2010
DOI: 10.1017/s0308210508000516
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On one-homogeneous solutions to elliptic systems with spatial variable dependence in two dimensions

Abstract: We extend the result [Ph02] of 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 1,2 one-homogeneous solutions, from which his treatment of Lipschitz solutions follows as a special case. A singular one-homogeneous solution to an ellipti… Show more

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Cited by 3 publications
(3 citation statements)
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“…It is in this sense that nontrivial one-homogeneous functions are singular. Secondly, it confirms that one of the hypotheses in the recent result [2][Theorem 2.1] is sharp. We restate that result here for the reader's benefit.…”
Section: Introductionsupporting
confidence: 78%
See 1 more Smart Citation
“…It is in this sense that nontrivial one-homogeneous functions are singular. Secondly, it confirms that one of the hypotheses in the recent result [2][Theorem 2.1] is sharp. We restate that result here for the reader's benefit.…”
Section: Introductionsupporting
confidence: 78%
“…We restate that result here for the reader's benefit. Theorem 1.1 (Theorem 2.1, [2]). Let u be a one-homogeneous function belonging to the class W 1,2 (B, R m ) and satisfying…”
Section: Introductionmentioning
confidence: 99%
“…See [8] for an example; see also [12], [19] and [11]. Indeed, it forms part of the hypotheses of the main results in [13], [21] and [22].…”
Section: Introductionmentioning
confidence: 97%