= Spherical mean of a biharmonic function
{title2=$\langle f\rangle_{S_a}=f(0)+a^2\Delta f(0)/6$}
For a smooth three-dimensional biharmonic function regular throughout a ball, its surface average has exactly this form. Taylor expansion and rotationally invariant spherical moments turn each even term into a multiple of an iterated <Laplacian>; terms from the fourth derivative onward vanish when $\Delta^2f=0$. This is the mean-value identity that converts the reciprocal surface average into <Faxén's first law>.
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