×

Solution to the gradient problem of C. E. Weil. (English) Zbl 1116.26007

The author gives a complete answer to the gradient problem of C. E. Weil by means of a counter-example. Precisely, he has found a differentiable function \(\;f:G\to R\;\), with \(\;G\subset R^2\) open, and an open set \(\Omega_1\subset R^2\;\) for which there exists a point \(p\in G\;\) such that \( \;\nabla f(p)\in \Omega_1\;\) but for a.e. \(\;q\in G\;\) the gradient \( \nabla f(q)\notin \Omega_1.\;\) This example also proves that the Denjoy-Clarkson property does not hold in higher dimensions.

MSC:

26B05 Continuity and differentiation questions
28A75 Length, area, volume, other geometric measure theory
37E99 Low-dimensional dynamical systems

References:

[1] Brucks, K. M. and Buczolich, Z.: Trajectory of the turning point is dense for a co-\sigma -porous set of tent maps. Fund. Math. 165 (2000), no. 2, 95-123. · Zbl 0966.37012
[2] Buczolich, Z.: The n-dimensional gradient has the 1-dimensional Denjoy- Clarkson property. Real Anal. Exchange, 18 (1992/93), no. 1, 221-224. · Zbl 0783.26010
[3] Buczolich, Z.: Level sets of functions f (x, y) with nonvanishing gradient. J. Math. Anal. Appl., 185 (1994), no. 1, 27-35. · Zbl 0876.58004 · doi:10.1006/jmaa.1994.1231
[4] Buczolich, Z.: Approximate continuity points of derivatives of functions of several variables. Acta Math. Hungar. 67 (1995), no. 3, 229-235. · Zbl 0851.26006 · doi:10.1007/BF01874334
[5] Buczolich, Z.: Another note on the gradient problem of C. E. Weil. Real Anal. Exchange, 22 (1996/97), no. 2, 775-784. · Zbl 0940.26011
[6] Buczolich, Z.: Functions of two variables with large tangent plane sets. J. Math. Anal. Appl., 220 (1998), no. 2, 562-570. · Zbl 0915.26005 · doi:10.1006/jmaa.1997.5848
[7] Clarkson, J. A.: A property of derivatives. Bull. Amer. Math. Soc. 53 (1947), 124-125. · Zbl 0032.27102 · doi:10.1090/S0002-9904-1947-08757-7
[8] Denjoy, A.: Sur une proprieté des fonctions dérivées. Enseignement Math. 18 (1916), 320-328. · JFM 46.0381.05
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.