On the geodetic boundary value problem for a fixed boundary surface—A satellite approach

Employing satellite-geometrical methods, the physical surface of the earth may be assumed to be known, while gravity measurements yield thelength of the gravity vector (including contributions from rotation). The problem then is to determine gravitational potential from such gravity observations. The corresponding linearized problem is an oblique derivative problem. The problem was discussed by Almqvist (1959), Koch (1970, 1971) and Koch and Pope (1972). Our presentation gives proofs for the existence (and uniqueness) of the solution in the non-linear case. The general implicit function theorem (in Banach spaces) is used to provewellposedness at least when data are close to given standard values (closeness is defined either in terms of Hölder or Sobolev norms). Iterative methods for solution by linear operators are given. The linearized problem is solved by harmonic reduction to an internal sphere in a generalization of the method by the first author for the Stokes problem. Also deflections of the vertical are treated.