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 ) ≔ ∫ D j g 0 ( x , y ) d y , g 0 ( x , y ) ≔ 1 4 π | x − y | , and D j ⊂ R 3 , j = 1 , 2 , D j ⊂ 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 | ≤ 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 i k | x − y | 4 π | x − y | , where k > 0 is a fixed constant.
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