Solution to the gradient problem of C.E. Weil

Buczolich, Zoltán

Rev. Mat. Iberoamericana, Tome 21 (2005) no. 2, p. 889-910 / Harvested from Project Euclid

Résumé

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.

Détail

Publié le : 2005-03-14
Classification:  gradient,  Denjoy-Clarkson property,  Lebesgue measure,  26B05,  28A75,  37E99