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 areaenclosed by the evolving curve while at the same time decreases its energy. We show theexistence of a unique local in time solution for the flow but also the existence of a global intime solution if the initial curve is close to the Wulff shape. In addition, we prove that theglobal solution converges to the Wulff shape as t → ∞. In the current setting, the anisotropyis not too strong so that the Wulff shape is given by a smooth curve. In the last section, weformulate the corresponding problem when the Wulff shape exhibits corners.