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

Journal Article


Abstract


  • We consider the parabolic polyharmonic diffusion and the 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 L 2, then the evolving curve converges exponentially in the C ∞ topology to a straight horizontal line segment. The same behaviour is shown for the L 2-gradient flow provided the energy of the initial curve is sufficiently small. In each case the smallness conditions depend only on m.

Publication Date


  • 2022

Citation


  • McCoy, J., Wheeler, G., & Wu, Y. (2022). High order curvature flows of plane curves with generalised Neumann boundary conditions. Advances in Calculus of Variations, 15(3), 497-513. doi:10.1515/acv-2020-0002

Scopus Eid


  • 2-s2.0-85100549495

Start Page


  • 497

End Page


  • 513

Volume


  • 15

Issue


  • 3

Abstract


  • We consider the parabolic polyharmonic diffusion and the 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 L 2, then the evolving curve converges exponentially in the C ∞ topology to a straight horizontal line segment. The same behaviour is shown for the L 2-gradient flow provided the energy of the initial curve is sufficiently small. In each case the smallness conditions depend only on m.

Publication Date


  • 2022

Citation


  • McCoy, J., Wheeler, G., & Wu, Y. (2022). High order curvature flows of plane curves with generalised Neumann boundary conditions. Advances in Calculus of Variations, 15(3), 497-513. doi:10.1515/acv-2020-0002

Scopus Eid


  • 2-s2.0-85100549495

Start Page


  • 497

End Page


  • 513

Volume


  • 15

Issue


  • 3