Is there any result like the mean value theorem for harmonic functions on ellipsoids (instead of sphere)?



Mean value theorem for harmonic functions on ellipsoid
Let me expand Aaron's answer: there is a mean value theorem with any centrally symmetric surface.
You integrate your harmonic function on the surface 
against the harmonic measure at the center, and you recover
the value of your function at the center. You can also generalize this to non centrally symmetric
surfaces, but the statement becomes a bit longer.


Harmonic measure on a surface can be defined by this property, 
and the fact is that it exists for all reasonable
surfaces. You can generalize even further, and dispose of the surface:-)
Just consider measures such that convolution with a harmonic function reproduces this harmonic
function. (They are called Jensen measures if I remember correctly).


EDIT: I remembered incorrectly: Jensen's measure at $x$ is a measure such that
$$u(x)\geq\int ud\mu$$ for all superharmonic functions. The measures I was writing about
apparently have no name.


