Asymptotics for the electric field when $M$-convex inclusions are close to the matrix boundary

TL;DR

This paper derives boundary asymptotic formulas for electric fields in composite materials with M-convex inclusions near the boundary, revealing singularities and weakening smoothness assumptions on inclusions.

Contribution

It develops a concise procedure to establish boundary asymptotics for perfect conductors of arbitrary shape, explicitly characterizing singularities of the blow-up factor.

Findings

Explicit boundary asymptotic formulas for electric fields near inclusions

Identification of singularities in the blow-up factor Q[]

Reduced smoothness requirements for inclusion boundaries

Abstract

In the perfect conductivity problem of composites, the electric field may become arbitrarily large as $\varepsilon$, the distance between the inclusions and the matrix boundary, tends to zero. The main contribution of this paper lies in developing a clear and concise procedure to establish a boundary asymptotic formula of the concentration for perfect conductors with arbitrary shape in all dimensions, which explicitly exhibits the singularities of the blow-up factor $Q[\varphi]$ introduced in [29] by picking the boundary data $\varphi$ of $k$-order growth. In particular, the smoothness of inclusions required for at least $C^{3,1}$ in [27] is weakened to $C^{2,\alpha}$, $0<\alpha<1$ here.