General rigidity principles for stable and minimal elastic curves

Proposition A.1.

Let p ∈ ( 1 , ∞ ) , L > 0 , and let γ : [ 0 , L ] → R 2 be an arclength parametrized p-elastica. Suppose that the signed curvature k ∈ C ⁢ ( [ 0 , L ] ) of γ has no zero. Then the second variation d 2 ⁢ B p of B p at γ in a direction η ∈ C 2 ⁢ ( [ 0 , L ] ; R 2 ) is given by

where I ⁢ [ γ ] ⁢ ( η ) , J ⁢ [ γ ] ⁢ ( η ) , and K ⁢ [ γ ] ⁢ ( η ) are defined by

Proof.

For η ∈ W 2 , p ⁢ ( 0 , L ; ℝ n ) and small ε ∈ ℝ , define γ ε ∈ W 2 , p ⁢ ( 0 , L ; ℝ n ) by

Then the curvature vector κ ε of γ ε is given by

Hence, the p-bending energy of γ ε is represented by

and its derivative along { γ ε } ε can be computed as

where

From this formula, the second derivative along { γ ε } ε can also be computed as

Noting that | γ ′ | ≡ 1 and ( γ ′ , γ ′′ ) = 0 , we have

Consequently, using γ ′ = T and γ ′′ = κ 0 = k ⁢ N , we obtain the desired formula as in (A.1). ∎

Remark A.2 (Ill-definedness for W 2 , p -variations: The singular case p < 2 ).

We have already noticed that for p ∈ ( 1 , 2 ) the second variation may diverge in the whole energy space W 2 , p . Indeed, 𝐊 ⁢ [ γ ] ⁢ ( η ) ≥ p ⁢ ( p - 1 ) ⁢ k * ⁢ ∫ 0 L | η ′′ | 2 ⁢ 𝑑 s holds for k * := min ⁡ | k | > 0 , while 𝐈 ⁢ [ γ ] ⁢ ( η ) and 𝐉 ⁢ [ γ ] ⁢ ( η ) are always convergent by Hölder’s inequality. This means that the second variation formula in Proposition A.1 makes sense for any η ∈ W 2 , p ⁢ ( 0 , L ; ℝ 2 ) if and only if p ≥ 2 .

Lemma A.3.

Let p ∈ ( 1 , ∞ ) , q ∈ ( 0 , 1 ) , and let cn p ⁡ ( ⋅ , q ) be the p-elliptic cosine function. Let U be a neighborhood of zero of cn p ⁡ ( ⋅ , q ) . Then

Proof.

First, recall that the p-elliptic cosine function cn p ⁡ ( ⋅ , q ) is defined by

where am 1 , p ⁡ ( ⋅ , q ) is the inverse function of

Let K 1 , p ⁢ ( q ) := F 1 , p ⁢ ( π 2 , q ) . Then we see that the set of zero points of cn p ⁡ ( ⋅ , q ) is given by

By periodicity, it suffices to consider a neighborhood of K 1 , p ⁢ ( q ) . For a sufficiently small δ > 0 , we have

where in the last equality we used the change of variables y = am 1 , p ⁡ ( x , q ) and set δ ′ > 0 by π 2 + δ ′ = am 1 , p ⁡ ( K 1 , p ⁢ ( q ) + δ , q ) . Since y ↦ 1 / 1 - q 2 ⁢ sin 2 ⁡ y is continuous, the integrability of | cn p ( ⋅ , q ) | p - 2 is reduced to that of | cos y | 3 - 6 p around y = π 2 . This is equivalent to 3 - 6 p > - 1 , and hence we obtain the desired conclusion. ∎

Remark A.4 (Ill-definedness for smooth variations: The highly singular case p ≤ 3 2 ).

Now we find that the second variation for wavelike p-elasticae with p ∈ ( 1 , 3 2 ] may not exist even if η is smooth. Since we still have | 𝐈 ⁢ [ γ ] ⁢ ( η ) | , | 𝐉 ⁢ [ γ ] ⁢ ( η ) | < ∞ , again we only need to take care of the term 𝐊 ⁢ [ γ ] ⁢ ( η ) . Let γ : [ 0 , L ] → ℝ 2 be an arclength parametrized wavelike p-elastica with some z ∈ ( 0 , L ) such that k ⁢ ( z ) = 0 . Consider η ∈ C 2 ⁢ ( 0 , L ; ℝ 2 ) such that | η ′′ ⁢ ( z ) | ≠ 0 ; then there is c * > 0 such that | η ′′ | ≥ c * holds in a neighborhood U of z. Then, by the change of variables we see that

where U ~ is a neighborhood of a zero of cn p ⁡ ( ⋅ , q ) . Thus we deduce by Lemma A.3 that 𝐊 ⁢ [ γ ] ⁢ ( η ) diverges if (and in fact only if) p ∈ ( 1 , 3 2 ] .