Boundedness and Regularizing Effects of O'HARA'S Knot Energies

Blatt, Simon

doi:10.1142/s0218216511009704

Cited by 45 publications

(69 citation statements)

References 9 publications

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“…Considering other curvature energies, such as general O'Hara knot energies [30], generalized versions of the tangent-point energy [6] and Menger curvature [23], one observes that these energies are also related to s p (1.2) for some s and some p with possibly p = 2, [3][4][5]25]. For several of these curvature energies, regularity even of minimizers is not understood, and arguments as in [19] seem not to work, since also the invariance class is not known.…”

Section: Integro-differential Harmonic Maps 507mentioning

confidence: 99%

Integro-Differential Harmonic Maps into Spheres

Schikorra

2014

Communications in Partial Differential Equations

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For s ∈ (0, 1) we introduce (integro-differential) harmonic maps v : Ω ⊂ R n → R N , which are defined as critical points of the Besov-Slobodeckij energywith the side-condition that v(Ω) ⊂ S N −1 , for the (N − 1)-spherethis are the classical fractional harmonic maps first considered by Da Lio and Rivière. For p s = 2 this is a new energy which has degenerate, non-local Euler-Lagrange equations. They are different from the n/s-harmonic maps introduced by Francesca Da Lio and the author, and have to be treated with new arguments, which might be of independent interest for further applications on geometric energies. For the critical case p s = n s we show Hölder continuity of these maps.

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Section: Integro-differential Harmonic Maps 507mentioning

confidence: 99%

Integro-Differential Harmonic Maps into Spheres

Schikorra

2014

Communications in Partial Differential Equations

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“…One of the most important ingredients in the proof of the long time existence result is the following quantitative version of the regularizing effects of Theorem 1.1 in [5]:…”

Section: Coercivity Of the Energymentioning

confidence: 99%

“…3.4 is a strengthening of the classification of curves of finite energy E α in [5] using fractional Sobolev spaces. For s…”

Section: Introductionmentioning

confidence: 99%

The gradient flow of O’Hara’s knot energies

Blatt

2017

Math. Ann.

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Jun O'Hara invented a family of knot energies E j, p , j, p ∈ (0, ∞), O'Hara in Topology Hawaii (Honolulu, HI, 1990). World Science Publication, River Edge 1992. We study the negative gradient flow of the sum of one of the energies E α = E α,1 , α ∈ (2, 3), and a positive multiple of the length. Showing that the gradients of these knot energies can be written as the normal part of a quasilinear operator, we derive short time existence results for these flows. We then prove long time existence and convergence to critical points.

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“…After a first attempt [Blatt & Reiter, 2008], the identification of the energy spaces [Blatt, 2012a] led to a significant improvement of the latter result [Blatt & Reiter, 2013a;.…”

Section: O'hara's Energiesmentioning

confidence: 99%

“…Full details of this argument can be found in [Blatt, 2012a]. In other words, an embedded arc-length parametrized curve γ belongs to W 1+α/2−1/(2p),2p if and only if its energy E α,p (γ) is finite.…”

Section: How Fractional Sobolev Spaces Come Into Playmentioning

confidence: 99%

How nice are critical knots? Regularity theory for knot energies

Blatt

Reiter

2014

J. Phys.: Conf. Ser.

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Abstract. In this note we report on some recent developments in geometric knot theory which aims at finding links between geometric properties of a given knotted curve and its knot type. The central object of this field are so-called knot energies which are defined on closed embedded curves.First we present three important examples of two-parameter knot energy families, namely O'Hara's energies, the (generalized) integral Menger curvature, and the (generalized) tangentpoint energies.Subsequently we outline the main steps that lead to C ∞ -regularity of stationary pointsespecially minimizers-in the non-degenerate sub-critical range of parameters.Particular attention is devoted to the appearing parallels between these energies which, surprisingly at first glance, are quite similar from an analyst's perspective.

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Boundedness and Regularizing Effects of O'HARA'S Knot Energies

Abstract: In this paper, we will give a necessary and sufficient condition under which O'Hara's Ej,p-energies are bounded. We show that a regular curve has bounded Ej,p-energy if and only if it is injective and belongs to a certain Sobolev–Slobodeckij space.

Cited by 45 publications

References 9 publications

Integro-Differential Harmonic Maps into Spheres

Integro-Differential Harmonic Maps into Spheres

The gradient flow of O’Hara’s knot energies

How nice are critical knots? Regularity theory for knot energies

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