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 $n$.

1. Hi Ngo, well done as usual!
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 $r^2$.

Comment by Fab — October 19, 2012 @ 17:44

• Thanks Fab.

Comment by Ngô Quốc Anh — October 19, 2012 @ 19:50

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