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Premise: I know this question would be better suited to MathSE, but since I endeavour to solve a free CQFT on a bounded domain, I'm confident I'll find a more exhaustive answer here.

I'm trying to obtain the Green's function on a planar square domain, i.e, I'm trying to solve the following BVP:

\begin{cases} \Delta G(\bar x,\bar y) = \delta^2(\bar x - \bar y),& \bar x, \bar y\in D\\ G(\bar x,\bar y) = 0,& \bar x\in \partial D \end{cases}

with $D = \{\bar x = (x_1,x_2)\in \mathbb{R}^2 / -L<x_1 < L, -L<x_1 < L\}$.
I've already solved the same problem on a disk with both the methods of images and conformal mapping as shown here; moreover, I know from this paper, page 5, what is to be the form of my solution and that it will involve Weierstrass sigma functions.
My question is thus the following; where can I find a resource in which this problem, if not solved, is at least adressed? All the undergraduate books I searched in, such as the Ahlfors or the Myint-U & Debnath do not cover this topic, and if they do they focus on much simpler examples.

Francesco Arnaudo
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1 Answers1

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Thanks to @Nephente advice, I'm now able to answer my question.

Finding the Green's function for the Laplacian in a 2D square can be considered as a particular case of the more general problem of finding it in a 2D rectangle. The latter can be solved in two ways; expanding the Green's function in terms of the Laplacian's eigenvalues, or using Riemann mapping theorem to conformally map the rectangular domain into a unit circle.

The first method is within the grasp of any average physics undergraduate student, and its full development can be found in Duffy's "Green's Functions with Applications", chapter 6.3; this book is the only one I found which exhaustively covers the topic for Dirichlet boundary conditions.

The second method is a little more tricky, and the math tools involved are beyond those traditionally given in undergraduate courses, since it requires a thorough knowledge of the properties of elliptic functions. A complete account of it can be found in Smirnov's "A course of higher mathematics", Vol. III, part 2, chapter 6.4.188. Though particularly heavy and rigorous, this book provides all the necessary preliminary knowledge to the understanding of the mapping in terms of Weierstrass elliptic functions.

To conclude, it is maybe worth mentioning that the first method was apparently first addressed in Harnack's work on potential theory, as stated by Duffy; I can't however guarantee for it, since I couldn't find a full translation of his works.

Francesco Arnaudo
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