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**On the behavior at infinity of solutions of elliptic systems with a finite energy integral.**
*(English)*
Zbl 0637.35030

The paper contains a study of the asymptotic behavior in the neighbourhood of infinity of the solution of elliptic partial differential equation of order 2m under the assumption that the weighted Dirichlet integral is bounded. The results provide solutions for systems of linear elasticity with finite energy. A class of solutions which are periodic in some independent variables has also been obtained by the authors.

The main results of the paper are presented in the form of three interesting theorems. The second theorem gives a complete asymptotic expansion at infinity of the solution. The first theorem is a special case of theorem 2 and it has been used in the proof of theorem 2. In the 3rd theorem asymptotic expansion in the neighbourhood of infinity has been obtained for the system: \[ \sum^{N}_{j=1}m_{kj}(\gamma_ z)u_ j(z)=f_ k(z),\quad k=1,2,...,N. \] This provides a solution of the system of linear elasticity if is assumed that u(z) is \(2\pi\)- periodic in y and the integral \[ \iint_{\Omega}\sum^{N}_{k,j=1}(\partial u_ k/\partial z_ j+\partial u_ j/\partial z_ k)^ 2 dz \] is finite.

The main results of the paper are presented in the form of three interesting theorems. The second theorem gives a complete asymptotic expansion at infinity of the solution. The first theorem is a special case of theorem 2 and it has been used in the proof of theorem 2. In the 3rd theorem asymptotic expansion in the neighbourhood of infinity has been obtained for the system: \[ \sum^{N}_{j=1}m_{kj}(\gamma_ z)u_ j(z)=f_ k(z),\quad k=1,2,...,N. \] This provides a solution of the system of linear elasticity if is assumed that u(z) is \(2\pi\)- periodic in y and the integral \[ \iint_{\Omega}\sum^{N}_{k,j=1}(\partial u_ k/\partial z_ j+\partial u_ j/\partial z_ k)^ 2 dz \] is finite.

Reviewer: K.N.Srivastava

### Keywords:

asymptotic behavior; neighbourhood of infinity; order 2m; weighted Dirichlet integral; linear elasticity; finite energy; periodic; asymptotic expansion at infinity
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\textit{V. A. Kondrat'ev} and \textit{O. A. Olejnik}, Arch. Ration. Mech. Anal. 99, 75--89 (1987; Zbl 0637.35030)

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### References:

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