Molodensky's problem

Problem

Let $\omega > 0$ be the known angular velocity of the Earth about the $x_3$-axis, and let $\Phi(x) = \tfrac12 \omega^2 (x_1^2 + x_2^2)$ be the centrifugal potential. Let $S^2$ be the unit sphere, used as a parameter set, and suppose the values $W^*(\xi) \in \R$ and $g^*(\xi) \in \R^3$ are given for every $\xi \in S^2$. Find a closed surface $S$ in $\R^3$, given by an embedding $x \colon S^2 \to \R^3$, and a function $V$ harmonic outside $S$ with $V(y) \to 0$ as $\norm{y} \to \infty$. They must be such that the geopotential $W = V + \Phi$ satisfies $$W(x(\xi)) = W^*(\xi) \quad \text{and} \quad \nabla W(x(\xi)) = g^*(\xi) \quad \text{for all } \xi \in S^2.$$ Here $W^*$ comes from levelling combined with gravity, and the direction of $g^*$ from astronomical latitude and longitude.