Stability analysis for the anisotropic curve shortening flow of planar networks

TL;DR

This paper investigates the stability of the anisotropic curve shortening flow in planar networks, demonstrating that flows starting near energy minima exist globally and converge, using a Lojasiewicz-Simon gradient inequality.

Contribution

The paper establishes a stability result for the anisotropic curve shortening flow of planar networks, applying a Lojasiewicz-Simon inequality to prove convergence to energy minima.

Findings

Flow exists globally for initial data close to energy minima.

Flow converges to a (possibly different) energy minimum.

Lojasiewicz-Simon inequality is key to the stability analysis.

Abstract

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.