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Construction of the maximal solution of Backus' problem in geodesy and geomagnetism

Díaz Díaz, Jesús Ildefonso and Díaz Díaz, Gregorio and Otero Juez, Jesús (2011) Construction of the maximal solution of Backus' problem in geodesy and geomagnetism. Studia Geophysica et Geodaetica, 55 (3). pp. 415-440. ISSN 0039-3169

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Abstract

The (simplified) Backus' Problem (BP) consists in finding a harmonic function u on the domain exterior to the three dimensional unit sphere S, such that u tends to zero at infinity and the norm of the gradient of u takes prescribed values g on S. Except for a change of sign, the solution is not unique in general. However, there is uniqueness of solutions in the class of functions with the additional property that the radial component of the gradient of u on S is nonpositive. This is the geodetically relevant case. If a solution u with this property exists, then u is the maximal solution of the problem (and -u the minimal one). In this paper we propose a method of successive approximations to get this particular solution of BP and prove the convergence for functions g close to a constant function.

Item Type:Article
Uncontrolled Keywords:Harmonic functions, fully nonlinear boundary problem, geodesy,geomagnetism
Subjects:Sciences > Mathematics > Geodesy
ID Code:17234
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