The hanging chain problem with respect to a circle

Abstract

Let $\s^1$ be a circle in Euclidean plane. We consider the problem of finding the shape of a planar curve which is an extremal of the potential energy that measures the distance to $\s^1$. We describe the shape of these curves distinguishing if the curves lie in the inside or outside of $\s^1$. We extend the problem for energies that are powers to the distance to $\s^1$.