Asymptotics of the solution of the Dirichlet problem for the system

G. V. Zhdanova

Sbornik. Mathematics, Tome 62 (1989) no. 1, pp. 1-27

Résumé

The asymptotics is found for a solution of the system of equations $$ A(\partial_x)\mathbf u(x)+\omega^2\rho\mathbf u(x)=0,\quad x\in D_\varepsilon,\qquad \mathbf u(x)=\mathbf f(x),\quad x\in S_\varepsilon, $$ of steady-state elastic vibrations of an isotropic medium. Here $x\in\mathbf R^3$, $\varepsilon>0$ is a small parameter, $S_\varepsilon$ is a bounded closed surface given in spheroidal coordinates by the equation $\xi=1+\varepsilon g(\eta,\varepsilon)$, and $D_\varepsilon$ is the exterior of $S_\varepsilon$. The vector-valued function $\mathbf u(x)$ satisfies a radiation condition. The asymptotics of the solution of the problem is found up to $O(\varepsilon^m)$, $m>0$ arbitrary, in the case where the boundary condition does not depend on the polar angle $\varphi$, and up to $O(\varepsilon^2\ln\varepsilon)$ in the case of boundary conditions which are not axially symmetric. The formulas obtained are valid everywhere near the body (including neighborhoods of the end points) and far from it. Bibliography: 12 titles.