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.
Similar content being viewed by others
References
Gromov M. L., Partial Differential Relations, Springer-Verlag, Berlin; Heidelberg (1986).
Muller S., Variational Models for Microstructure and Phase Transitions, Max-Plank-Institut für Mathematik in den Naturwissenschaften, Leipzig (1998) (Lecture Notes; No. 2; http://www. mis.mpg.de/jump/publications.html).
Panov E. Yu., Weak Solutions of the Cauchy Problem for Quasilinear Conservation Laws, Diss. Kand. Fiz.-Mat. Nauk (1991).
Korobkov M. V. and Panov E. Yu., “On isentropic solutions of first-order quasilinear equations,” Mat. Sb., 197, No. 5, 99–124 (2006).
Korobkov M. V., “An analog of Sard’s theorem for C 1-smooth functions of two variables,” Siberian Math. J., 47, No. 5, 889–897 (2006).
Malý J., “The Darboux property for gradients,” Real Anal. Exchange, 22, No. 1, 167–173 (1996/97).
Korobkov M. V., “On one generalization of the Darboux theorem to the multidimensional case,” Siberian Math. J., 41, No. 1, 100–112 (2000).
Saks S., Theory of the Integral [Russian translation], Izdat. Inostr. Lit., Moscow (1949).
Natanson I. P., The Theory of Functions of a Real Variable [in Russian], Nauka, Moscow (1974).
Author information
Authors and Affiliations
Additional information
__________
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.
Rights and permissions
About this article
Cite this article
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
Received:
Issue Date:
DOI: https://doi.org/10.1007/s11202-007-0065-6