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The Mean Value Property for harmonic functions tells us that the value of a harmonic function evaluated at the center of $D(P,r)$ equals its weighted integral over $\partial D(P,r)$. I am wondering if there is a mean value property for domains other than disks - such as perhaps polygons.

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  • $\begingroup$ In short yes. There is a homotopic version of cauchy's integral theorem that would allow you to do that, if I remember correctly. $\endgroup$ – DaveNine Apr 28 '15 at 7:06
  • $\begingroup$ @DaveNine If this is true could you plz provide a sketch of proof. The standard way to prove the mean value property is to take the real parts of the Cauchy integral fomula. While this simplifies nicely for disks it seemingly doesn't for other domains. $\endgroup$ – wellfedgremlin Apr 28 '15 at 14:04
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You are looking for Jensen measures (or slightly more general, Arens-Singer measures) for subharmonic (resp. harmonic) functions.

A Jensen measure with barycenter $x$ is (modulo some technicalities on where the measure is supported and exactly what class of functions it's acting on) is a positive (usually) Borel measure $\mu$ such that $$ u(x) \le \int u\,d\mu $$ for all subharmonic functions. In particular, if $u$ is harmonic, then $u$ and $-u$ are both subharmonic, so $$ u(x) = \int u\,d\mu. $$ There are many Jensen measures. For example, if $\Omega$ is any (at least regular) domain in $\mathbb{R}^n$ then for each $x \in \Omega$, there is at least one Jensen measure for $x$ supported on $\partial \Omega$. (Let's require the defining inequalities to hold for functions that are subharmonic on a neighbourhood of $\bar\Omega$, to simplify the technicalities.)

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