Stability analysis for the anisotropic curve shortening flow of planar networks

In this article we study the anisotropic curve shortening flow for a planar network of three curves with fixed endpoints and which meet in a triple junction. We show that the anisotropic curvature energy fulfills a Lojasiewicz-Simon gradient inequality and use this knowledge to derive stability results for the flow. Precisely, in our main theorem we show that for any initial data, which are $C^{2,\alpha }$-close to a (local) energy minimizer, the flow exists globally and converges to a possibly different energy minimum.