High order curvature flows of plane curves with generalised Neumann boundary conditions

Publication Name

Advances in Calculus of Variations


We consider the parabolic polyharmonic diffusion and the L2{L^{2}}-gradient flow for the square integral of the m-th arclength derivative of curvature for regular closed curves evolving with generalised Neumann boundary conditions. In the polyharmonic case, we prove that if the curvature of the initial curve is small in L2, then the evolving curve converges exponentially in the C∞ topology to a straight horizontal line segment. The same behaviour is shown for the L2-gradient flow provided the energy of the initial curve is sufficiently small. In each case the smallness conditions depend only on m.

Open Access Status

This publication is not available as open access



Link to publisher version (DOI)