A curvature flow in the plane with a nonlocal term

TL;DR

This paper investigates a geometric flow of planar curves influenced by curvature and a nonlocal term, proving long-term existence and asymptotic behavior under convexity conditions.

Contribution

It introduces a novel curvature flow with a nonlocal term and establishes its long-term existence and asymptotic properties under convexity assumptions.

Findings

Flow exists for all time under convexity

Curve becomes asymptotically circular

Long-term behavior characterized by convergence to a circle

Abstract

We study the geometric flow of a planar curve driven by its curvature and the normal derivative of its capacity potential. Under a convexity condition that is natural to our problem, we establish long term existence and large time asymptotics of this flow.