Harmonic extension from the exterior of a cylinder

Gardiner, Stephen; Render, Hermann

doi:10.1090/proc/15172

Harmonic extension from the exterior of a cylinder
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by Stephen J. Gardiner and Hermann Render

Proc. Amer. Math. Soc. 149 (2021), 1077-1089

DOI: https://doi.org/10.1090/proc/15172

Published electronically: January 13, 2021

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Abstract:

Let $h$ be a harmonic function on an annular cylinder. It was recently shown that, if $h$ vanishes on the outer cylindrical boundary component, then it has a harmonic extension to a larger annular cylinder formed by radial reflection. This paper shows that a different kind of extension result holds if $h$ instead vanishes on the inner boundary component. As a corollary it is shown that any harmonic function on the exterior of an infinite cylinder which vanishes on the boundary has a harmonic extension to the complement of the axis of the cylinder.

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