| 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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|
