Question. Let $f: \mathbf{R} \to \mathbf{R}$ be an analytic function. Is there a harmonic function $u$ on the circular cylinder $D \times \mathbf{R} \subset \mathbf{R}^3$ so that $u = f$ along the axis $\{ (0,0) \} \times \mathbf{R}$?

• The problem is obviously ill-posed in the sense of Hadamard because it is very underdetermined. Although it doesn't make sense as a 'Dirichlet problem', I think a (positive or negative) answer is possible.
• Probabilistic arguments seem tricky because the axis is too small for hitting times of Brownian motion to be defined.