Inverse problem of potential theory

TL;DR

This paper advances the inverse potential problem by removing geometric restrictions, introducing a new approach, and providing counterexamples for a related problem with oscillatory kernels.

Contribution

It extends Novikov's uniqueness result by eliminating star-shapedness, proposes a novel method, and constructs counterexamples for a modified kernel case.

Findings

Removed star-shape assumption for domain uniqueness

Developed a new approach to inverse potential problems

Constructed counterexamples for oscillatory kernel case

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, $$u_j(x):=\int_{D_j}g_0(x,y)dy, \quad g_0(x,y):=\frac 1 {4\pi |x-y|},$$ and $D_j\subset \mathbb{R}^3$, $j=1,2,$ $D_j\subset B_R$, are bounded, connected, smooth domains, star-shaped with respect to a common point, then $D_1=D_2$. Here $B_R:= \{x: |x|\le R\}$. 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 $g=\frac {e^{ik|x-y|}}{4\pi |x-y|}$, where $k>0$ is a fixed constant.