Commit f96e088a by Felix Hilsky

### 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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