... ... @@ -73,17 +73,64 @@ Q = s(n \otimes n - \tfrac13 \operatorname{Id}) \quad (s = -3 λ_1 = -3 λ_2, n ## 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) ### Notation - $P : 𝕊^2 → 𝒬$ removes orientation. Orientable = in the image of $P$. Same for $Q$ only defined on $∂Ω$ - $𝒬 := \{Q = s(n \otimes n - \frac13 \operatorname{ID}) | n ∈ 𝕊^2\}$ - $𝒬_2 := \{Q = s((n_1, n_2, 0) \otimes (n_1, n_2, 0) - \frac13 \operatorname{ID}) | n = (n_1, n_2, 0) ∈ 𝕊^2\}$ - $C^k$ 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 - 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 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*: *Proof sketch*: ## 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 ## Questions - What is the third index $k$ for $Q$ on page 3 (= 495)? - Probably it's a derivative of $Q_{ij}$ in direction $k$. - And why is it written as a product instead of a square$^2$? - To make spectral representation of $Q$ fit to $Q = s(n \otimes n - \tfrac13 \operatorname{Id})$, we need $s = -3 λ_1 = -3 λ_2, n = ê_3$. How can $s$ then depend on $a, b, c$ (see formula (3) on page 3 (=495))? If that's the case, $Q$ is further reduced to the choice of $n$ but that's what we assumed as our model at the beginning. So apparently that's fine. - in Proposition 1 it should be $\det Q = +\frac{2s^3}{27}$ since the $-$ exists in exactly two eigenvalues: $\det Q = λ_1 λ_2 λ_3 = (-\frac s3)(-\frac s3)(+\frac {2s}3) = +\frac{2s^3}{27}$. Correct? ## 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 $n(x) = ± m(x)$
 ... ... @@ -18,3 +18,5 @@ ## Nach Themenwahl * hypersetup in tex/header/preamble.tex - Gute Quelle lesen zum $\operatorname{tr}$-Operator an Rändern inkl. $W^{1-\frac1p,p}(Ω, 𝕊^2)$ bzw. $W^{1-\frac{1p},p}(∂Ω, 𝒬)$ (s. unter Def. 1 auf S. 7 (=499) und Proposition 3 von B&Z).