Inverse problem of potential theory

A. G. Ramm

paper

Inverse problem of potential theory

arXiv:1801.04237

Abstract

P. Novikov in 1938 has proved that if $u_{1} (x) = u_{2} (x)$ for $∣ x ∣ > R$ , where $R > 0$ is a large number, $u_{j} (x) := \int_{D_{j}} g_{0} (x, y) d y, g_{0} (x, y) := \frac{1}{4 π ∣ x - y ∣},$ and $D_{j} \subset R^{3}$ , $j = 1, 2,$ $D_{j} \subset B_{R}$ , are bounded, connected, smooth domains, star-shaped with respect to a common point, then $D_{1} = D_{2}$ . Here $B_{R} := {x : ∣ x ∣ \leq R}$ . Our basic results are: a) the removal of the assumption about star-shapeness of $D_{j}$ , b) a new approach to the problem, c) the construction of counter-examples for a similar problem in which $g_{0}$ is replaced by $g = \frac{e ^{ik ∣ x - y ∣}}{4 π ∣ x - y ∣}$ , where $k > 0$ is a fixed constant.