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Necessary and sufficient conditions for a curve to be the gradient range of a C 1-smooth function

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Abstract

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 ϕ under which the differential equation \(\frac{{\partial v}}{{\partial t}} = \varphi \left( {\frac{{\partial v}}{{\partial x}}} \right)\) has nontrivial C 1-smooth solutions.

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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 48, No. 4, pp. 789–810, July–August, 2007.

Original Russian Text Copyright © 2007 Korobkov M. V. and Panov E. Yu.

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Korobkov, M.V., Panov, E.Y. Necessary and sufficient conditions for a curve to be the gradient range of a C 1-smooth function. Sib Math J 48, 629–647 (2007). https://doi.org/10.1007/s11202-007-0065-6

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  • DOI: https://doi.org/10.1007/s11202-007-0065-6

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