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# Fundamental solution of the Laplacian on flat tori and boundary value problems for the planar Poisson equation in rectangles

- Malik Mamode
^{1}Email author

**2014**:221

https://doi.org/10.1186/s13661-014-0221-4

© Mamode; licensee Springer 2014

**Received:**11 July 2014**Accepted:**16 September 2014**Published:**1 October 2014

## Abstract

The fundamental solution of the Laplacian on flat tori is obtained using Eisenstein’s approach to elliptic functions via infinite series over lattices in the complex plane. Most boundary value problems stated for the planar Poisson equation in a rectangle for which series-only representations of solution were known, may thus be solved explicitly in closed-form using the method of images. Moreover, the fundamental solution of *n*-Laplacian on flat tori may also be simply derived by a convolution power.

**PACS Codes:** 02.30.Em, 02.30.Jr.

## Keywords

- 2D Poisson equation
- Green function
- horizon
- boundary value problem
- flat torus
- analytical solution

## 1 Introduction

The Poisson equation is certainly the simplest and the most famous partial differential equation of elliptic type which arises in many areas of mathematical physics, for example for steady state problems in electrostatics, heat conduction and fluid flow. The studies of boundary value problems (BVPs) for Poisson equation in finite or semi-infinite connected domains are ancient and well documented in many reference books [1]–[3]. Different analytical methods to construct solutions of these problems exist such as separation of variables, eigenfunction expansion, *etc.* and may depend on the nature of boundary conditions and the shape of domains. Among these methods, the method of source function, also called the Green function method, is the only one giving convenient and compact analytical representation of solutions in two- or three-dimensional space. In this method, the BVPs are reformulated into integral equations that involve the boundary conditions and the related Green function. But, as is well known, the actual implementation of the source function method in applications is often intricate owing to the difficulty to obtain a closed-form analytical expression for the Green function. This is notably the case for the BVPs for Poisson equation in rectangular domains for which series-only representations of Green functions are used [4], although it is possible in principle to circumvent the difficulty by converting known closed-form results obtained for half-plane or circular domain into the rectangle via conformal transformations. Thus, using a conformal Schwarz-Christoffel transformation mapping the half-plane into the rectangle, one can easily find the closed-form solution of homogeneous Dirichlet problem in terms of Jacobi elliptic functions (see for instance [1], [2], [5]), but to date a large number of other BVPs for rectangular Poisson equation still have no analytical closed-form solutions.

We show in the present paper that the solutions of these BVPs amount to solve the Poisson equation on flat tori in the distributional sense *i.e.* in the space of doubly periodic Schwartz distributions on the Euclidean plane ${\mathbb{E}}^{2}={\mathbb{R}}^{2}\text{or}\mathbb{C}$[6], [7]. For this, the fundamental solution of the Laplacian is first obtained in Section 2 in terms of elliptic functions using Eisenstein’s approach via infinite series over rectangular lattices in ℂ. Secondly, using the method of images for instance, the analytical closed-form representation of Green functions for most BVPs stated for planar Poisson equation in the rectangle may easily be derived as shown in Section 3.

## 2 Fundamental solution and concept of horizon

where *h* is an arbitrary harmonic function in ${\mathbb{R}}^{2}$*i.e.* a solution of $\mathrm{\Delta}h=0$. Nevertheless, the fundamental solution is that for which it is usually set $h=\mathrm{constant}$ and this may be justified in a heuristic manner as follows. Equation (1) may be seen as establishing a causal connection between a point source located at the origin, and the change produced at point $(x,y)$ to a certain property *G* of the free space. In that sense, in the absence of source, this spatial property governed by (1) does not change and has to remain uniform: $\mathrm{\Delta}h=0$ thus must necessarily imply $h=\mathrm{constant}$. In addition, since the fundamental solution may also reflect some geometrical properties of the space like isotropy, it must possess circular symmetry *i.e.* must depend only on one variable $r=\sqrt{{x}^{2}+{y}^{2}}$ and it is easy to show that the only harmonic functions with circular symmetry are again constant functions.

where ${r}_{0}$ is an arbitrary constant length we shall call in the sequel *horizon*. It is common, not to say always, the case, to set *a priori* for convenience ${r}_{0}=1$. But, in the light of our concern, such a simplification has two inconveniences: (i) it overshadows the scale invariance property of solution of (1) by the change of coordinates $(x,y)$ into $(\epsilon x,\epsilon y)$ for any $\epsilon >0$ and the consequences for existence of the solution of BVPs (see below); (ii) the concept of horizon (which may be chosen as far as wanted) restores the property that *G* tends to 0 as ${({x}^{2}+{y}^{2})}^{1/2}\to {r}_{0}$ which is, here at finite distance, the property retained for the fundamental solution of Laplace equation in $(n>2)$-dimensional space for which the horizon is located at infinity.

*i.e.*may be given by the double convolution product,

*up to a constant*provided that

We focus our attention in the rest of this note to the case where the source term *F* is doubly periodic in the $x,y$-variables. We shall see indeed that the BVPs for the rectangle, not surprisingly, are relevant to that case related to the tessellation of the Euclidean plane ${\mathbb{E}}^{2}$ with an infinite number of identical rectangular tiles in both $x,y$-directions.

*F*is of period 2

*a*and 2

*b*in the

*x*- and

*y*-variable, respectively. It is thus always possible to define the distribution ${F}_{0}$ on the rectangle $\{(x,y)|-a<x<a,-b<y<b\}$ such that ‘by gluing together the pieces’ [7]

provided that the previous series is convergent, *which is unfortunately not the case for any fixed*${r}_{0}$.

*V*should be unchanged owing to the compatibility condition (4), if the term ${r}_{0}^{2}$ is replaced by

*viz.*the center of fundamental rectangle. In this way, it follows that the double infinite product (6) may be seen as the modulus square of

*is always not absolutely convergent*and has not the rearrangement property

*i.e.*its limit $\varphi (z)$ (if it exists) is dependent of the order of the factors. Using thus the Eisenstein convention [8] which groups together the factors of opposite rank according to the following calculation process:

*a*, while under translation along the imaginary axis,

*up to a constant*, is thus explicitly given by

*i.e.*${\mathrm{\Psi}}_{a,b}$ is the fundamental solution (Green function) for the Laplacian on ${\mathrm{\Omega}}_{a,b}$. Indeed, ${\varphi}_{e}(z)$ is zero at $z=0$ so that $\mathrm{\Delta}{\mathrm{\Psi}}_{a,b}$ has a singularity at the center of the torus and,

*ζ*-Weierstrass elliptic function, which is meromorphic, having a simple pole at the origin (and no others on the integration domain) with residue 1 [9], it follows from the residue theorem that

### Remarks

- (i)Let us notice that interchanging the roles of variables
*x*and*y*in what precedes yields an equivalent formulation of the fundamental solution,${\mathrm{\Psi}}_{a,b}(x,y)=\frac{1}{2\pi}log\left\{{e}^{-\pi {x}^{2}/4ab}\right|{\vartheta}_{1}(\frac{\pi}{2b}(y+ix)|i\frac{a}{b})\left|\right\},$(12)

it follows by induction that, for any integer $n\ge 1$,

## 3 Applications

Having found the fundamental solution for the Laplacian (10), it is straightforward using the image method (by an appropriate periodic distribution of images) to obtain the exact expression of the Green function for most BVPs stated for Poisson equation in the rectangle.

*i.e.*the solution of the equation $\mathrm{\Delta}{G}_{D}=\delta (x-{x}_{0})\delta (y-{y}_{0})$ on the rectangle $\{(x,y)|0<x<a,0<y<b\}$ with the homogeneous conditions ${G}_{D}=0$ on the sides, or equivalently the periodic distribution solution of

*i.e.*by setting $z=x+iy$ and ${z}_{0}={x}_{0}+i{y}_{0}$,

This result is to be compared with that obtained in [5]. Here, it is worth to note that in addition to its simplicity, the complex formulation of the Green function together with the properties of functions of a complex variable and of conformal transformation may provide us with several methods for extending as desired the Dirichlet problem to many other geometries [10].

*i.e.*the solution of the equation $\mathrm{\Delta}{G}_{N}=\delta (x-{x}_{0})\delta (y-{y}_{0})-\frac{1}{ab}$ on the rectangle $\{(x,y)|0<x<a,0<y<b\}$ with the homogeneous conditions $\frac{\partial {G}_{N}}{\partial n}=0$ on the sides ($\partial /\partial n$ denotes as usual the outward normal derivative; notice that here the compatibility condition (4) is nothing but the Gauss theorem), or equivalently the periodic distribution solution of

*i.e.*

## 4 Perspectives and conclusions

The search of analytical closed-form solutions for BVPs for planar Poisson equation - but this is also true for many other problems of mathematical physics - are nowadays little tackled for the benefit of numerical computational methods while the existence of a compact analytical closed-form solution for such problems is always a great advance for a better understanding of underlying physics. The present work has developed a simple method for analytically solving most BVPs stated for the planar Poisson equation in a rectangular domain for which, to our knowledge, only series representation of solutions were available. This construction, based on the Green function method, amounts to solve the Poisson equation on flat tori in the distributional sense having found the fundamental solution of the Laplacian in terms of elliptic functions. When it is possible, the complex formulation of obtained results will allow one to extend - via conformal transformations - the number of problems posed on regions of more complicated shape and with a mixed setting of different kinds of boundary conditions. Although it was not the primary object of this note, having obtained the fundamental solution of *n*-Laplacian on the flat torus will also be of great interest for explicitly solving BVPs for two-dimensional *n*-harmonic equation in rectangles as, for instance, the biharmonic equation and the linear clamped plate boundary value problem in mechanics.

## Declarations

### Acknowledgements

The author would like to thank the referees for their valuable remarks and suggestions.

## Authors’ Affiliations

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## Copyright

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