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

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