Abstract
We establish longtime existence for a projected Sobolev gradient flow of generalized integral Menger curvature in the Hilbert case and provide
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
for ℓperiodic functions
For
where
Since the restriction of any ℓperiodic Sobolev function
where we used the ℓperiodicity of the integrand to shift the domain of the inner integration by
As a first application of inequality (A.2), we show that the wellknown Morrey embedding into classical Hölder spaces transfers to the periodic setting.
To be more precise, the periodic Hölder seminorm for an ℓperiodic function
and the periodic Hölder space
Theorem A.2 (Morrey embedding)
Let
Proof
It suffices to treat the case
for any ℓperiodic function
By virtue of the nonperiodic Morrey embedding [15, Theorem 8.2] and (A.2), the righthand side may be bounded from above by the expression
We frequently make use of the following Poincaré inequality.
Proposition A.3 (Poincaré inequality)
For every
Proof
By Jensen’s inequality, we have
As a first application, we deal with the composition of Sobolev–Slobodeckiǐ functions.
Let
for some constant
Notice that both sides in (A.6) may be infinite as we do not assume that
Proof
In order to prove (A.6), we extend 𝑔 onto all of ℝ, first by setting
where we have set
With
Now we use the transformation
due to periodicity of
The
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
Proof
For brevity, we write
of which the
which implies by means of the Morrey embedding (A.4) that the Sobolev norm
The numerator of the integrand in the seminorm
which may be bounded by
Integrating this against
For
since
Notice that target dimension 𝑛 enters this constant through the Morrey embedding constant
We conclude this section with an estimate on the fractional Sobolev norm of the multiplicative inverse of a nonvanishing Sobolev–Slobodeckiǐ function.
Let
Proof
The estimate
follows immediately from the pointwise lower bound on ℎ.
The integrand of the seminorm
B Arc length and bilipschitz constants
Whenever we speak about the arc length parametrization of a curve
Notice that
Recall from the introduction the definition of the bilipschitz constant (1.15) and the notation
Let
where
Proof
The upper bound on
at points
If
which can be bounded from below by
Lemma B.2 (Seminorm of arc length)
Let
Note that the righthand side of this estimate might be infinite since we did not assume that 𝛾 is of class
Proof
If the seminorm on the righthand side is infinite, there is nothing to prove.
If, on the other hand,
The bilipschitz constant
Let
for all
Proof
It suffices to prove the first inequality in (B.4), and for that, assume without loss of generality that
If
which is bounded from below by
then we use the ℓperiodicity to replace the term
The following corollary quantifies the fact that the set
Let
for all
Notice that, neither in this corollary nor in the preceding Lemma B.3, anything is required about the
Proof
By the Morrey embedding, Theorem A.2, the curve 𝛾 is of class
so that we can apply Lemma B.3 to finish the proof. Similarly, again by Morrey’s embedding (A.4) and Poincaré’s inequality (A.5), one finds
Acknowledgements
Parts of Section 2 are contained in J. Knappmann’s Ph.D. thesis [23], and some of the content of Sections 3–6 will appear in D. Steenebrügge’s Ph.D. thesis [33].

Communicated by: Frank Duzaar
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