Asymptotic behavior of a solution to a boundary value problem in a perforated domain with oscillating boundary. (English. Russian original) Zbl 0918.35043

Sib. Math. J. 39, No. 4, 621-644 (1998); translation from Sib. Mat. Zh. 39, No. 4, 730-754 (1998).
The aim of this article is to study a model problem for the Poisson equation in a perforated domain with very rapidly oscillating outer boundary in the presence of small dissipation on the boundaries of holes. This model appears in material technology for studying the macroscopic behavior of micro-inhomogeneous perforated media and bodies with rough surfaces. The authors study a particular case of such a medium in which perforation as well as oscillation of the boundary are locally periodic and their structures are assumed to be adjusted. In the study of locally periodic perforation, a difficulty arises: the geometry of cavities is not fixed. The method of compensated compactness or the method of two-scale convergence can be applied for constructing a limit problem, but it provides no estimates for the error. In order to overcome this obstacle, the authors use the technique of asymptotic expansion which requires the data to be regular but makes it possible to estimate the convergence rate. Namely, two terms of the interior asymptotic expansion guarantee an estimate of order \(\sqrt\varepsilon\) in the \(H^1\)-norm. The estimate for the residual is improved by constructing a boundary layer corrector that yields to an estimate of order \(\varepsilon\).


35J25 Boundary value problems for second-order elliptic equations
35B40 Asymptotic behavior of solutions to PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35C20 Asymptotic expansions of solutions to PDEs
35A20 Analyticity in context of PDEs
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
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