Abstract:
The class of solenoidal vector fields whose lines lie in planes parallel to R2 is constructed by the method of mappings. This class exhausts the set of all smooth planar-helical solutions of Gromeka's problem in some domain D⊂R3. In the case of domains D with cylindrical boundaries whose generators are orthogonal to R2, it is shown that the choice of a concrete solution from the constructed class is reduced to the Dirichlet problem with respect to two functions that are harmonically conjugate in D2=D∩R2; i.e., Gromeka's nonlinear problem is reduced to linear boundary value problems. As an example, a concrete solution of the problem for an axially symmetric layer is presented. The solution is based on solving Dirichlet problems in the form of series uniformly convergent in ¯¯¯¯¯D2 in terms of wavelet systems that form bases of various spaces of functions harmonic in D2.
Citation:
V. P. Vereshchagin, Yu. N. Subbotin, N. I. Chernykh, “The class of solenoidal planar-helical vector fields”, Trudy Inst. Mat. i Mekh. UrO RAN, 16, no. 4, 2010, 128–143; Proc. Steklov Inst. Math. (Suppl.), 273, suppl. 1 (2011), S171–S187
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\by V.~P.~Vereshchagin, Yu.~N.~Subbotin, N.~I.~Chernykh
\paper The class of solenoidal planar-helical vector fields
\serial Trudy Inst. Mat. i Mekh. UrO RAN
\yr 2010
\vol 16
\issue 4
\pages 128--143
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\jour Proc. Steklov Inst. Math. (Suppl.)
\yr 2011
\vol 273
\issue , suppl. 1
\pages S171--S187
\crossref{https://doi.org/10.1134/S008154381105018X}
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Linking options:
https://www.mathnet.ru/eng/timm648
https://www.mathnet.ru/eng/timm/v16/i4/p128
This publication is cited in the following 3 articles:
Li Q., Liu X., Song R., Ma X., “An Image-Guiding System For Orthognathic Assisted Robot Based on Three Dimensional-Digital Imaging Correlation: System Establishment and Accuracy Evaluation”, 2017 Chinese Automation Congress (Cac), IEEE, 2017, 90–94
V. P. Vereschagin, Yu. N. Subbotin, N. I. Chernykh, “Postanovka i reshenie kraevoi zadachi v klasse ploskovintovykh vektornykh polei”, Tr. IMM UrO RAN, 18, no. 1, 2012, 123–138
“Sovmestnaya nauchnaya deyatelnost Yu. N. Subbotina i N. I. Chernykh”, Tr. IMM UrO RAN, 17, no. 3, 2011, 4–7