# Harmonic functions which vanish on coaxial cylinders

## TL;DR

This paper studies harmonic functions on annular cylinders that vanish on boundaries and shows they extend to the entire space minus the axis, introducing a new estimate for zeros of Bessel functions.

## Contribution

It extends previous results by analyzing harmonic functions on annular cylinders and provides a novel estimate for zeros of cross product Bessel functions.

## Key findings

- Harmonic functions vanish on boundary extend to space minus axis
- New estimate for zeros of cross product Bessel functions
- Harmonic extension properties on annular cylinders

## Abstract

It was recently established that a function which is harmonic on an infinite cylinder and vanishes on the boundary necessarily extends to an entire harmonic function. This paper considers harmonic functions on an annular cylinder which vanish on both the inner and outer cylindrical boundary components. Such functions are shown to extend harmonically to the whole of space apart from the common axis of symmetry. One of the ingredients in the proof is a new estimate for the zeros of cross product Bessel functions.