Characterization of subdifferentials of a singular convex functional in Sobolev spaces of order minus one

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 energyʼs gradient which formally involves 4th order spacial derivatives of the surfaceʼs 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.