We establish long-time existence for a projected Sobolev gradient flow of generalized integral Menger curvature in the Hilbert case and provide -bounds in time for the solution that only depend on the initial curve. The self-avoidance property of integral Menger curvature guarantees that the knot class of the initial curve is preserved under the flow, and the projection ensures that each curve along the flow is parametrized with the same speed as the initial configuration. Finally, we describe how to simulate this flow numerically with substantially higher efficiency than in the corresponding numerical gradient descent or other optimization methods.
Funding source: Deutsche Forschungsgemeinschaft
Award Identifier / Grant number: 320021702/GRK2326
Award Identifier / Grant number: 282535003
Funding statement: D. Steenebrügge is partially supported by the Graduiertenkolleg Energy, Entropy, and Dissipative Dynamics (EDDy) of the Deutsche Forschungsgemeinschaft (DFG) – project no. 320021702/GRK2326. H. Schumacher and H. von der Mosel gratefully acknowledge support from DFG project 282535003: Geometric curvature functionals: energy landscape and discrete methods. This work is also partially funded by the Excellence Initiative of the German federal and state governments.
A Periodic Sobolev–Slobodeckiǐ spaces
For fixed , and , define the seminorm
for ℓ-periodic functions that are locally 𝜚-integrable on ℝ, i.e., for .
For , the function space
where , is called the periodic Sobolev–Slobodeckiǐ space with fractional differentiability of order and integrability 𝜚. Here, denotes the usual Sobolev space of ℓ-periodic functions whose weak derivatives of order are locally 𝜚-integrable, where we have set .
Since the restriction of any ℓ-periodic Sobolev function to its fundamental domain is contained in the Sobolev space , it suffices to compare the seminorm (A.1) to the standard Gagliardo seminorm (which we denote by ) used in the literature on (non-periodic) Sobolev–Slobodeckiǐ spaces in order to transfer known results to the periodic setting. Indeed, for , the absolute value in the integrand of (A.1) equals the ℓ-periodic distance so that we rewrite the seminorm (A.1) by substituting and estimate, for ,
where we used the ℓ-periodicity of the integrand to shift the domain of the inner integration by .
As a first application of inequality (A.2), we show that the well-known Morrey embedding into classical Hölder spaces transfers to the periodic setting. To be more precise, the periodic Hölder seminorm for an ℓ-periodic function and is defined as (cf. [18, Definition 3.2.5])
and the periodic Hölder space for consists of those functions whose Hölder norm satisfies . Notice that the second equality in (A.3) follows from the ℓ-periodicity of 𝑓 since, for any distinct with, say, , we find such that , or , namely, if , or if .
Theorem A.2 (Morrey embedding)
Let and . If and , then there is a positive constant such that
It suffices to treat the case ; the full statement follows by induction. Moreover,
for any ℓ-periodic function so that we can focus on the periodic Hölder seminorm. Rewrite the numerator in (A.3) as , where we used the shift for some fixed . Hence the right-hand side of (A.3) for may be rephrased by setting and estimated from above as
By virtue of the non-periodic Morrey embedding [15, Theorem 8.2] and (A.2), the right-hand side may be bounded from above by the expression for some constant . Changing variables via translations by and using the ℓ-periodicity of the respective integrands in the -norm, one can easily check that for each , which concludes the proof. ∎
We frequently make use of the following Poincaré inequality.
Proposition A.3 (Poincaré inequality)
For every , and , there is a constant such that
By Jensen’s inequality, we have
As a first application, we deal with the composition of Sobolev–Slobodeckiǐ functions.
Let , , , with on , and . Then 𝑔 can be extended to a function such that and , satisfying
for some constant , where .
Notice that both sides in (A.6) may be infinite as we do not assume that . We should mention also that we have suppressed the respective domains , or , or , in our notation for the seminorms of 𝐺 and , or of 𝑓 in these inequalities.
In order to prove (A.6), we extend 𝑔 onto all of ℝ, first by setting for and consecutively for , and then by consecutively for , to find , for all . We then bound the numerator in the integrand of the seminorm for and from above as , which leads to
where we have set . Note that 𝑓 is of class due to Theorem A.2. From now on, we may assume without loss of generality that the seminorm is finite; otherwise, there is nothing to prove for (A.6). Substituting and using periodicity, we can rewrite the double integral as
With , we obtain for the inverse function , for all ,
Now we use the transformation , with
The -norm of in (A.7) may be bounded by according to the Morrey embedding, Theorem A.2, and the Poincaré inequality, Proposition A.3, and combining this with (A.10) leads to (A.6) with the constant . ∎
The following result provides an estimate for the Sobolev–Slobodeckiǐ norms of products of functions of fractional Sobolev regularity.
Proposition A.5 (Bounds on inner product)
Let , , and . Then there is a constant such that
For brevity, we write for , and then for any two functions . By virtue of the Morrey embedding, Theorem A.2, the functions are both of class . Then, according to the Leibniz rule, we have, for every ,
of which the -norm can be estimated as
which implies by means of the Morrey embedding (A.4) that the Sobolev norm may be bounded from above by
The numerator of the integrand in the seminorm can be estimated from above by means of (A.11) similarly to before by
which may be bounded by
Integrating this against and using the Morrey embedding inequality (A.4), we find
For , we observe that
Notice that target dimension 𝑛 enters this constant through the Morrey embedding constant ; cf. Theorem A.2. ∎
We conclude this section with an estimate on the fractional Sobolev norm of the multiplicative inverse of a non-vanishing Sobolev–Slobodeckiǐ function.
Let , , , and with for almost all . Then the multiplicative inverse , is in , and we have
follows immediately from the pointwise lower bound on ℎ. The integrand of the seminorm may be estimated as , which implies , and together with , we obtain the desired estimate for . ∎
B Arc length and bilipschitz constants
Whenever we speak about the arc length parametrization of a curve with a.e. on ℝ and with length , we refer to the mapping obtained from the inverse of the strictly increasing arc length function of class defined as
Notice that , for all , as well as for all , and therefore, the composition is 𝐿-periodic and 1-Lipschitz because of
Recall from the introduction the definition of the bilipschitz constant (1.15) and the notation for the subset of all regular -curves such that their restriction to the fundamental interval is injective.
Let and with length and with arc length parametrization . Then the bilipschitz constants of 𝛾 and Γ are related by
The upper bound on in (B.2) follows immediately from the estimate
at points where exists. Taking the infimum over all such establishes the upper bound in (B.2). For the proof of the lower bound in (B.2), assume without loss of generality that so that, by monotonicity, . Then can be written as and estimated from below by
If , then the distance on the right-hand side of (B.3) equals , which proves the claim in this case. If, on the other hand, , the latter can be written as
Lemma B.2 (Seminorm of arc length)
Let , and . Then
Note that the right-hand side of this estimate might be infinite since we did not assume that 𝛾 is of class .
If the seminorm on the right-hand side is infinite, there is nothing to prove. If, on the other hand, , then we estimate
The bilipschitz constant , defined in (1.15), of a closed embedded, i.e., immersed and injective, -curve is strictly positive since, under this regularity assumption, the quotient in (1.15) converges to as . This means that this positive quotient possesses a continuous extension onto all of , which is bounded by a positive constant from below by periodicity and continuity. We now quantify a neighbourhood of such a curve 𝛾, in which the bilipschitz constant remains under control.
Let , and let be injective on with on ℝ. Then
for all with .
It suffices to prove the first inequality in (B.4), and for that, assume without loss of generality that . We have
If , then we can estimate the right-hand side from below by
which is bounded from below by . If, on the other hand,
The following corollary quantifies the fact that the set of all regular embedded Sobolev–Slobodeckiǐ loops is an open subset of that fractional Sobolev space if that space embeds into .
Let , and , and suppose . Then , and one has, for the bilipschitz constant and the minimal velocity ,
Notice that, neither in this corollary nor in the preceding Lemma B.3, anything is required about the -distance between the curves, so in principle, 𝛾 and 𝜂 could be very far apart in .
By the Morrey embedding, Theorem A.2, the curve 𝛾 is of class so that . Moreover, again by Theorem A.2, in combination with the Poincaré inequality, Proposition A.3, we estimate, using our assumption on 𝜂,
Communicated by: Frank Duzaar
 S. Bartels and P. Reiter, Stability of a simple scheme for the approximation of elastic knots and self-avoiding inextensible curves, Math. Comp. 90 (2021), no. 330, 1499–1526. 10.1090/mcom/3633Search in Google Scholar
 S. Bartels, P. Reiter and J. Riege, A simple scheme for the approximation of self-avoiding inextensible curves, IMA J. Numer. Anal. 38 (2018), no. 2, 543–565. 10.1093/imanum/drx021Search in Google Scholar
 S. Blatt, Note on continuously differentiable isotopies, Report no. 34, Institute for Mathematics, RWTH Aachen, 2009, http://www.instmath.rwth-aachen.de/Preprints/blatt20090825.pdf. Search in Google Scholar
 H. Cartan, Differential Calculus, Hermann, Paris, 1971. Search in Google Scholar
 E. Denne and J. M. Sullivan, Convergence and isotopy type for graphs of finite total curvature, Discrete Differential Geometry, Oberwolfach Semin. 38, Birkhäuser, Basel (2008), 163–174. 10.1007/978-3-7643-8621-4_8Search in Google Scholar
 P. Drábek and J. Milota, Methods of Nonlinear Analysis, 2nd ed., Birkhäuser Adv. Texts Basler Lehrbücher, Birkhäuser/Springer, Basel, 2013. Search in Google Scholar
 A. Gilsbach, On symmetric critical points of knot energies, PhD thesis, RWTH Aachen University, 2018, https://publications.rwth-aachen.de/record/726186/. Search in Google Scholar
 T. Hermes, Analysis of the first variation and a numerical gradient flow for integral Menger curvature, PhD thesis, RWTH Aachen University, 2012, https://publications.rwth-aachen.de/record/82904/. Search in Google Scholar
 K. Hoffman and R. Kunze, Linear Algebra, 2nd ed., Prentice-Hall, Englewood Cliffs, 1971. Search in Google Scholar
 J. Knappmann, On the second variation of integral Menger curvature, PhD thesis, RWTH Aachen University, 2020, https://publications.rwth-aachen.de/record/802770/. Search in Google Scholar
 J. Nocedal and S. J. Wright, Numerical Optimization, 2nd ed., Springer Ser. Oper. Res. Financ. Eng., Springer, New York, 2006. Search in Google Scholar
 S. Redon, A. Kheddar and S. Coquillart, Fast continuous collision detection between rigid bodies, Comput. Graph. Forum 21 (2002), no. 3, 279–287. 10.1111/1467-8659.t01-1-00587Search in Google Scholar
 P. Reiter, All curves in a -neighbourhood of a given embedded curve are isotopic, Report no. 4, Institute for Mathematics, RWTH Aachen, 2005, https://www.instmath.rwth-aachen.de/Preprints/reiter20051017.pdf. Search in Google Scholar
 W. Rudin, Functional Analysis, McGraw-Hill Ser. High. Math., McGraw-Hill, New York, 1973. Search in Google Scholar
 D. Steenebrügge, Parametrization-controlling gradient flows and regularity of critical points for knot energies, PhD thesis, RWTH Aachen University, 2022. Search in Google Scholar
 P. Strzelecki, M. Szumańska and H. von der Mosel, Regularizing and self-avoidance effects of integral Menger curvature, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 9 (2010), no. 1, 145–187. 10.2422/2036-2145.2010.1.06Search in Google Scholar
 J. H. von Brecht and R. Blair, Dynamics of embedded curves by doubly-nonlocal reaction-diffusion systems, J. Phys. A 50 (2017), no. 47, Article ID 475203. 10.1088/1751-8121/aa9109Search in Google Scholar
 E. Zeidler, Nonlinear Functional Analysis and its Applications. I: Fixed-Point Theorems, Springer, New York, 1993. Search in Google Scholar
© 2022 Walter de Gruyter GmbH, Berlin/Boston