Commit 2186dcaa by Felix Hilsky

### move compatibility section

parent ce4bb1ef
 ... ... @@ -75,11 +75,6 @@ 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)  ## Compatibility - Oseen-Frank and de Gennes are compatible if the oriented line field (= unit vector field) can be oriented (without changing regularity) - otherwise Oseen-Frank might miss a global minimizer because it is not orientable - Question if orientable calculable with integer programming problem (p. 4 = 496) (= (linear) optimization problem with only integer coefficiants) ### Notation - $P : 𝕊^2 → 𝒬$ removes orientation. Orientable = in the image of $P$. Same for $Q$ only defined on $∂Ω$ - $𝒬 := \{Q = s(n ⊗ n - \frac13 \operatorname{ID}) | n ∈ 𝕊^2\}$ ... ... @@ -172,6 +167,16 @@ Let $Q ∈ W^{1,p}(Ω, 𝒬), 1 ≤ p ≤ ∞$ be non-orientable. Then there e - 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 ## Compatibility - Oseen-Frank and de Gennes are compatible if the oriented line field (= unit vector field) can be oriented (without changing regularity) - otherwise Oseen-Frank might miss a global minimizer because it is not orientable - Question if orientable calculable with integer programming problem (p. 4 = 496) (= (linear) optimization problem with only integer coefficiants) - In chapter 3, orienting a line field is translated into lifting $Q$ from $ℝP^2$ to covering space $𝕊^2$. - Classical algebraic topology tools only consider continous maps, but we have $W^{1,p}(Ω)`$ maps and want to preserve this regularity ## Questions ## Errors in the paper ... ...
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