$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

# 7.1: Fundamental Solution

$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

Here we consider particular solutions of the Laplace equation in $$\mathbb{R}^n$$ of the type

$$u(x)=f(|x-y|),$$

where $$y\in\mathbb{R}^n$$ is fixed and $$f$$ is a function which we will determine such that $$u$$ defines a solution if the Laplace equation.

Set $$r=|x-y|$$, then

\begin{eqnarray*}
u_{x_i}&=&f'(r)\frac{x_i-y_i}{r}\\
u_{x_ix_i}&=&f''(r)\frac{(x_i-y_i)^2}{r^2}+f'(r)\left(\frac{1}{r}-\frac{(x_i-y_i)^2}{r^3}\right)\\
\triangle u&=&f''(r)+\frac{n-1}{r}f'(r).
\end{eqnarray*}

Thus a solution of $$\triangle u=0$$ is given by

$$f(r)=\left\{\begin{array}{r@{\quad:\quad}l} c_1\ln r+c_2&n=2\\ c_1r^{2-n}+c_2&n\ge3 \end{array}\right.$$

with constants $$c_1$$, $$c_2$$.

Definition. Set $$r=|x-y|$$. The function

$$s(r):=\left\{\begin{array}{r@{\quad:\quad}l} -\frac{1}{2\pi}\ln r&n=2\\ \frac{r^{2-n}}{(n-2)\omega_n}&n\ge3 \end{array}\right.$$

is called singularity function associated to the Laplace equation. Here is $$\omega_n$$ the area of the n-dimensional unit sphere which is given by$$\omega_n=2\pi^{n/2}/\Gamma(n/2)$$, where

$$\Gamma(t):=\int_0^\infty\ e^{-\rho}\rho^{t-1}\ d\rho,\ \ t>0,$$

is the Gamma function.

Definition. A function

$$\gamma(x,y)=s(r)+\phi(x,y)$$

is called fundamental solution associated to the Laplace equation if $$\phi\in C^2(\Omega)$$ and $$\triangle_x\phi=0$$ for each fixed $$y\in\Omega$$.

Remark. The fundamental solution $$\gamma$$ satisfies for each fixed $$y\in\Omega$$ the relation

$$-\int_\Omega\ \gamma(x,y)\triangle_x\Phi(x)\ dx=\Phi(y)\ \ \mbox{for all}\ \Phi\in C_0^2(\Omega),$$

see an exercise. This formula follows from considerations similar to the next section.

In the language of distribution, this relation can be written by definition as

$$-\triangle_x\gamma(x,y)=\delta(x-y),$$

where $$\delta$$ is the Dirac distribution, which is called $$\delta$$-function.

## Contributors

• Integrated by Justin Marshall.