chapter 4 of B&Z skimmed

notes/paper-notes/q4-Orientiation-Equivalence.md

@@ -28,3 +28,12 @@
 ## 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
+- $`P : S^2 → 𝒬`$ removes orientation. Orientable = in the image of $`P`$. Same for $`Q`$ only defined on $`∂Ω`$
+- 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 $`p < 2`$
+- chapter 4: different cases of $`Q`$ orientable on boundary $`⇒`$/$`⇔`$/$`⇐`$ on $`Ω`$. Sometimes $`Q: → 𝒬_2`$, sometimes $`Q`$ continous (on boundary)
+
+## 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