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

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.