Minimal elastic networks

Dall’Acqua, Anna; Novaga, Matteo; Pluda, Alessandra

doi:10.1512/iumj.2020.69.8036

Cited by 12 publications

(13 citation statements)

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“…In fact, since bothp andp tend to p 0 as l ↓ 0, W(γ n ) is almost equal to W(γ n ) for each n ∈ N. This observation implies that l/L determines which of the curvesγ + 1 andγ + 0 has the second smallest elastic energy. (2) In the case l = 0, the minimization problem is considered by [12] and the minimizes are uniquely determined as the half-fold figure-eight up to the reflection (see [30,31]). We expect thatγ + 0 andγ + 0 tend to the half-fold figure-eight letting l → 0 formally.…”

Section: Characterization Of Critical Pointsmentioning

confidence: 99%

The critical points of the elastic energy among curves pinned at endpoints

Yoshizawa¹

2022

DCDS

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<p style='text-indent:20px;'>In this paper we find curves minimizing the elastic energy among curves whose length is fixed and whose ends are pinned. Applying the shooting method, we can identify all critical points explicitly and determine which curve is the global minimizer. As a result we show that the critical points consist of wavelike elasticae and the minimizers do not have any loops or interior inflection points.</p>

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Section: Characterization Of Critical Pointsmentioning

confidence: 99%

The critical points of the elastic energy among curves pinned at endpoints

Yoshizawa¹

2022

DCDS

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“…The short time existence result can be easily adapted to open curves (see Remark 3.15), but present some extra difficulties in the case of networks, that we explain in Remark 3. 16.…”

Section: Asymptotic Analysismentioning

confidence: 99%

“…Suppose now that P = Q. Then by a generalization of the Gauss-Bonnet Theorem (see [16,Corollary A.2]) to not necessarily embedded curves with coinciding endpoints it holds γ t |k| ds ≥ π and so repeating the chain of inequalities (4.6) one gets (4.5).…”

Section: Lemma 43 Let N T =mentioning

confidence: 99%

A Survey of the Elastic Flow of Curves and Networks

2021

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We collect and present in a unified way several results in recent years about the elastic flow of curves and networks, trying to draw the state of the art of the subject. In particular, we give a complete proof of global existence and smooth convergence to critical points of the solution of the elastic flow of closed curves in $${\mathbb {R}}^2$$ R 2 . In the last section of the paper we also discuss a list of open problems.

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“…We mention that inequality (13) is already present in [DaNoPl18], proved with a different method in the setting of networks, but in the following we will need the specific approach used in the proof of Lemma 2.1. Also, we are going to prove that inequality (12) is true (up to changing the constant) in an analogous sense in the setting of varifolds as stated in the inequality (15) in Subsection 2.3.…”

Section: Preliminary Estimatesmentioning

confidence: 99%

A varifold perspective on the $p$-elastic energy of planar sets

Pozzetta

2019

Preprint

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Under suitable regularity assumptions, the p-elastic energy of a planar set E ⊂ R 2 is defined to bewhere k ∂E is the curvature of the boundary ∂E. In this work we use a varifold approach to investigate this energy, that can be well defined on varifolds with curvature. First we show new tools for the study of 1-dimensional curvature varifolds, such as existence and uniform bounds on the density of varifolds with finite elastic energy. Then we characterize a new notion of L 1 -relaxation of this energy by extending the definition of regular sets by an intrinsic varifold perspective, also comparing this relaxation with the classical one of [BeMu04], [BeMu07]. Finally we discuss an application to the inpainting problem, examples and qualitative properties of sets with finite relaxed energy.

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Minimal elastic networks

Cited by 12 publications

References 0 publications

The critical points of the elastic energy among curves pinned at endpoints

The critical points of the elastic energy among curves pinned at endpoints

A Survey of the Elastic Flow of Curves and Networks

A varifold perspective on the $p$-elastic energy of planar sets

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