Near-boundary expansion of Green’s function associated with clamped plates

The Green’s function $G\left ( {P,P’} \right )$ associated with a clamped plate of arbitrary shape is considered, when $P’$ is at a distance $O\left ( \epsilon \right )$ from a regular point $O$ of the boundary. First an outer expansion of $G$ is described, valid when $P$ is not near $P’$. Then an inner expansion of $G$ is constructed when both $P$ and $P’$ are near 0. The leading term of the inner expansion is just the Green’s function ${G_s}$, for the halfplane bounded by the tangent to the boundary at $O$, and ${\epsilon ^{ - 2}}G$ differs from ${\epsilon ^{ - 2}}{G_s}$ by $O\left ( \epsilon \right )$. The first two terms of the inner expansion agree with the first two terms of the expansion of ${G_c}$ , the Green’s function for the interior of the osculating circle of the boundary at 0, if the boundary is convex at $O$. If it is concave, ${G_c}$ is the Green’s function for the exterior of the osculating circle. Moreover, ${\epsilon ^{ - 2}}G$ differs from ${\epsilon ^{ - 2}}{G_c}$ by $O\left ( {{\epsilon ^2}} \right )$. A two-term inner expansion is explicitly given.