Commit 0474407a by Felix Hilsky

### 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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