Some constructions with solutions of variable coefficient elliptic equations (Q1312628)

The author considers the following inverse problem: Suppose that \(\Omega\) is a bounded domain in \(\mathbb{R}^ d\) with a smooth boundary \(\partial\Omega\), and \(f\) and \(g\) are smooth real-valued functions on \(\partial\Omega\). When does there exist a divergence-form elliptic equation div\((A\nabla u)=0\) with a solution \(u\) such that \(u|_{\partial\Omega}=f\), \(A\nabla u\cdot n|_{\partial\Omega}=g\)? (Here \(n\) is the exterior unit normal, \(A\) and \(u\) are supposed to be smooth.) Further, a characterization of such pairs of functions \(f\), \(g\) on the boundary of \(\Omega\) (more generally: on the boundary of a compact manifold) and some applications of related questions in potential theory are given.