... ... @@ -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) ... ...