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. For this computationally demanding problem with millions of data points, one classically just uses the derivatives to radial direction because they commute with the Laplace operator Delta. We will investigate in what extent this approach can also be used for other derivatives. We classify all first and purely second-order operators D which fulfill Delta Dv=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 D, i.e. integration.