| import Mathlib |
| import OAI.Analysis.Conductivity.Sources.AngularGraph |
| import OAI.Analysis.Conductivity.Geometry.CylinderPointEnergy |
|
|
| namespace OAI |
|
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| section |
|
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| noncomputable section |
| namespace ScalarConductivity |
| open Set MeasureTheory Filter Topology UnitAddTorus Matrix |
| open scoped ENNReal |
| local instance endPointEnergyMeasureSpaceUnitAddCircle : MeasureSpace UnitAddCircle := ⟨AddCircle.haarAddCircle⟩ |
| local instance endPointEnergyIsProbabilityMeasure : IsProbabilityMeasure (volume : Measure UnitAddCircle) := |
| inferInstanceAs (IsProbabilityMeasure AddCircle.haarAddCircle) |
|
|
| def sourceEndSmoothDensity (s : Fin 3 → ℝ) (f : spectralTraceGraph (torusRate s)) |
| (κ a b : ℝ) (φ : (Fin 3 → ℝ) → ℝ) (z : ℝ × UnitAddTorus (Fin 2)) : ℝ := |
| fullEndFlatCovector s f κ z.1 z.2 ⬝ᵥ |
| (flatCylinderMatrix s*ᵥ(fun k : Fin 3 => (sourceAngularJet φ a b k.succ z).re)) |
|
|
| lemma sourceEndSmoothDensity_cylinder (s : Fin 3 → ℝ) |
| (hs : ∀ x y : ℝ,(1/2)*(x^2+y^2) ≤ s 0*x^2+2*s 1*x*y+s 2*y^2) |
| (f : spectralTraceGraph (torusRate s)) (κ a b R : ℝ) (hR : 0≤R) |
| {φ : (Fin 3 → ℝ) → ℝ} (hφ : ContDiff ℝ (↑(⊤:ℕ∞)) φ) : |
| Integrable (sourceEndSmoothDensity s f κ a b φ) ((FiniteAxisMeasure R).prod volume) ∧ |
| (∫ z,sourceEndSmoothDensity s f κ a b φ z ∂(FiniteAxisMeasure R).prod volume)= |
| (flatCylinderEnergy s (cylinderEndJet s hs κ R hR f) |
| (sourceAngularLpJet hφ a b R)).re := by |
| let u := cylinderEndJet s hs κ R hR f |
| let v := sourceAngularLpJet hφ a b R |
| have hv (k : Fin 3) : ∀ᵐ z∂(FiniteAxisMeasure R).prod volume,(v k.succ z).im=0 := by |
| filter_upwards [sourceAngularLpJet_ae hφ a b R k.succ] with z hz |
| change (sourceAngularLpJet hφ a b R k.succ z).im=0 |
| rw [hz] |
| exact sourceAngularJet_im φ a b k.succ z |
| obtain ⟨hi,he⟩ := flatCylinderEnergy_real_integral s u v hv |
| have hve : ∀ᵐ z∂(FiniteAxisMeasure R).prod volume, |
| (fun k : Fin 3 => (v k.succ z).re)= |
| (fun k : Fin 3 => (sourceAngularJet φ a b k.succ z).re) := by |
| filter_upwards [ae_all_iff.mpr (fun k : Fin 3 => sourceAngularLpJet_ae hφ a b R k.succ)] with z hz |
| exact funext (fun k => congrArg Complex.re (hz k)) |
| have hd : (fun z => (fun k : Fin 3 => (u k.succ z).re) ⬝ᵥ |
| (flatCylinderMatrix s*ᵥ(fun k : Fin 3 => (v k.succ z).re)))=ᵐ[(FiniteAxisMeasure R).prod volume] |
| sourceEndSmoothDensity s f κ a b φ := by |
| filter_upwards [cylinderEndJet_covector_ae s hs κ R hR f,hve] with z hu hv |
| change (fun k : Fin 3 => (cylinderEndJet s hs κ R hR f k.succ z).re) ⬝ᵥ |
| (flatCylinderMatrix s*ᵥ(fun k : Fin 3 => (v k.succ z).re))=_ |
| rw [hu,hv] |
| rfl |
| exact ⟨hi.congr hd,(integral_congr_ae hd).symm.trans he.symm⟩ |
|
|
| lemma sourceEndSmoothDensity_green (s : Fin 3 → ℝ) |
| (hs : ∀ x y : ℝ,(1/2)*(x^2+y^2) ≤ s 0*x^2+2*s 1*x*y+s 2*y^2) |
| (f : spectralTraceGraph (torusRate s)) (κ a b R : ℝ) (hR : 0<R) |
| {φ : (Fin 3 → ℝ) → ℝ} (hφ : ContDiff ℝ (↑(⊤:ℕ∞)) φ) : |
| (∫ z,sourceEndSmoothDensity s f κ a b φ z ∂(FiniteAxisMeasure R).prod volume)= |
| inner ℝ (spectralGraphWeight (torusRate s) f) |
| (spectralGraphWeight (torusRate s) (smoothCollarTrace s b hφ))- |
| κ*spectralGraphMean (torusRate s) (smoothCollarTrace s b hφ)+ |
| cylinderTerminalFlux s κ R f (smoothCollarTrace s (a*R+b) hφ) := by |
| rw [(sourceEndSmoothDensity_cylinder s hs f κ a b R hR.le hφ).2] |
| exact cylinderEndJet_real_green s hs κ hR f |
| ⟨(_,(_,_)),sourceAngularLpJet_graph s hφ a b R⟩ |
|
|
| end ScalarConductivity |
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| end |
| end |
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| end OAI |
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