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Journal article
Representation formulae for higher order curvature flows
Journal of Differential Equations, Vol.344, pp.1-43
2023
Abstract
In [25], Smoczyk showed that expansion of convex curves and hypersurfaces by the reciprocal of the harmonic mean curvature gives rise to a linear second order equation for the evolution of the support function, with corresponding representation formulae for solutions. In this article we consider L2(dθ)-gradient flows for a class of higher-order curvature functionals. These give rise to higher order linear parabolic equations for which we derive similar representation formulae for their solutions. Solutions exist for all time under natural conditions on the initial curve and converge exponentially fast in the smooth topology to multiply-covered circles. We consider both closed, embedded convex curves and closed, convex curves of higher rotation number. We give some corresponding remarks where relevant on open convex curves. We also consider corresponding ‘globally constrained’ flows which preserve the length or enclosed area of the evolving curve and a higher order approach to the Yau problem of evolving one convex planar curve to another. In an Appendix, we give some related second order results, including a version of the Yau problem for star-shaped curves.
Details
- Title
- Representation formulae for higher order curvature flows
- Authors/Creators
- James A. McCoy - University of Newcastle AustraliaPhil Schrader - College of Science, Health, Engineering and Education, Mathematics and Statistics, Murdoch University, Murdoch, WA 6150, AustraliaGlen Wheeler - University of Wollongong
- Publication Details
- Journal of Differential Equations, Vol.344, pp.1-43
- Publisher
- Elsevier Inc
- Grant note
- DP180100431 / Australian Research Council (https://doi.org/10.13039/501100000923) University of Newcastle College of Engineering, Science and Environment
- Identifiers
- 991005599161707891
- Copyright
- © 2022 The Authors.
- Murdoch Affiliation
- Murdoch University
- Language
- English
- Resource Type
- Journal article
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- 9 Mathematics
- 9.50 Applied Statistics & Probability
- 9.50.415 Harmonic Maps
- Web Of Science research areas
- Mathematics
- ESI research areas
- Mathematics