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 $\nabla \alpha =\mathrm{rot}u,\mathrm{div}u=0$ holds into $\Omega$. The space $\mathcal{Q}(\Omega )$ of harmonic fields is a subspace of the Banach algebra $\mathcal{C}\left(\Omega \right)$ of continuous quaternion fields with the point-wise multiplication $q{q}^{\prime }=\{\alpha {\alpha }^{\prime }-u\cdot u^{\prime },\alpha u^{\prime }+{\alpha }^{\prime }u+u\wedge u^{\prime }\}$. We prove a Stone-Weierstrass type theorem: the subalgebra $\vee \mathcal{Q}(\Omega )$ generated by harmonic fields is dense in $\mathcal{C}\left(\Omega \right)$. Some results on 2-jets of harmonic functions and the uniqueness sets of harmonic fields are provided.

Cited by in corpus (1)

• On the EIT problem for nonorientable surfaces