13.2 – Laplacegleichung und Poissongleichung

a )

Bestimmen Sie alle Lösungen der Laplacegleichung, die nur von $r = \left| x \right|$ abhängen.

b )

Zeigen Sie, dass $u\left( x \right) = - \frac{1}{{2\pi }}\log \left( {\left| x \right|} \right)$ die Fundamentallösung der Poissongleichung in $\mathbb R^2$ ist.

Lösung

a )

$u = u\left( r \right),\quad r = \sqrt {x_1^2+\ldots+x_d^2}$

$\frac{{\partial u}}{{\partial {x_i}}} = \frac{{du}}{{dr}}\frac{{\partial r}}{{\partial {x_i}}} = {u^\prime } \cdot \frac{1}{r} \cdot \frac{1}{2} \cdot 2 \cdot {x_i} = {u^\prime }\frac{{{x_i}}}{r}$

$\frac{{{\partial ^2}u}}{{\partial x_i^2}} = u^{\prime\prime} \frac{{{x_i}}}{r}\frac{{{x_i}}}{r}+{u^\prime }\left[ {\frac{{r \cdot 1-{x_i} \cdot \frac{{2{x_i}}}{{2r}}}}{{{r^2}}}} \right]$

$= u^{\prime\prime} \frac{{x_i^2}}{{{r^2}}}+{u^\prime }\left[ {\frac{1}{r}-\frac{{x_i^2}}{{{r^3}}}} \right]$

$-\Delta u = -u^{\prime\prime} \underbrace {\left[ {\frac{{\sum {x_i^2} }}{{{r^2}}}} \right]}_{ = 1}-{u^\prime }\left[ {\underbrace {\frac{{\sum 1 }}{r}}_{\frac{d}{r}}-\underbrace {\frac{{\sum {x_i^2} }}{{{r^3}}}}_{\frac{1}{r}}} \right] = -u^{\prime\prime} -\left[ {\frac{{d-1}}{r}} \right]{u^\prime } = 0$