Short time existence for the elastic flow of clamped curves

Spener, Adrian

doi:10.1002/mana.201600304

Cited by 22 publications

(49 citation statements)

References 17 publications

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“…The author would like to thank Prof. Dr. Oliver Schnürer for drawing his attention to the mistake in the statement and the proof of [, Theorem 1.1].…”

Section: Acknowledgementsmentioning

confidence: 99%

“…Proof We start with the second part of the statement and

k = 1

since the corresponding ODE is easier. As in [, (2.6)] we introduce the function

ξ false(t, x false) = \frac{⟨ \dot{u} false(t, x false), \partial_{x} u false(t, x false) ⟩}{false| \partial_{x} {u false(t, x false) |}^{2}},

where

u = f + ϕ N \in C^{1} (false[0, T false); C^{1} (\bar{I}; R^{d + 1})) \cap C (false[0, T false); C^{2} (\bar{I}; R^{d + 1}))

and u is analytic in the interior of its domain. Under the assumptions of the original paper, the function ξ was not necessarily Lipschitz continuous in x , thus the statements on uniqueness and regularity of the solution to the ODE below were wrong.…”

mentioning

confidence: 99%

“…The general case with

k \geq 2

follows from [, Theorem 9.5 and Remark 9.6]. From here onwards we can follow the proof from . For the other direction, we let

\overset{u}{true}

be given as in [, Beginning of the Proof of Theorem 1.1 on p. 2054] with the regularity under the new assumptions and consider the function

0true F (t, x, y) = - \frac{false⟨ \overset{\tilde{u}}{true} (t, y), \partial_{x} f (x) false⟩}{false⟨ \partial_{x} \overset{u}{true} (t, y), \partial_{x} f (x) false⟩} .

As stated below [, (2.1)], the function

F (t, x, y)

is well defined for t and

| x - y |

small enough.…”

mentioning

confidence: 99%

“…From here onwards we can follow the proof from . For the other direction, we let

\overset{u}{true}

be given as in [, Beginning of the Proof of Theorem 1.1 on p. 2054] with the regularity under the new assumptions and consider the function

0true F (t, x, y) = - \frac{false⟨ \overset{\tilde{u}}{true} (t, y), \partial_{x} f (x) false⟩}{false⟨ \partial_{x} \overset{u}{true} (t, y), \partial_{x} f (x) false⟩} .

As stated below [, (2.1)], the function

F (t, x, y)

is well defined for t and

| x - y |

small enough. Moreover, from the regularity assumption on

\overset{u}{true}

we find that

F \in C^{k - 1} (false[0, T false); C^{k} (\bar{I}))

, whence by [, Theorem 9.5 and Remark 9.6] there exists a unique solution

Φ \in C^{k} (false[0, T false) \times \bar{I}; R)

to the problem

({, \begin{matrix} \dot{normalΦ} false(t, x false) = F false(t, x, normalΦ (x, t) false), & t > 0, x \in \bar{I}, \\ Φ \end{matrix})

…”

mentioning

confidence: 99%

“…The corrected statement of this theorem reads as follows.Theorem (Corrected version of [3, Theorem 1.1]). Let 0 ∶ → ℝ +1 be an immersion satisfying [3, Assumption 1.3] and let ∈ ℝ. If the elastic flow [3, (1.2)] has a unique (up to reparametrisation) solution in…”

mentioning

confidence: 99%

See 4 more Smart Citations

Short time existence for the elastic flow of clamped curves

2018

Mathematische Nachrichten

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“…The author would like to thank Prof. Dr. Oliver Schnürer for drawing his attention to the mistake in the statement and the proof of [, Theorem 1.1].…”

Section: Acknowledgementsmentioning

confidence: 99%

“…Proof We start with the second part of the statement and

k = 1

since the corresponding ODE is easier. As in [, (2.6)] we introduce the function

ξ false(t, x false) = \frac{⟨ \dot{u} false(t, x false), \partial_{x} u false(t, x false) ⟩}{false| \partial_{x} {u false(t, x false) |}^{2}},

where

u = f + ϕ N \in C^{1} (false[0, T false); C^{1} (\bar{I}; R^{d + 1})) \cap C (false[0, T false); C^{2} (\bar{I}; R^{d + 1}))

mentioning

confidence: 99%

“…The general case with

k \geq 2

follows from [, Theorem 9.5 and Remark 9.6]. From here onwards we can follow the proof from . For the other direction, we let

\overset{u}{true}

be given as in [, Beginning of the Proof of Theorem 1.1 on p. 2054] with the regularity under the new assumptions and consider the function

0true F (t, x, y) = - \frac{false⟨ \overset{\tilde{u}}{true} (t, y), \partial_{x} f (x) false⟩}{false⟨ \partial_{x} \overset{u}{true} (t, y), \partial_{x} f (x) false⟩} .

As stated below [, (2.1)], the function

F (t, x, y)

is well defined for t and

| x - y |

small enough.…”

mentioning

confidence: 99%

“…From here onwards we can follow the proof from . For the other direction, we let

\overset{u}{true}

be given as in [, Beginning of the Proof of Theorem 1.1 on p. 2054] with the regularity under the new assumptions and consider the function

0true F (t, x, y) = - \frac{false⟨ \overset{\tilde{u}}{true} (t, y), \partial_{x} f (x) false⟩}{false⟨ \partial_{x} \overset{u}{true} (t, y), \partial_{x} f (x) false⟩} .

As stated below [, (2.1)], the function

F (t, x, y)

is well defined for t and

| x - y |

small enough. Moreover, from the regularity assumption on

\overset{u}{true}

we find that

F \in C^{k - 1} (false[0, T false); C^{k} (\bar{I}))

, whence by [, Theorem 9.5 and Remark 9.6] there exists a unique solution

Φ \in C^{k} (false[0, T false) \times \bar{I}; R)

to the problem

({, \begin{matrix} \dot{normalΦ} false(t, x false) = F false(t, x, normalΦ (x, t) false), & t > 0, x \in \bar{I}, \\ Φ \end{matrix})

…”

mentioning

confidence: 99%

mentioning

confidence: 99%

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Short time existence for the elastic flow of clamped curves

2018

Mathematische Nachrichten

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Existence and convergence of the length-preserving elastic flow of clamped curves

Rupp,

Spener

2024

J. Evol. Equ.

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We study the evolution of curves with fixed length and clamped boundary conditions moving by the negative $$L^2$$ L 2 -gradient flow of the elastic energy. For any initial curve lying merely in the energy space we show existence and parabolic smoothing of the solution. Applying previous results on long-time existence and proving a constrained Łojasiewicz–Simon gradient inequality we furthermore show convergence to a critical point as time tends to infinity.

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Elastic flow of curves with partial free boundary

Diana

2024

Nonlinear Differ. Equ. Appl.

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We consider a curve with boundary points free to move on a line in $${{{\mathbb {R}}}}^2$$ R 2 , which evolves by the $$L^2$$ L 2 -gradient flow of the elastic energy, that is, a linear combination of the Willmore and the length functional. For this planar evolution problem, we study the short and long-time existence. Once we establish under which boundary conditions the PDE’s system is well-posed (in our case the Navier boundary conditions), employing the Solonnikov theory for linear parabolic systems in Hölder space, we show that there exists a unique flow in a maximal time interval [0, T). Then, using energy methods we prove that the maximal time is $$T= + \infty $$ T = + ∞ .

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Short time existence for the elastic flow of clamped curves

Cited by 22 publications

References 17 publications

Short time existence for the elastic flow of clamped curves

Short time existence for the elastic flow of clamped curves

Existence and convergence of the length-preserving elastic flow of clamped curves

Elastic flow of curves with partial free boundary

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