Commit f96e088a authored by Felix Hilsky's avatar Felix Hilsky
Browse files

understood lemma 1 proof a bit further

parent d6ca49e1
......@@ -41,6 +41,8 @@
= n^T s (n n^T - \frac 13 \operatorname{Id}) n
= s (n^T n n^T n - \frac13 n^T \operatorname{Id}n)
= s(1 · 1 - \frac13 · 1) \frac23 s
```
```math
= ∫_{𝕊^2} n^T p p^T n - \frac13 n^T n dμ(p)
= ∫_{𝕊^2} (n · p)^2 - \frac 13 dμ(p)
= ⟨\cos^2 θ - \frac13⟩
......@@ -102,7 +104,7 @@ Q = s(n \otimes n - \tfrac13 \operatorname{Id}) \quad (s = -3 λ_1 = -3 λ_2, n
*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 $`x_2`$, $`x_3`$.
In several dimensions this is [true in each direction](https://math.aalto.fi/~jkkinnun/files/sobolev_spaces.pdf#section.1.14) 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.
......@@ -124,16 +126,30 @@ Trace operator is introduced by Evans but with continuity estimate for $`\operat
---
*Proposition 4*: Orientability is preserved by weak convergence.
For that: *Lemma 1*: (regularity is preserved) ($`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
*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 $`gh ∈ W^{1,1} ∩ L^{∞}`$. (Here used with $`g = n_i`$, $`h=n_{j,k}`$) With $`Q = P(n)`$ we get $`Q ∈ W^{1,1} ∩ L^{∞}`$ and also $`∇Q ∈ L^p`$ since components of $`∇Q`$ are products (and sums) of $`n_i`$ (bounded by $`1`$) and $`n_{j,k} ∈ L^p`$.
Formula is straight calculation (!! $`n_j n_j`$ means $`\sum_{j=1}^3 n_jn_j`$!!).
For converse ($`Q ∈ W^{1,p}`$ given), using the formula to define a $`n_{i, k}`$ candidate doesn't help since when trying to use this, we need the derivative of $`n_j`$ again which we want to show that it exists:
```math
Q ∈
\tilde n_{ik} := \sum_{j} Q_{ij,k} n_j
```
```math
\text{For } φ ∈ C^{∞}_c: ∫_{Ω} \tilde n_{ik} φ
= ∫ \sum_j s^{-1} Q_{ij,k} n_j φ
= ∫ - \sum_j Q_{ij} \underbrace{(n_j φ)_{,k}}_{\text{exists?!??}}
```
That's why the authors take an ugly route via normal differentiability almost everywhere (using the assumption about continuity line-wise). Aditionally the continuity is necessary because otherwise the direction of $`n`$ can swap everywhere funnily.
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).
---
*Proposition 4*: Orientability is preserved by weak convergence.
## Energy functionals
- specific energy functionals are regarded (p. 11/503) because they were looked at before and Oseen-Frank was successful for them
......
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