Bifurcations of spherically asymmetric solutions to an evolution equation for curves

Abstract

We show that a certain non-local curvature flow for planar curves has non-trivial self-similar solutions with $n$-fold rotational symmetry, bifurcated from a trivial circular solution. Moreover, we show that the trivial solution is stable with respect to perturbations which keep the geometric center and the enclosed area, and that, for $n$ different from 3, the $n$-fold symmetric solution is stable with respect to perturbations which satisfy the same conditions as above and have the same symmetry as the solutions.