Boundedness and Regularizing Effects of O Hara s Knot Energies

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1 March, 2 2:24 WSPC/INSTRUCTION FILE OHarasEnergies Boundedness and Regularizing Effects of O Hara s Knot Energies Simon Blatt Departement Mathematik ETH Zürich Rämistrasse CH-84 Zürich Switzerland simon.blatt@math.ethz.ch Keywords: Knot energies, geometric knot theory, fractional Sobolev spaces, regularity Mathematics Subject Classification 2: 57M25, 46E35 Abstract In this small note, we will give a necessary and sufficient condition under which O Hara s E j,p -energies are bounded. We show that a regular curve has bounded E j,p - energy if and only if it is injective and belongs to a certain Sobolev-Slobodeckij space.. Introduction The search for nice representatives of a given knot class led to the invention of a variety of new energies which are subsumed under the term knot energies. These new energies were needed for example due to the fact that other well known candidates like the elastic energy cannot be minimized within a given knot class cf. [vdm98]) or at least their gradient flow can leave the given knot class. One of the first families of geometric knot-energies were the E j,p -energies introduced and investigated by Jun O Hara in [O H9,O H92a,O H92b,O H94]. For a closed regular curve C, R/Z, R n ) and j, p, ) they are defined by E j,p γ) := R/Z) 2 γv) γu) j u v j γ u) γ v) dudv..) Note that these energies are known to be infinite for all smooth closed curves if jp and fail to be self-repulsive for jp < 2 [BR8,O H92b]. Although there are some deep results about the regularity of local minimizers and the regularity of stationary points of those energies [FHW94,He,Rei9] and a few results on the gradient flow of the Möbius-energy [Bla9], no necessary and sufficient criterion is known for the boundedness of these energies so far. The only results in this direction are that these energies are bounded for embedded regular curves in C,α for α > jp 2)/p + 2) [O H94, Proposition.4] and that on

2 March, 2 2:24 WSPC/INSTRUCTION FILE OHarasEnergies 2 Simon Blatt the other hand boundedness of the energy implies that the curve is in C,α for α = jp 2)/+4) [BR8, Theorem.], [O H94, Theorem.]. This small note will fill this gap and thereby extend the above mentioned results. It turns out, that periodic Sobolev-Slobodeckij spaces are the right setting for this task. A detailed discussion of these spaces can be found for example in [Ada75, Tay96,Tar7]. For s, ) and q [, ) we set W s,q R/Z, R n ) := {f L q R/Z, R n ) : and equip this space with the norm f W s,q R/Z,R n ) := f W q + Furthermore, we let /2 u /2 u f u + w) fu) q < }. w +qs fu + w) fu) q w +qs ) /q. W +s,q R/Z, R n ) := {f W,q R/Z, R n ) : f W s,q R/Z, R n )}. Theorem.. Let γ C, R/Z, R n ) be an embedded regular curve parametrized by arc-length and j, p, ) with jp 2 and s := jp 2 p+2 < and p. Then E j,p γ) < if and only if γ W +s, R/Z, R n ). Moreover, there is a C = Cj, p) such that ) γ W C E j,p γ) + γ s, L.. In the forthcoming paper [Bla], Theorem. will play a key role in the proof of long time existence of the gradient flow of the energies E α := E α,, α 2, 3). Furthermore, it is to be expected that this result is of great importance in the study of the regularity of stationary points and local minimizers of these energies. Combining Theorem. with standard embedding theorems for Sobolev spaces into Hölder spaces, one immediately gets the following extension of the main Theorem. in [BR8] and Theorem. in [O H94]: Corollary.2. Let γ C, R/Z, R n ) be an embedded regular curve parametrized by arc-length with E j,p R/Z, R n ) < for some j, p, ) with jp 2 and s := jp <. Then γ C,α R/Z, R n ) where α := jp 2 This shows that the Hölder exponent α = jp 2)/ + 4) in Theorem. in [BR8] and Theorem. in [O H94] was not sharp. Theorem. also sheds new light on the first part of Theorem. in [BR8]. There it is shown that there are curves with finite E 2/p,p -energy which are not differentiable. In view of our new theorem, this can be seen as consequence of the fact that there are embedded curves parametrized by arc-length in W +/, which are not differentiable.

3 March, 2 2:24 WSPC/INSTRUCTION FILE OHarasEnergies Boundedness and Regularizing Effects of O Hara s Knot Energies 3 2. Preliminaries Let us first prove bilipschitz-estimates for injective curves in W +s,. Lemma 2.. For every embedded regular curve γ C, R/Z, R n arametrized by arc-length and every j, p), ) 2 with jp 2, s := jp <, and p the following holds: If γ W +s,, then γ is bilipschitz, i.e. there is a constant C < such that s t C γs) γt) s, t R/Z. Proof. Let 2 > δ > be such that B rz) B r) / γ u + w) γ u) /2 for all z R/Z and all r δ. For z R/Z and r δ we hence get 2r B rz) γ x) 2r 4r 2 B rz) B rz) B rz) 2r) jp 2 γ y)dy dx 4r 2 B rz) B rz) 2δ) jp B rz) B rz) γ x) γ y) dxdy γ x) γ y) x y jp dxdy γ x) γ y) dxdy Since 2r B γ rz) y)dy and jp 2 we deduce that inf a R n a 2r B rz) γ y) a dy 2.

4 March, 2 2:24 WSPC/INSTRUCTION FILE OHarasEnergies 4 Simon Blatt For x, y R/Z with x y 2δ let r := x y 2 and z R/Z be the midpoint of the shorter arc between x and y. Then γx) γy) = sup γ t), a dt a R n Hence, a B rz) = sup a R n a B rz) inf a R n a γ t), γ t) + a γ t)) 2r B rz) γx) γy) x y 2 γ t) a dt x y x y 2 for all x, y R/Z with x y 2δ. Since γ is embedded and x, y) γy) γx) y x defines a continuous positive function on I δ := {x, y) R/Z) 2 : x y 2δ}, we furthermore have γy) γx) γx) γy) min{ : x, y) I δ } x y. y x for all x, y) I δ where min{ γy) γx) y x of the lemma. : x, y) I δ } >. This completes the proof Lemma 2.2. For q there is a constant C = Cq) such that for all a, b, c X, X ), X, X ) a normed vector space, and ε > we have a + b + c q X q )ε) a q X Cε q ) b q X + c q X ). Especially, there are constants < c, C < such that a + b + c q X c a q X C b q X + c q X ). Proof. Using the mean value theorem and the Cauchy Schwartz inequality, we get for x, y R x + y q x q q x q y q )ε) x q ε q ) y q. Combining this with a + b + c X a X b + c X and putting C = 2 q, one gets a + b + c q X q )ε) a q X ε q ) b + c q X q )ε) a Cε q ) b q X + c q X ).

5 March, 2 2:24 WSPC/INSTRUCTION FILE OHarasEnergies Boundedness and Regularizing Effects of O Hara s Knot Energies 5 3. Proof of Theorem. In this section C < and c > are constants whose value may change from line to line. Let us first prove that E j,p γ) is bounded for every embedded regular curve γ W +s, R/Z, R n ). Using the definition of E j,p γ) we see E j,p γ) = = /2 Lemma 2. /2 C γu + w) γu) j w ) jp γu+w) γu) j γu + w) γu) /2 a j j+) a)j+) a 2 ) = C /2 γ = C/2 p Jensen s inequality γu+w) γu) ) j p C /2 /2 C /2 γu+w) γu) 2 w 2 γ u + tw), γ u + sw) dsdt γ u + tw) γ u + sw) 2 dsdt γ u + tw) γ u + sw) dsdt. Using Fubini s lemma to change the order of integration and successively substituting ũ = u + tw, w = s t)w, we get E j,p γ) C C C /2 s t /2 R/Z s t /2 /2 γ ũ) γ ũ + s t)w) dwdũdsdt s t jp γ ũ) γ ũ + w) w jp d wdũdsdt γ ũ) γ ũ + w) w jp d wdũ <

6 March, 2 2:24 WSPC/INSTRUCTION FILE OHarasEnergies 6 Simon Blatt as γ W +s, R/Z, R n ). Now, let us assume that γ C, R/Z, R n ) is a curve parametrized by arc length with E j,p γ) M. From now on C = Cj, p) <, c = cj, p) > are constants which only depend on j and p but are still allowed to change from line to line. One calculates E j,p γ) = γ /2 /2 a j a) /2 a 2 ) γ = c/2 w ) jp γu+w) γu) j γu + w) γu) γu+w) γu) ) j p c /2 /2 γu+w) γu) 2 w 2 γ u + tw) γ u + sw) 2 dsdt w jp = cẽj,p γ ) where Ẽ j,p γ ) := /2 γ u + tw) γ u + sw) 2 dsdt. w jp We will finish the proof of the theorem, by showing that for all functions f C, R/Z, R n ) we have f W s, R/Z,R n ) CẼj,p f ) + C f L. 3.) Of course we can assume without loss of generality that the right hand side is finite. To prove this inequality, let us first assume that f C R/Z, R n ). We get for < ε < /2 f u + tw) f u + sw) 2 dsdt Lemma 2.2 ci ε f) CI ε 2f) + I ε 3f))

7 March, 2 2:24 WSPC/INSTRUCTION FILE OHarasEnergies Boundedness and Regularizing Effects of O Hara s Knot Energies 7 where I ε f) := I ε 2f) := I ε 3f) := /2 /2 /2 ε ε ε ε f u) f u + w) 2 dsdt ε f u + w) f u + sw) 2 dsdt ε f u + tw) f u) 2 dsdt. Note that I ε 2f) = I ε 3f) I ε f) = ε /2 f u) f u + w) w jp d wdu, and I ε 3f) = ε p Hölder-inequality /2 ε w:=tw = Cε ε Cε R/Z Cε jp I ε f). ε f u) f u + tw) 2 dt /2 ε t/2 t/2 t jp dt ε f u) f u + tw) dt ) t jp f u) f u + w) w jp d wdtdu ε/2 R/Z ε/2 f u) f u + w) w jp d wdu Hence, Ẽ j,p f) c Cε jp )ε cε ε/2 R/Z ε/2 ε/2 R/Z ε/2 f u) f u + w) w jp d wdu f u) f u + w) w jp d wdu

8 March, 2 2:24 WSPC/INSTRUCTION FILE OHarasEnergies 8 Simon Blatt if ε > is small enough. With J ε := [ /2, /2] [ ε/2, ε/2] and fixing ε > small enough, this leads to /2 = R/Z J ε f u + w) f u) f u + w) f u) + C f L + C /2 ε/2 R/Z ε/2 f u + w) f u) f u + tw) f u + sw) 2 dsdt which proves Equation 3.) for smooth f. For general f C, R/Z) for which the right hand side of inequality 3.) is finite, we choose a function φ C R, [, )) with support in B /2 ) and φ =. We set φε z) := ε φz/ε) and define the smoothened functions f εx) := /2 /2 fx+z)φz)dz for ε <. It is well known that f ε C R/Z, R n ) and f ε f in W,q for all q, ). Furthermore, /2 = f εu + tw) f εu + sw) 2 dsdt /2 Jensen s inequality Fubini & Jensen = /2 /2 /2 /2 /2 φ εz)f u + tw + z) f u + sw + z))dz 2 dsdt /2 φ ε z) /2 /2 /2 φ εz) f u + tw + z) f u + sw + z) 2 dzdsdt f u + tw + z) f u + sw + z) 2 dsdt dz f u + tw) f u + sw) 2 dsdt, and another application of Jensen s inequality implies f ε L f L. From 3.) for smooth functions and the last two inequalities, we hence get f ε Ẽj,p ) W C f) + f s, L. 3.2)

9 March, 2 2:24 WSPC/INSTRUCTION FILE OHarasEnergies Boundedness and Regularizing Effects of O Hara s Knot Energies 9 Thus there is a subsequence of f ε converging weakly in W s,. The limit of the subsequence is f as we already know that f ε f in W,q for all q [, ). Hence, f W +s,. Since the norm W s, is lower semicontinuous with respect to weak convergence, we deduce from 3.2) that 3.) also holds for f. Bibliography [Ada75] Robert A. Adams. Sobolev spaces. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 975. Pure and Applied Mathematics, Vol. 65. [Bla9] Simon Blatt. The gradient flow of the Möbius energy near local minima. November 29. [Bla] Simon Blatt. The gradient flow of O Haras knot energies. In preparation, 2. [BR8] Simon Blatt and Philipp Reiter. Does finite knot energy lead to differentiability? J. Knot Theory Ramifications, 7):28 3, 28. [FHW94] Michael H. Freedman, Zheng-Xu He, and Zhenghan Wang. Möbius energy of knots and unknots. Ann. of Math. 2), 39): 5, 994. [He] Zheng-Xu He. The Euler-Lagrange equation and heat flow for the Möbius energy. Comm. Pure Appl. Math., 534):399 43, 2. [O H9] Jun O Hara. Energy of a knot. Topology, 32):24 247, 99. [O H92a] Jun O Hara. Energy functionals of knots. In Topology Hawaii Honolulu, HI, 99), pages World Sci. Publ., River Edge, NJ, 992. [O H92b] Jun O Hara. Family of energy functionals of knots. Topology Appl., 482):47 6, 992. [O H94] Jun O Hara. Energy functionals of knots. II. Topology Appl., 56):45 6, 994. [Rei9] Philipp Reiter. Repulsive knot energies and pseudodifferential calculus: rigorous analysis and regularity theory for O Hara s knot energy family E α), α [2, 3). PhD thesis, RWTH Aachen, 29. [Tar7] Luc Tartar. An introduction to Sobolev spaces and interpolation spaces, volume 3 [Tay96] [vdm98] of Lecture Notes of the Unione Matematica Italiana. Springer, Berlin, 27. Michael E. Taylor. Partial differential equations. I, volume 5 of Applied Mathematical Sciences. Springer-Verlag, New York, 996. Basic theory. Heiko von der Mosel. Minimizing the elastic energy of knots. Asymptot. Anal., 8-2):49 65, 998.