On algebraic and uniqueness properties of 3d harmonic quaternion fields

Abstract

Let $\Omega$ be a smooth compact oriented 3-dimensional Riemannian manifold with boundary. A quaternion field is a pair $q=\{\alpha,u\}$ of a function $\alpha$ and a vector field $u$ on $\Omega$. A field $q$ is {\it harmonic} if $\alpha, u$ are continuous in $\Omega$ and $\nabla\alpha={\rm rot\,}u,\,{\rm div\,}u=0$ holds into $\Omega$. The space ${\mathscr Q}(\Omega)$ of harmonic fields is a subspace of the Banach algebra $\mathscr C\left(\Omega\right)$ of continuous quaternion fields with the point-wise multiplication $qq'=\{\alpha\alpha'-u\cdot u',\,\alpha u'+\alpha'u+u\wedge u'\}$. We prove a Stone-Weierstrass type theorem: the subalgebra $\vee{\mathscr Q}(\Omega)$ generated by harmonic fields is dense in $\mathscr C\left(\Omega\right)$. Some results on 2-jets of harmonic functions and the uniqueness sets of harmonic fields are provided.