Commit 91ad41e5 authored by Felix Hilsky's avatar Felix Hilsky
Browse files

use unicode instead of LaTeX commands

parent 0474407a
......@@ -5,7 +5,7 @@
- *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*: $`Q = s(n \otimes n - \tfrac13\operatorname{Id})`$
- *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:
......@@ -25,11 +25,11 @@
- $`M = M^T`$, $`\operatorname{Tr} M = Σ_{i=1}^3 ∫_{𝕊^2} p_i^2 dμ(p) = ∫_{𝕊^2} 1 dμ(p) = 1`$ ($`μ`$ 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
$`M_0 = (4π)^{-1} ∫_{𝕊^2} p \otimes p dA = \frac 13 \operatorname{Id}`$
$`M_0 = (4π)^{-1} ∫_{𝕊^2} p p dA = \frac 13 \operatorname{Id}`$
- $`∫_{𝕊^2} p_1 p_2 dμ(p) = 0`$ 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 $`Q = M - M_0 = ∫_{𝕊^2}(p \otimes p - \frac13 \operatorname{Id}) dμ(p)`$
- $`Q`$ symmetric, $`\operatorname{Q} = 0`$ $`⇒`$ (spectral theorem) $`Q = λ_1 ê_1 \otimes ê_2 + λ_2 ê_2 \otimes ê_2 - (λ_1 + λ_2)ê_3 \otimes ê_3`$ ($`ê_{1,2,3}`$ orthonormal eigenvector basis, $`λ_1, λ_2`$ eigenvectors)
- 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)
......@@ -71,7 +71,7 @@ minimizers of $`ℱ_{\text{LG}}`$ for $`L → 0`$ are "suitably approximated" by
```
*if* $`Q ∈ W^{1,2}`$ with $`Q`$ uniaxial almost everywhere, i.e.
```math
Q = s(n \otimes n - \tfrac13 \operatorname{Id}) \quad (s = -3 λ_1 = -3 λ_2, n = ê_3)
Q = s(n n - \tfrac13 \operatorname{Id}) \quad (s = -3 λ_1 = -3 λ_2, n = ê_3)
```
## Compatibility
......@@ -81,8 +81,8 @@ Q = s(n \otimes n - \tfrac13 \operatorname{Id}) \quad (s = -3 λ_1 = -3 λ_2, n
### Notation
- $`P : 𝕊^2 → 𝒬`$ removes orientation. Orientable = in the image of $`P`$. Same for $`Q`$ only defined on $`∂Ω`$
- $`𝒬 := \{Q = s(n \otimes n - \frac13 \operatorname{ID}) | n ∈ 𝕊^2\}`$
- $`𝒬_2 := \{Q = s((n_1, n_2, 0) \otimes (n_1, n_2, 0) - \frac13 \operatorname{ID}) | n = (n_1, n_2, 0) ∈ 𝕊^2\}`$
- $`𝒬 := \{Q = s(n n - \frac13 \operatorname{ID}) | n ∈ 𝕊^2\}`$
- $`𝒬_2 := \{Q = s((n_1, n_2, 0) (n_1, n_2, 0) - \frac13 \operatorname{ID}) | n = (n_1, n_2, 0) ∈ 𝕊^2\}`$
- $`C^k`$ 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)
......
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