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

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 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.

References

[1] M.Abel and K.Jarosz. Noncommutative uniform algebras. Studia Mathematica, 162 (3) (2004), 213–218.
[2] M.I.Belishev. The Calderon problem for two-dimensional manifolds by the BC-method. SIAM J.Math.Anal., 35 (1): 172–182, 2003.
[3] M.I.Belishev. Some remarks on impedance tomography problem for 3d–manifolds. CUBO A Mathematical Journal, 7, no 1: 43–53, 2005.
[4] M.I. Belishev. Boundary Control Method and Inverse Problems of Wave Propagation. En- cyclopedia of Mathematical Physics, v.1, 340–345. eds. J.-P.Francoise, G.L.Naber and Tsou S.T., Oxford: Elsevier, (ISBN 978-0-1251-2666-3), 2006.
[5] M.I.Belishev. Geometrization of Rings as a Method for Solving Inverse Problems. Sobolev Spaces in Mathematics III. Applications in Mathematical Physics, Ed. V.Isakov., Springer, 2008, 5–24.
[6] M.I.Belishev. Algebras in reconstruction of manifolds. Spectral Theory and Partial Differential Equations, G.Eskin, L.Friedlander, J.Garnett Eds. Contemporary Mathematics, AMS, 640 (2015), 1–12. http://dx.doi.org/10.1090/conm/640 . ISSN: 0271-4132.
[7] M.I.Belishev. Boundary Control Method. Encyclopedia of Applied and Computational Math- ematics, Volume no: 1, Pages: 142–146. DOI: 10.1007/978-3-540-70529-1. ISBN 978-3-540- 70528-4
[8] M.I.Belishev. On algebras of three-dimensional quaternionic harmonic fields. Zapiski Nauch. Semin. POMI, 451 (2016), 14–28 (in Russian). English translation: M.I.Belishev. On algebras of three-dimensional quaternion harmonic fields. Journal of Mathematical Sciences, 226(6):701n ̃710, 2017.
[9] M.I.Belishev. Boundary control and tomography of Riemannian manifolds (BC-method). Russian Mathematical Surveys, 2017, 72:4, 581–644. https://doi.org/10.4213/rm 9768
[10] M.I.Belishev, V.A.Sharafutdinov. Dirichlet to Neumann operator on differential forms. Bul- letin de Sciences Math ́ematiques, 132 (2008), No 2, 128–145.
[11] M.I.Belishev, A.F.Vakulenko. On algebras of harmonic quaternion fields in R3. Algebra i Analiz, 31 (2019), No 1, 1–17 (in Russian). English translation: arXiv:1710.00577v3 [math.FA] 11 Oct 2017.
[12] L.Bers, F.John, M.Schechter. Partial Differential Equations. New Ypork-Landon-Sydney, 1964.
[13] K.Jarosz. Function representation of a noncommutative uniform algebra. Proceedings of the AMS, 136 (2) (2007), 605–611.
[14] J. Holladay. A note on the Stone-Weierstrass theorem for quaternions. Proc. Amer. Math. Soc., 8 (1957), 656n ̃657. MR0087047 (19:293d).
[15] S.H.Kulkarni and B.V.Limaye. Real Function Algebras, Monographs and Textbooks in Pure and Applied Math., 168, Marcel Dekker, Inc., New York, 1992. MR1197884 (93m:46059
[16] R.Leis. Initial boundary value problems in mathematical physics. Teubner, Stuttgart, 1972.
[17] C.Miranda. Equazioni alle derivate parziali di tipo ellittico. Springer-Verlag, Berlin, Goettingen, Heidelberg, 1955.
[18] M.Mitrea, M.Taylor. Boundary Layer Methods for Lipschitz Domains in Riemannian Manifolds. Journal of Functional Analysis, 163 (1999), 181–251.
[19] M.A.Naimark. Normed Rings. WN Publishing, Gronnongen, The Netherlands, 1970.
[20] R.Narasimhan. Analysis on real and complex manifolds. Masson and Cie, editier - Paris North-Holland Publishing Company, Amsterdam, 1968.
[21] G.Schwarz. Hodge decomposition - a method for solving boundary value problems. Lecture notes in Math., 1607. Springer–Verlag, Berlin, 1995.
Published
2019-08-14
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