Commit 318f5f66 authored by Felix Hilsky's avatar Felix Hilsky
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Proposition 2 read carefully

parent 8055fe7e
......@@ -8,3 +8,8 @@
- 2022-01-10: --
- 2022-01-11: "Riemannian metrics" Kapitel begonnen zu lesen. Abgestorben bei erster Übung
- 2022-01-12: Gespräch mit Hanne über neues Thema Flüssigkristalle. Aufschreiben des Arbeitsauftrags in meinem Verständnis
- 2022-01-13: Paper von B&Z überflogen, grobe Notizen gemacht
- 2022-01-14: --
- 2022-01-15: Kapitel Introduction von B&Z großteils verstanden
- 2022-01-16: Kapitel Introduction von B&Z beendet, Kapitel Notation (2) begonnen
- 2022-01-17: Proposition 2 B&Z richtig verstanden
......@@ -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).
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