Ngô Quốc Anh

November 9, 2010

Characterization of biharmonic functions

Filed under: Uncategorized — Ngô Quốc Anh @ 16:30

It is well-known that if B_R(x) is a ball with center x and radius R which is completely contained in the open set \Omega\subset\mathbb R^n, then the value u(x) of a harmonic function (i.e. \Delta u =0) u :\Omega\to\mathbb R at the center of the ball is given by the average value of u on the surface of the ball; this average value is also equal to the average value of u in the interior of the ball. In other words

\displaystyle u(x) = \frac{1}{n\omega_n R^{n-1}}\int_{\partial B_R(x)} ud\sigma = \frac{1}{\omega_n R^n}\int_{B_R(x)} u(y)dy

where \omega_n is the volume of the unit ball in n dimensions and \sigma is the n-1 dimensional surface measure.

For a biharmonic function u (i.e. \Delta^2u=0) we have a similar result due to Pizzetti.

Theorem. For any n \in \mathbb N, any solution u of

\Delta^2 u =0 in B_R(x) \subset \mathbb R^n

satisfies

\displaystyle u(x)-\frac{1}{\omega_n R^n}\int_{B_R(x)} u(y)dy=\frac{R^2}{2(n+2)}\Delta u(x).

Proof. We may assume that B_R(x)=B_R(0)=B_R. For 0<r<R let G_r be the fundamental solution of the operator \Delta^2 on B_r satisfying

G_r=\Delta G_r=0 on \partial B_r.

Note that

\displaystyle G_r(x)=\frac{1}{r^{n-4}}G_1\left(\frac{x}{r}\right).

If n=4 we have

\displaystyle G_r(x)=c_0\left(\log\frac{r}{|x|}-\frac{r^2-|x|^2}{4r^2}\right).

Applying the mean value formula to the harmonic function \Delta u, for some constants c_1, c_2 we have

\displaystyle\begin{gathered} 0 = \int_{{B_r}} {{G_r}{\Delta ^2}udx} \hfill \\ \quad= u(0) + \int_{\partial {B_r}} {\left( {\frac{\partial }{{\partial n}}{G_r}\Delta u + \frac{\partial }{{\partial n}}\Delta {G_r}u} \right)d\sigma } \hfill \\ \quad= u(0) - \frac{1}{{n{\omega _n}{r^{n - 1}}}}\int_{\partial {B_r}} {({c_1}{r^2}\Delta u + {c_2}u)d\sigma } \hfill \\ \quad= u(0) - {c_1}{r^2}\Delta u(0) - \frac{{{c_2}}}{{n{\omega _n}{r^{n - 1}}}}\int_{\partial {B_r}} {ud\sigma } \hfill \\ \end{gathered}

that is, for some constants c_3, c_4,

\displaystyle n{r^{n - 1}}u(0) = {c_3}{r^{n + 1}}\Delta u(0) + {c_4}\int_{\partial {B_r}} {ud\sigma } .

Integrating over 0<r<R and dividing by R^n we obtain the identity

\displaystyle u(0) = {c_5}{R^2}\Delta u(0) + \frac{{{c_6}}}{{{\omega _n}{R^n}}}\int_{{B_R}} {udx}

with uniform constants c_5, c_6 for all biharmonic functions u on B_R. Inserting a harmonic function u, we obtain the value c_6=1, whereas the choice u(x)=|x|^2 yields c_5=\frac{1}{2(n+2)}.

I haven’t seen such a characterization of triharmonic functions, say function u such that \Delta^3 u=0.

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