Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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


Author: Chien Heng Wu
Journal: Quart. Appl. Math. 34 (1976), 39-45
MSC: Primary 73.41
DOI: https://doi.org/10.1090/qam/455712
MathSciNet review: 455712
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Abstract: 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.


References [Enhancements On Off] (What's this?)

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DOI: https://doi.org/10.1090/qam/455712
Article copyright: © Copyright 1976 American Mathematical Society


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