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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 | 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
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