No Generalization of Mean Value Property for harmonic functions?

Question

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.

Answer 1

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.)