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I would appreciate any insights on this matter;

Let us consider the mean value property for harmonic functions in three dimensional euclidean space. This suggests that the value of a harmonic function at the center of a sphere is the average of values of this function over the entire sphere.

Now suppose I have an ellipsoid which is very skewed in one direction. Can one write the mean value property with respect to a different measure? Furthermore are there any inequalities relating the average of the harmonic function over the whole ellipsoid and the value at the center?

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    $\begingroup$ I think your "different measure" is harmonic measure. $\endgroup$
    – Robert Israel
    Jun 14, 2016 at 17:03
  • $\begingroup$ Is there a very rough estimate available in the case of an ellipsoid for the harmonic measure at the center? $\endgroup$
    – Ali
    Jun 14, 2016 at 17:14

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