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Let $u$ be a harmonic function on $\mathbb R^n$. We know that if $B$ is a ball centered at $x$, then

$$u(x) = \frac{1}{|B|} \int_B u(y) dy = \frac{1}{|\partial B|} \int_{\partial B} u(z) dS(z).$$

I am wondering if there is an analagous result with the ball $B$ replaced by a different set, perhaps an $n$-cube or an ellipsoid. Is such a generalize mean value formula possible?

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up vote 3 down vote accepted

Not quite, but for any (sufficiently nice, certainly for ellipsoids or cubes) domain $D$ and point $x \in D$ there is a "harmonic measure" $\omega_{x,D}$ on $\partial D$ such that the value of a harmonic function $u(x)$ is the average of the boundary values with respect to the measure $\omega_x$, i.e., $$ u(x) = \int_{\partial D} u(z) \, d\omega_{x,D}(z).$$ It is in general not possible to write down explicit expressions for harmonic measure, even for relatively simple domains (like cubes or ellipsoids), but there are good estimates and numerical techniques for harmonic measure computations.

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