**Question.** What is a good reference (textbook, article etc.) to learn more about harmonic functions on finite (and infinite) cylinders?

I am trying to gain a better understanding of the behavior of harmonic functions $u$ defined on cylindrical domains $C_{R,L}$ of the form \begin{equation} C_{R,L} := D_R \times (-L,L) \subset \mathbf{R}^3, \end{equation} where $D_R$ is the disc of radius $R > 0$, and $L$ may be finite or infinite.

I have asked a couple of questions around this recently, but I've come to realize that I don't know the background theory well enough.

*Remark.*
I should say that ultimately I am trying to gain a better feel for related, but non-linear problems, so I would prefer a recommendation that emphasizes methods that can be generalized.

For clarity, I am listing some examples of the questions that I am specifically interested in.

- The
*solvability*of the Dirichlet problem on the infinite cylinder $D_R \times \mathbf{R}$ with periodic boundary data $f: \partial D_R \times \mathbf{R} \to \mathbf{R}$. - The
*behaviour*of these solutions: for example, I would expect them to have a mix of 'oscillatory regimes', which might make way for 'exponentially growing regimes' as $z \to \pm \infty$. - The
*construction of solutions*in the slab $\mathbf{R}^2 \times (-L,L)$ via the passage to the limit $R \to \infty$. Which estimates and methods are used to establish existence and non-vanishing of the limit?

Green, Brown, and Probabilityis a really nice introduction, I think. $\endgroup$