A speed preserving Hilbert gradient flow for generalized integral Menger curvature

For k ∈ N ∪ { 0 } , the function space

where ∥ f ∥ W k + s , ϱ ⁢ ( R / ℓ ⁢ Z , R n ) := ( ∥ f ∥ W k , ϱ ⁢ ( ( 0 , ℓ ) , R n ) ϱ + [ f ( k ) ] ϱ ) 1 ϱ , is called the periodic Sobolev–Slobodeckiǐ space with fractional differentiability of order k + s and integrability 𝜚. Here, W k , ϱ ⁢ ( R / ℓ ⁢ Z , R n ) ⊂ W loc k , ϱ ⁢ ( R , R n ) denotes the usual Sobolev space of ℓ-periodic functions whose weak derivatives f ( i ) of order i ∈ { 0 , … , k } are locally 𝜚-integrable, where we have set f ( 0 ) := f .

Theorem A.2 (Morrey embedding)

Let k ∈ N ∪ { 0 } and ℓ ∈ ( 0 , ∞ ) . If ϱ ∈ ( 1 , ∞ ) and s ∈ ( 1 ϱ , 1 ) , then there is a positive constant C E = C E ⁢ ( n , k , s , ϱ , ℓ ) such that

Proof

It suffices to treat the case k = 0 ; the full statement follows by induction. Moreover,

for any ℓ-periodic function f : R → R n so that we can focus on the periodic Hölder seminorm. Rewrite the numerator in (A.3) as | f ∘ τ x - ℓ 2 ⁢ ( w + ℓ 2 ) - f ∘ τ x - ℓ 2 ⁢ ( ℓ 2 ) | , where we used the shift τ a ⁢ ( x ) := x + a for some fixed a ∈ R . Hence the right-hand side of (A.3) for α := s - 1 ϱ ∈ ( 0 , 1 ) may be rephrased by setting z := w + ℓ 2 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 sup x ∈ [ 0 , ℓ ] ⁡ C ⁢ ∥ f ∘ τ x - ℓ 2 ∥ W s , ϱ ⁢ ( [ 0 , ℓ ] , R n ) ≤ sup x ∈ [ 0 , ℓ ] ⁡ C ⁢ ∥ f ∘ τ x - ℓ 2 ∥ W s , ϱ ⁢ ( R / ℓ ⁢ Z , R n ) for some constant C = C ⁢ ( n , s , ϱ , ℓ ) . Changing variables via translations by x - ℓ 2 and using the ℓ-periodicity of the respective integrands in the W s , ϱ -norm, one can easily check that ∥ f ∘ τ x - ℓ 2 ∥ W s , ϱ ⁢ ( R / ℓ ⁢ Z , R n ) = ∥ f ∥ W s , ϱ ⁢ ( R / ℓ ⁢ Z , R n ) for each x ∈ R , which concludes the proof. ∎

Proposition A.3 (Poincaré inequality)

For every ℓ ∈ ( 0 , ∞ ) , s ∈ ( 0 , 1 ) and ϱ ∈ [ 1 , ∞ ) , there is a constant C P = C P ⁢ ( s , ϱ , ℓ ) such that

Proof

By Jensen’s inequality, we have

Let L ∈ ( 0 , ∞ ) , ϱ ∈ ( 1 , ∞ ) , s ∈ ( 1 ϱ , 1 ) , g ∈ C 1 ⁢ ( [ 0 , L ] ) with | g ′ | > 0 on [ 0 , L ] , ℓ := g ⁢ ( L ) - g ⁢ ( 0 ) and f ∈ W 1 + s , ϱ ⁢ ( R / ℓ ⁢ Z , R n ) . Then 𝑔 can be extended to a function G ∈ C 1 ⁢ ( R ) such that f ∘ G ∈ C 1 ⁢ ( R / L ⁢ Z , R n ) and G ′ ∈ C 0 ⁢ ( R / L ⁢ Z ) , satisfying

for some constant C = C ⁢ ( n , s , ϱ , ℓ ) , where v g := min [ 0 , L ] ⁡ | g ′ ⁢ ( ⋅ ) | > 0 .

Proof

In order to prove (A.6), we extend 𝑔 onto all of ℝ, first by setting G ⁢ ( x ) := g ⁢ ( x ) for x ∈ ( 0 , L ] and G ⁢ ( x ) := g ⁢ ( x - L ) + ℓ consecutively for x ∈ ( L , 2 ⁢ L ] , ( 2 ⁢ L , 3 ⁢ L ] , … , and then by G ⁢ ( x ) := g ⁢ ( x + L ) - ℓ consecutively for x ∈ ( - L , 0 ] , ( - 2 ⁢ L , - L ] , ( - 3 ⁢ L , - 2 ⁢ L ] , … , to find G ⁢ ( x + L ) = G ⁢ ( x ) + ℓ , G ′ ⁢ ( x + L ) = G ′ ⁢ ( x ) for all x ∈ R . We then bound the numerator in the integrand of the seminorm [ ( f ∘ G ) ′ ] s , ϱ for x ∈ R and | w | ≤ L 2 from above as | ( f ∘ G ) ′ ⁢ ( x + w ) - ( f ∘ G ) ′ ⁢ ( x ) | ϱ ≤ 2 ϱ - 1 ⁢ ( | G ′ ⁢ ( x + w ) | ϱ ⁢ | f ′ ⁢ ( G ⁢ ( x + w ) ) - f ′ ⁢ ( G ⁢ ( x ) ) | ϱ ⁢ | f ′ ⁢ ( G ⁢ ( x ) ) | ϱ ⁢ | G ′ ⁢ ( x + w ) - G ′ ⁢ ( x ) | ϱ ) , which leads to

where we have set α := 1 + ϱ ⁢ s . Note that 𝑓 is of class C 1 ⁢ ( R / ℓ ⁢ Z , R n ) due to Theorem A.2. From now on, we may assume without loss of generality that the seminorm [ G ′ ] s , ϱ is finite; otherwise, there is nothing to prove for (A.6). Substituting u ⁢ ( w ) := x + w and using periodicity, we can rewrite the double integral as

With | G ⁢ ( u ) - G ⁢ ( x ) | R / ℓ ⁢ Z ≤ | u - x | R / L ⁢ Z ⁢ ∥ G ′ ∥ C 0 ⁢ ( R / L ⁢ Z ) , we obtain for the inverse function G - 1 : R → R , for all x , u ∈ R ,

Now we use the transformation Φ : ( R / L ⁢ Z ) 2 → ( R / ℓ ⁢ Z ) 2 , ( u , x ) ↦ ( u ~ , x ~ ) := ( G ⁢ ( u ) , G ⁢ ( x ) ) with

due to periodicity of G ′ and G ′ | [ 0 , L ] = g ′ to rewrite and estimate (A.8) by means of (A.9) as

The C 0 -norm of f ′ in (A.7) may be bounded by C E ⁢ C P ⁢ [ f ′ ] s , ϱ 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 C := ( 2 ϱ - 1 ⁢ ( ( C E ⁢ C P ) ϱ + 1 ) ) 1 ϱ . ∎

Proposition A.5 (Bounds on inner product)

Let k ∈ N ∪ { 0 } , ℓ ∈ ( 0 , ∞ ) , ϱ ∈ ( 1 , ∞ ) and s ∈ ( 1 ϱ , 1 ) . Then there is a constant C = C ⁢ ( n , k , s , ϱ , ℓ ) ∈ ( 0 , ∞ ) such that

Proof

For brevity, we write ⟨ ξ , ζ ⟩ := ⟨ ξ , ζ ⟩ R n for ξ , ζ ∈ R n , and then ⟨ F , G ⟩ := ⟨ F ⁢ ( ⋅ ) , G ⁢ ( ⋅ ) ⟩ for any two functions F , G : R → R n . By virtue of the Morrey embedding, Theorem A.2, the functions f , g are both of class C k . Then, according to the Leibniz rule, we have, for every 0 ≤ i ≤ k ,

of which the L ϱ -norm can be estimated as

which implies by means of the Morrey embedding (A.4) that the Sobolev norm ∥ ⟨ f , g ⟩ ∥ W k , ϱ ϱ may be bounded from above by

The numerator of the integrand in the seminorm [ ⟨ f , g ⟩ ( k ) ] s , ϱ can be estimated from above by means of (A.11) similarly to before by

which may be bounded by

Integrating this against | w | - ( 1 + s ⁢ ϱ ) and using the Morrey embedding inequality (A.4), we find

For i < k , we observe that

since ϱ - 1 - s ⁢ ϱ > ϱ - 2 > - 1 and by Morrey’s embedding (A.4). Analogous estimates hold for the seminorms of f ( j ) , which combine with (A.12) to the inequality as claimed with constant

Notice that target dimension 𝑛 enters this constant through the Morrey embedding constant C E ; cf. Theorem A.2. ∎

Let ℓ ∈ ( 0 , ∞ ) , ϱ ∈ ( 1 , ∞ ) , s ∈ ( 1 ϱ , 1 ) , c > 0 and h ∈ W s , ϱ ⁢ ( R / ℓ ⁢ Z ) with | h ⁢ ( x ) | ≥ c for almost all x ∈ R . Then the multiplicative inverse 1 h : R → R , x ↦ 1 h ⁢ ( x ) is in W s , ϱ ⁢ ( R / ℓ ⁢ Z ) , and we have

Proof

The estimate

follows immediately from the pointwise lower bound on ℎ. The integrand of the seminorm [ 1 h ] s , ϱ may be estimated as | w | - 1 - s ⁢ ϱ ⁢ | h ⁢ ( x + w ) | - ϱ ⁢ | h ⁢ ( x ) | - 1 ⁢ ϱ ⁢ | h ⁢ ( x + w ) - h ⁢ ( x ) | ϱ ≤ c - 2 ⁢ ϱ ⁢ | w | - 1 - s ⁢ ϱ ⁢ | h ⁢ ( x + w ) - h ⁢ ( x ) | ϱ , which implies [ 1 h ] s , ϱ ϱ ≤ c - 2 ⁢ ϱ ⁢ [ h ] s , ϱ ϱ , and together with ∥ h ∥ L ϱ ⁢ ( R / ℓ ⁢ Z ) ϱ ≥ c ϱ ⁢ ℓ , we obtain the desired estimate for ∥ 1 h ∥ W s , ϱ ⁢ ( R / ℓ ⁢ Z ) . ∎

Let ℓ ∈ ( 0 , ∞ ) and γ ∈ W ir 1 , 1 ⁢ ( R / ℓ ⁢ Z , R n ) with length L := L ⁢ ( γ ) ∈ ( 0 , ∞ ) and with arc length parametrization Γ ∈ W 1 , ∞ ⁢ ( R / L ⁢ Z , R n ) . Then the bilipschitz constants of 𝛾 and Γ are related by

where v γ = ess ⁢ inf [ 0 , ℓ ] ⁡ | γ ′ ⁢ ( ⋅ ) | ≥ 0 .

Proof

The upper bound on BiLip ⁡ ( γ ) in (B.2) follows immediately from the estimate

at points x ∈ R / ℓ ⁢ Z where γ ′ ⁢ ( u ) exists. Taking the infimum over all such x ∈ R / ℓ ⁢ Z establishes the upper bound in (B.2). For the proof of the lower bound in (B.2), assume without loss of generality that 0 ≤ y ≤ x < ℓ so that, by monotonicity, 0 ≤ L ⁢ ( γ , y ) ≤ L ⁢ ( γ , x ) < L . Then | γ ⁢ ( x ) - γ ⁢ ( y ) | can be written as and estimated from below by