  We establish two geometric inequalities, respectively, for harmonic functions
in exterior Dirichlet problems, and for Green's functions in interior Dirichlet
problems, where the boundary surfaces are smooth and convex. Both inequalities
involve integrals over the mean curvature and the Gaussian curvature on an
equipotential surface, and the normal derivative of the harmonic potential
thereupon. These inequalities generalize a geometric conservation law for
equipotential curves in dimension two, and offer solutions to two free boundary
problems in three-dimensional electrostatics.
