Revista Matemática Iberoamericana

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Volume 21, Issue 3, 2005, pp. 889–910
DOI: 10.4171/RMI/439

Published online: 2005-12-31

Solution to the gradient problem of C. E. Weil

Zoltán Buczolich[1]

(1) Eötvös Loránd University, Budapest, Hungary

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.

Keywords: Gradient, Denjoy-Clarkson property, Lebesgue measure

Buczolich Zoltán: Solution to the gradient problem of C. E. Weil. Rev. Mat. Iberoam. 21 (2005), 889-910. doi: 10.4171/RMI/439