Preface

This section concerns about Laplace equations in infinite domain.

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Introduction to Linear Algebra with Mathematica

Glossary

Laplace equation in infinite domain

We can identify the n-dimensional Eucledian space ℝn with \( \displaystyle \mathbb{R}^{n-1} \times \mathbb{R} , \) writing a typical point z \in ℝn as \( \displaystyle z = (x,y) , \) where x ∈ ℝn-1 and y ∈ ℝ. The upper half-space is the set
\[ R_{+}^n = \left\{ (x,y)\,:\ x\in \mathbb{R}^{n-1} , \quad y> 0 \right\} . \]
We identify ℝn-1 with ℝn-1 × {0}; with this convention we then have the boundary \( \displaystyle \partial \mathbb{R}^{n}_{+} = \mathbb{R}^{n-1} . \)

A harmonic function u (so Δ u = 0) in an unbounded domain E is called regular if it satisfies the following condition at infinity:
  • For n = 2 (two dimensional space ℝ²), u(x, y) is bounded at infinity when (x, y) ↦ ∞.
  • For n > 2, the harmonic function u(x, y) approaches zero as (x, y) ↦ ∞.

When domain is unbounded, the main technique to solve Laplace's equation is the Fourier transformation

\begin{equation} \label{Fourier.1} \hat{f} (k ) =ℱ_{x \to k}\left[ f(x) \right] (k ) = f^F (k ) = \int_{-\infty}^{\infty} f(x)\,e^{{\bf j} k\cdot x} \,{\text d}x \qquad ({\bf j}^2 = -1) . \end{equation}
The Fourier transformation gives the spectral representation of the derivative operator j ∂x. It means that the Fourier transformation represent this differential operator as a multiplication operator:
\begin{equation} \label{Fourier.2} ℱ_{x \to k} \left[ {\bf j}\,\frac{{\text d} f(x)}{{\text d} x} \right] = k\,ℱ\left[ f(x) \right] (k) = k \,f^F (k ) . \end{equation}
Recall that the inverse Fourier transform is also expressed via an improper integral, but now in special sense:
\begin{equation} \label{Fourier.3} ℱ^{-1}_{k \to x} \left[ f^F (k ) \right] = \mbox{(P.V.) }\frac{1}{2\pi} \int_{-\infty}^{\infty} f^F (k)\,e^{-{\bf j} k\, x} \,{\text d}k = \lim_{R\to \infty} \frac{1}{2\pi} \int_{-R}^{R} {\text d}k\, e^{- {\bf j}kx} f^F (k) = \frac{f(x+0) + f(x-0)}{2} \qquad ({\bf j}^2 = -1). \end{equation}
Here P.V. stands for Cauchy "principle value", which is used to define the inverse Fourier transform. In particular,
\begin{equation} \label{Fourier.4} ℱ^{-1}_{k \to x} \left[ f^F (-k ) \right] = \int_{-\infty}^{\infty} f^F (k)\,e^{-{\bf j} k\, (-x)} \,{\text d}k = \int_{\infty}^{\infty} f^F (k)\,e^{{\bf j} k\, x} \,{\text d}k = f(-x) . \end{equation}

 

Poisson equation

The Green function for Poisson's equation in infinite Euclidean space ℝn
\begin{equation} \label{Poisson.1} \Delta u = \frac{\partial^2 u}{\partial x_1^2} + \frac{\partial^2 u}{\partial x_2^2} + \cdots + \frac{\partial^2 u}{\partial x_n^2} = f({\bf x}), \qquad {\bf x} \in \mathbb{R}^n , \end{equation}
is
\begin{equation} \label{Poisson.2} G_n (r) = \begin{cases} \frac{\Gamma \left( \frac{n}{2} \right)}{2 \left( n-2 \right) \pi^{n/2}} \, r^{2-n} , & \quad n> 2, \\ \frac{1}{2\pi} \,\ln \left( \frac{1}{r} \right) , & \quad n = 2, \end{cases} \qquad r = \sqrt{x_1^2 + \cdots + x_n^2} . \end{equation}
In particular,
\[ G_3 (r) = \frac{1}{4\pi} \,\frac{1}{r} , \qquad G_4 (r) = \frac{1}{4\pi} \,\frac{1}{r^2} . \]
Then the solution of Poisson's equation \eqref{Poisson.1} becomes
\begin{equation} \label{Poisson.3} u({\bf x}) = \int_{\mathbb{R}^n} G_n \left( {\bf x} - {\bf y} \right) f({\bf y})\,{\text d}{\bf y} . \end{equation}

 

Semi-infinite domain

The Dirichlet problem for the half-plane y > 0 and, in general, for half-hyper-space xn > 0 does not have a unique solution in the space of continuous functions C². For example, the problem
\[ \Delta u = u_{xx} + u_{yy} = 0 \quad (x\in \mathbb{R}, \ y>0), \qquad u(x,0) = 0 , \]
has two solutions u ≡ 0 and u = y. Nevertheless, for problems solvable with the aid of the Fourier transform, a natural class of solutions is usually taken from the subspace of square integrable functions 𝔏². More precisely, we require that function u(x, y) has two derivatives and
\[ u, \ u_x, \ u_{xx} , \ u_{yy} \in 𝔏²\left( \mathbb{R}^2_{+} \right) , \qquad \mbox{with} \quad \mathbb{R}^2_{+} = (-\infty < x < \infty , \ y> 0) . \]

 

  2. Grigoryan, V, Parial Differential Equations, 2010, University of California, Santa Barbara.

 

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