Journal of Statistical Physics. 95(5-6):1221–1244, JUNE 1999

Issn Print: 0022-4715

Publication Date: June 1999

# A Variational Approach to Crystalline Triple-Junction Motion

### Abstract

A variational description is given for curves with triple junctions for the motion with normal velocity v=M(C + κΦ), where κΦ stands for the crystalline curvature as determined by the curves and by the crystalline (polygonal Wulff shape) surface free energy functions Φ for each interface, C is constant on each interface, and M is a compatible normal-dependent mobility function for each interface. This variational formulation is based on the idea that the motion should be gradient flow, in the L2 inner product, for the sum of the surface free energy and the bulk free energy. If the surface free energy functions Φ are identically zero, the motion is that given by Taylor (1995). If the surface free energy functions are positive and crystalline, then the motion is that given by Taylor (1993). Finally, if the surface free energy functions are written as Φ = εΦ0, then the limiting motion as ε ↓ 0 is in general different from the motion for ε = 0 [and hence different from that given by Taylor (1993); the limiting motion is presumably that given by Reitich and Mete Soner (1996)].