期刊
ADVANCES IN CALCULUS OF VARIATIONS
卷 15, 期 3, 页码 497-513出版社
WALTER DE GRUYTER GMBH
DOI: 10.1515/acv-2020-0002
关键词
Curvature flow; high order parabolic equation; Neumann boundary condition
资金
- Australian Research Council [DP180100431]
- University of Wollongong Faculty of Engineering and Information Sciences Postgraduate research scholarship
The study focuses on the evolution of regular closed curves, showing that when the curvature or energy of the initial curve is small enough, the evolving curve converges to a straight horizontal line segment. The smallness conditions depend only on m.
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-infinity 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.
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