On algebraic and uniqueness properties of harmonic quaternion fields on 3d manifolds

  • M.I. Belishev Saint-Petersburg Department of the Steklov Mathematical Institute, St-Petersburg State University, Russia.
  • A.F. Vakulenko Saint-Petersburg Department of the Steklov Mathematical Institute, St-Petersburg State University, Russia.
Keywords: 3d quaternion harmonic fields, real uniform Banach algebras, Stone- Weierstrass type theorem on density, uniqueness theorems


Let Ω be a smooth compact oriented 3-dimensional Riemannian manifold with boundary. A quaternion field is a pair q = {α, u} of a function α and a vector field u on Ω. A field q is harmonic if α, u are continuous in Ω and ∇α = rot u, div u = 0 holds into Ω. The space 𝒞(Ω) of harmonic fields is a subspace of the Banach algebra 𝒬 (Ω) of continuous quaternion fields with the point-wise multiplication qq′ = {αα′ − u · u ′ , αu′ + α ′u + u ∧ u ′ }. We prove a Stone-Weierstrass type theorem: the subalgebra ∨𝒞(Ω) generated by harmonic fields is dense in 𝒬 (Ω). Some results on 2-jets of harmonic functions and the uniqueness sets of harmonic fields are provided. Comprehensive study of harmonic fields is motivated by possible applications to inverse problems of mathematical physics.


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How to Cite
M.I. Belishev, & A.F. Vakulenko. (2019). On algebraic and uniqueness properties of harmonic quaternion fields on 3d manifolds. CUBO, A Mathematical Journal, 21(1), 01–19. https://doi.org/10.4067/S0719-06462019000100001