q4-Orientiation-Equivalence.md

# Orientability and Energy Minimization in Liquid Crystal Models

## Overview

- *uniaxial nematic liquid crystals* werden modelliert
- *Oseen-Frank*: unit vector field $`n`$ → nicht als $`ℝP^2`$ gesehen, sondern mit $`n ≠ -n`$ (ignoriert Symmetrie)
  - wird als Standardsichtweise dargestellt
- *Landau-de Gennes*: $`Q = s(n ⊗ n - \tfrac13\operatorname{Id})`$
- Resultat:
  - Theorien sind gleich für einfach zusammenhängende Urbilder und $`W^{1,2}`$
  - Unterschiede in anderen Fällen sind beschrieben:
  - for simple energy functional, holes, various boundary conditions, difference cases are characterised (i.e. $`⇔`$)

## de Gennes

- closer to physics reality
- $`Q`$-tensors are generally more [complex](https://arxiv.org/pdf/1409.3542.pdf), in our case just of the simple form („constrained”)

### Derivation of $`Q`$-Tensors
(from chapter 1 Introduction)

- each point has preferred direction but can have any direction → probability measure $`μ(x, ·) : ℒ(𝕊^2) → [0,1]`$ ($`ℒ`$ being all Lebesque sets) modelling very small area around $`x`$
- symmetry modeled as $`μ(x, A) = μ(x, -A) ⇒ ⟨p⟩ = ∫_{𝕊^2} p dμ(p) = 0`$ (first moment or average)
- Tensor of second moments: $`M_{ij} = ∫_{𝕊^2} p_ip_j dμ(p)`$ ($`i, j = 1, 2, 3`$)
  - $`M = M^T`$, $`\operatorname{Tr} M = Σ_{i=1}^3 ∫_{𝕊^2} p_i^2 dμ(p) = ∫_{𝕊^2} 1 dμ(p) = 1`$ ($`μ`$ is probability measure)
  - $`e · M e = ∫_{𝕊^2} (e · p)^2 dμ(p) = ⟨\cos^2(θ)⟩`$ ($`θ`$ = angle between $`p`$ and $`e`$) for $`|e| =1`$)
  - if $`μ`$ is isotropic (no preferred direction): $`μ_0`$ with $`dμ_0(p) = (4π)^{-1}dA`$ with
    $`M_0 = (4π)^{-1} ∫_{𝕊^2} p ⊗ p dA = \frac 13 \operatorname{Id}`$
    - $`∫_{𝕊^2} p_1 p_2 dμ(p) = 0`$ since what happens on one half-sphere is countered by the other half-sphere ($`p_2`$, $`p_3`$ are same but $`p_1`$ is of opposite sign)
    - $`∫_{𝕊^2} p_i^2 dμ(p)`$ are equal ($`i=1,2,3`$) and in sum $`= \operatorname{Tr} M_0 = 1`$
- Def. *de Gennes order-parameter tensor $`Q`$*: difference of second moment tensor to isotropic case $`Q = M - M_0 = ∫_{𝕊^2}(p ⊗ p - \frac13 \operatorname{Id}) dμ(p)`$
  - $`Q`$ symmetric, $`\operatorname{Q} = 0`$ $`⇒`$ (spectral theorem) $`Q = λ_1 ê_1 ⊗ ê_2 + λ_2 ê_2 ⊗ ê_2 - (λ_1 + λ_2)ê_3 ⊗ ê_3`$ ($`ê_{1,2,3}`$ orthonormal eigenvector basis, $`λ_1, λ_2`$ eigenvectors)
    - Eigenvectors for different non-zero eigenvalues are orthogonal: $`(λ_1e_1) · (λ_2^{-1} e_2) = (Q e_1)^T (Q^{-1}e_2) = e_1 · Q^T Q^{-1} e_2 = e_1 · e_2 ⇒ e_1 ⊥ e_2`$
  - two eigenvalues equal and non-zero: *uniaxial* (considered here)
  - otherwise *biaxial* (more complicated)
- Constraint on $`s`$:
  - $`s`$ defined by limiting process $`L → 0`$ and the constants $`a, b, c`$
    - To make spectral representation of $`Q`$ fit to $`Q = s(n ⊗ n - \tfrac13 \operatorname{Id})`$, we need $`s = -3 λ_1 = -3 λ_2, n = ê_3`$. With $`s`$ depending on $`a, b, c`$ (see formula (3) on page 3 (=495)), $`Q`$ is further reduced to the choice of $`n`$. That is what we assumed as our model at the beginning.
  - but also $`Q`$ has an original definition via $`μ`$. Together:
  ```math
  Qn · n
  = n^T s (n n^T - \frac 13 \operatorname{Id}) n
  = s (n^T n n^T n - \frac13 n^T \operatorname{Id}n)
  = s(1 · 1 - \frac13 · 1) \frac23 s
  ```
  ```math
  = ∫_{𝕊^2} n^T p p^T n - \frac13 n^T n dμ(p)
  = ∫_{𝕊^2} (n · p)^2 - \frac 13 dμ(p)
  = ⟨\cos^2 θ - \frac13⟩
  ⇒ s = \frac32 ⟨\cos^2 θ - \frac13⟩
  ```
  - $`⇒ -\frac12 \leq s \leq 1`$: $`s`$ says how much the molecules agree, are „in order“ (max: $`s = 1`$: perfectly ordered $`p∥n`$; $`s = -\frac12`$: $`p ⊥ n`$; $`s=0`$: $`Q = 0`$, isotropic)
  - $`⇒`$ s is *scalar order parameter associated to the tensor $`Q`$*
  - usual assumption: $`s`$ is constant $`⇒`$ space of $`Q`$-Tensors "is" $`ℝP^2`$

### Simpliest energy functional
(p. 495 = 3)
```math
ℱ_{\text{LG}}[Q] = ∫_{Ω} \left(\sum_{i,j,k = 1}^3 \frac L2 Q_{ij,k} Q_{ij,k} - \frac a2 \operatorname{Tr} Q^2 - \frac b3 \operatorname{Tr} Q^3 + \frac c4 (\operatorname{Tr} Q^2)^2 \right) dx
```
$`a, b, c`$ constants, $`L`$ „*elastic constant*”. Physics is interested in $`L → 0`$

## Oseen-Frank
- simpler, but sometimes wrong
- (here:) with orientation ($`n ∈ 𝕊^2`$, not $`n ∈ ℝP^2`$)
- problem: *„fake defects”*: „non-orientable line field” (places where vector field has to be rough but would be OK if we took away the orientation

### Simpliest energy functional
minimizers of $`ℱ_{\text{LG}}`$ for $`L → 0`$ are "suitably approximated" by minimizers of
```math
ℱ_{\text{OF}}[Q] = ∫_{Ω} \sum_{i, j, k = 1}^3 Q_{ij,k} Q_{ij,k} dx
```
*if* $`Q ∈ W^{1,2}`$ with $`Q`$ uniaxial almost everywhere, i.e.
```math
Q = s(n ⊗ n - \tfrac13 \operatorname{Id}) \quad (s = -3 λ_1 = -3 λ_2, n = ê_3)
```

## Notation
- $`P : 𝕊^2 → 𝒬`$ removes orientation. Orientable = in the image of $`P`$. Same for $`Q`$ only defined on $`∂Ω`$
- $`𝒬 := \{Q = s(n ⊗ n - \frac13 \operatorname{ID}) | n ∈ 𝕊^2\}`$
- $`𝒬_2 := \{Q = s((n_1, n_2, 0) ⊗ (n_1, n_2, 0) - \frac13 \operatorname{ID}) | n = (n_1, n_2, 0) ∈ 𝕊^2\}`$
- $`C^k`$ and Lipschitz domains are defined as having a graph as the boundary (as usual)
- $`W^{1,p}`$ defined via embedding
- $`W^{1,p}_{φ} = W^{1,p}`$ with $`φ`$ on the boundary (use $`\operatorname{Tr}`$-Operator to define boundary value)
- $`P : 𝕊^2 → 𝒬`$ removes orientation. Orientable = in the image of $`P`$. Same for $`Q`$ only defined on $`∂Ω`$
  - for $`Q ∈ W^{1,p}(Ω, 𝒬)`$ $`ℒ^d`$-almost everywhere
  - for $`Q ∈ W^{1-\frac1p, p}(∂Ω, 𝒬)`$ orientable to $`n ∈ W^{1-\frac1p, p}(Ω, 𝕊^2)`$ $`ℋ^{d-1}`$-almost everywhere
- $`v_{,k}`$ is the variable $`v`$ differentiated in the direction $`k`$. $`Q_{ij,k}`$ is the $`i-j`$'th component of the matrix field $`Q`$ differentiated in the direction $`k`$. This is not mentioned anywhere!!
- Indeces which appear twice, are summed over, without mentioning it once!! Even when both are lower indeces.
- $`b:𝒬 → ℝP^2`$ (and $`b:𝒬_2 → ℝP^1`$) is isometry between $`𝒬`$ and $`RP^2`$ by
  ```math
  b(s(n ⊗ n - \tfrac13 \operatorname{Id})) = \{n, -n\} ∈ ℝP^2
  ```
  $`⇒ 𝒬`$ is Riemannian manifold (p. 12 (=504) section "Orientability Issues"

  With this identification, $`P`$ is a covering map (details p. 12)
- Jordan curve: continous, injective (non-self-intersecting) loop in $`ℝ^2`$ (Jordan curve theorem: it has an outside and an inside)
- in chapter 3.1 firstly for continous $`Q`$ since that's standard from topology
  - *Theorem 1*: $`𝒬_2`$ (one-dim), orientable on boundary of holes, $`Q`$ continous $`⇒`$ orientable (long proof with a lot of fiddling!)
- chapter 3.2 Theorem 2: $`Ω`$ simply connected, $`Q ∈ W^{1, p}`$, $`p \geq 2`$ (!) $`⇒`$ orientable with Sobolev-Seminorm estimate. Counterexample for $`p < 2`$
- chapter 4: different cases of $`Q`$ orientable on boundary $`⇒`$/$`⇔`$/$`⇐`$ on $`Ω`$. Sometimes $`Q: → 𝒬_2`$, sometimes $`Q`$ continous (on boundary)

### Chapter 2 Propositions
(choose better subsection title!)

*Proposition 2*: $`Q ∈ W^{1,p}(Ω, 𝒬)`$ ($`1 \leq p \leq ∞`$) can have only two orientations.

*Proof*:
In one dimension weakly differentiable means that $`f(x) = f(a) + ∫_a^x f'(y) dy`$.
(Which is in turn equivalent to absolutely continous.)
In several dimensions this is [true in each direction](https://math.aalto.fi/~jkkinnun/files/sobolev_spaces.pdf#section.1.14) but only for almost all $`x_2`$, $`x_3`$.
So: $`n`$, $`m`$ have representatives such that for almost everywhere $`x_2`$, $`x_3`$: $`n`$, $`m=τn`$ are  absolutely continous, so also $`τ = τ·1 = τn·n = m·n`$.

Absolute continuity implies continuity, so $`τ`$ has a representative that is for those almost all $`x_2, x_3`$, constant.
Now use this in Fubini on a small ball in $`Ω`$ around $`y`$:
```math
∫_{B_{ε}(y)} ∂_1 τ(x) φ(x) dx = ∫_{B_{ε}(y)} τ(x) ∂_1 φ(x) dx = ∫_{B_{ε}(y)} ∂_1(τφ)(x) dx = 0
```
(first =: Fubini, second =: $`τ`$ is constant along $`x_1`$ for almost all $`x_2, x_3`$, third =: first integrated over $`x_1`$ direction, this is $`± φ(x_{1, \max}, x_2, x_3) - (± φ(x_{1, \min}, x_2, x_3)) = 0 - 0 = 0`$, then integrated over $`x_2`$, $`x_3`$, it stays 0)

Hence $`∇τ = 0`$ (since $`x_2`$, $`x_3`$ are analogous) weakly but this means that $`τ`$ is constant on $`B_{ε}(y)`$ for every $`y ∈ Ω`$ and therefore constant on $`Ω`$. Done. 

---

Trace operator is introduced by Evans but with continuity estimate for $`\operatorname{Tr} : W^{1,p} → L^p`$. Here we need the stronger statement that $`\operatorname{Tr}`$ maps into the [Sobolev-Slobodeckij space](https://en.wikipedia.org/wiki/Trace_operator#Characterization_using_Sobolev%E2%80%93Slobodeckij_spaces) $`W^{1-1/p, p}`$.

*Proposition 3*: If $`Q = P(n)`$ (orientable) on $`Ω`$, then $`\operatorname{Tr} Q P(\operatorname{Tr} n)`$ ($`Q`$ on $`∂Ω`$ is orientable).

*Proof sketch*: Approximate $`n`$ with differentable functions on and around $`Ω`$, then use continuity of $`\operatorname{Tr}`$ and $`P`$ (show it!) in $`W^{1,p}(Ω)`$ and $`L^p(∂Ω)`$ to show $`P(\operatorname{Tr} Q) = \operatorname{Tr} P(Q)`$. Show with integral approximation property that $`\operatorname{Tr} n`$ lies in $`𝕊^2`$.

---

For Proposition 4: *Lemma 1*: (regularity is preserved) ($`Ω`$ bounded)
* $`n ∈ W^{1,p}(Ω, 𝕊^2) ⇒ Q = P(n) ∈ W^{1,p}(Ω, 𝒬)`$
* $`Q ∈ W^{1,p}(Ω, 𝒬) ∧ Q = P(n)`$ and $`n`$ is measurable and continous along almost every line parallel to the coordinate axes. Then $`n ∈ W^{1,p}(Ω, 𝒬)`$ and $`\sum_{j=1}^3 Q_{ij,k} n_j = sn_{i,k}`$.

*Proof of Lemma 1*: Since $`Ω`$ is bounded, $`W^{1,p}(Ω) ⊆ W^{1,1}(Ω)`$, so from $`n ∈ W^{1,p}`$ we get $`n ∈ W^{1,1}`$. Also $`|n(x)|=1`$ for all $`x`$, so $`n ∈ L^{∞}`$. For $`g,h∈W^{1,1} ∩ L^{∞}`$, we have $`gh ∈ W^{1,1} ∩ L^{∞}`$. (Here used with $`g = n_i`$, $`h=n_{j,k}`$) With $`Q = P(n)`$ we get $`Q ∈ W^{1,1} ∩ L^{∞}`$ and also $`∇Q ∈ L^p`$ since components of $`∇Q`$ are products (and sums) of $`n_i`$ (bounded by $`1`$) and $`n_{j,k} ∈ L^p`$.

Formula is straight calculation (!! $`n_j n_j`$ means $`\sum_{j=1}^3 n_jn_j`$!!).

For converse ($`Q ∈ W^{1,p}`$ given), using the formula to define a $`n_{i, k}`$ candidate doesn't help since when trying to use this, we need the derivative of $`n_j`$ again which we want to show that it exists:
```math
ñ_{ik} := \sum_{j} Q_{ij,k} n_j
```
```math
\text{For } φ ∈ C^{∞}_c: ∫_{Ω} ñ_{ik} φ
= ∫ \sum_j s^{-1} Q_{ij,k} n_j φ
= ∫ - \sum_j Q_{ij} \underbrace{(n_j φ)_{,k}}_{\text{exists?!??}}
```
That's why the authors take an ugly route via normal differentiability almost everywhere (using the assumption about continuity line-wise). Aditionally the continuity is necessary because otherwise the direction of $`n`$ can swap everywhere funnily.

For the theorems about differentiability almost everywhere parallel to the axis, see [ACL characterization from Nikodym](https://math.aalto.fi/~jkkinnun/files/sobolev_spaces.pdf#page=40).

The set of lines parallel to axes is a Lebesgue-zero set (assumption) and the set of points where $`Q`$ is not differentable in the axes directions is a zero set (Nikodym). Hence for almost all $`x ∈ Ω`$, $`n`$ is continous along the line $`(x + ℝe_k) ∩ Ω`$ and $`Q`$ is differentable at $`x`$ in the direction $`e_k`$. Then calculate a lot (start with difference quotient $`Δ_t Q_{ij,k}`$, multiply with $`\frac12(n_j(x+te_k)+n_j(x))`$ and sum over $`j`$) and reach
```math
s · \lim_{t → 0} \frac{n_i (x+te_k) - n_i(x)}{t} = Q_{ij, k}(x) n_j(x)
```
$`∇n ∈ L^p`$ since $`∇Q ∈ L^p`$ and $`n ∈ L^{∞}`$.

---

*Proposition 4*: Orientability is preserved by weak convergence.

*Proof of Proposition 4*: [Uniform boundedness principle](https://en.wikipedia.org/wiki/Uniform_boundedness_principle#Theorem) implies that from $`Q^{(k)} ⇀ Q`$ follows $`\lVert Q^{(k)} \rVert`$ is bounded and with $`|n(x)|`$ bounded by $`1`$ and the formula $`\rVert n^{(k)} \rVert_{W^{1,p}}`$ is bounded.

From boundedness we get a weakly convergent subsequence $`n^{(k_l)} ⇀ n`$, which also has $`n^{(k_l)}(x) → n(x)`$ almost everywhere, so $`P(n) = Q`$.

---

*Lemma 2*: non-orientability is a stable property with respect to the $`W^{1,p}(Ω, ℝ^9)`$ norm:
Let $`Q ∈ W^{1,p}(Ω, 𝒬), 1 ≤ p ≤ ∞`$ be non-orientable. Then there exists $`ε > 0`$, depending on $`Q`$, so that for all $`Q̃ ∈ W^{1,p}(Ω, 𝒬) with `$\lVert Q̃−Q \rVert_{W^{1,p}(M,ℝ^9)} < ε$ the line field Q̃ is also non-orientable.

*Proof of Lemma 2*: obvious, given Proposition 4

## Energy functionals

- specific energy functionals are regarded (p. 11/503) because they were looked at before and Oseen-Frank was successful for them
  - conversion between energy functionals possible


## Compatibility

- Oseen-Frank and de Gennes are compatible if the oriented line field (= unit vector field) can be oriented (without changing regularity)
  - otherwise Oseen-Frank might miss a global minimizer because it is not orientable
  - Question if orientable calculable with integer programming problem (p. 4 = 496) (= (linear) optimization problem with only integer coefficiants)
- In chapter 3, orienting a line field is translated into lifting $`Q`$ from $`ℝP^2`$ to covering space $`𝕊^2`$.
  - Classical algebraic topology tools only consider continous maps, but we have $`W^{1,p}(Ω)`$ maps and want to preserve this regularity


## Questions

## Errors in the paper
p.5 (=497) Proposition 1 (iii): sign of $`\operatorname{det} Q`$ is wrong. Must be positive, since exactly two of the eigenvalues are negative

p. 7 (=499) Proposition 2: it says $`Q(x) ∈ W^{1,p}(Ω, 𝒬)`$ but $`(x)`$ must be erased ($`Q(x) ∈ 𝒬`$)
Also the statement of the proposition is trivial and not what the colloquial formulation says: it should be "... with $`P(n) = P(m) = Q`$ and $`n ≠ m`$ [as $`W^{1,p}`$-generalized functions, so on a non-0-Lebesgue set], we have $`m = -n`$ almost everywhere in $`Ω`$. If $`n ≠ m`$ almost everywhere, it follows straight away that $`m = -n`$ almost everywhere since pointwise $`n(x) = ± m(x)`$

~~p. 8 (=500) Proof Proposition 3: "Mollify $`\overline n`$ o get $`\overline n^{(j)} ∈ C^1(\overline{Ω}, ℝ^3)`$ should be $`∈ C^1(B, ℝ^3)`$. By mollifying we cannot get a bigger domain and it would be weird to first extend $`n`$ to then ignore this extension when mollifying again.~~ ($`\overline{Ω} ⊆ B!`$)