The class of solenoidal planar-helical vector fields

The class of solenoidal vector fields whose lines lie in planes parallel to $R^2$ 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 $D\subset R^3$. In the case of domains $D$ with cylindrical boundaries whose generators are orthogonal to $R^2$, 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 $D^2=D\cap R^2$; 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 $\overline D^2$ in terms of wavelet systems that form bases of various spaces of functions harmonic in $D^2$.