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Mathematics LibreTexts

7.1: Fundamental Solution

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.

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