On a uniqueness property of harmonic functions (Q934534)

Let \(\Omega \) be a domain in \(\mathbb{R}^n\) that is (axially) symmetric about the \(x_1\)-axis. It is shown that, if \(u\) is a harmonic function on \( \Omega \) that vanishes on an interval of the \(x_1\)-axis contained in \( \Omega \), then \(u\) vanishes at all points of the \(x_1\)-axis that lie in \( \Omega \). In the special case where \(\Omega \) is a spherical shell, this answers a question of Nadirashvili. The authors then extend their ideas to deal with solutions of more general analytic elliptic equations.