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

Published online by Cambridge University Press:  21 May 2010

Jonathan J. Bevan
Jonathan J. Bevan
Affiliation:
Department of Mathematics, University of Surrey, Guildford, Surrey GU2 7XH, UK ([email protected])
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Abstract

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.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 140 , Issue 3 , June 2010 , pp. 449 - 475
Copyright
Copyright © Royal Society of Edinburgh 2010

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