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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Author:Frank Bauer
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
Date of the Publication (Server):2005/03/22
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:Kaiserslautern - Fachbereich Mathematik
DDC-Cassification:5 Naturwissenschaften und Mathematik / 510 Mathematik
MSC-Classification (mathematics):30-XX FUNCTIONS OF A COMPLEX VARIABLE (For analysis on manifolds, see 58-XX) / 30Exx Miscellaneous topics of analysis in the complex domain / 30E25 Boundary value problems [See also 45Exx]
35-XX PARTIAL DIFFERENTIAL EQUATIONS / 35Gxx General higher-order equations and systems / 35G15 Boundary value problems for linear higher-order equations
35-XX PARTIAL DIFFERENTIAL EQUATIONS / 35Jxx Elliptic equations and systems [See also 58J10, 58J20] / 35J99 None of the above, but in this section
65-XX NUMERICAL ANALYSIS / 65Nxx Partial differential equations, boundary value problems / 65N99 None of the above, but in this section
Licence (German):Standard gemäß KLUEDO-Leitlinien vor dem 27.05.2011