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1996/1997 A note on the gradient problem
P. Holický, J. Malý, C. E. Weil, L. Zajíček
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Real Anal. Exchange 22(1): 225-235 (1996/1997).


C. E. Weil formulated the following problem: “Assume that \(f\) is a differentiable real-valued function of \(N\) real variables, \(N \geq 2\), and let \(g = \nabla f\) denote its gradient, which is a function from \(^N\) to \(\mathbb{R}^N\). Let \(G \subset \mathbb{R}^N\) be a nonempty open set and let \(g^{-1}(G) \ne \emptyset.\) Does \(g^{-1}(G)\) have positive \(N\)-dimensional Lebesgue measure?” For \(N = 1\) the answer is yes as was first proved by Denjoy in 1916. Z. Buczolich gave a partial answer to this problem showing that \(g^{-1}(G)\) has positive one-dimensional Hausdorff measure. In other words, he proved that the gradient has the “one-dimensional Denjoy-Clarkson property”. In the present article, we prove the Buczolich result using a quite different method. Moreover, our method gives improvements and generalizations.


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P. Holický. J. Malý. C. E. Weil. L. Zajíček. "A note on the gradient problem." Real Anal. Exchange 22 (1) 225 - 235, 1996/1997.


Published: 1996/1997
First available in Project Euclid: 1 June 2012

zbMATH: 0879.26041
MathSciNet: MR1433610

Primary: 26B05

Keywords: gradient , measure

Rights: Copyright © 1996 Michigan State University Press

Vol.22 • No. 1 • 1996/1997
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