Inverse problem of potential theory

الأصل · EN

P. Novikov in 1938 has proved that if u₁(x)=u₂(x) for |x|>R, where R>0 is a large number, uⱼ(x):=∫Dⱼg₀(x,y)dy, g₀(x,y):= 1 4π|x-y|, and Dⱼ⊂ R³, j=1,2, Dⱼ⊂ Bᵣ, are bounded, connected, smooth domains, star-shaped with respect to a common point, then D₁=D₂. Here Bᵣ:= {x: |x|≤ R}. Our basic results are: a) the removal of the assumption about star-shapeness of Dⱼ, b) a new approach to the problem, c) the construction of counter-examples for a similar problem in which g₀ is replaced by g= eik|x-y|4π|x-y|, where k>0 is a fixed constant.