While I was working on some theorems in PDEs, I encountered the following boundary value problem
$$\matrix{ {{\nabla ^2}H = 0} \hfill & {{\rm{in}}} \hfill & \Omega \hfill \cr {{\nabla ^2}H = 0} \hfill & {{\rm{on}}} \hfill & {r = a} \hfill \cr {{\nabla ^2}H = 0} \hfill & {{\rm{on}}} \hfill & {z = - \ell } \hfill \cr {{\nabla ^2}H = 0} \hfill & {{\rm{on}}} \hfill & {z = \ell } \hfill \cr } $$
where $H:\mathbb{R}^3\to\mathbb{R}$ is an infinitely differentiable scalar field $C^{\infty}(\mathbb{R}^3)$. The domain $\Omega$ is a cylinder defined as
$$\Omega = \left\{ {(r,\varphi ,z)|0 \le r \lt a,0 \lt \varphi < 2\pi , - \ell \lt z \lt \ell } \right\}$$
What can we say about the uniqueness of $H$? Specifically, as the Laplacian is prescribed over the boundary so the answer is not unique. But what is the degree of non-uniqueness?
Can we simply say that $H$ can be any harmonic function?