On one-homogeneous solutions to elliptic systems with spatial variable dependence in two dimensions

We extend a result from Phillips by showing that one-homogeneous solutions of certain elliptic systems in divergence form either do not exist or must be affine. The result is novel in two ways. Firstly, the system is allowed to depend (in a sufficiently smooth way) on the spatial variable x. Secondl...

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Bibliographic Details
Published in:Proceedings of the Royal Society of Edinburgh. Section A. Mathematics 2010-06, Vol.140 (3), p.449-475
Main Author: Bevan, Jonathan J.
Format: Article
Language:English
Subjects:
Hypothesis testing
Mathematics
Variables
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Summary:We extend a result from Phillips by showing that one-homogeneous solutions of certain elliptic systems in divergence form either do not exist or must be affine. The result is novel in two ways. Firstly, the system is allowed to depend (in a sufficiently smooth way) on the spatial variable x. Secondly, Phillips's original result is shown to apply to W one-homogeneous solutions, from which his treatment of Lipschitz solutions follows as a special case. A singular one-homogeneous solution to an elliptic system violating the hypotheses of the main theorem is constructed using a variational method.
ISSN:0308-2105
1473-7124
DOI:10.1017/S0308210508000516