Curvature diffusion of planar curves with generalised Neumann boundary conditions inside cones

2023/12/29 by Mashniah Gazwani, Gazwani, Mashniah, James McCoy +1 · 1 citation
Mathematics · #35G61 #53E40 #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2312.17490

openalex publication_date 2023/12/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study families of smooth immersed regular planar curves α: [-1,1 ]× [0,T )→ ℝ² satisfying the fourth order nonlinear curve diffusion flow with generalised Neumann boundary conditions inside cones. We show that if the initial curve has sufficiently small oscillation of curvature then this remains so under the flow. Such families of evolving curves either exist for a finite time, when an end of the curve has reached the tip of the cone or the curvature has become unbounded in L², or they exist for all time and converge exponentially in the C^∞- topology to a circular arc that, together with the cone boundary encloses the same area as that of the initial curve and cone boundary. The same kind of result is possible for the higher order polyharmonic curve diffusion flows with appropriate boundary conditions; in particular in the sixth order case the smallness condition on the oscillation of curvature is exactly the same as for curve diffusion.

Cited by

• Existence of self-similar solutions for the surface diffusion flow with nonlinear boundary conditions in the half space

Related

• Higher order curvature flows of plane curves with generalised Neumann boundary conditions
• On the curve diffusion flow of closed plane curves
• Length-constrained, length-penalised and free elastic flows of planar curves inside cones
• Curve diffusion and straightening flows on parallel lines
• Short Time Existence for the Curve Diffusion Flow with a Contact Angle