Statement and solution of a boundary value problem in the class of planar-helical vector fields

V. P. Vereshchagin   ; Yu. N. Subbotin   ; N. I. Chernykh

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 18 (2012) no. 1, pp. 123-138

Résumé

The problem is solved on the selection of a particular vector field from the class $\mathfrak L_\mathrm{ph}(D)$ of all vector fields smooth in some domain $D\subset R^3$. The class $\mathfrak L_\mathrm{ph}(D)$ consists of fields that are solenoidal in $D$ and such that the lines of each field form a family of smooth curves lying in planes parallel to some fixed plane $R^2\subset R^3$ and coincide everywhere in $D$ with the vortex lines of the field. Additional conditions are formulated in the form of boundary conditions for the selected field on certain specially chosen lines belonging to the boundary $\partial D$ under some not very restricting conditions on the domain $D$ and on its projection $D^2$ to the plane $R^2$. As a result, the selection of a particular field from the class $\mathfrak L_\mathrm{ph}(D)$ is reduced to solving a boundary value problem, a part of which is the problem on finding a pair of functions that are harmonically conjugate in $D^2$ and continuous in the closure $\overline{D^2}$ and take given continuous values on the boundary of the domain $D^2$. An algorithm for solving the boundary value problem is proposed. The solution of the boundary value problem is considered in detail for the case of the domain $D$ whose projection to the plane $R^2$ is an open unit disk $K$. We use an approach based on representing the components of the field as expansions on a system of harmonic wavelets converging uniformly in the closure $\overline K$. The vector field found for such a domain can then be extended to any domain $D$ whose projection $D^2$ is a conformal image of a unit disk.

Détail

Keywords: scalar fields, vector fields, tensor fields, curl, wavelets, Dirichlet problem.

Bibliographie

[1] Vereschagin V. P., Subbotin Yu. N., Chernykh N. I., “Klass solenoidalnykh ploskovintovykh vektornykh polei”, Tr. In-ta matematiki i mekhaniki UrO RAN, 16, no. 4, 2010, 128–143

[2] Lavrentev M. A., Shabat B. V., Metody teorii funktsii kompleksnogo peremennogo, Nauka, M., 1958, 680 pp. | MR

[3] Subbotin Yu. N., Chernykh N. I., “Vspleski v prostranstve garmonicheskikh funktsii”, Izv. RAN. Ser. mat., 64:1 (2000), 145–174 | MR | Zbl

[4] Subbotin Yu. N., Chernykh N. I., “Garmonicheskie vspleski i asimptotika resheniya zadachi Dirikhle v kruge s malym otverstiem”, Mat. modelirovanie, 14:5 (2002), 17–30 | MR | Zbl

[5] Subbotin Yu. N., Chernykh N. I., “Vspleski periodicheskie, garmonicheskie i analiticheskie v kruge s netsentralnym otverstiem”, Tr. Mezhdunar. letnei mat. shkoly S. B. Stechkina po teorii funktsii, Izd-vo TulGU, Tula, 2007, 129

[6] Meyer Y., Ondelettes et operateurs, v. I, Ondelettes, Herman, Paris, 1990 | MR | Zbl

[7] Offin D., Oskolkov K., “A note on orthonormal polynomial bases and wavelets”, Constr. Approx., 9 (1963), 319–325 | DOI | MR

[8] Khardi G. Kh., Rogozinskii V. V., Ryady Fure, Fizmatgiz, M., 1959, 156 pp.