On the behavior at infinity of solutions of elliptic systems with a finite energy integral
type: Item
instance of: scholarly article
title: On the behavior at infinity of solutions of elliptic systems with a finite energy integral
zbMATH Open document ID: 0637.35030
DOI: 10.1007/BF00251392
published in: Archive for Rational Mechanics and Analysis
publication date: 1987-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 35J30
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 4038329
zbMATH Keywords: asymptotic behavior
zbMATH Keywords: neighbourhood of infinity
zbMATH Keywords: order 2m
zbMATH Keywords: weighted Dirichlet integral
zbMATH Keywords: linear elasticity
zbMATH Keywords: finite energy
zbMATH Keywords: periodic
zbMATH Keywords: asymptotic expansion at infinity
author: V. A. Kondrat'ev
author: Olga Arsenevna Oleĭnik
MaRDI profile type: Publication
cites work: Isolated Singularities of Solutions of Linear Elliptic Equations
cites work: On the asymptotic behaviour at infinity of solutions in linear elasticity
cites work: Interior estimates for elliptic systems of partial differential equations
full work available at URL: https://doi.org/10.1007/bf00251392
OpenAlex ID: W2122030649
description: scientific article; zbMATH DE number 4038329