  Subdifferentials of a singular convex functional representing the surface
free energy of a crystal under the roughening temperature are characterized.
The energy functional is defined on Sobolev spaces of order -1, so the
subdifferential mathematically formulates the energy's gradient which formally
involves 4th order spacial derivatives of the surface's height. The
subdifferentials are analyzed in the negative Sobolev spaces of arbitrary
spacial dimension on which both a periodic boundary condition and a Dirichlet
boundary condition are separately imposed. Based on the characterization
theorem of subdifferentials, the smallest element contained in the
subdifferential of the energy for a spherically symmetric surface is calculated
under the Dirichlet boundary condition.
