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

• Autores: M. I. Belishev, A. F. Vakulenko
• Localización: Cubo: A Mathematical Journal, ISSN 0716-7776, ISSN-e 0719-0646, Vol. 21, Nº. 1, 2019, págs. 1-19
• Idioma: inglés
• DOI: 10.4067/S0719-06462019000100001
• Enlaces
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• Resumen
  • español

    Resumen Sea Ω una variedad Riemanniana 3-dimensional suave con borde, orientada y compacta. Un campo cuaterniónico es un par q = {α, u} dado por una función α y un campo de vectores u en Ω. Un campo q es armónico si α, u son continuos en Ω y ∇α = rotu, div u = 0 vale en todo Ω. El espacio (Ω) de campos armónicos es un subespacio del álgebra de Banach (Ω) de campos cuaterniónicos continuos con la multiplicación punto a punto qq′ = {αα′ − u · u ′ , αu′ + α ′u + u ∧ u ′ }. Probamos un teorema de tipo Stone-Weierstrass: la subálgebra ∨(Ω) generada por campos armónicos es densa en 𝒞(Ω). Se entregan también algunos resultados acerca de 2-jets de funciones armónicas y los conjuntos de unicidad campos armónicos.

  • English

    Abstract 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 (Ω) of harmonic fields is a subspace of the Banach algebra 𝒞 (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 ∨(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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