Split Operators for Oblique Boundary Value Problems

  • In the field of gravity determination a special kind of boundary value problem respectively ill-posed satellite problem occurs; the data and hence side condition of our PDE are oblique second order derivatives of the gravitational potential. In mathematical terms this means that our gravitational potential \(v\) fulfills \(\Delta v = 0\) in the exterior space of the Earth and \(\mathscr D v = f\) on the discrete data location which is on the Earth's surface for terrestrial measurements and on a satellite track in the exterior for spaceborne measurement campaigns. \(\mathscr D\) is a first order derivative for methods like geometric astronomic levelling and satellite-to-satellite tracking (e.g. CHAMP); it is a second order derivative for other methods like terrestrial gradiometry and satellite gravity gradiometry (e.g. GOCE). Classically one can handle first order side conditions which are not tangential to the surface and second derivatives pointing in the radial direction employing integral and pseudo differential equation methods. We will present a different approach: We classify all first and purely second order operators \(\mathscr D\) which fulfill \(\Delta \mathscr D v = 0\) if \(\Delta v = 0\). This allows us to solve the problem with oblique side conditions as if we had ordinary i.e. non-derived side conditions. The only additional work which has to be done is an inversion of \(\mathscr D\), i.e. integration.

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Metadaten
Author:Frank Bauer
URN (permanent link):urn:nbn:de:hbz:386-kluedo-13720
Serie (Series number):Schriften zur Funktionalanalysis und Geomathematik (17)
Document Type:Preprint
Language of publication:English
Year of Completion:2005
Year of Publication:2005
Publishing Institute:Technische Universität Kaiserslautern
Tag:Ableitung höherer Ordnung; Split-Operator
Boundary Value Problem ; Higher Order Differentials as Boundary Data ; Split Operator
GND-Keyword:Randwertproblem / Schiefe Ableitung
Faculties / Organisational entities:Fachbereich Mathematik
DDC-Cassification:510 Mathematik
MSC-Classification (mathematics):30E25 Boundary value problems [See also 45Exx]
35G15 Boundary value problems for linear higher-order equations
35J99 None of the above, but in this section
65N99 None of the above, but in this section

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