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

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Yes and No. No: There is no point equidistant to every point on an ellipsoid so there is no point whose value will be given by the mean of the boundary values. Yes: If you weight the boundary values properly you can recover any interior value you like. –  Aaron Hoffman Apr 11 '13 at 14:13
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up vote 2 down vote accepted

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.

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Alexandre, why do you impose the centrality condition? –  R W Apr 11 '13 at 23:04
R.W.: I mentioned that one can do without. With centrality condition you can take the averages over all surfaces similar to the given surface and having center at x, to recover u(x). Without centrality, we have to restrict ourselves to shifts and homotheties of the given surface. –  Alexandre Eremenko Apr 12 '13 at 12:35
OK - but if you formulate the claim in terms of harmonic measures, then there is no need for any assumptions like that - right? –  R W Apr 12 '13 at 13:27
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