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Mathematics LibreTexts

7.1: Fundamental Solution

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


    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

    \triangle u&=&f''(r)+\frac{n-1}{r}f'(r).

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

    c_1\ln r+c_2&n=2\\

    with constants \(c_1\), \(c_2\).

    Definition. Set \(r=|x-y|\). The function

    -\frac{1}{2\pi}\ln r&n=2\\

    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


    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


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