Open Access
December, 2005 Solution to the gradient problem of C.E. Weil
Zoltán Buczolich
Rev. Mat. Iberoamericana 21(3): 889-910 (December, 2005).


In this paper we give a complete answer to the famous gradient problem of C. E. Weil. On an open set $G\subset \mathbb{R}^{2}$ we construct a differentiable function $f:G\to\mathbb{R}$ for which there exists an open set $\Omega_{1}\subset\mathbb{R}^{2}$ such that $\nabla f(\mathbf{p})\in \Omega_{1}$ for a $\mathbf{p}\in G$ but $\nabla f(\mathbf{q})\not\in\Omega_{1}$ for almost every $\mathbf{q}\in G$. This shows that the Denjoy-Clarkson property does not hold in higher dimensions.


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Zoltán Buczolich. "Solution to the gradient problem of C.E. Weil." Rev. Mat. Iberoamericana 21 (3) 889 - 910, December, 2005.


Published: December, 2005
First available in Project Euclid: 11 January 2006

zbMATH: 1116.26007
MathSciNet: MR2231014

Primary: 26B05
Secondary: 28A75 , 37E99

Keywords: Denjoy-Clarkson property , gradient , Lebesgue measure

Rights: Copyright © 2005 Departamento de Matemáticas, Universidad Autónoma de Madrid

Vol.21 • No. 3 • December, 2005
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