Geodesic
The asymptotics of the Green function for a~second-order elliptic equation near the boundary of the domain
Izvestiya. Mathematics , Tome 23 (1984) no. 3, pp. 579-594.

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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. Bibliography: 12 titles. 
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A. M. Il'in; B. I. Suleimanov. The asymptotics of the Green function for a~second-order elliptic equation near the boundary of the domain. Izvestiya. Mathematics , Tome 23 (1984) no. 3, pp. 579-594. http://geodesic.mathdoc.fr/item/IM2_1984_23_3_a9/

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