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Asymptotic analysis of an array of closely spaced absolutely conductive inclusions. (English) Zbl 1116.35310

Summary: We consider the conductivity problem in an array structure with square closely spaced absolutely conductive inclusions of the high concentration, i.e., the concentration of inclusions is assumed to be close to 1. The problem depends on two small parameters: \(\varepsilon\), the ratio of the period of the micro-structure to the characteristic macroscopic size, and \(\delta\), the ratio of the thickness of the strips of the array structure and the period of the micro-structure. The complete asymptotic expansion of the solution to problem is constructed and justified as both \(\varepsilon\) and \(\delta\) tend to zero. This asymptotic expansion is uniform with respect to \(\varepsilon\) and \(\delta\) in the area \(\{\varepsilon=O(\delta^\alpha),\delta=O(\varepsilon^\beta)\}\) for any positive \(\alpha,\beta\).

MSC:

35C20 Asymptotic expansions of solutions to PDEs
78M35 Asymptotic analysis in optics and electromagnetic theory
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
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