| import Mathlib |
| import OAI.Analysis.Conductivity.Scalarization.GeneralModeEndingScalarization |
| import OAI.Analysis.Conductivity.Fourier.FourierFiniteEndingNormal |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| namespace ScalarConductivity |
| open Set Filter Topology TopologicalSpace MeasureTheory Matrix |
| open scoped Matrix.Norms.Elementwise |
|
|
| local instance generalModeFiniteEndNormalMeasurableSpace : MeasurableSpace Mat3 := |
| inferInstanceAs (MeasurableSpace (Fin 3 → Fin 3 → ℝ)) |
| local instance generalModeFiniteEndNormalBorelSpace : BorelSpace Mat3 := |
| inferInstanceAs (BorelSpace (Fin 3 → Fin 3 → ℝ)) |
|
|
| theorem general_mode_finite_ending_raw_normal {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) |
| {h₀ : Fin 2 → ℤ} (hh₀ : h₀≠0) |
| {a b pa pb : (Fin 2 → ℤ) → ℝ} {A B ga gb : ℝ} |
| (hA : 0≤A) (hB : 0≤B) (ha : ∀ h,|a h|≤A) (hb : ∀ h,|b h|≤B) |
| (hga : 0<ga) (hgb : 0<gb) |
| (hra : ∀ h,a h≠0 → ga≤torusRate s h) |
| (hrb : ∀ h,b h≠0 → torusRate s h₀+gb≤torusRate s h) |
| {δ : ℝ} (hδ : 0<δ) : |
| ∃ (Q : Matrix (Fin 2) (Fin 2) ℤ) (θ : Fin 2 → ℝ) (k : ℤ), |
| Q.det=1 ∧ 0<k ∧ h₀ ᵥ* Q=![0,k] ∧ |
| ∃ p T c C : ℝ,p∈Ioc 0 (1/2) ∧ p<δ ∧ 7<T ∧ 0<c ∧ c≤C ∧ |
| ∃ (v : Coord3 → Fin 2 → ℝ) (E : Coord3 → Mat3) |
| (hsym : ∀ x,(E x).IsSymm), |
| ContDiff ℝ 2 v ∧ Measurable E ∧ |
| AngularPeriodic (2*Real.pi) v ∧ AngularPeriodic (2*Real.pi) E ∧ |
| (∀ x w,c*(w ⬝ᵥ w)≤w ⬝ᵥ (E x*ᵥw) ∧ w ⬝ᵥ (E x*ᵥw)≤C*(w ⬝ᵥ w)) ∧ |
| (∀ x,x 0∈Icc 0 (1/2) → v x= |
| ![x 0+flatFourier s (normalizedFourierCoefficients s 0 |
| (Real.sqrt (angularNormalization Q)*(10+1/p)) a) pa |
| (angularNormalizedLift Q (angularRealTranslation θ x))/Real.sqrt (angularNormalization Q), |
| flatPhaseMode s h₀ (pb h₀) (angularNormalizedLift Q (angularRealTranslation θ x))+ |
| flatFourier s (normalizedFourierCoefficients s (torusRate s h₀) |
| (Real.sqrt (angularNormalization Q)*(10+1/p)) b) pb |
| (angularNormalizedLift Q (angularRealTranslation θ x))]) ∧ |
| (∀ x,x 0∈Icc 0 (1/2) → E x=flatBackgroundTensor (normalizedAngularTensor s Q)) ∧ |
| (∀ x,T≤x 0 → v x=![x 0,0]) ∧ |
| (∀ x,7≤x 0 → E x*ᵥPi.single 0 1=Pi.single 0 1) ∧ |
| (∀ᵐ x : Coord3,0<x 0 → x∈regularRegion v (fun y => ⟨E y,hsym y⟩) {y : Coord3 | 0<y 0}) ∧ |
| (∃ G : Coord3 → Matrix (Fin 3) (Fin 2) ℝ, ContDiff ℝ 1 G ∧ |
| ∀ᵐ x : Coord3,E x*gradientColumns (fderiv ℝ v x)=G x) ∧ |
| ∀ (j : Fin 2) (ψ : SmoothScalar Coord3),HasCompactSupport ψ.val → |
| tsupport ψ.val ⊆ {x : Coord3 | 0<x 0} → |
| Integrable (fun x => ∑ i,(E x*gradientColumns (fderiv ℝ v x)).col j i* |
| (smoothDirection (Pi.single i 1) ψ).val x) ∧ |
| (∫ x,∑ i,(E x*gradientColumns (fderiv ℝ v x)).col j i* |
| (smoothDirection (Pi.single i 1) ψ).val x)=0 := by |
| obtain ⟨k,Q,hk,hQ,he⟩ := leading_frequency_SL2_alignment hh₀ |
| let m := Real.sqrt (angularNormalization Q) |
| have hm : 0 < m := Real.sqrt_pos.mpr (angularNormalization_pos Q) |
| let e := angularFrequencyEquiv Q hQ |
| let θ : Fin 2 → ℝ := ![0,-pb h₀/(k:ℝ)] |
| let a' : (Fin 2 → ℤ) → ℝ := fun h => a (e.symm h)/m |
| let b' : (Fin 2 → ℤ) → ℝ := b ∘ e.symm |
| let pa' := angularShiftPhase (pa ∘ e.symm) θ |
| let pb' := angularShiftPhase (pb ∘ e.symm) θ |
| have hphase : pb' ![0,k]=0 := by |
| have he0 : e.symm ![0,k]=h₀ := by |
| rw [←he] |
| exact e.symm_apply_apply h₀ |
| dsimp [pb',angularShiftPhase] |
| rw [he0] |
| exact alignedPhaseTranslation_zero hk (pb h₀) |
| have hmode (x : Coord3) : |
| flatPhaseMode s h₀ (pb h₀) (angularNormalizedLift Q (angularRealTranslation θ x))= |
| flatPhaseMode (normalizedAngularTensor s Q) ![0,k] 0 x := by |
| rw [flatPhaseMode_angular_conjugacy s hQ,he,flatPhaseMode_angular_translation] |
| rw [show pb h₀+((![0,k] : Fin 2 → ℤ) 0:ℝ)*θ 0+ |
| ((![0,k] : Fin 2 → ℤ) 1:ℝ)*θ 1=0 from alignedPhaseTranslation_zero hk (pb h₀)] |
| have hab : ∀ h,|a' h|≤A/m := by |
| intro h |
| dsimp [a'] |
| rw [abs_div,abs_of_pos hm] |
| exact (div_le_div_iff_of_pos_right hm).2 (ha _) |
| have hbb : ∀ h,|b' h|≤B := fun h => hb (e.symm h) |
| have hgapA := angular_conjugacy_gap s hQ (a:=a) (lam:=0) hga |
| (fun h hh => by simpa only [zero_add] using hra h hh) |
| have hgapB := angular_conjugacy_gap s hQ hgb hrb |
| have hrate : torusRate (normalizedAngularTensor s Q) ![0,k]=m*torusRate s h₀ := by |
| rw [←he,torusRate_angular_conjugacy s hQ] |
| have hra' : ∀ h,a' h≠0 → m*ga≤torusRate (normalizedAngularTensor s Q) h := by |
| intro h hh |
| have hh' : (a ∘ e.symm) h≠0 := by |
| intro hz |
| apply hh |
| dsimp [a'] |
| rw [show a (e.symm h)=0 from hz,zero_div] |
| simpa only [mul_zero,zero_add] using hgapA.2 h hh' |
| have hrb' : ∀ h,b' h≠0 → torusRate (normalizedAngularTensor s Q) ![0,k]+m*gb≤ |
| torusRate (normalizedAngularTensor s Q) h := by |
| intro h hh |
| rw [hrate] |
| exact hgapB.2 h hh |
| obtain ⟨p,T,c,C,hp,hpδ,hT,hc,hcC,v,E,hv,hEm,hvp,hEp,⟨hsym,hreg⟩,hbound,hvin,hEin,hterm,hnormal,_hrank,hG,hpde⟩ := |
| aligned_fourier_finite_ending_with_normal (pa:=pa') |
| (normalizedAngularTensor_lower hs hQ) hk (div_nonneg hA hm.le) hB hab hbb |
| (mul_pos hm hga) (mul_pos hm hgb) hra' hrb' hphase hδ |
| refine ⟨Q,θ,k,hQ,hk,he,p,T,c,C,hp,hpδ,hT,hc,hcC,v,E,hsym,hv,hEm,hvp,hEp,hbound,?_,hEin,hterm,hnormal,hreg,hG,hpde⟩ |
| intro x hx |
| rw [hvin x hx,hmode] |
| congr 1 |
| · congr 1 |
| rw [normalizedFourierCoefficients_div,flatFourier_div_coefficients] |
| congr 1 |
| simpa only [pa',e,Function.comp_def,mul_zero] using normalized_flatFourier_angular_conjugacy s hQ a pa 0 (10+1/p) θ x |
| · rw [hrate] |
| congr 2 |
| simpa only [b',pb',m,e,Function.comp_apply] using normalized_flatFourier_angular_conjugacy s hQ b pb (torusRate s h₀) (10+1/p) θ x |
|
|
| end ScalarConductivity |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| namespace ScalarConductivity |
| open Set Filter Topology MeasureTheory Matrix |
| open scoped Matrix.Norms.Elementwise |
|
|
| def angularInverseLift (A : Matrix (Fin 2) (Fin 2) ℤ) (x : Coord3) : Coord3 := |
| ![x 0/Real.sqrt (angularNormalization A), |
| (A 1 1:ℝ)*x 1-(A 0 1:ℝ)*x 2,-(A 1 0:ℝ)*x 1+(A 0 0:ℝ)*x 2] |
|
|
| def angularNormalizedLinear (A : Matrix (Fin 2) (Fin 2) ℤ) : Coord3 →ₗ[ℝ] Coord3 where |
| toFun := angularNormalizedLift A |
| map_add' x y := by ext i; fin_cases i <;> simp [angularNormalizedLift] <;> ring |
| map_smul' c x := by ext i; fin_cases i <;> simp [angularNormalizedLift] <;> ring |
|
|
| def angularInverseLinear (A : Matrix (Fin 2) (Fin 2) ℤ) : Coord3 →ₗ[ℝ] Coord3 where |
| toFun := angularInverseLift A |
| map_add' x y := by ext i; fin_cases i <;> simp [angularInverseLift] <;> ring |
| map_smul' c x := by ext i; fin_cases i <;> simp [angularInverseLift] <;> ring |
|
|
| lemma real_angular_det_one {A : Matrix (Fin 2) (Fin 2) ℤ} (hA : A.det=1) : |
| (A 0 0:ℝ)*(A 1 1:ℝ)-(A 0 1:ℝ)*(A 1 0:ℝ)=1 := by |
| have h : A 0 0*A 1 1-A 0 1*A 1 0=1 := by simpa only [Matrix.det_fin_two] using hA |
| exact_mod_cast h |
|
|
| lemma angularNormalizedLift_inverse {A : Matrix (Fin 2) (Fin 2) ℤ} (hA : A.det=1) (x : Coord3) : |
| angularNormalizedLift A (angularInverseLift A x)=x := by |
| have hd := real_angular_det_one hA |
| have hm : Real.sqrt (angularNormalization A)≠0 := (Real.sqrt_pos.mpr (angularNormalization_pos A)).ne' |
| ext i |
| fin_cases i |
| · change Real.sqrt (angularNormalization A) * (x 0 / Real.sqrt (angularNormalization A)) = x 0 |
| field_simp [hm] |
| · change (A 0 0:ℝ)*((A 1 1:ℝ)*x 1-(A 0 1:ℝ)*x 2)+ |
| (A 0 1:ℝ)*(-(A 1 0:ℝ)*x 1+(A 0 0:ℝ)*x 2)=x 1 |
| nlinarith [congrArg (fun z : ℝ => z*x 1) hd] |
| · change (A 1 0:ℝ)*((A 1 1:ℝ)*x 1-(A 0 1:ℝ)*x 2)+ |
| (A 1 1:ℝ)*(-(A 1 0:ℝ)*x 1+(A 0 0:ℝ)*x 2)=x 2 |
| nlinarith [congrArg (fun z : ℝ => z*x 2) hd] |
|
|
| lemma angularInverseLift_normalized {A : Matrix (Fin 2) (Fin 2) ℤ} (hA : A.det=1) (x : Coord3) : |
| angularInverseLift A (angularNormalizedLift A x)=x := by |
| have hd := real_angular_det_one hA |
| have hm : Real.sqrt (angularNormalization A)≠0 := (Real.sqrt_pos.mpr (angularNormalization_pos A)).ne' |
| ext i |
| fin_cases i |
| · simp [angularNormalizedLift,angularInverseLift,hm] |
| · change (A 1 1:ℝ)*((A 0 0:ℝ)*x 1+(A 0 1:ℝ)*x 2)- |
| (A 0 1:ℝ)*((A 1 0:ℝ)*x 1+(A 1 1:ℝ)*x 2)=x 1 |
| nlinarith [congrArg (fun z : ℝ => z*x 1) hd] |
| · change -(A 1 0:ℝ)*((A 0 0:ℝ)*x 1+(A 0 1:ℝ)*x 2)+ |
| (A 0 0:ℝ)*((A 1 0:ℝ)*x 1+(A 1 1:ℝ)*x 2)=x 2 |
| nlinarith [congrArg (fun z : ℝ => z*x 2) hd] |
|
|
| def angularNormalizedEquiv (A : Matrix (Fin 2) (Fin 2) ℤ) (hA : A.det=1) : Coord3 ≃L[ℝ] Coord3 := |
| (LinearEquiv.ofLinearMap (angularNormalizedLinear A) (angularInverseLinear A) |
| (by apply LinearMap.ext; intro x; exact angularNormalizedLift_inverse hA x) |
| (by apply LinearMap.ext; intro x; exact angularInverseLift_normalized hA x)).toContinuousLinearEquiv |
|
|
| @[simp] lemma angularNormalizedEquiv_apply (A : Matrix (Fin 2) (Fin 2) ℤ) (hA : A.det=1) (x : Coord3) : |
| angularNormalizedEquiv A hA x=angularNormalizedLift A x := rfl |
|
|
| @[simp] lemma angularNormalizedEquiv_symm_apply (A : Matrix (Fin 2) (Fin 2) ℤ) (hA : A.det=1) (x : Coord3) : |
| (angularNormalizedEquiv A hA).symm x=angularInverseLift A x := rfl |
|
|
| def angularLiftMatrix (A : Matrix (Fin 2) (Fin 2) ℤ) : Mat3 := |
| !![Real.sqrt (angularNormalization A),0,0; |
| 0,(A 0 0:ℝ),(A 0 1:ℝ);0,(A 1 0:ℝ),(A 1 1:ℝ)] |
|
|
| lemma operatorMatrix_angularNormalizedEquiv (A : Matrix (Fin 2) (Fin 2) ℤ) (hA : A.det=1) : |
| operatorMatrix (angularNormalizedEquiv A hA : Coord3 →L[ℝ] Coord3)=angularLiftMatrix A := by |
| ext i j |
| fin_cases i <;> fin_cases j <;> |
| simp [operatorMatrix,angularLiftMatrix,angularNormalizedEquiv,angularNormalizedLinear, |
| angularNormalizedLift] |
|
|
| end ScalarConductivity |
|
|
| end |
| end |
|
|
| end OAI |
|
|
