= Side boundary data do not determine a bounded harmonic function in a half-cylinder
{title2=$e^{-j_{0,1}z}J_0(j_{0,1}r)$}
In the unit half-cylinder $z\ge0$, specifying only the curved-side boundary values does not determine a bounded <harmonic function>. The displayed nonzero function, where $j_{0,1}$ is a positive zero of the <Bessel function of the first kind> $J_0$, is regular at the axis, bounded, harmonic, and zero on $r=1$. It may be added to any particular solution without changing the side data, and even decays at infinity. Boundary information at the base $z=0$ is still required for uniqueness.
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