Abstract
We investigate the case of a medium with two inclusions or inhomogeneities with nearly touching corner singularities. We present two different asymptotic models to describe the phenomenon under specific geometrical assumptions. These asymptotic expansions are analysed and compared in a common framework. We conclude by a representation formula to characterise the detachment of the corners and we provide the possible extensions of the geometrical hypotheses.
Highlights
The aim of the paper is to investigate the asymptotic behavior of the solution to the conductivity problem in a domain with two nearly touching inclusions with corner singularity
We provide a common framework for two different methods to derive the asymptotic expansions of the solution at δ = 0
It is important to derive the asymptotic expansion in the ideal case where no singularity appears at any order
Summary
The aim of the paper is to investigate the asymptotic behavior of the solution to the conductivity ( called thermal) problem in a domain with two nearly touching inclusions with corner singularity. We provide a common framework for two different methods to derive the asymptotic expansions of the solution at δ = 0 These expansions are mainly based upon the solution to the limit problem δ = 0 and the appropriate so-called profile terms. As we can see in (1.3), the singularities of u0 are canceled through the cutoff function ψ (at distance O(δ) from the origin), and replaced by adapted counterparts through the profiles terms, rescaled to fit the δ-depending domain Πδ It is worth noting whatever the method, the zeroth–order terms of the expansion of uδ are given by (1.3). The goal of the paper is to push forward the expansion for each method to provide different ways to approximate uδ
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