The asymptotics of the Green function for a second-order elliptic equation near the boundary of the domain

The Green function $G(x,\xi)$ of the first boundary value problem in a bounded domain with smooth boundary is investigated. It is assumed that the point $\xi$, where the function has a singularity, tends to the boundary of the domain. Under this condition an asymptotic expansion of $G(x,\xi)$ is constructed up to any power of the distance from $\xi$ to the boundary. Asymptotic expansions different in form are constructed for points $x$ near $\xi$ and for points far from $\xi$. The final construction and the justification for the asymptotics of $G(x,\xi)$ is carried out by the method of matching asymptotic expansions.
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