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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
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