Commit 2186dcaa authored by Felix Hilsky's avatar Felix Hilsky
Browse files

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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