Vector fields with zero flux through spheres of fixed radius

The classical property of a periodic function on the real axis is the possibility of its representation by a trigonometric Fourier series. The natural analogue of the periodicity condition in the Euclidean space Rn is the constancy of the integrals of the function over all balls (or spheres) of a fixed radius. Functions with the specified property can be expanded in a series in special eigenfunctions of the Laplace operator. This fact admits a generalization to vector fields in Rn, having zero flow through spheres of fixed radius. In this case, Smith's representation arises for them as the sum of a solenoidal vector field and an infinite number of potential vector fields. Potential vector fields satisfy the Helmholtz equation related to the zeros of the Bessel function Jn/2. The purpose of this paper is to obtain local analogs of the Smith theorem. We study vector fields A with zero flow through spheres of fixed radius on domains O in Euclidean space that are invariant with respect to rotations...