Remarks on the reflection principle for harmonic functions (Q913034)

The authors prove three theorems which show why various extensions of the reflection principle fail for harmonic functions in \({\mathbb{R}}^ m\), \(m\geq 3\). One of their results is the following. Let \(\gamma\) be a real analytic curve in a plane in \({\mathbb{R}}^ 3\). Consider a cylinder \(\Gamma ={\mathbb{R}}\times \gamma\). Denote by \(H_{\Gamma}(U)\) the set of functions harmonic in a neighbourhood U of \(\Gamma\) and vanishing on \(\Gamma\). Theorem. If \(\Gamma\) is not a piece of a plane then for every a and b in U there exists \(f\in H_{\Gamma}(U)\) such that \(f(a)=0\) and \(f(b)=1\).