Commit 0474407a authored by Felix Hilsky's avatar Felix Hilsky
Browse files

notes about proof Lemma 1 and proposition 4 and lemma 2

parent a2ce0f04
......@@ -126,7 +126,7 @@ Trace operator is introduced by Evans but with continuity estimate for $`\operat
---
For that: *Lemma 1*: (regularity is preserved) ($`Ω`$ bounded)
For Proposition 4: *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 $`\sum_{j=1}^3 Q_{ij,k} n_j = sn_{i,k}`$.
......@@ -147,10 +147,26 @@ That's why the authors take an ugly route via normal differentiability almost ev
For the theorems about differentiability almost everywhere parallel to the axis, see [ACL characterization from Nikodym](https://math.aalto.fi/~jkkinnun/files/sobolev_spaces.pdf#page=40).
The set of lines parallel to axes is a Lebesgue-zero set (assumption) and the set of points where $`Q`$ is not differentable in the axes directions is a zero set (Nikodym). Hence for almost all $`x ∈ Ω`$, $`n`$ is continous along the line $`(x + ℝe_k) ∩ Ω`$ and $`Q`$ is differentable at $`x`$ in the direction $`e_k`$. Then calculate a lot (start with difference quotient $`Δ_t Q_{ij,k}`$, multiply with $`\frac12(n_j(x+te_k)+n_j(x))`$ and sum over $`j`$) and reach
```math
s · \lim_{t → 0} \frac{n_i (x+te_k) - n_i(x)}{t} = Q_{ij, k}(x) n_j(x)
```
$`∇n ∈ L^p`$ since $`∇Q ∈ L^p`$ and $`n ∈ L^{∞}`$.
---
*Proposition 4*: Orientability is preserved by weak convergence.
*Proof of Proposition 4*: [Uniform boundedness principle](https://en.wikipedia.org/wiki/Uniform_boundedness_principle#Theorem) implies that from $`Q^{(k)} ⇀ Q`$ follows $`\lVert Q^{(k)} \rVert`$ is bounded and with $`|n(x)|`$ bounded by $`1`$ and the formula $`\rVert n^{(k)} \rVert_{W^{1,p}}`$ is bounded.
From boundedness we get a weakly convergent subsequence $`n^{(k_l)} ⇀ n`$, which also has $`n^{(k_l)}(x) → n(x)`$ almost everywhere, so $`P(n) = Q`$.
---
*Lemma 2*: non-orientability is a stable property with respect to the $`W^{1,p)(Ω, ℝ^9)`$ norm:
Let $`Q ∈ W^{1,p}(Ω, 𝒬), 1 ≤ p ≤ ∞`$ be non-orientable. Then there exists $`ε > 0`$, depending on $`Q`$, so that for all $`Q̃ ∈ W^{1,p}(Ω, 𝒬) with `$\lVert Q̃−Q \rVert_{W^{1,p}(M,ℝ^9)} < ε$ the line field Q̃ is also non-orientable.
*Proof of Lemma 2*: obvious, given Proposition 4
## 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
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