A note on the gradient problem

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.