LONG TIME BEHAVIOR OF SURFACE DIFFUSION OFANISOTROPIC SURFACE ENERGY

We investigate the surface diffusion flow of smooth curves with anisotropic surface energy.

This geometric flow is the H−1-gradient flow of an energy functional. It preserves the area

enclosed by the evolving curve while at the same time decreases its energy. We show the

existence of a unique local in time solution for the flow but also the existence of a global in

time solution if the initial curve is close to the Wulff shape. In addition, we prove that the

global solution converges to the Wulff shape as t → ∞. In the current setting, the anisotropy

is not too strong so that the Wulff shape is given by a smooth curve. In the last section, we

formulate the corresponding problem when the Wulff shape exhibits corners.