Commit f4b8db2a authored by Felix Hilsky's avatar Felix Hilsky
Browse files

read a little bit further

parent 33c73e7f
......@@ -83,10 +83,12 @@ Q = s(n \otimes n - \tfrac13 \operatorname{Id}) \quad (s = -3 λ_1 = -3 λ_2, n
- $`𝒬_2 := \{Q = s((n_1, n_2, 0) \otimes (n_1, n_2, 0) - \frac13 \operatorname{ID}) | n = (n_1, n_2, 0) ∈ 𝕊^2\}`$
- $`C^k`$ 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)
- $`W^{1,p}_{φ} = W^{1,p}`$ 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
- $`v_{,k}`$ is the variable $`v`$ differentiated in the direction $`k`$. $`Q_{ij,k}`$ is the $`i-j`$'th component of the matrix field $`Q`$ differentiated in the direction $`k`$. This is not mentioned anywhere!!
- Indeces which appear twice, are summed over, without mentioning it once!! Even when both are lower indeces.
- 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 $`p < 2`$
......@@ -114,11 +116,24 @@ Hence $`∇τ = 0`$ (since $`x_2`$, $`x_3`$ are analogous) weakly but this means
---
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) $`W^{1-1/p, p}`$.
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*:
*Proposition 3*: If $`Q = P(n)`$ (orientable) on $`Ω`$, then $`\operatorname{Tr} Q P(\operatorname{Tr} n)`$ ($`Q`$ on $`∂Ω`$ is orientable).
*Proof sketch*:
*Proof sketch*: Approximate $`n`$ with differentable functions on and around $`Ω`$, then use continuity of $`\operatorname{Tr}`$ and $`P`$ (show it!) in $`W^{1,p}(Ω)`$ and $`L^p(∂Ω)`$ to show $`P(\operatorname{Tr} Q) = \operatorname{Tr} P(Q)`$. Show with integral approximation property that $`\operatorname{Tr} n`$ lies in $`𝕊^2`$.
---
*Proposition 4*: Orientability is preserved by weak convergence.
For that: *Lemma 1*: (regularity is preserved) ($`Ω`$ bounded)
* $`n ∈ W^{1,p}(Ω, 𝕊^2) ⇒ Q = P(n) ∈ W^{1,p}(Ω, 𝒬)`$
* $`Q ∈ W^{1,p}(Ω, 𝒬) ∧ Q = P(n)`$ and $`n`$ is measurable and continous along almost every line parallel to the coordinate axes. Then $`n ∈ W^{1,p}(Ω, 𝒬)`$ and $`\sum_{j=1}^3 Q_{ij,k} n_j = sn_{i,k}`$.
*Proof of Lemma 1*: Since $`Ω`$ is bounded, $`W^{1,p}(Ω) ⊆ W^{1,1}(Ω)`$, so from $`n ∈ W^{1,p}`$ we get $`n ∈ W^{1,1}`$. Also $`|n(x)|=1`$ for all $`x`$, so $`n ∈ L^{∞}`$. For $`g,h∈W^{1,1} ∩ L^{∞}`$, we have $`gh ∈ W^{1,1} ∩ L^{∞}`$. With
```math
Q ∈
```
## Energy functionals
- specific energy functionals are regarded (p. 11/503) because they were looked at before and Oseen-Frank was successful for them
......@@ -136,3 +151,5 @@ p.5 (=497) Proposition 1 (iii): sign of $`\operatorname{det} Q`$ is wrong. Must
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 $`n(x) = ± m(x)`$
~~p. 8 (=500) Proof Proposition 3: "Mollify $`\overline n`$ o get $`\overline n^{(j)} ∈ C^1(\overline{Ω}, ℝ^3)`$ should be $`∈ C^1(B, ℝ^3)`$. By mollifying we cannot get a bigger domain and it would be weird to first extend $`n`$ to then ignore this extension when mollifying again.~~ ($`\overline{Ω} ⊆ B!`$)
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