q4-Orientiation-Equivalence.md 2.57 KB
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# 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*: $`Q = s(n \otimes n - \tfrac13\operatorname{Id}`$
- 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”)
- each point has preferred direction but can have any direction → probability measure $`μ(x, ·) : ℒ(S^2) → [0,1]`$ ($`ℒ`$ being all Lebesque sets)
- symmetry modeled as $`μ(x, A) = μ(x, -A) ⇒ ⟨p⟩ = ∫_{S^2} p dμ(p) = 0`$ (first moment or average)
- ... (some more steps to reach the $`Q`$ form from above)
  - including $`-\frac12 \leq s \leq 1`$: $`s`$ says how much the molecules agree, are „in order“
  - usual assumption: $`s`$ is constant $`⇒`$ space of $`Q`$-Tensors "is" $`ℝP^2`$

## Oseen-Frank
- simpler, but sometimes wrong
- (here:) with orientation ($`n ∈ S^2`$, not $`n ∈ ℝP^2`$)
- problem: *„fake defects”*: „non-orientable line field”

## 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
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- $`P : S^2 → 𝒬`$ removes orientation. Orientable = in the image of $`P`$. Same for $`Q`$ only defined on $`∂Ω`$
- 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)

## 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