## Asymptotics of the solution of the Dirichlet problem for a system of the theory of elasticity in the exterior of a thin body of revolution.(Russian)Zbl 0658.73003

This paper is concerned with the asymptotics of the solution to the Dirichlet problem for a system of the theory of elasticity in the exterior domain $$D_{\epsilon}$$ whose boundary $$S_{\epsilon}$$ converges to $$r=0$$, $$| z| \leq 1$$ as $$\epsilon$$ $$\to 0$$. More precisely, the following problem $$A(\partial_ x)u(x)+\omega^ 2\rho u(x)=0$$, $$x\in D_{\epsilon}$$, $$u(x)=f(x)$$ on $$S_{\epsilon}$$, and at infinity: $$u(x)=(\exp (ik_ 1R)/R)f_ 1(\theta,\phi,\epsilon)+(\exp (ik_ tR)/R)f_ t(\theta,\phi,\epsilon)+O(R^{-2})$$ as $$R\to \infty$$, where $$A(\partial_ x)=\| A_{ij}(\partial_ x)\|_{3\times 3}$$, $$A_{ij}(\partial_ x)=\delta_{ij}\mu \Delta +(\lambda +\mu)(\partial^ 2/\partial x_ i\partial x_ j)$$, $$x\in R^ 3$$, $$x=(x_ 1,x_ 2,x_ 3)$$ $$u(x)=\sum^{3}_{k=1}u_ ki_ k$$, $$k^ 2_ 1=\omega^ 2\rho /(\lambda +2\mu)$$, $$k^ 2_ t=\omega^ 2\rho /\mu$$, is considered. The main results of this paper is to give an approximation to the accurate solution and to obtain an error estimate which is about $$\epsilon^ 2| \ln \epsilon |$$.
Reviewer: S.Zheng

### MSC:

 74B05 Classical linear elasticity 35C20 Asymptotic expansions of solutions to PDEs 74K15 Membranes
Full Text: