Harmonic functions which vanish on coaxial cylinders (Q2330806)

The paper under review deals with the extension of harmonic functions vanishing on coaxial cylinders. In [\textit{S. J. Gardiner} and \textit{H. Render}, J. Math. Anal. Appl. 433, No. 2, 1870--1882 (2016; Zbl 1327.31012)] for any \(a>0\) it was shown that a harmonic function in a finite cylider \(B'\times (-a,a)\) vanishing on \(\partial B' \times (-a,a)\) has a harmonic extension to the strip \(\mathbb{R}^{N-1}\times (-a,a)\). Here, the authors consider the same problem in annular cylinders. They define the set \[ \Omega_b \equiv A'_b \times \mathbb{R}\, , \] where \[ A'_b = \{x' \in \mathbb{R}^{N-1}: 1 < \|x'\| < b \} \] with \(b>1\). They show that if \(h\) is a harmonic function on \(\Omega_b\) vanishing on \(\partial \Omega_b\) then \(h\) has a harmonic extension to \((\mathbb{R}^{N-1}\setminus \{0'\})\times \mathbb{R}\), where \(0'\) denotes the origin in \(\mathbb{R}^{N-1}\). The proof relies on an analysis of the Green function and on an estimate for the zeros of cross product Bessel functions.