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159357f | 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 69 70 71 72 | import Mathlib
import OAI.Analysis.Conductivity.Sources.AngularGraph
import OAI.Analysis.Conductivity.Geometry.CylinderPointEnergy
namespace OAI
section
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
end
end
end OAI
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