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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 | import Mathlib
import OAI.Analysis.Conductivity.Fourier.RectangularTorusAtlas
import OAI.Analysis.Conductivity.Geometry.TorusLipschitzTrace
import OAI.Analysis.Conductivity.Fourier.RectangularRayTrace
namespace OAI
noncomputable section
namespace ScalarConductivity
open Set MeasureTheory Filter Topology UnitAddTorus
open scoped NNReal ENNReal
lemma rectangularRay_abs {w : ℝ} (hw : 0<w) (hw' : w≤1)
(θ : UnitAddCircle) (i : Fin 2) : |rectangularRay w θ i|≤2 := by
have hd := rayDenominator_lower hw hw' θ
have hp : 0<rayDenominator w θ := by linarith
rw [rectangularRay,abs_div,abs_of_pos hp]
apply (div_le_iff₀ hp).mpr
linarith [rayDirection_abs θ i]
lemma rectangularRay_coordinate_bound {w : ℝ} (hw : 0<w) (hw' : w≤1)
(x : UnitAddTorus (Fin 2)) (j k i : Fin 2) (t : ℝ) :
|rectangularRay w ((x+torusCoordinateShift j t) k) i-rectangularRay w (x k) i|≤
((2+4/w)*(2*Real.pi))*|t| := by
by_cases h : k=j
· subst k
simpa [torusCoordinateShift,mul_assoc] using rectangularRay_difference hw hw' (x j) t i
· simp only [Pi.add_apply,torusCoordinateShift,ite_eq_right h,add_zero,sub_self,abs_zero]
positivity
def sourceRayCoordinates (x : UnitAddTorus (Fin 2)) : Fin 4 → ℝ :=
![rectangularRay 1 (x 0) 0,rectangularRay 1 (x 0) 1,
rectangularRay sourceRadialWidth (x 1) 0,rectangularRay sourceRadialWidth (x 1) 1]
def sourceRayBox : Set (Fin 4 → ℝ) := Icc (fun _ => -2) (fun _ => 2)
lemma sourceRayCoordinates_mem (x : UnitAddTorus (Fin 2)) : sourceRayCoordinates x∈sourceRayBox := by
have hw : 0<sourceRadialWidth := by norm_num [sourceRadialWidth,sourceHole]
have hw' : sourceRadialWidth≤1 := by norm_num [sourceRadialWidth,sourceHole]
have h (i : Fin 4) : |sourceRayCoordinates x i|≤2 := by
fin_cases i
· exact rectangularRay_abs (by norm_num) (by norm_num) _ _
· exact rectangularRay_abs (by norm_num) (by norm_num) _ _
· exact rectangularRay_abs hw hw' _ _
· exact rectangularRay_abs hw hw' _ _
exact ⟨fun i => (abs_le.mp (h i)).1,fun i => (abs_le.mp (h i)).2⟩
lemma continuous_sourceRayCoordinates : Continuous sourceRayCoordinates := by
have hc1 := continuous_rectangularRay (w:=1) (by norm_num) (by norm_num)
have hcw := continuous_rectangularRay (w:=sourceRadialWidth)
(by norm_num [sourceRadialWidth,sourceHole]) (by norm_num [sourceRadialWidth,sourceHole])
apply continuous_pi
intro i
fin_cases i <;> dsimp [sourceRayCoordinates]
· exact (continuous_apply (0 : Fin 2)).comp (hc1.comp (continuous_apply (0 : Fin 2)))
· exact (continuous_apply (1 : Fin 2)).comp (hc1.comp (continuous_apply (0 : Fin 2)))
· exact (continuous_apply (0 : Fin 2)).comp (hcw.comp (continuous_apply (1 : Fin 2)))
· exact (continuous_apply (1 : Fin 2)).comp (hcw.comp (continuous_apply (1 : Fin 2)))
def sourceRayConstant : ℝ := 6*(2*Real.pi)+(2+4/sourceRadialWidth)*(2*Real.pi)
lemma sourceRayConstant_nonneg : 0≤ sourceRayConstant := by
have hw : 0<sourceRadialWidth := by norm_num [sourceRadialWidth,sourceHole]
unfold sourceRayConstant
positivity
lemma sourceRayCoordinates_difference (j : Fin 2) (t : ℝ) (x : UnitAddTorus (Fin 2)) :
‖sourceRayCoordinates (x+torusCoordinateShift j t)-sourceRayCoordinates x‖≤ sourceRayConstant*|t| := by
have hw : 0<sourceRadialWidth := by norm_num [sourceRadialWidth,sourceHole]
have hw' : sourceRadialWidth≤1 := by norm_num [sourceRadialWidth,sourceHole]
have h1 : (2+4/(1:ℝ))*(2*Real.pi)≤ sourceRayConstant := by
dsimp [sourceRayConstant]
norm_num
positivity
have h2 : (2+4/sourceRadialWidth)*(2*Real.pi)≤ sourceRayConstant := by
dsimp [sourceRayConstant]
linarith [Real.pi_pos]
apply (pi_norm_le_iff_of_nonneg (mul_nonneg sourceRayConstant_nonneg (abs_nonneg _))).mpr
intro i
change |sourceRayCoordinates (x+torusCoordinateShift j t) i-sourceRayCoordinates x i|≤_
fin_cases i
· exact (rectangularRay_coordinate_bound (w:=1) (by norm_num) (by norm_num) x j 0 0 t).trans
(mul_le_mul_of_nonneg_right h1 (abs_nonneg _))
· exact (rectangularRay_coordinate_bound (w:=1) (by norm_num) (by norm_num) x j 0 1 t).trans
(mul_le_mul_of_nonneg_right h1 (abs_nonneg _))
· exact (rectangularRay_coordinate_bound hw hw' x j 1 0 t).trans
(mul_le_mul_of_nonneg_right h2 (abs_nonneg _))
· exact (rectangularRay_coordinate_bound hw hw' x j 1 1 t).trans
(mul_le_mul_of_nonneg_right h2 (abs_nonneg _))
def sourceAngularPolynomial (t : ℝ) (z : Fin 4 → ℝ) : Fin 3 → ℝ :=
![sourceLength*(sourceRadialCenter+(1-t)*z 2)*z 0,
(sourceRadialCenter+(1-t)*z 2)*z 1,(1-t)*z 3]
def sourceAngularCollar (t : ℝ) (x : UnitAddTorus (Fin 2)) : Fin 3 → ℝ :=
sourceAngularPolynomial t (sourceRayCoordinates x)
lemma contDiff_sourceAngularPolynomial (t : ℝ) :
ContDiff ℝ (↑(⊤ : ℕ∞)) (sourceAngularPolynomial t) := by
apply contDiff_pi.mpr
intro i
fin_cases i <;> dsimp [sourceAngularPolynomial] <;> fun_prop
lemma continuous_sourceAngularCollar (t : ℝ) : Continuous (sourceAngularCollar t) :=
(contDiff_sourceAngularPolynomial t).continuous.comp continuous_sourceRayCoordinates
theorem sourceAngularTrace_summable (s : Fin 3 → ℝ) (t : ℝ) {f : (Fin 3 → ℝ) → ℝ}
(hf : ContDiff ℝ (↑(⊤ : ℕ∞)) f) :
Summable (fun h : Fin 2 → ℤ => torusRate s h*
‖mFourierCoeff (fun x => (f (sourceAngularCollar t x):ℂ)) h‖^2) := by
let G : (Fin 4 → ℝ) → ℂ := fun z => (f (sourceAngularPolynomial t z):ℂ)
have hG : ContDiff ℝ (↑(⊤ : ℕ∞)) G := by
exact Complex.ofRealCLM.contDiff.comp (hf.comp (contDiff_sourceAngularPolynomial t))
obtain ⟨K,hK⟩ := hG.contDiffOn.exists_lipschitzOnWith (by simp)
(convex_Icc _ _) isCompact_Icc
let F : C(UnitAddTorus (Fin 2),ℂ) :=
⟨fun x => G (sourceRayCoordinates x),hG.continuous.comp continuous_sourceRayCoordinates⟩
have hdiff (j : Fin 2) (u : ℝ) (x : UnitAddTorus (Fin 2)) :
‖F (x+torusCoordinateShift j u)-F x‖≤((K:ℝ)*sourceRayConstant)*|u| := by
have hh := hK.dist_le_mul _ (sourceRayCoordinates_mem (x+torusCoordinateShift j u))
_ (sourceRayCoordinates_mem x)
rw [dist_eq_norm,dist_eq_norm] at hh
apply hh.trans
simpa only [mul_assoc] using mul_le_mul_of_nonneg_left (sourceRayCoordinates_difference j u x) K.coe_nonneg
exact torus_coordinate_trace_summable s F (fun _ => (K:ℝ)*sourceRayConstant)
(fun _ => mul_nonneg K.coe_nonneg sourceRayConstant_nonneg) hdiff
end ScalarConductivity
end
end OAI
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