Abstract
We prove the existence of an n-dimensional completely integrable Pfaff system with multidimensional time of dimension m ⩾ 2, with bounded infinitely differentiable coefficients, and with the set of lower characteristic vectors of its solutions having positive Lebesgue m-measure.
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References
Gaishun, I.V., Vpolne razreshimye mnogomernye differentsial’nye uravneniya (Completely Integrable Multidimensional Differential Equations), Minsk: Navuka i Tekhnika, 1983.
Gaishun, I.V., Lineinye uravneniya v polnykh proizvodnykh (Linear Total Differential Equations), Minsk: Navuka i Tekhnika, 1989.
Grudo, E.I., Characteristic Vectors and Sets of Functions of Two Variables and Their Fundamental Properties, Differ. Uravn., 1976, vol. 12, no. 12, pp. 2115–2128.
Izobov, N.A., On the Existence of Linear Pfaffian Systems Whose Set of Lower Characteristic Vectors Has Positive Plane Measure, Differ. Uravn., 1997, vol. 33, no. 12, pp. 1623–1630.
Izobov, N.A., Vvedenie v teoriyu pokazatelei Lyapunova (Introduction to the Theory of Lyapunov Exponents), Minsk: BGU, 2006.
Izobov, N.A., On the Existence of a Linear Pfaff System with Countably Many Distinct Characteristic Sets of Solutions, Differ. Uravn., 1998, vol. 34, no. 6, pp. 735–743.
Izobov, N.A., On the Countability of the Number of Solutions of a Two-Dimensional Linear Pfaff System with Distinct Characteristic Sets, Ukrain. Mat. Zh., 2007, vol. 59, no. 2, pp. 172–189.
Izobov, N.A., The Set of Lower Exponents of Positive Measure, Differ. Uravn., 1968, vol. 4, no. 6, pp. 1147–1149.
Izobov, N.A., Krasovskii, N.G., and Platonov, A.S., The Existence of Linear Pfaff Systems with Lower Characteristic Set of Positive Measure in the Space R 3, Differ. Uravn., 2008, vol. 44, no. 10, pp. 1311–1318.
Natanson, I.P., Teoriya funktsii veshchestvennoi peremennoi (Theory of Functions of a Real Variable), Moscow: Nauka, 1974.
Gelbaum, B. and Olmsted, J., Counterexamples in Analysis, San-Francisco: Dover, 1964. Translated under the title Kontrprimery v analize, Moscow: Mir, 1967.
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Original Russian Text © N.A. Izobov, S.G. Krasovskii, A.S. Platonov, 2009, published in Differentsial’nye Uravneniya, 2009, Vol. 45, No. 5, pp. 635–646.
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Izobov, N.A., Krasovskii, S.G. & Platonov, A.S. On the existence of linear Pfaff systems with lower characteristic sets of positive Lebesgue m-measure. Diff Equat 45, 650–661 (2009). https://doi.org/10.1134/S0012266109050036
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DOI: https://doi.org/10.1134/S0012266109050036