On the global solvability of the fixed gravimetric boundary value problem

Abstract

We consider the so called fixed gravimetric boundary value problem (Backus probleml, where from the knowledge of the modulus of the gravity vector on the known physical surface of the Earth we want to find the Earth's outer potential. This is a nonlinear oblique boundary value problem for a linear elliptic equation (Laplace's equationl, and nowdays the general theory of such problems is not as developed as for the linear case. In this paper, we propose a possible existence program for the fixed gravimetric boundary value problem based on the establishment of a priori estimates in a Holder space. Under certain hypothesis and simplifications, we obtain maximun modulus and gradient bounds for the solutions of this boundary value problem.