On prescribing the number of singular points in a Cosserat-elastic solid

Theorem 1.

For every N ∈ N there exist smooth boundary data g 0 = ( φ 0 , R 0 ) ∈ C ∞ ⁢ ( ∂ ⁡ B 3 , R 3 × S ) with deg ⁡ ( R 0 ) = 0 such that each (restricted) minimizer f = ( φ , R ) of the Cosserat energy J B 3 ⁢ ( ⋅ ) in the class H g 0 1 ⁢ ( B 3 , R 3 × S ) ≔ { g ∈ H 1 ⁢ ( B 3 , R 3 × S ) : g | ∂ B 3 = g 0 } must have at least N singularities in its micro-rotational part R.

Remark 1.

The property deg ⁡ ( R 0 ) = 0 emphasizes that the singularities, which we are about to enforce, do not appear simply due to elementary topological reasons, see the discussion in [3, p. 15] for example, in regard to harmonic mappings u : Ω → S 2 . But, as 𝒮 ≃ ℝ ⁢ P 2 is a non-orientable manifold, the concept of the classical Brouwer-degree deg ⁡ ( ψ ) of a mapping ψ between orientable manifolds, which is used for the deformation component φ, needs to be modified for the micro-rotational component R. Inspired by observations in [16] and [12, Section 4], we define the ( mod ⁡ 2 ) -degree as follows.

Definition 1.

Let Ω ⊂ ℝ 3 be a bounded, simply connected and locally path-connected set and let f be a map with components f = ( φ , R ) : Ω → ℝ 3 × 𝒮 .

1. For R ∈ C 0 ⁢ ( Ω , 𝒮 ) , there exists a lift n : Ω → S 2 , which means F ∘ n = R . Then the ( mod ⁡ 2 ) -degree of R is given by

   deg ⁡ ( R ) ≔ deg ⁡ ( n ) ⁢ mod ⁡  2 .

2. For an isolated singularity a ∈ Sing ⁡ ( R ) , we define

   deg a ⁡ ( R ) ≔ deg ⁡ ( R | S r 2 ( a ) ) = deg a ⁡ ( n ( a ) ) ⁢ mod ⁡  2 ,

   where S r 2 ⁢ ( a ) ⊂ Ω is an arbitrary sphere of radius r > 0 around a such that the corresponding ball B r 3 ¯ ⁢ ( a ) does not contain any other singularities of f, and n ( a ) denotes the lift of R existing on B r 3 ¯ ⁢ ( a ) ∖ { a } .

Definition 2.

Let Ω ⊂ ℝ 3 and f = ( φ , R ) : Ω → ℝ 3 × 𝒮 be as in Definition 1. A pair of singularities ( P , N ) of R is called a dipole for R if there is an open bounded cylinder Z r 3 ⁢ ( q ) ⊂ Ω , rotationally symmetric (of radius r > 0 ) around the line segment [ P , N ] such that

1. [ P , N ] ⊂ Z r 3 ⁢ ( q ) and Z r 3 ⁢ ( q ) is centered at the center q of [ P , N ] ,

2. Z r 3 ⁢ ( q ) does not contain any further singularities of f,

3. deg P ⁡ ( R ) = 1 = deg N ⁡ ( R ) and the lift n ( q ) of R (existing on Z r 3 ¯ ⁢ ( q ) ∖ { P , N } ) has a classical dipole ( P , N ) , i.e. deg P ⁡ ( n ( q ) ) = d = - deg N ⁡ ( n ( q ) ) for a d ∈ ℤ ∖ { 0 } .

Theorem 2.

Let Ω ⊂ R 3 be a bounded, simply connected and locally path-connected set. Let P , N be two distinct points in Ω such that the line segment [ P , N ] lies fully in Ω. For any smooth mapping f = ( φ , R ) ∈ C ∞ ⁢ ( Ω , R 3 × S ) , there exists a sequence of mappings

with the following three properties. First, the pair ( P , N ) is a dipole for each R m , i.e. in particular it holds

Second, all mappings f m agree with f outside of a small neighborhood K m of [ P , N ] which itself fulfils K m → [ P , N ] , m → ∞ , in Hausdorff-distance. Third,

Proof of Theorem 2.

This proof is divided into three steps: First, we present a construction, that was used by F. Béthuel in [1] to remove a dipole from a given map with a controlled amount of (Dirichlet) energy. Working in the other direction, it gives rise to a sequence of Lipschitz mappings with the desired singularities of ( mod ⁡ 2 ) -degree 1 inserted. Additionally, the mappings of the sequence exhibit further singularities of degree 0. Second, we calculate the estimates for the Cosserat energy needed. During the last step, we use some approximation results from [2] to replace each Lipschitz mapping of the sequence with an approximation in order to get rid of the additional singularities of degree 0 and to gain the desired smoothness (except in P , N ) without affecting the Cosserat energy.

Step 1 (Construction). In [1, Lemma 2], F. Béthuel uses a cuboid construction together with a cube lemma [1, Lemma 3] plus some calculations from the two-dimensional case in [4]. We can use exactly the same cuboid construction, together with the following modified cube lemma for the Cosserat energy which itself will be proved after having completed the proof of Theorem 2.

Following the notation from [1] for the cuboid construction, with

1. d ≔ | P - N | ,

2. a m ≔ d 2 ⁢ ( m - 1 ) , m ∈ ℕ > 1 ,

3. K m the cuboid around [ P , N ] : K m ≔ [ - a m , a m ] 2 × [ - a m , d + a m ] ,

4. K m divided into m cubes: C m j ≔ [ - a m , a m ] 2 × [ ( - 1 + 2 ⁢ j ) ⁢ a m , ( 1 + 2 ⁢ j ) ⁢ a m ] , j = 0 , … , m - 1 ,

5. c j the barycenter of C m j and

6. π m j : C m j → ∂ ⁡ C m j the radial retraction with center c j , given by

   π m j ⁢ ( x ) = x - c j | x - c j | ∞ ⋅ a m + c j ,

   where | x - c j | ∞ = max i = 1 , 2 , 3 ⁡ ( | x i - ( c j ) i | ) ,

we iteratively use Lemma 1 (for m fixed) on the boundaries of each of the (m-1) single lower cubes C m j , j = 0 , … , m - 2 . For the boundary of the uppermost cube C m m - 1 , we set f m , α to be equal to f on all faces except for the bottom one. As this bottom face of the last cube C m m - 1 simultaneously is the upper face of the cube C m m - 2 , we set f m , α to have the same values there as constructed by using Lemma 1 on C m m - 2 , in order to obtain matching boundary values. Extending everything to the cubes’ interiors by means of radial retraction π m j , the whole process implies that for each m ≥ m 0 ≫ 1 , there exists a sequence of Lipschitz mappings ( f ~ m , α ) α ∈ H 1 ⁢ ( Ω , ℝ 3 × 𝒮 ) (with α ↘ 0 ) given by

where

with Sing ( f ~ m , α ) = { P = c 0 , c 1 , … , c m - 2 , c m - 1 = N } . For this sequence, we have

as well as

due to the fact that during the construction, changes of the original mapping only happen on little discs (of radius α < a m ) on the upper faces of the lower m - 1 cubes C m 0 , … , C m m - 2 , so that the values on ∂ ⁡ K m remain unaffected. The degree of R ~ m , α in the inner singularities c 1 , … , c m - 2 vanishes, because for the corresponding cubes’ boundaries, the original map was changed both on the upper and the lower face. Also, by definition and known properties of the classical topological degree for the deformation component, we have deg c j ⁡ ( φ ~ m , α ) = deg ⁡ ( φ ) ⋅ deg ⁡ ( π m j ) = 0 ⋅ 1 = 0 .

Step 2 (Calculation of Cosserat energy cost). Many of the calculations in this step follow ideas and estimates carried out in [19], where in the context of removing dipoles from given maps u : Ω → S 2 , [1, Lemma 2] was generalized to the case that S 2 is equipped with an arbitrary Riemannian metric.

So similar to [19], in order to calculate the Cosserat energy of f ~ m , α on K m , we start by dividing each cube C m j into disjoint sets

and the rest

Note that different constants appearing in the following estimates are always denoted by the same γ ∈ ℝ ≥ 0 . They only depend on a m , Lipschitz constants of π m | S ρ 2 ( c j ) j (see below) and the suprema of | D ⁢ φ | 2 and | D ⁢ R | 2 for the original smooth ( φ , R ) in the compact set K m .

Starting with B a m 3 ⁢ ( c j ) , we can estimate the Cosserat energy in terms of

because of (2.1) and because the original f = ( φ , R ) is smooth in all of Ω. Therefore, | R T ⋅ D ⁢ φ - I | 2 and | D ⁢ R | 2 are bounded on K m , thus for each j = 0 , … , m - 1 ,

In the set G m j , because each x ∈ G m j gets projected by π m j onto the boundary of C m j outside of the small discs B α 2 × { ( - 1 + 2 ⁢ j ) ⁢ a m ; ( 1 + 2 ⁢ j ) ⁢ a m } , we have f ~ m , α ⁢ ( x ) = ( f m , α ∘ π m j ) ⁢ ( x ) = ( f ∘ π m j ) ⁢ ( x ) . Hence,

as ( R ∘ π m j ) ∈ 𝒮 ⊂ SO ⁢ ( 3 ) , D ⁢ φ and I 3 , as well as D ⁢ R are bounded and π m j is Lipschitz in C m j ∖ B a m 3 . Similarly, we have

by construction, and therefore

Along the same line of reasoning, which is possible because of (2.2), we find

and

We now proceed with estimates on A m j ( j ≠ 0 ) , and note that D m j ( j ≠ m - 1 ) can be treated analogously by symmetry. While using the Cube Lemma (Lemma 1) in the construction’s background, we changed the original lift n : Ω → S 2 of R into a Lipschitz mapping n m , α : Ω → S 2 in order to get the new R m , α = F ∘ n m , α having m singularities. Here again, F denotes the two-fold covering map between S 2 and 𝒮 as introduced in the explanation just above ((${\mathcal{P}}$)). The deformation part of the Cosserat energy on A m j ( j ≠ 0 ) thus is bounded once again by γ ⋅ a m 3 with the same argument as for estimate (2.5). Additionally, for the micro-rotational part we have the following, notating R ~ m , α = R m , α ∘ π m j = F ∘ n m , α ∘ π m j = F ∘ n ~ m , α and with the fact that n ~ m , α is constant in ( x 1 , x 2 , 2 ⁢ j ⁢ a m - x 3 ) -direction. It holds

and thus

As x 1 2 + x 2 2 ≤ ( x 3 - 2 ⁢ j ⁢ a m ) 2 in this regime, it directly follows that

using (2.3) and R m , α = F ∘ n m , α in combination with the fact, that the covering map F is homothetic.

With the transformation to polar coordinates ( η , ϑ , ξ ) for the cone ( π m j ) - 1 ⁢ ( B α 2 2 × { ( - 1 + 2 ⁢ j ) ⁢ a m } ) translated in x 3 -direction, with η denoting the radius, ϑ ∈ [ 0 , 2 ⁢ π ) denoting the angle and ξ denoting the translated height-level of the disc of points ( η , ϑ ) , i.e.

we finally get the estimate

since η ≤ α 2 < a m . Hence for j = 1 , … , m - 1 , (2) becomes

The last inequality holds because of α < a m and the estimate

as 1 + x ≤ 1 + 2 ⁢ x + x ≤ ( 1 + x ) 2 for non-negative x.

As mentioned above, we also find

for j = 0 , … , m - 2 by symmetry.

Combining (2.4)–(2.8), (2) and (2.11), we have

In other words, for any ε > 0 and any m ≥ m 0 , there is a number α ~ and a corresponding mapping f ~ m , α ~ , which we simply call f ~ m , with

Hence, for each ε > 0 we have a number m ( = m ⁢ ( ε ) ) ≥ m 0 and a mapping f ~ m that fulfils

As a consequence, for any sequence ( ε m ) m ∈ ℕ with ε m ↘ 0 , we have constructed a sequence of Lipschitz mappings f ~ m ∈ H 1 ⁢ ( Ω , ℝ 3 × 𝒮 ) such that

1. f ~ m = ( φ ~ m , R ~ m ) = { ( φ , R ) = f in  ⁢ Ω ∖ K m , ( φ ∘ π m j , R m ∘ π m j ) in  ⁢ K m ,

   where K m → [ P , N ] , m → ∞ , in Hausdorff-distance,

2. Sing ( f ~ m ) = { P = c 0 , c 1 , … , c m - 2 , c m - 1 = N } with

   { deg c j ⁡ ( φ ~ m ) = 0 , j = 0 , … , m - 1 , deg c j ⁡ ( R ~ m ) = 0 , j = 1 , … , m - 2 , deg P ⁡ ( R ~ m ) = 1 = deg N ⁡ ( R ~ m ) ,

3. lim sup m → ∞ ⁡ 𝒥 Ω ⁢ ( f ~ m ) ≤ 𝒥 Ω ⁢ ( f ) + 64 ⁢ π ⋅ | P - N | .

Step 3 (Approximation). Finally, we need suitable approximation arguments to achieve smoothness except in P , N for each f ~ m without affecting the Cosserat energy estimates. Also, ( P , N ) is not yet a dipole for R ~ m according to Definition 2. Luckily, we are able to use several methods developed in [2]. During the construction in Step 1, we changed the original smooth mapping f in the cuboid K m only. Since ℝ 3 is contractible, we can use [2, Theorem 1 bis]. Therefore it is possible to approximate the changed deformation component φ ~ m | K m in H 1 -topology with mappings

while the boundary values remain φ m , s = φ ~ m = φ in ∂ ⁡ K m . Replacing φ m , s by a subsequence, we may assume φ m , s → φ ~ m , s → ∞ , pointwise almost everywhere.

For the micro-rotational component R ~ m | K m = ( R m ∘ π m j ) ∈ H 1 ⁢ ( K m , 𝒮 ) we first note again that

with n ~ m ∈ H 1 ⁢ ( K m , S 2 ) ,

for each j = 1 , … , m - 2 , having smooth boundary values n ~ m | ∂ K m = n | ∂ K m ∈ C ∞ ⁢ ( ∂ ⁡ K m , S 2 ) due to the underlying construction making use of the smooth lift n of the original smooth R. Applying [2, Theorem 2 bis], since S 2 is a compact manifold without boundary and K m is dividable into cubes (cf. “cubeulation” in [2]), there exists a sequence

with

1. n m , t → n ~ m , t → ∞ , in H 1 ⁢ ( K m , S 2 ) ,

2. deg P ⁡ ( n m , t ) = 2 ⁢ k + 1 = - deg N ⁡ ( n m , t ) and deg c j ⁡ ( n m , t ) = 0 , j = 1 , … , m - 2 ,

3. n m , t | ∂ K m = n ~ m | ∂ K m = n | ∂ K m .

Now we can use the technique from the proof of [2, Lemma 1 bis] to get rid of those singularities c 1 , … , c m - 2 , in which the homotopy class of n m , t is trivial. For n m , t in Q m ≔ ⋃ j = 1 m - 2 C m j , each n m , t | Q m (subject to their own boundary values g ≔ n m , t | ∂ Q m ) can be approximated in H 1 -topology by mappings

that agree with n m , t outside of ⋃ j = 1 m - 2 B 1 / s 3 ⁢ ( c j ) .

That is why for each n ~ m : K m → S 2 , there exists a sequence of mappings

by defining

with n m , t , s → n ~ m in H 1 ⁢ ( K m , S 2 ) , deg P ⁡ ( n m , t , s ) = 2 ⁢ k + 1 = - deg N ⁡ ( n m , t , s ) and smooth values on the boundary of K m which are given by n m , t , s | ∂ K m = n ~ m | ∂ K m = n | ∂ K m .

Finally, we project everything back from S 2 to 𝒮 . For each of the mappings R ~ m | K m = F ∘ n ~ m : K m → 𝒮 , we thus have a sequence of mappings

which approximate R ~ m | K m in H 1 -topology, because (after passing to an a.e.-pointwise convergent subsequence) it holds that

and

Summarizing, for any sequence ( ε m ) m ∈ ℕ with ε m ↘ 0 , there exists a sequence of Sobolev mappings ( f ~ m ) m with

as well as another sequence of mappings ( f m , t , s ) s ,

with f m , t , s → f ~ m , s → ∞ , in H 1 ⁢ ( Ω , ℝ 3 × 𝒮 ) . Moreover, R m , t , s has a dipole ( P , N ) . Hence for each ε m , by dominated convergence, we get the existence of a mapping f m with the desired properties and

meaning we have found the sequence ( f m ) m ∈ ℕ of mappings, which are smooth except for a dipole in the micro-rotation and whose Cosserat energy fulfils