Document Zbl 1164.26326

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Necessary and sufficient conditions for a curve to be the gradient range of a \(C^1\)-smooth function. (Russian, English) Zbl 1164.26326

Sib. Mat. Zh. 48, No. 4, 789-810 (2007); translation in Sib. Math. J. 48, No. 4, 629-647 (2007).

Summary: We find some necessary and sufficient conditions for a plane curve to be the gradient range of a \(C^1\)-smooth function of two variables. As one of the consequences we give the necessary and sufficient conditions on a continuous function \(\varphi\) under which the differential equation \(\frac{\partial v}{\partial t}=\varphi\bigl( \frac{\partial v}{\partial x}\bigr)\) has nontrivial \(C^1\)-smooth solutions.

MSC:

26B35	Special properties of functions of several variables, Hölder conditions, etc.
58C05	Real-valued functions on manifolds

Keywords:

smooth function; gradient range; curve

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