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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.
We analyze the regular oblique boundary problem for the Poisson equation on a C^1-domain with stochastic inhomogeneities. At first we investigate the deterministic problem. Since our assumptions on the inhomogeneities and coefficients are very weak, already in order to formulate the problem we have to work out properties of functions from Sobolev spaces on submanifolds. An further analysis of Sobolev spaces on submanifolds together with the Lax-Milgram lemma enables us to prove an existence and uniqueness result for weak solution to the oblique boundary problem under very weak assumptions on coefficients and inhomogeneities. Then we define the spaces of stochastic functions with help of the tensor product. These spaces enable us to extend the deterministic formulation to the stochastic setting. Under as weak assumptions as in the deterministic case we are able to prove the existence and uniqueness of a stochastic weak solution to the regular oblique boundary problem for the Poisson equation. Our studies are motivated by problems from geodesy and through concrete examples we show the applicability of our results. Finally a Ritz-Galerkin approximation is provided. This can be used to compute the stochastic weak solution numerically.
