It is well-known that if is a ball with center and radius which is completely contained in the open set , then the value of a harmonic function (i.e. ) at the center of the ball is given by the average value of on the surface of the ball; this average value is also equal to the average value of in the interior of the ball. In other words
where is the volume of the unit ball in dimensions and is the dimensional surface measure.
For a biharmonic function (i.e. ) we have a similar result due to Pizzetti.
Theorem. For any , any solution of
Proof. We may assume that . For let be the fundamental solution of the operator on satisfying
If we have
Applying the mean value formula to the harmonic function , for some constants we have
that is, for some constants ,
Integrating over and dividing by we obtain the identity
with uniform constants for all biharmonic functions on . Inserting a harmonic function , we obtain the value , whereas the choice yields .
I haven’t seen such a characterization of triharmonic functions, say function such that .