| import Mathlib |
| import OAI.Analysis.Conductivity.Branching.AlignedEndingRegularity |
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| namespace OAI |
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| noncomputable section |
| namespace ScalarConductivity |
| open Set Filter Topology Real MeasureTheory Matrix |
| open scoped Matrix.Norms.Elementwise |
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| def delayedEndingSymmetricTensor (lam k J L K R : ℝ) (x : Coord3) : Symmetric3 := |
| alignedEndingSymmetricTensor lam k J L K (x-Pi.single 0 R) |
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| def axialAmplitudeEquiv (a : ℝ) (ha : a≠0) : (Fin 2 → ℝ) ≃L[ℝ] (Fin 2 → ℝ) := |
| ContinuousLinearEquiv.piCongrRight (fun i : Fin 2 => |
| (LinearEquiv.smulOfNeZero ℝ ℝ (if i=0 then 1 else a) (by split_ifs <;> simp_all)).toContinuousLinearEquiv) |
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| lemma axialAmplitudeEquiv_apply (a : ℝ) (ha : a≠0) (v : Fin 2 → ℝ) (i : Fin 2) : |
| axialAmplitudeEquiv a ha v i=(if i=0 then 1 else a)*v i := rfl |
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| lemma RegularPatch.amplitude {u : Coord3 → Fin 2 → ℝ} {A : Coord3 → Symmetric3} |
| {O : Set Coord3} (h : RegularPatch u A O) (a : ℝ) (ha : a≠0) : |
| RegularPatch (fun x => axialAmplitudeEquiv a ha (u x)) A O := by |
| refine ⟨h.1,(axialAmplitudeEquiv a ha).contDiff.contDiffOn.comp h.2.1 (fun _ _ => mem_univ _),h.2.2.1,?_⟩ |
| exact h.2.2.2.linearCoordinates h.1 h.2.1 (ContinuousLinearEquiv.refl ℝ Coord3) (axialAmplitudeEquiv a ha) |
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| lemma delayedEnding_axialPair (lam k J L K R : ℝ) (x : Coord3) : |
| axialAmplitudeEquiv (exp (-lam*R)) (exp_pos _).ne' |
| (axialPair (alignedEndingValue lam k J L K) (x-Pi.single 0 R))+![R,0]= |
| axialPair (delayedEndingValue lam k J L K R) x := by |
| ext i |
| fin_cases i <;> simp [axialAmplitudeEquiv_apply,axialPair,delayedEndingValue] |
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| lemma RegularPatch.delayedEnding {lam k J L K R : ℝ} {O : Set Coord3} |
| (h : RegularPatch (axialPair (alignedEndingValue lam k J L K)) |
| (alignedEndingSymmetricTensor lam k J L K) O) : |
| RegularPatch (axialPair (delayedEndingValue lam k J L K R)) |
| (delayedEndingSymmetricTensor lam k J L K R) ((fun x : Coord3 => x-Pi.single 0 R) ⁻¹' O) := by |
| have hh := (h.amplitude (exp (-lam*R)) (exp_pos _).ne').translate (Pi.single 0 R) ![R,0] |
| convert hh using 1 |
| · funext x |
| exact (delayedEnding_axialPair _ _ _ _ _ _ _).symm |
| · rfl |
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| lemma delayedEnding_regularRegion_ae {lam k J L K R : ℝ} (hk : k≠0) (hL : 0<L) (hK : 0<K) |
| {U : Set Coord3} (hU : IsOpen U) : |
| ∀ᵐ x : Coord3,x∈U → x∈regularRegion (axialPair (delayedEndingValue lam k J L K R)) |
| (delayedEndingSymmetricTensor lam k J L K R) U := by |
| have he := (measurePreserving_add_right (volume : Measure Coord3) (-Pi.single 0 R)).quasiMeasurePreserving.ae |
| (alignedEnding_regularRegion_ae (lam:=lam) (J:=J) hk hL hK isOpen_univ) |
| simp only [←sub_eq_add_neg] at he |
| filter_upwards [he] with x hx |
| intro hxU |
| obtain ⟨O,hOU,hO,hxO⟩ := mem_regularRegion_iff.mp (hx (mem_univ _)) |
| have hh : x∈regularRegion (axialPair (delayedEndingValue lam k J L K R)) |
| (delayedEndingSymmetricTensor lam k J L K R) univ := by |
| apply mem_regularRegion_iff.mpr |
| refine ⟨(fun y : Coord3 => y-Pi.single 0 R) ⁻¹' O,subset_univ _,?_,?_⟩ |
| · exact hO.delayedEnding (R:=R) |
| · exact hxO |
| exact mem_regularRegion_congr_nhds hh (Filter.EventuallyEq.rfl) (Filter.EventuallyEq.rfl) hU hxU |
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| end ScalarConductivity |
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| end |
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| end OAI |
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