On finding the asymptotes to the solutions of short-wave diffraction problems

A. Ya. Povzner ; I. V. Sukharevskii

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 1 (1961) no. 2, pp. 224-245

Résumé

We use the following notation: $x,y,s$ a are the radius vectors of points in the three-dimensional region $D$ or on the boundary $S$ of this region; $|x-s|$ is the distance between the points $x$ and $s$; $\partial s$, $\partial s_j$ is an element of area of the surface $S$; $\mathbf n$ is the orthonormal to $S$ going out of $D$; $u^+(x)$ is the limit of the function $u(y)$ as the point y of $D$ tends to the point $x$ on the surface $S$; $(\partial u/\partial n)^+$ is the boundary value of the normal derivative passing into $S$ from the region $D$; $u^-$, $(\partial u/\partial n)^-$; have analogous meanings in passing into $S$ from the other side of the surface.