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

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.