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
Full-text PDF Free Access
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Abstract: Consider the following boundary value problem in the exterior space ${\hat S^{d - 1}} = \left \{ {x \in {^d}:\left | x \right | > 1} \right \}$ 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.
Abramowitz, M., Stegun, I. A. (eds): Handbook of mathematical functions, Dover Publications, New York 1972.
- George E. Backus, Application of a non-linear boundary-value problem for Laplace’s equation to gravity and geomagnetic intensity surveys, Quart. J. Mech. Appl. Math. 21 (1968), 195–221. MR 227444, DOI https://doi.org/10.1093/qjmam/21.2.195
Backus, G. E.: Non-uniqueness of the external geomagnetic field determined by surface intensity measurements, J. Geophys. Res. 75, 6339–6341 (1970).
Bloxham, J., Jackson, A.: Fluid flow near the surface of earth’s outer core, Reviews of Geophysics 29, 1, 97–120 (1991).
- R. P. Boas Jr., Expansions of analytic functions, Trans. Amer. Math. Soc. 48 (1940), 467–487. MR 2594, DOI https://doi.org/10.1090/S0002-9947-1940-0002594-3
- R. Courant and D. Hilbert, Methods of mathematical physics. Vol. I, Interscience Publishers, Inc., New York, N.Y., 1953. MR 0065391
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.
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).
Kakeya, S.: On the limits of the roots of an algebraic equation with positive coefficients, Tôhoku Math. J. 2, 140–142 (1912).
- Oliver Dimon Kellogg, Foundations of potential theory, Die Grundlehren der Mathematischen Wissenschaften, Band 31, Springer-Verlag, Berlin-New York, 1967. Reprint from the first edition of 1929. MR 0222317
Kono, M.: Uniqueness problems in the spherical harmonic analysis of the geomagnetic field direction data, J. Geomag. Geoelectr. 28, 11–29 (1976).
Merrill, R. T., McElhinny, M. W.: The Earth’s Magnetic Field (Its History, Origin and Planetary Perspective), Academic Press, London 1983.
- Béla de Sz. Nagy, Expansion theorems of Paley-Wiener type, Duke Math. J. 14 (1947), 975–978. MR 23452
- Raymond E. A. C. Paley and Norbert Wiener, Fourier transforms in the complex domain, American Mathematical Society Colloquium Publications, vol. 19, American Mathematical Society, Providence, RI, 1987. Reprint of the 1934 original. MR 1451142
- Murray H. Protter and Hans F. Weinberger, Maximum principles in differential equations, Springer-Verlag, New York, 1984. Corrected reprint of the 1967 original. MR 762825
Proctor, M. R. E., Gubbins, D.: Analysis of geomagnetic directional data, Geophys. J. Int. 100, 69–77 (1990).
- John Riordan, Combinatorial identities, Robert E. Krieger Publishing Co., Huntington, N.Y., 1979. Reprint of the 1968 original. MR 554488
- J. Stoer and R. Bulirsch, Einführung in die numerische Mathematik. II, 2nd ed., Heidelberger Taschenbücher [Heidelberg Paperbacks], vol. 114, Springer-Verlag, Berlin-New York, 1978 (German). In reference to lectures by F. L. Bauer. MR 514972
- D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskiĭ, Quantum theory of angular momentum, World Scientific Publishing Co., Inc., Teaneck, NJ, 1988. Irreducible tensors, spherical harmonics, vector coupling coefficients, $3nj$ symbols; Translated from the Russian. MR 1022665
Abramowitz, M., Stegun, I. A. (eds): Handbook of mathematical functions, Dover Publications, New York 1972.
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).
Backus, G. E.: Non-uniqueness of the external geomagnetic field determined by surface intensity measurements, J. Geophys. Res. 75, 6339–6341 (1970).
Bloxham, J., Jackson, A.: Fluid flow near the surface of earth’s outer core, Reviews of Geophysics 29, 1, 97–120 (1991).
Boas, R. P.: Expansions of analytic functions, Transactions of the American Mathematical Society 48, 467–487 (1940).
Courant, R., Hilbert, D.: Methods of Mathematical Physics, vol II, chapter IV, §1, Interscience Publishers, New York 1962.
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.
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).
Kakeya, S.: On the limits of the roots of an algebraic equation with positive coefficients, Tôhoku Math. J. 2, 140–142 (1912).
Kellogg, O. D.: Foundations of Potential Theory, Berlin, Heidelberg, New York 1967.
Kono, M.: Uniqueness problems in the spherical harmonic analysis of the geomagnetic field direction data, J. Geomag. Geoelectr. 28, 11–29 (1976).
Merrill, R. T., McElhinny, M. W.: The Earth’s Magnetic Field (Its History, Origin and Planetary Perspective), Academic Press, London 1983.
de Sz. Nagy, B.: Expansion Theorems of Paley-Wiener-Type, Duke Math. J. 14, 975–978 (1947).
Paley, R., Wiener, N.: Fourier Transforms in the Complex Domain, The American Mathematical Society, New York 1973.
Protter, R. P., Weinberger, H. R.: Maximum Principles in Differential Equations, Springer, New York 1984.
Proctor, M. R. E., Gubbins, D.: Analysis of geomagnetic directional data, Geophys. J. Int. 100, 69–77 (1990).
Riordan, J.: Combinatorial Identities, Huntington, New York 1979.
Stoer, J., Bulirsch, R.: Einführung in die Numerische Mathematik II, 2nd ed., Berlin, Heidelberg, New York 1978.
Varshalovich, D. A., Moskalev, A. N., Khersonskii, V. K.: Quantum Theory of Angular Momentum, World Scientific, Singapore, New Jersey, Hong Kong 1988.
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