Split operators for oblique boundary value problems

Abstract

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 Δ. 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 ΔD v = 0 if Δv = 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.

Keywords:

• Boundary value problem
• Higher order differentials as boundary data
• Split operator

AMS 2000 Mathematics Subject Classifications::

• 30E25
• 35G15
• 35J99
• 65N99