Ngô Quốc Anh

September 20, 2012

Poly-Laplacian of rotationally symmetric functions in R^3

Filed under: Các Bài Tập Nhỏ, PDEs — Ngô Quốc Anh @ 5:24

In $\mathbb R^3$, it is known that for any rotationally symmetric function $f$, i.e. $f$ depends only on the radius $r$, the following holds

$\displaystyle \Delta f= f'' + \frac{2}{r} f' = \frac{1}{r^2}(r^2 f')' .$

By a simple calculation, it is easy to have

$\displaystyle\begin{gathered} {\Delta ^2}f = \Delta \left( {f'' + \frac{2}{r}f'} \right) \hfill \\ \qquad= {\left( {f'' + \frac{2}{r}f'} \right)^\prime }^\prime + \frac{2}{r}{\left( {f'' + \frac{2}{r}f'} \right)^\prime } \hfill \\ \qquad= {f^{(4)}} + {\left( { - \frac{2}{{{r^2}}}f' + \frac{2}{r}f''} \right)^\prime } + \frac{2}{r}{f^{(3)}} - \frac{4}{{{r^3}}}f' + \frac{4}{{{r^2}}}f'' \hfill \\ \qquad= {f^{(4)}} + \left( {\frac{4}{{{r^3}}}f' - \frac{2}{{{r^2}}}f'' - \frac{2}{{{r^2}}}f'' + \frac{2}{r}{f^{(3)}}} \right) + \frac{2}{r}{f^{(3)}} - \frac{4}{{{r^3}}}f' + \frac{4}{{{r^2}}}f'' \hfill \\ \qquad= {f^{(4)}} + \frac{4}{r}{f^{(3)}}. \hfill \\ \end{gathered}$

In other words, there holds

$\displaystyle {\Delta ^2}f = \frac{1}{{{r^4}}}({r^4}{f^{(3)}})'.$

For each natural number $n$, since

$\displaystyle\begin{gathered} \Delta \left( {{f^{(2n)}} + \frac{2n}{r}{f^{(2n - 1)}}} \right) = {\left( {{f^{(2n)}} + \frac{2n}{r}{f^{(2n - 1)}}} \right)^\prime }^\prime + \frac{2}{r}{\left( {{f^{(2n)}} + \frac{2n}{r}{f^{(2n - 1)}}} \right)^\prime } \hfill \\ \qquad\qquad\qquad\qquad\quad= {f^{(2n + 2)}} + \frac{2n+2}{r}{f^{(2n + 1)}} \hfill \\ \end{gathered}$

by induction, one can prove that

$\displaystyle {\Delta ^n}f = {f^{(2n)}} + \frac{2n}{r}{f^{(2n - 1)}} = \frac{1}{{{r^{2n}}}}({r^{2n}}{f^{(2n - 1)}})'$

for any natural number $n$.

With this nice formulas at the hand you can easily give examples of (radial symmetric) functions which are biharmonic but not harmonic, for instance you can think about $r^2$.