  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.
