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

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