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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 | import Mathlib
import OAI.Analysis.Conductivity.Fourier.EndPoissonJets
import OAI.Analysis.Conductivity.Sources.AngularCoefficient
import OAI.Analysis.Conductivity.Geometry.TorusDirectionalDerivative
namespace OAI
section
noncomputable section
namespace ScalarConductivity
open Set MeasureTheory Filter Topology UnitAddTorus
open scoped ENNReal NNReal
local instance angularJetMeasureSpaceUnitAddCircle : MeasureSpace UnitAddCircle := ⟨AddCircle.haarAddCircle⟩
local instance angularJetIsProbabilityMeasure : IsProbabilityMeasure (volume : Measure UnitAddCircle) :=
inferInstanceAs (IsProbabilityMeasure AddCircle.haarAddCircle)
lemma measurable_torusDirectionalDerivative_joint
{f : ℝ × UnitAddTorus (Fin 2) → ℂ} (hf : Continuous f) (j : Fin 2) :
Measurable (fun z : ℝ × UnitAddTorus (Fin 2) =>
torusDirectionalDerivative (fun θ => f (z.1,θ)) j z.2) := by
let F : ℝ × TorusRealPlane → ℂ := fun z => f (z.1,torusRealProjection z.2)
have hc : Continuous F := hf.comp
(continuous_fst.prodMk (continuous_torusRealProjection.comp continuous_snd))
have hm := (measurable_lineDeriv (𝕜:=ℝ) hc (v:=(0,Pi.single j 1))).comp
(measurable_fst.prodMk (measurable_torusRealRepresentative.comp measurable_snd))
convert hm using 1
funext z
unfold torusDirectionalDerivative lineDeriv
congr 2
ext t
simp [F]
def sourceAngularTest (f : (Fin 3 → ℝ) → ℝ) (a b : ℝ)
(z : ℝ × UnitAddTorus (Fin 2)) : ℂ :=
(f (sourceAngularCollar (a*z.1+b) z.2) : ℂ)
lemma continuous_sourceAngularTest {f : (Fin 3 → ℝ) → ℝ}
(hf : ContDiff ℝ (↑(⊤ : ℕ∞)) f) (a b : ℝ) :
Continuous (sourceAngularTest f a b) :=
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))))
theorem sourceAngularTest_lipschitz {f : (Fin 3 → ℝ) → ℝ}
(hf : ContDiff ℝ (↑(⊤ : ℕ∞)) f) (a b R : ℝ) :
∃ K : ℝ≥0, ∀ t∈Icc (0:ℝ) R, ∀ j u θ,
‖sourceAngularTest f a b (t,θ+torusCoordinateShift j u)-
sourceAngularTest f a b (t,θ)‖≤(K:ℝ)*|u| := by
let G : ℝ × (Fin 4 → ℝ) → ℂ := fun z =>
(f (sourceAngularPolynomial (a*z.1+b) z.2) : ℂ)
have hG : ContDiff ℝ (↑(⊤ : ℕ∞)) G :=
Complex.ofRealCLM.contDiff.comp (hf.comp
(contDiff_sourceAngularPolynomial_joint.comp
((contDiff_const.mul contDiff_fst |>.add contDiff_const).prodMk contDiff_snd)))
obtain ⟨K,hK⟩ := hG.contDiffOn.exists_lipschitzOnWith (by simp)
((convex_Icc (0:ℝ) R).prod (convex_Icc _ _))
(isCompact_Icc.prod (isCompact_Icc (a:=(fun _ : Fin 4 => (-2:ℝ))) (b:=fun _ => (2:ℝ))))
refine ⟨K*⟨sourceRayConstant,sourceRayConstant_nonneg⟩,?_⟩
intro t ht j u θ
have hh := hK.dist_le_mul (t,sourceRayCoordinates (θ+torusCoordinateShift j u))
⟨ht,sourceRayCoordinates_mem _⟩ (t,sourceRayCoordinates θ) ⟨ht,sourceRayCoordinates_mem _⟩
rw [Prod.dist_eq,dist_self,max_eq_right (dist_nonneg),dist_eq_norm,dist_eq_norm] at hh
apply hh.trans
change _ ≤ (K:ℝ)*sourceRayConstant*|u|
simpa only [mul_assoc] using
mul_le_mul_of_nonneg_left (sourceRayCoordinates_difference j u θ) K.coe_nonneg
def sourceAngularJet (f : (Fin 3 → ℝ) → ℝ) (a b : ℝ)
(j : Fin 4) (z : ℝ × UnitAddTorus (Fin 2)) : ℂ :=
![sourceAngularTest f a b z,sourceAngularAxial f a b z.1 z.2,
torusDirectionalDerivative (fun θ => sourceAngularTest f a b (z.1,θ)) 0 z.2/(2*Real.pi),
torusDirectionalDerivative (fun θ => sourceAngularTest f a b (z.1,θ)) 1 z.2/(2*Real.pi)] j
lemma measurable_sourceAngularJet {f : (Fin 3 → ℝ) → ℝ}
(hf : ContDiff ℝ (↑(⊤ : ℕ∞)) f) (a b : ℝ) (j : Fin 4) :
Measurable (sourceAngularJet f a b j) := by
fin_cases j
· exact (continuous_sourceAngularTest hf a b).measurable
· exact (continuous_sourceAngularAxial hf a b).measurable
· exact (measurable_torusDirectionalDerivative_joint (continuous_sourceAngularTest hf a b) 0).div_const _
· exact (measurable_torusDirectionalDerivative_joint (continuous_sourceAngularTest hf a b) 1).div_const _
end ScalarConductivity
end
end
section
noncomputable section
namespace ScalarConductivity
open Set MeasureTheory Filter Topology UnitAddTorus
open scoped ENNReal NNReal
local instance angularJetFiniteCylinderMeasureSpace : MeasureSpace UnitAddCircle :=
⟨AddCircle.haarAddCircle⟩
local instance angularJetFiniteCylinderProbabilityMeasure :
IsProbabilityMeasure (volume : Measure UnitAddCircle) :=
inferInstanceAs (IsProbabilityMeasure AddCircle.haarAddCircle)
lemma finiteCylinder_ae_mem (R : ℝ) :
∀ᵐ z ∂(FiniteAxisMeasure R).prod (volume : Measure (UnitAddTorus (Fin 2))),
z.1∈Icc (0:ℝ) R :=
Measure.quasiMeasurePreserving_fst.ae
((ae_restrict_mem measurableSet_Ioc).mono fun _ ht => Ioc_subset_Icc_self ht)
lemma continuous_memLp_finiteCylinder {f : ℝ × UnitAddTorus (Fin 2) → ℂ}
(hf : Continuous f) (R : ℝ) :
MemLp f 2 ((FiniteAxisMeasure R).prod volume) := by
obtain ⟨C,hC⟩ := ((isCompact_Icc (a:=(0:ℝ)) (b:=R)).prod
(isCompact_univ : IsCompact (univ : Set (UnitAddTorus (Fin 2))))).exists_bound_of_continuousOn hf.continuousOn
apply MemLp.of_bound hf.aestronglyMeasurable C
filter_upwards [finiteCylinder_ae_mem R] with z hz
exact hC z ⟨hz,mem_univ _⟩
lemma sourceAngularJet_memLp {f : (Fin 3 → ℝ) → ℝ}
(hf : ContDiff ℝ (↑(⊤ : ℕ∞)) f) (a b R : ℝ) (j : Fin 4) :
MemLp (sourceAngularJet f a b j) 2 ((FiniteAxisMeasure R).prod volume) := by
obtain ⟨K,hK⟩ := sourceAngularTest_lipschitz hf a b R
have hang (k : Fin 2) : MemLp (fun z : ℝ × UnitAddTorus (Fin 2) =>
torusDirectionalDerivative (fun θ => sourceAngularTest f a b (z.1,θ)) k z.2/(2*Real.pi))
2 ((FiniteAxisMeasure R).prod volume) := by
apply MemLp.of_bound
(((measurable_torusDirectionalDerivative_joint (continuous_sourceAngularTest hf a b) k).div_const _).aestronglyMeasurable) ((2:ℝ)*K/‖(2*Real.pi:ℂ)‖)
filter_upwards [finiteCylinder_ae_mem R] with z hz
rw [norm_div]
apply div_le_div_of_nonneg_right _ (norm_nonneg _)
exact torusDirectionalDerivative_norm
⟨fun θ => sourceAngularTest f a b (z.1,θ),
(continuous_sourceAngularTest hf a b).comp (continuous_const.prodMk continuous_id)⟩
K (hK z.1 hz) k z.2
fin_cases j
· exact continuous_memLp_finiteCylinder (continuous_sourceAngularTest hf a b) R
· exact continuous_memLp_finiteCylinder (continuous_sourceAngularAxial hf a b) R
· exact hang 0
· exact hang 1
def sourceAngularLpJet {f : (Fin 3 → ℝ) → ℝ}
(hf : ContDiff ℝ (↑(⊤ : ℕ∞)) f) (a b R : ℝ) : FiniteCylinderJets R :=
WithLp.toLp 2 (fun j => (sourceAngularJet_memLp hf a b R j).toLp _)
lemma sourceAngularLpJet_ae {f : (Fin 3 → ℝ) → ℝ}
(hf : ContDiff ℝ (↑(⊤ : ℕ∞)) f) (a b R : ℝ) (j : Fin 4) :
sourceAngularLpJet hf a b R j =ᵐ[(FiniteAxisMeasure R).prod volume]
sourceAngularJet f a b j :=
(sourceAngularJet_memLp hf a b R j).coeFn_toLp
lemma mFourierCoeff_div_const (f : UnitAddTorus (Fin 2) → ℂ) (h : TorusModes) (c : ℂ) :
mFourierCoeff (fun θ => f θ/c) h=mFourierCoeff f h/c := by
simp only [mFourierCoeff,smul_eq_mul,←mul_div_assoc,integral_div]
lemma torusDirectionalDerivative_scaled_fourier (f : C(UnitAddTorus (Fin 2),ℂ)) (K : ℝ≥0)
(hLip : ∀ j t θ, ‖f (θ+torusCoordinateShift j t)-f θ‖≤(K:ℝ)*|t|)
(j : Fin 2) (h : TorusModes) :
mFourierCoeff (fun θ => torusDirectionalDerivative f j θ/(2*Real.pi)) h=
(Complex.I*(h j))*mFourierCoeff f h := by
rw [mFourierCoeff_div_const,torusDirectionalDerivative_fourier f K hLip]
have hp : (Real.pi:ℂ)≠0 := Complex.ofReal_ne_zero.mpr Real.pi_ne_zero
field_simp
lemma sourceAngularJet_fourier {f : (Fin 3 → ℝ) → ℝ}
(hf : ContDiff ℝ (↑(⊤ : ℕ∞)) f) (a b R : ℝ) (h : TorusModes)
{t : ℝ} (ht : t∈Icc (0:ℝ) R) (j : Fin 4) :
mFourierCoeff (fun θ => sourceAngularJet f a b j (t,θ)) h=
![sourceAngularCoefficient f a b h t,deriv (sourceAngularCoefficient f a b h) t,
(Complex.I*(h 0))*sourceAngularCoefficient f a b h t,
(Complex.I*(h 1))*sourceAngularCoefficient f a b h t] j := by
obtain ⟨K,hK⟩ := sourceAngularTest_lipschitz hf a b R
have ha (k : Fin 2) := torusDirectionalDerivative_scaled_fourier
⟨fun θ => sourceAngularTest f a b (t,θ),
(continuous_sourceAngularTest hf a b).comp (continuous_const.prodMk continuous_id)⟩
K (hK t ht) k h
fin_cases j
· rfl
· exact (sourceAngularCoefficient_hasDerivAt hf a b h t).deriv.symm
· exact ha 0
· exact ha 1
end ScalarConductivity
end
end
end OAI
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