The Class of Solenoidal Planar-Helical Vector Fields

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 planarhelical 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 specific solution from the constructed class is reduced to the Dirichlet problem with respect to two functions that are harmonic conjugates in D2 = D∩R2; i.e., Gromeka's nonlinear problem is reduced to linear boundary value problems. As an example, a specific solution of the problem for an axisymmetric layer is presented. The solution is based on solving Dirichlet problems in the form of series uniformly convergent in D̄2 in terms of wavelet systems that form bases of various spaces of functions harmonic in D2. © 2011 Pleiades Publishing, Ltd.