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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 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 | import Mathlib
import OAI.Analysis.Conductivity.Variational.CentralSmoothTrace
import OAI.Analysis.Conductivity.Variational.SmoothCompactParameter
namespace OAI
section
noncomputable section
namespace ScalarConductivity
open Set MeasureTheory Filter Topology UnitAddTorus
open scoped ENNReal
local instance angularCoefficientMeasureSpace : MeasureSpace UnitAddCircle :=
⟨AddCircle.haarAddCircle⟩
local instance angularCoefficientProbabilityMeasure :
IsProbabilityMeasure (volume : Measure UnitAddCircle) :=
inferInstanceAs (IsProbabilityMeasure AddCircle.haarAddCircle)
lemma contDiff_sourceAngularPolynomial_joint :
ContDiff ℝ (↑(⊤ : ℕ∞)) (fun z : ℝ × (Fin 4 → ℝ) => sourceAngularPolynomial z.1 z.2) := by
apply contDiff_pi.mpr
intro i
fin_cases i <;> dsimp [sourceAngularPolynomial] <;> fun_prop
def sourceAngularCoefficient (f : (Fin 3 → ℝ) → ℝ) (a b : ℝ)
(h : TorusModes) (t : ℝ) : ℂ :=
mFourierCoeff (fun θ => (f (sourceAngularCollar (a*t+b) θ) : ℂ)) h
lemma sourceAngularCoefficient_smooth {f : (Fin 3 → ℝ) → ℝ}
(hf : ContDiff ℝ (↑(⊤ : ℕ∞)) f) (a b : ℝ) (h : TorusModes) :
ContDiff ℝ (↑(⊤ : ℕ∞)) (sourceAngularCoefficient f a b h) := by
let F : ℝ × ((Fin 4 → ℝ) × ℂ) → ℂ := fun z =>
z.2.2 * (f (sourceAngularPolynomial (a*z.1+b) z.2.1) : ℂ)
have hF : ContDiff ℝ (↑(⊤ : ℕ∞)) F := by
apply ContDiff.mul (contDiff_snd.comp contDiff_snd)
exact Complex.ofRealCLM.contDiff.comp (hf.comp
(contDiff_sourceAngularPolynomial_joint.comp
((contDiff_const.mul contDiff_fst |>.add contDiff_const).prodMk
(contDiff_fst.comp contDiff_snd))))
have hr : Continuous (fun θ : UnitAddTorus (Fin 2) =>
(sourceRayCoordinates θ,mFourier (-h) θ)) :=
continuous_sourceRayCoordinates.prodMk (mFourier (-h)).continuous
exact contDiff_compact_parameter hF hr
def sourceAngularAxial (f : (Fin 3 → ℝ) → ℝ) (a b t : ℝ)
(θ : UnitAddTorus (Fin 2)) : ℂ :=
((deriv (fun u => f (sourceAngularCollar (a*u+b) θ)) t : ℝ) : ℂ)
lemma continuous_sourceAngularAxial {f : (Fin 3 → ℝ) → ℝ}
(hf : ContDiff ℝ (↑(⊤ : ℕ∞)) f) (a b : ℝ) :
Continuous (fun z : ℝ × UnitAddTorus (Fin 2) => sourceAngularAxial f a b z.1 z.2) := by
let F : ℝ × (Fin 4 → ℝ) → ℝ := fun z =>
f (sourceAngularPolynomial (a*z.1+b) z.2)
have hF : ContDiff ℝ (↑(⊤ : ℕ∞)) F :=
hf.comp (contDiff_sourceAngularPolynomial_joint.comp
((contDiff_const.mul contDiff_fst |>.add contDiff_const).prodMk contDiff_snd))
have he (t : ℝ) (θ : UnitAddTorus (Fin 2)) :
sourceAngularAxial f a b t θ =
(fderiv ℝ F (t,sourceRayCoordinates θ) (1,0) : ℂ) := by
have h := (hF.differentiable (by simp)).differentiableAt.hasFDerivAt
(x:=(t,sourceRayCoordinates θ))
have hg := h.comp_hasDerivAt t ((hasDerivAt_id t).prodMk
(hasDerivAt_const t (sourceRayCoordinates θ)))
simpa only [sourceAngularAxial,sourceAngularCollar,F,Function.comp_def,id_eq] using
congrArg (fun u : ℝ => (u:ℂ)) hg.deriv
simp_rw [he]
exact Complex.continuous_ofReal.comp
((hF.continuous_fderiv (by simp)).comp
(continuous_fst.prodMk (continuous_sourceRayCoordinates.comp continuous_snd))
|>.clm_apply continuous_const)
lemma sourceAngularCoefficient_hasDerivAt {f : (Fin 3 → ℝ) → ℝ}
(hf : ContDiff ℝ (↑(⊤ : ℕ∞)) f) (a b : ℝ) (h : TorusModes) (t : ℝ) :
HasDerivAt (sourceAngularCoefficient f a b h)
(mFourierCoeff (sourceAngularAxial f a b t) h) t := by
have hF : Continuous (fun z : ℝ × UnitAddTorus (Fin 2) =>
mFourier (-h) z.2 * (f (sourceAngularCollar (a*z.1+b) z.2) : ℂ)) := by
apply ((mFourier (-h)).continuous.comp continuous_snd).mul
exact Complex.continuous_ofReal.comp (hf.continuous.comp
(contDiff_sourceAngularPolynomial_joint.continuous.comp
((continuous_const.mul continuous_fst |>.add continuous_const).prodMk
(continuous_sourceRayCoordinates.comp continuous_snd))))
have hD := ((mFourier (-h)).continuous.comp continuous_snd).mul
(continuous_sourceAngularAxial hf a b)
apply hasDerivAt_compact_parameter hF hD
intro u θ
have hg : ContDiff ℝ (↑(⊤ : ℕ∞)) (fun v => f (sourceAngularCollar (a*v+b) θ)) :=
hf.comp (contDiff_sourceAngularPolynomial_joint.comp
((contDiff_const.mul contDiff_id |>.add contDiff_const).prodMk contDiff_const))
simpa only [sourceAngularAxial,Function.comp_def,Pi.mul_apply] using
(((hg.differentiable (by simp)).differentiableAt (x:=u)).hasDerivAt.ofReal_comp).const_mul
(mFourier (-h) θ)
end ScalarConductivity
end
end
end OAI
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