| Abstract |
We shall discuss the problem of uniqueness for functions \(u\) harmonic in a domain \(G\) in \(R^n\) and vanishing on some parts of the intersection \(V\) (not necessarily connected) of \(G\) with a line \(m\). The question originated more than a decade ago with N. Nadirashvili (private communication). For example, let \(G\) be a spherical shell, i.e., the region between two concentric spheres, and \(m\) is a line through the origin. Does \(u\) vanish on both segments along which \(m\) intersects \(G\) if it does so on one of them? To illustrate the cunning depth of the question note that if you let \(G\) to be the annulus with a sector cut out, the function \(u=\arg z\) in the plane does vanish on the positive part of the real axis, but not on the whole intersection. What happens if \(G\) is a spherical shell but m does NOT pass through the center? What if we replace harmonic functions by polyharmonic functions, or, more generally, solutions of analytic elliptic equations, or even worse, by linear combinations of Riesz potentials that satisfy no PDE altogether? The answers are by no means obvious and, in many cases, may be judged as surprising.
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