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Properties of the \(C^1\)-smooth functions with nowhere dense gradient range. (Russian, English) Zbl 1164.26325

Sib. Mat. Zh. 48, No. 6, 1272-1284 (2007); translation in Sib. Math. J. 48, No. 6, 1019-1028 (2007).
Summary: One of the main results of the present article is as follows.
Theorem. Let \(v:\Omega\to\mathbb R\) be a \(C^1\)-smooth function on a domain \(\Omega\subset\mathbb R^2\). Suppose that \(\text{Int}\nabla v(\Omega)=\varnothing\). Then, for every point \(z\in\Omega\), there is a straight line \(L\ni z\) such that \(\nabla(v)=\text{const}\) on the connected component of the set \(L\cap\Omega\) containing \(z\).
Also, we prove that, under the conditions of the theorem, the range of the gradient \(\nabla v(\Omega)\) is locally a curve and this curve has tangents in the weak sense and the direction of these tangents is a function of bounded variation.

MSC:

26B35 Special properties of functions of several variables, Hölder conditions, etc.
28A75 Length, area, volume, other geometric measure theory
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