q4-Orientiation-Equivalence.md 2.57 KB
 Felix Hilsky committed Jan 13, 2022 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 # 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  Felix Hilsky committed Jan 14, 2022 31 32 33 34 35 36 37 38 39 - $P : S^2 → 𝒬$ 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