On the Isoperimetric Inequality and Surface Diffusion Flow for Multiply Winding Curves

Miura, Tatsuya; Okabe, Shinya

doi:10.1007/s00205-020-01591-7

Cited by 4 publications

(3 citation statements)

References 41 publications

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“…A natural question is on the stability of this homothetic solution. This correlates well with other work on fourth-order flows, for instance stability of circles under curve diffusion [7,14,19], the elastic flow [9], and Chen's flow [4,6].…”

Section: Introductionsupporting

confidence: 87%

The gradient flow for entropy on closed planar curves

O'Donnell¹,

Wheeler²,

Wheeler³

2021

Preprint

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In this paper we consider the steepest descent L 2 -gradient flow of the entropy functional. The flow expands convex curves, with the radius of an initial circle growing like the square root of time. Our main result is that, for any initial curve (either immersed locally convex of class C 2 or embedded of class W 2,2 bounding a convex domain), the flow converges smoothly to a round expanding multiply-covered circle.

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Section: Introductionsupporting

confidence: 87%

The gradient flow for entropy on closed planar curves

O'Donnell¹,

Wheeler²,

Wheeler³

2021

Preprint

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“…It would be worth mentioning that the core of our relaxation technique is in the same spirit of some recent studies in different contexts, although the details are quite independent; an isoperimetric inequality for multiply-winding curves by [33], and on a Li-Yau type inequality in terms of the bending energy [32] as well as the p-bending energy [35]. The main characteristic here is that local minimality is reduced to global minimality.…”

Section: Introductionmentioning

confidence: 99%

Polar tangential angles and free elasticae

Miura

2021

Mathematics in Engineering

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A new stabilization phenomenon induced by degenerate diffusion is discovered in the context of pinned planar p-elasticae. It was known that in the non-degenerate regime p ∈ (1, 2], including the classical case of Euler's elastica, there are no local minimizers other than unique global minimizers.Here we prove that, in stark contrast, in the degenerate regime p ∈ (2, ∞) there emerge uncountably many local minimizers with diverging energy.

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“…Up to now global existence is ensured only for perturbations of circles, see e.g. [13,14,35] (and also [30] for a multiply-covered case). In particular, Wheeler's result [35] gives an explicit (but non-optimal) quantitative sufficient condition for all-time embeddedness.…”

Section: Introductionmentioning

confidence: 99%

Optimal thresholds for preserving embeddedness of elastic flows

Miura¹,

Müller²,

Rupp³

2021

Preprint

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We consider elastic flows of closed curves in Euclidean space. We obtain optimal energy thresholds below which elastic flows preserve embeddedness of initial curves for all time. The obtained thresholds take different values between codimension one and higher. The main novelty lies in the case of codimension one, where we obtain the variational characterization that the thresholding shape is a minimizer of the bending energy (normalized by length) among all nonembedded planar closed curves of unit rotation number. It turns out that a minimizer is uniquely given by a nonclassical shape, which we call "elastic two-teardrop".

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On the Isoperimetric Inequality and Surface Diffusion Flow for Multiply Winding Curves

Cited by 4 publications

References 41 publications

The gradient flow for entropy on closed planar curves

The gradient flow for entropy on closed planar curves

Polar tangential angles and free elasticae

Optimal thresholds for preserving embeddedness of elastic flows

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