The asymptotics for the perfect conductivity problem with stiff \(C^{1,\alpha}\)-inclusions
Consider bounded domains \(D_1 \subset D \subset \mathbb R^n\) (\(n \geq 2\)) with the boundaries \(\partial D_1\) and \(\partial D\) being of \(C^{1,\alpha}\) (\(0 < \alpha < 1\)). Let \(\nu\) denote the outward unit normal to \(\partial D_1\). The authors discuss the boundary value problem \(\Delta u =0\) in \(D\backslash D_1\), \(u=C\) on \(\partial D_1\), \(\int_{\partial D_1} \frac{\partial u}{\partial \nu} ds =0\), \(u=\phi\) on \(\partial D\) with a given function \(\phi\) and an undetermined constant \(C\). Let \(\varepsilon\) denote the distance between \(\partial D_1\) and \(\partial D\). The authors investigate the behavior of \(u\) and \(\nabla u\) as \(\varepsilon \to 0\) when the curves become touching at one isolated point. The boundary and interior fields are estimated.
blow-up phenomena in highcontrast fiber-reinforced composites
scientific article; zbMATH DE number 7358768
scientific article