition is also imposed at the surface of one of the holes. Although in these two cases, the homogenized problem is the same, the asymptotic behavior of solutions is rather different. In particular, the presence of perturbation in the boundary condition in the second problem results in logarithmic rate of convergence, while for non-perturbed problem the rate of convergence is of power-law type. (2000). 35B25 · 35B27 · 35J25. Mathematics Subject Classification

Spectra of open waveguides in periodic media

“…In this case the Douglis-Nirenberg system (4.10) decouples into two second-order equations and a fourth-order equation. In Section 4.3 we show that our scheme works also in the case of high-order differential equations including the Kirchhoff plate (see also [13]). Remark 9 Lipschitz domains occur for example in the grating of quantum waveguides with long-haul thickening as in Fig.…”

Scalar boundary value problems on junctions of thin rods and plates

“…Both the problems permit separation of variables and are rather standard in the asymptotic analysis of rods and perforated plates, even isolated (cf., respectively, the monographs [30,20,45,31] and the papers [33,7]). We here present only some comprehensible pieces of information on them which will be used for asymptotic structures in Section 3.…”

“…Under conditions (4.16) and (4.17), the solution of problem (1.8), (1.9), (1.19), (1.11)-(1.14) with α = 0 satisfies∇ x (u 0 − U # 0 ); L 2 (Ω • (h)) + j ∇ x (u j − U # j ); L 2 (Ω j (h)) ≤ ch 3/2 N ,(5 7). …”

Convergence Rates in Homogenization of the Mixed Boundary Value Problems

“…In [8] the 1 weak convergence of solutions was obtained in homogenization problems of multi-level-junction type. In 2011, Cardone [9] considered the homogenization with mixed boundary value problems in a thin periodically perforated plate and obtained the logarithmic rate of convergence of solutions. In the monograph [10], the 1 convergence rate was shown by the method of potentials for the solutions to the Dirichler-Fourier mixed boundary value problem in the perforated domain.…”