### notation

parent 45615500
 ... ... @@ -75,7 +75,7 @@ minimizers of $ℱ_{\text{LG}}$ for $L → 0$ are "suitably approximated" by Q = s(n ⊗ n - \tfrac13 \operatorname{Id}) \quad (s = -3 λ_1 = -3 λ_2, n = ê_3)  ### Notation ## Notation - $P : 𝕊^2 → 𝒬$ removes orientation. Orientable = in the image of $P$. Same for $Q$ only defined on $∂Ω$ - $𝒬 := \{Q = s(n ⊗ n - \frac13 \operatorname{ID}) | n ∈ 𝕊^2\}$ - $𝒬_2 := \{Q = s((n_1, n_2, 0) ⊗ (n_1, n_2, 0) - \frac13 \operatorname{ID}) | n = (n_1, n_2, 0) ∈ 𝕊^2\}$ ... ... @@ -94,6 +94,7 @@ Q = s(n ⊗ n - \tfrac13 \operatorname{Id}) \quad (s = -3 λ_1 = -3 λ_2, n = ê $⇒ 𝒬$ is Riemannian manifold (p. 12 (=504) section "Orientability Issues" With this identification, $P$ is a covering map (details p. 12) - Jordan curve: continous, injective (non-self-intersecting) loop in $ℝ^2$ (Jordan curve theorem: it has an outside and an inside) - 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 $p < 2`$ ... ...
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