Asymptotic analysis for the electric field concentration with geometry of the core-shell structure

Abstract

<p style='text-indent:20px;'>In the perfect conductivity problem arising from composites, the electric field may become arbitrarily large as <inline-formula><tex-math id="M1">\begin{document}$ \varepsilon $\end{document}</tex-math></inline-formula>, the distance between the inclusions and the matrix boundary, tends to zero. In this paper, by making clear the singular role of the blow-up factor <inline-formula><tex-math id="M2">\begin{document}$ Q[\varphi] $\end{document}</tex-math></inline-formula> introduced in [<xref ref-type="bibr" rid="b27">27</xref>] for some special boundary data of even function type with <inline-formula><tex-math id="M3">\begin{document}$ k $\end{document}</tex-math></inline-formula>-order growth, we prove the optimality of the blow-up rate in the presence of <inline-formula><tex-math id="M4">\begin{document}$ m $\end{document}</tex-math></inline-formula>-convex inclusions close to touching the matrix boundary in all dimensions. Finally, we give closer analysis in terms of the singular behavior of the concentrated field for eccentric and concentric core-shell geometries with circular and spherical boundaries from the practical application angle.</p>