An inverse problem of the logarithmic potential theory for the lemniscates (Q1202898)

Considering two connected bounded domains \(D_ 1\), \(D_ 2\) such that their boundaries are smooth Jordan curves and the complements of \(D_ 1\), \(D_ 2\), \(D_ 1D_ 2\) are connected, the author shows that if \(D_ 1=\{(x,y)\in\mathbb{R}^ 2;\;P(x,y)=1\}\), where \(P(x,y)\) is an elliptic polynomial, then the equality of the logarithmic potentials of \(D_ 1\) and \(D_ 2\) for \(| z|\gg 1\) implies \(D_ 1=D_ 2\).