| import Mathlib |
| import OAI.Analysis.Conductivity.Sobolev.AxialPairRank |
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| namespace OAI |
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| noncomputable section |
| namespace ScalarConductivity |
| open Real Set Filter Topology MeasureTheory Matrix |
| open scoped Matrix.Norms.Elementwise |
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| 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]⟩) |
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| 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) |
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| 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⟩ |
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| 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) |
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| 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] |
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| end ScalarConductivity |
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| end |
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| end OAI |
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