Publication: Construction of the maximal solution of Backus' problem in geodesy and geomagnetism
Loading...
Full text at PDC
Publication Date
2011-07
Advisors (or tutors)
Editors
Journal Title
Journal ISSN
Volume Title
Publisher
Springer
Citations
Exportar
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.
Description
UCM subjects
Unesco subjects
Keywords
Citation
Backus G.E., 1968. Application of a non-linear boundary-value problem for the Laplace’s equation to gravity and geomagnetic intensity surveys. Q. J. Mech. Appl. Math., XXI, 195−221.
Backus G.E., 1970. Non-uniqueness of the external geomagnetic field determined by surface intensity measurements. J. Geophys. Res., 75, 6339−6341.
Bjerhammar A. and Svensson L., 1983. On the geodetic boundary value problem for a fixed boundary surface - A satellite approach. Bull. Geod., 57, 382−393.
Campbell W.H., 1997. Introduction to Geomagnetic Felds. Cambridge University Press,Cambridge, U.K.
Čunderlík R., Mikula K. and Mojzeš M., 2008. Numerical solution of the linearized fixed gravimetric boundary-value problem. J. Geodesy, 82, 15−29.
Čunderlík R. and Mikula K., 2010. Direct BEM for high-resolution gravity field modelling. Stud.Geophys. Geod., 54, 219−238.
Dautray R. and Lions J.L. (Eds.), 1987. Analyse mathématique et calcul numérique. Tome 2: L’opérateur de Laplace. Masson, Paris, France.
Díaz G., Díaz J.I. and Otero J., 2006. On an oblique boundary value problem related to the Backus problem in geodesy. Nonlinear Anal.-Real World Appl., 7, 147−166.
Freeden W. and Michel V., 2004. Multiscale Potential Theory: with Applications to Geosciences. Birkhäuser, Boston.
Gilbarg D. and Trudinger N.S., 1983. Elliptic Partial Differential Equations of Second Order, 2nd Ed. Springer-Verlag, Berlin/Heidelberg/New York.
Grafarend E.W., 1989. The Geoid and the Gravimetric Boundary Value Problem. Report No. 18. Department of Geodesy, The Royal Institute of Technology, Stockholm, Sweden.
Heck B., 1989. On the non-linear geodetic boundary value problem for a fixed boundary surface. Bull. Geod., 63, 57−67.
Holota P., 1997. Coerciveness of the linear gravimetric boundary-value problem and a geometrical interpretation. J. Geodesy, 71, 640−651.
Holota P., 2005. Neumann’s boundary-value problem in studies on Earth gravity field: weak solution. In: Holota P. and Slaboch V. (Eds.), 50 Years of the Research Institute of Geodesy, Topography and Cartography. Research Institute of Geodesy, Topography and Cartography, Prague, Volume 50, Publication No. 36, 49−69(http://www.vugtk.cz/odis/sborniky/sb2005).
Hörmander L., 1976. The boundary problems of physical geodesy. Arch. Ration. Mech. Anal., 62, 1−52.
Kenig C.E. and Nadirashvili N.S., 2001. On optimal estimates for some oblique derivative problems. J. Funct. Anal., 187, 70−93.
Koch K.R. and Pope A.J., 1972. Uniqueness and existence for the geodetic boundary value problem using the known surface of the earth. Bull. Geod., 46, 467−476.
Kufner A., John O. and Fučik S., 1977. Function Spaces. Noordhoff International Publishing, Leyden, The Netherlands.
Jorge M.C., 1987. Local existence of the solution to a nonlinear inverse problem in gravitation. Q. Appl. Math., XLV, 287−292.
Maergoiz I.D., 1973. Limits on the gradient of a harmonic function at the boundary of a domain. Sib. Math. J., 14, 890−903.
Sacerdote F. and Sansò F., 1989. On the analysis of the fixed-boundary gravimetric boundary-value problem. In: Sacerdote F. and Sansò F. (Eds.), 2nd Hotine-Marussi Symposium on Mathematical Geodesy, Politecnico di Milano, Milano, Italy, 507−516.