# Reference for harmonic functions in cylinders

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 $$$$C_{R,L} := D_R \times (-L,L) \subset \mathbf{R}^3,$$$$ where $$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?
• Just a remark: judging by your other question, you may be more interested in the general theory. I do not really know the literature well, but Chung's Green, Brown, and Probability is a really nice introduction, I think. Commented Oct 12, 2022 at 17:38

First, some general background. For a bounded domain, the boundary value problem is solved by the Poisson formula, however an explicit form of the Poisson kernel for a cylinder of finite length is probably too complicated for a practical use. For unbounded domains, one needs an additional growth restriction to ensure uniqueness. For example, that the boundary data and the solution for a cylinder are bounded, but this condition can be substantially relaxed. Poisson formula for a slab is relatively simple and it is written in

MR4238605 Madych, W. R. Harmonic functions in slabs and half-spaces. Harmonic analysis and applications, 325–359, Springer Optim. Appl., 168, Springer, Cham, 2021.

For the circular cylinder you may look into

MR3595961 D. Khavinson, E. Lundberg, and H. Render, Dirichlet's problem with entire data posed on an ellipsoidal cylinder, Potential Anal. 46 (2017), no. 1, 55–62.

MR1094495 Yoshida, H. Harmonic majorization of a subharmonic function on a cone or on a cylinder. Pacific J. Math. 148 (1991), no. 2, 369–395.

Qiao, Lei, Asymptotic behavior of Poisson integrals in a cylinder and its application to the representation of harmonic functions, Bull. Sci. Math. 144 (2018), 39–54.

MR2904138 Ancona, Alano, On positive harmonic functions in cones and cylinders, Rev. Mat. Iberoam. 28 (2012), no. 1, 201–230.

Look also at the reference list in these papers.

For the infinite cylinder with periodic $$L^1$$ boundary conditions, there is a unique periodic solution. It is also unique among Dirichlet solutions which grow slower than $$x^\omega$$, where $$\omega$$ is the smallest Dirichlet eigenvalue of $$D$$. Similar results are available for the slab.

• Thanks Alexandre, I'll take a look at the articles you've listed. Commented Oct 11, 2022 at 15:42