Spherical mean of a biharmonic function ($⟨f{⟩}_{S_a}=f(0)+a^2\Delta f(0)/6$)

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 ${\Delta }^{2}f=0$. This is the mean-value identity that converts the reciprocal surface average into Faxén's first law.