Mathematics – Analysis of PDEs
Scientific paper
2011-04-19
Mathematics
Analysis of PDEs
26 pages
Scientific paper
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.
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