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[SOLVED] Mean value

dwsmith

Well-known member
Feb 1, 2012
1,673
For the Dirichlet problem on a sphere of radius a with the boundary condition
$$
u(a,\theta,\phi) = f(\theta,\phi),
$$
show that the value of $u$ at the origin $(r = 0)$ is equal to the average value of $f$ over the surface of the sphere.

I know that the max and min occur on the boundaries and that is because the origin is the mean value but I don't know how to show it is the mean value.