On some problems of vector analysis

In this paper we give an explicit method for the construction of a vector field $\vec v$ in a domain $\Omega\subset\mathbb R^m$, $m\geqslant2$ which has the prescribed divergence $f=\operatorname{div}\vec v$ and boundary values $\vec\alpha=\vec v|_{\partial\Omega}$ The differentiability properties of $\vec v$ are determined in a “proper way” by the smoothness of $f$, $\vec\alpha$ and $\partial\Omega$. As a by-product of our construction we obtain the solutions for some other problems of vector analysis which are of self-dependent interest.