File size: 2,972 Bytes
a458723 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 | import Mathlib
import OAI.Analysis.Conductivity.Sobolev.AxialPairRank
namespace OAI
noncomputable section
namespace ScalarConductivity
open Real Set Filter Topology MeasureTheory Matrix
open scoped Matrix.Norms.Elementwise
lemma TwoFieldRankRegular.translate {u : Coord3 → Fin 2 → ℝ} {O : Set Coord3}
(h : TwoFieldRankRegular u O) (c : Coord3) (d : Fin 2 → ℝ) :
TwoFieldRankRegular (fun x => u (x-c)+d) ((fun x => x-c) ⁻¹' O) := by
rcases h with h|h|⟨κ,hκ⟩
· left
intro x hx
rw [fderiv_add_const,fderiv_comp_sub]
exact h (x-c) hx
· right; left
obtain ⟨v,l,κ,hv,hl,hd,he⟩ := h
refine ⟨fun x => v (x-c),l,κ+d,?_,hl,?_,?_⟩
· exact hv.comp (contDiff_id.sub contDiff_const).contDiffOn (fun _ hx => hx)
· intro x hx
rw [fderiv_comp_sub]
exact hd (x-c) hx
· intro x hx j
simp only [Pi.add_apply,he (x-c) hx j]
ring
· exact Or.inr (Or.inr ⟨κ+d,fun x hx => by dsimp only; rw [hκ (x-c) hx]⟩)
lemma RegularPatch.translate {u : Coord3 → Fin 2 → ℝ} {A : Coord3 → Symmetric3} {O : Set Coord3}
(h : RegularPatch u A O) (c : Coord3) (d : Fin 2 → ℝ) :
RegularPatch (fun x => u (x-c)+d) (fun x => A (x-c)) ((fun x => x-c) ⁻¹' O) := by
refine ⟨h.1.preimage (continuous_id.sub continuous_const),?_,?_,h.2.2.2.translate c d⟩
· exact (h.2.1.comp (contDiff_id.sub contDiff_const).contDiffOn (fun _ hx => hx)).add contDiffOn_const
· exact h.2.2.1.comp (contDiff_id.sub contDiff_const).contDiffOn (fun _ hx => hx)
lemma mem_regularRegion_translate {u : Coord3 → Fin 2 → ℝ} {A : Coord3 → Symmetric3}
{U : Set Coord3} {x c : Coord3} (h : x-c∈regularRegion u A U) (d : Fin 2 → ℝ) :
x∈regularRegion (fun y => u (y-c)+d) (fun y => A (y-c)) ((fun y => y-c) ⁻¹' U) := by
obtain ⟨O,hOU,hO,hxO⟩ := mem_regularRegion_iff.mp h
exact mem_regularRegion_iff.mpr ⟨(fun y => y-c) ⁻¹' O,preimage_mono hOU,hO.translate c d,hxO⟩
lemma mem_regularRegion_congr_nhds {u v : Coord3 → Fin 2 → ℝ} {A B : Coord3 → Symmetric3}
{V U : Set Coord3} {x : Coord3} (h : x∈regularRegion u A V)
(hu : v =ᶠ[𝓝 x] u) (hA : B =ᶠ[𝓝 x] A) (hU : IsOpen U) (hx : x∈U) :
x∈regularRegion v B U := by
obtain ⟨O,hOV,hO,hxO⟩ := mem_regularRegion_iff.mp h
have hh : {y | y∈O ∧ y∈U ∧ v y=u y ∧ B y=A y}∈𝓝 x := by
filter_upwards [hO.1.mem_nhds hxO,hU.mem_nhds hx,hu,hA] with y hy0 hyU hyu hyA
exact ⟨hy0,hyU,hyu,hyA⟩
obtain ⟨W,hWsub,hW,hxW⟩ := mem_nhds_iff.mp hh
refine mem_regularRegion_iff.mpr ⟨W,fun y hy => (hWsub hy).2.1,?_,hxW⟩
exact (hO.mono hW (fun y hy => (hWsub hy).1)).congr
(fun y hy => (hWsub hy).2.2.1) (fun y hy => (hWsub hy).2.2.2)
lemma axialPair_translate (f : Coord3 → ℝ) (c x : Coord3) :
axialPair f (x-c)+![c 0,0]=axialPair (fun y => f (y-c)) x := by
ext i
fin_cases i <;> simp [axialPair]
end ScalarConductivity
end
end OAI
|