Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



A non-standard boundary value problem related to geomagnetism

Authors: Ralf Kaiser and Michael Neudert
Journal: Quart. Appl. Math. 62 (2004), 423-457
MSC: Primary 35Q60; Secondary 35R25, 35R30, 86A25, 93B30
DOI: https://doi.org/10.1090/qam/2086038
MathSciNet review: MR2086038
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Abstract: Consider the following boundary value problem in the exterior space $ {\hat S^{d - 1}} = \left\{ {x \in {^d}:\left\vert x \right\vert > 1} \right\}$ of a sphere $ {S^{d - 1}}$ in two and three dimensions $ \left( d = 2, 3 \right)$: Given a vector field $ D:{S^{d - 1}} \to {^d}$ we ask for all harmonic vector fields $ B:{\hat S^{d - 1}} \to {^d}$ which decay at least as fast as a dipole field at infinity and are parallel to D on $ {S^{d - 1}}$ i.e. there is $ f:{S^{d - 1}} \to $ such that $ B = fD$. For $ d = 3$, this problem is related to the problem of reconstructing the geomagnetic field outside the earth from directional data measured on the earth's surface. The question for uniqueness or non-uniqueness is of particular interest here.

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  • 1. Abramowitz, M., Stegun, I. A. (eds): Handbook of mathematical functions, Dover Publications, New York 1972.
  • 2. Backus, G. E.: Application of a non-linear boundary-value problem for Laplace's equation to gravity and geomagnetic intensity surveys, Quart. J. Mech. Appl. Math. 21, 195-221 (1968). MR 0227444
  • 3. Backus, G. E.: Non-uniqueness of the external geomagnetic field determined by surface intensity measurements, J. Geophys. Res. 75, 6339-6341 (1970).
  • 4. Bloxham, J., Jackson, A.: Fluid flow near the surface of earth's outer core, Reviews of Geophysics 29, 1, 97-120 (1991).
  • 5. Boas, R. P.: Expansions of analytic functions, Transactions of the American Mathematical Society 48, 467-487 (1940). MR 0002594
  • 6. Courant, R., Hilbert, D.: Methods of Mathematical Physics, vol II, chapter IV, §1, Interscience Publishers, New York 1962. MR 0065391
  • 7. Hide, R.: Frozen vector fields and the inverse problem of inferring motions in the electrically-conducting fluid core of a planet from observations of secular changes in its main magnetic field, in: The Physics of the Planets, The Royal Astronomical Society, 1986.
  • 8. Hulot, G., Khokhlov, A., Le Mouël, J. L.: Uniqueness of mainly dipolar magnetic fields recovered from directional data, Geophys. J. Int. 129, 347-354 (1997).
  • 9. Kakeya, S.: On the limits of the roots of an algebraic equation with positive coefficients, Tôhoku Math. J. 2, 140-142 (1912).
  • 10. Kellogg, O. D.: Foundations of Potential Theory, Berlin, Heidelberg, New York 1967. MR 0222317
  • 11. Kono, M.: Uniqueness problems in the spherical harmonic analysis of the geomagnetic field direction data, J. Geomag. Geoelectr. 28, 11-29 (1976).
  • 12. Merrill, R. T., McElhinny, M. W.: The Earth's Magnetic Field (Its History, Origin and Planetary Perspective), Academic Press, London 1983.
  • 13. de Sz. Nagy, B.: Expansion Theorems of Paley-Wiener-Type, Duke Math. J. 14, 975-978 (1947). MR 0023452
  • 14. Paley, R., Wiener, N.: Fourier Transforms in the Complex Domain, The American Mathematical Society, New York 1973. MR 1451142
  • 15. Protter, R. P., Weinberger, H. R.: Maximum Principles in Differential Equations, Springer, New York 1984. MR 762825
  • 16. Proctor, M. R. E., Gubbins, D.: Analysis of geomagnetic directional data, Geophys. J. Int. 100, 69-77 (1990).
  • 17. Riordan, J.: Combinatorial Identities, Huntington, New York 1979. MR 554488
  • 18. Stoer, J., Bulirsch, R.: Einführung in die Numerische Mathematik II, 2nd ed., Berlin, Heidelberg, New York 1978. MR 514972
  • 19. Varshalovich, D. A., Moskalev, A. N., Khersonskii, V. K.: Quantum Theory of Angular Momentum, World Scientific, Singapore, New Jersey, Hong Kong 1988. MR 1022665

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DOI: https://doi.org/10.1090/qam/2086038
Article copyright: © Copyright 2004 American Mathematical Society

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