Inverse problem of potential theory

P. Novikov in 1938 has proved that if u(1) (x) = u(2)(x) for vertical bar x vertical bar > R, where R > 0 is a large number, u(j)(x) := integral(Dj) g0(x, y)dy, g0(x, y) := 1/4 pi vertical bar x - y vertical bar, and D-j subset of R-3, j = 1,2, D-j subset of B-R, are bounded, connected, smooth domains, star-shaped with respect to a common point, then D-1 = D-2. Here B-R := {x : vertical bar x vertical bar <= 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 = e(ik)vertical bar x - y vertical bar/4 pi vertical bar x - y vertical bar, where k > 0 is a fixed constant. (C) 2017 Elsevier Ltd. All rights reserved.