Real Analysis Exchange

A note on the gradient problem

P. Holický, J. Malý, C. E. Weil, and L. Zajíček

Full-text: Open access

Abstract

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.

Article information

Source
Real Anal. Exchange, Volume 22, Number 1 (1996), 225-235.

Dates
First available in Project Euclid: 1 June 2012

Permanent link to this document
https://projecteuclid.org/euclid.rae/1338515217

Mathematical Reviews number (MathSciNet)
MR1433610

Zentralblatt MATH identifier
0879.26041

Subjects
Primary: 26B05: Continuity and differentiation questions

Keywords
gradient measure

Citation

Holický, P.; Malý, J.; Zajíček, L.; Weil, C. E. A note on the gradient problem. Real Anal. Exchange 22 (1996), no. 1, 225--235. https://projecteuclid.org/euclid.rae/1338515217


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References

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