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

TL;DR

This paper characterizes the subdifferentials of a convex surface energy functional in negative Sobolev spaces, providing explicit formulas and analysis for different boundary conditions relevant to crystal surface roughening.

Contribution

It offers a novel mathematical characterization of subdifferentials for a singular convex functional in Sobolev spaces of order -1, including explicit calculations for symmetric surfaces.

Findings

Explicit subdifferential formulas for periodic and Dirichlet boundary conditions.

Calculation of the minimal element in the subdifferential for spherically symmetric surfaces.

Analysis of subdifferentials in arbitrary spatial dimensions.

Abstract

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.