On the gradient problem of C. E. Weil (Q2433624)

The gradient problem is about the multidimensional version of the Denjoy-Clarkson property, stating that if \((\alpha,\beta)\) is any open interval, then its inverse image \((f')^{-1}(\alpha,\beta)\) by the derivative \(f'\) is either empty, or of positive one-dimensional Lebesgue measure. In [Rev. Mat. Iberoam. 21, No.~3, 889--910 (2005; Zbl 1116.26007)] the author gives the solution to Weil's gradient problem. In the present paper, he is going through the road leading to the final solution of this difficult problem. In a first step, the \(n\)-dimensional Lebesgue measure is replaced by the one-dimensional Hausdorff measure, making the Denjoy-Clarkson property to hold in the multidimensional case as well. In a second step, a ``paradoxical convexity'' property of the counterexample functions to the gradient problem is proposed. In a third step, attention is directed towards differentiable functions with many tangent planes (the discussion involves here the Sobolev lemma and the Morse-Sard theorem). In a fourth step, the level set structure of functions defined on the plane is explored. In a fifth step, a generalization to higher dimensions is obtained of the following result belonging to G. Petruska: Derivatives take every value on the set of their approximate continuity points. Now, the author is ready to state the final result. Let \(G= (-1,1)\times (-1,1)\), \(\Omega_0= [-1/2,1/2]\times [0,2]\), \(\Omega_1= (-0.49, 0.49)\times (0,1.99)\), and \(\Omega_2= [-0.51,0.51]\times [0,2.01]\). For \(f: G\to \mathbb{R}\) differentiable, let \(\nabla f= (\partial_1 f,\partial_2 f)\). There exists a differentiable function \(f: G\to \mathbb{R}\) such that \(\nabla f(0,0)= (0,1)\) and \(\nabla f(p)\not\in\Omega_1\) for almost every \(p\in G\). The final pages are devoted to comments about the main ideas of the construction. Reference is made to a geometric construction we cannot reproduce here and where \(\Omega_0\) and \(\Omega_2\) play also their role. References include 25 titles, six of which are previous articles by the author. Reviewer's remark: The whole construction proposed by the author is an example of mathematical deepness and beauty.