Application of analytic continuation to the inverse potential problem (Q1317957)

Let \(D\) be a bounded domain in \(\mathbb{R}^ n\) and \(V_ D\) be the Newtonian potential of \(D\), that is, the solution of \(\Delta V_ D({\mathbf x})=- I_ D({\mathbf x})\) where \(I_ D\) is the characteristic function of \(D\). The inverse potential problem is to determine conditions on the domains \(D_ 1\) and \(D_ 2\) such that \(V_{D_ 1}= V_{D_ 2}\) for all large \({\mathbf x}\) implies \(D_ 1= D_ 2\). In the main lemma, the author proves, under geometric conditions on \(D_ 1\) that are not too restrictive, that \(V_{D_ 1}= V_{D_ 2}\) for large \({\mathbf x}\) implies \(\nabla V_{D_ 1} ({\mathbf x}^*)= \nabla V_{D_ 2}({\mathbf x}^*)\) for some \({\mathbf x}^*\in D_ 1\setminus D_ 2\). He then describes how one can continue the potential analytically across a boundary. For certain types of boundary, this extension can be calculated explicitly and, in conjunction with the lemma, this leads to a condition under which \(V_{D_ 1}= V_{D_ 2}\) for all large \({\mathbf x}\) implies \(D_ 1= D_ 2\) if \(D_ 1\) is a convex polyhedron in \(\mathbb{R}^ n\). The method also yields a similar result for the inverse problem for the complex potential when \(D_ 1\) is a lemniscate in \(\mathbb{C}\).