A reflection result for harmonic functions which vanish on a cylindrical surface

Suppose that a harmonic function h on a finite cylinder U vanishes on the curved part A of the boundary. It was recently shown that h then has a harmonic continuation to the infinite strip bounded by the hyperplanes containing the flat parts of the boundary. This paper examines what can be said if the above function h is merely harmonic near A (and inside U). It is shown that h then has a harmonic extension to a larger domain formed by radial reflection.