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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 | import Mathlib
import OAI.Analysis.Conductivity.Branching.ChildCentralL2
namespace OAI
noncomputable section
namespace ScalarConductivity
section
open Set MeasureTheory Filter Topology
lemma child_band_indicator_join {a : ℝ} (ha : 0<a) (y : Fin 3 → ℝ)
(hy : y∈sourceClosedCollarBand (-2*centralThickness) 0)
(hne : sourceCollarTime y≠-centralThickness) (q p : (Fin 3 → ℝ) → ℝ) :
(sourceClosedCollarBand (-2*centralThickness) (-centralThickness)).indicator q y+
(sourceClosedCollarBand (-centralThickness) 0).indicator p y=
if 0<a*(sourceCollarTime y-(-centralThickness)) then p y else q y := by
rcases lt_or_gt_of_ne hne with ht | ht
· have hp : ¬0<a*(sourceCollarTime y-(-centralThickness)) :=
not_lt_of_ge (mul_nonpos_of_nonneg_of_nonpos ha.le (sub_nonpos.mpr ht.le))
have hc : y∈sourceClosedCollarBand (-2*centralThickness) (-centralThickness) := ⟨hy.1,ht.le⟩
have he : y∉sourceClosedCollarBand (-centralThickness) 0 := fun h => not_le_of_gt ht h.1
simp only [indicator_of_mem hc,indicator_of_notMem he,add_zero,ite_eq_right hp]
· have hp : 0<a*(sourceCollarTime y-(-centralThickness)) := mul_pos ha (sub_pos.mpr ht)
have hc : y∉sourceClosedCollarBand (-2*centralThickness) (-centralThickness) := fun h => not_le_of_gt ht h.2
have he : y∈sourceClosedCollarBand (-centralThickness) 0 := ⟨ht.le,hy.2⟩
simp only [indicator_of_notMem hc,indicator_of_mem he,zero_add,ite_eq_left hp]
variable (s : Fin 3 → ℝ)
(hs : ∀ u v : ℝ,(1/2)*(u^2+v^2) ≤ s 0*u^2+2*s 1*u*v+s 2*v^2)
{a : ℝ} (ha : 0<a)
(k : Fin 2)
{χ : (Fin 3 → ℝ) → ℝ} (hχ : ContDiff ℝ (↑(⊤:ℕ∞)) χ) (hc : HasCompactSupport χ)
def childLocalComponentCLM (i : Fin 4) : CentralAmbient s →L[ℝ]
Lp ℝ 2 (volume : Measure (Fin 3 → ℝ)) :=
Fin.cases ((compactMultiplierCLM hχ.continuous hc).comp (childJoinedComponentCLM s hs ha k 0))
(fun j => ((compactMultiplierCLM ((hχ.continuous_fderiv (by simp)).clm_apply continuous_const)
(hc.fderiv_apply ℝ (Pi.single j 1))).comp (childJoinedComponentCLM s hs ha k 0))+
((compactMultiplierCLM hχ.continuous hc).comp (childJoinedComponentCLM s hs ha k j.succ))) i
def childLocalJetCLM : CentralAmbient s →L[ℝ] JetSpace :=
physicalFourJetCLM.comp (ContinuousLinearMap.pi (childLocalComponentCLM s hs ha k hχ hc))
end
open Set MeasureTheory Filter Topology
variable (s : Fin 3 → ℝ)
(hs : ∀ u v : ℝ,(1/2)*(u^2+v^2) ≤ s 0*u^2+2*s 1*u*v+s 2*v^2)
{a : ℝ} (ha : 0<a)
(k : Fin 2)
{χ : (Fin 3 → ℝ) → ℝ} (hχ : ContDiff ℝ (↑(⊤:ℕ∞)) χ) (hc : HasCompactSupport χ)
(hχs : tsupport χ⊆sourceClosedCollarBand (-2*centralThickness) 0)
lemma childJoinedComponentCLM_smooth_value (f : centralSmoothFunctions) :
childJoinedComponentCLM s hs ha k 0 (centralEmbedL s f)=ᵐ[volume] (fun y =>
(sourceClosedCollarBand (-2*centralThickness) (-centralThickness)).indicator
(fun y => f (sourcePairCoordinates (sourceChildCoordinates (actualChildSign k) y))) y+
(sourceClosedCollarBand (-centralThickness) 0).indicator
(fun y => (attachedEndPoissonField s (centralSmoothTrace s f k.succ) a (-centralThickness) 0 y).re) y) := by
apply (Lp.coeFn_add _ _).trans
filter_upwards [childCentralComponentCLM_smooth_value s k f,childEndValueCLM_ae s hs ha (centralSmoothTrace s f k.succ)] with y hy hz
change (childCentralComponentCLM s k 0 (centralEmbedL s f)) y+
(childEndValueCLM s hs ha (centralSmoothTrace s f k.succ)) y=_
rw [hy,hz]
lemma childJoinedComponentCLM_smooth_gradient (f : centralSmoothFunctions) (i : Fin 3) :
childJoinedComponentCLM s hs ha k i.succ (centralEmbedL s f)=ᵐ[volume] (fun y =>
(sourceClosedCollarBand (-2*centralThickness) (-centralThickness)).indicator
(fun y => fderiv ℝ (fun y => f (sourcePairCoordinates (sourceChildCoordinates (actualChildSign k) y))) y (Pi.single i 1)) y+
(sourceClosedCollarBand (-centralThickness) 0).indicator
(fun y => fderiv ℝ (fun y => (attachedEndPoissonField s (centralSmoothTrace s f k.succ) a (-centralThickness) 0 y).re) y (Pi.single i 1)) y) := by
apply (Lp.coeFn_add _ _).trans
filter_upwards [childCentralComponentCLM_smooth_gradient s k f i,childEndGradientCLM_ae s hs ha (centralSmoothTrace s f k.succ) i] with y hy hz
change (childCentralComponentCLM s k i.succ (centralEmbedL s f)) y+
(childEndGradientCLM s hs ha i (centralSmoothTrace s f k.succ)) y=_
rw [hy,hz]
include hχs in
lemma childLocalComponentCLM_smooth_value (f : centralSmoothFunctions) :
childLocalComponentCLM s hs ha k hχ hc 0 (centralEmbedL s f)=ᵐ[volume]
localJoinedValue (fun y => a*(sourceCollarTime y-(-centralThickness))) χ
(fun y => f (sourcePairCoordinates (sourceChildCoordinates (actualChildSign k) y)))
(fun y => (attachedEndPoissonField s (centralSmoothTrace s f k.succ) a (-centralThickness) 0 y).re) := by
have he := compactMultiplierCLM_ae hχ.continuous hc (childJoinedComponentCLM s hs ha k 0 (centralEmbedL s f))
filter_upwards [he,childJoinedComponentCLM_smooth_value s hs ha k f,
sourceColevel_ae_ne (show -centralThickness∈Icc (-(1:ℝ)/100) (1/100) by norm_num [centralThickness])] with y hy hv hn
simp only [childLocalComponentCLM, Fin.cases_zero, ContinuousLinearMap.comp_apply, localJoinedValue]
rw [hy,hv]
by_cases hys : y∈tsupport χ
· rw [child_band_indicator_join ha y (hχs hys) hn]
· rw [image_eq_zero_of_notMem_tsupport hys]
simp
include hχs in
lemma childLocalComponentCLM_smooth_gradient (f : centralSmoothFunctions) (i : Fin 3) :
childLocalComponentCLM s hs ha k hχ hc i.succ (centralEmbedL s f)=ᵐ[volume]
localJoinedGradient (fun y => a*(sourceCollarTime y-(-centralThickness))) χ
(fun y => f (sourcePairCoordinates (sourceChildCoordinates (actualChildSign k) y)))
(fun y => (attachedEndPoissonField s (centralSmoothTrace s f k.succ) a (-centralThickness) 0 y).re) i := by
have he₀ := compactMultiplierCLM_ae ((hχ.continuous_fderiv (by simp)).clm_apply continuous_const)
(hc.fderiv_apply ℝ (Pi.single i 1)) (childJoinedComponentCLM s hs ha k 0 (centralEmbedL s f))
have he₁ := compactMultiplierCLM_ae hχ.continuous hc (childJoinedComponentCLM s hs ha k i.succ (centralEmbedL s f))
apply (Lp.coeFn_add _ _).trans
filter_upwards [he₀,he₁,childJoinedComponentCLM_smooth_value s hs ha k f,
childJoinedComponentCLM_smooth_gradient s hs ha k f i,
sourceColevel_ae_ne (show -centralThickness∈Icc (-(1:ℝ)/100) (1/100) by norm_num [centralThickness])] with y h₀ h₁ hv hg hn
change (compactMultiplierCLM ((hχ.continuous_fderiv (by simp)).clm_apply continuous_const)
(hc.fderiv_apply ℝ (Pi.single i 1)) (childJoinedComponentCLM s hs ha k 0 (centralEmbedL s f))) y+
(compactMultiplierCLM hχ.continuous hc (childJoinedComponentCLM s hs ha k i.succ (centralEmbedL s f))) y=_
rw [h₀,h₁,hv,hg]
simp only [localJoinedGradient]
by_cases hys : y∈tsupport χ
· rw [child_band_indicator_join ha y (hχs hys) hn,
child_band_indicator_join ha y (hχs hys) hn]
· rw [image_eq_zero_of_notMem_tsupport hys,fderiv_of_notMem_tsupport ℝ hys]
simp
include hχs in
lemma childLocalJetCLM_smooth_ae (f : centralSmoothFunctions) :
childLocalJetCLM s hs ha k hχ hc (centralEmbedL s f)=ᵐ[ballMeasure] (fun x =>
WithLp.toLp 2 (Fin.cases
(localJoinedValue (fun y => a*(sourceCollarTime y-(-centralThickness))) χ
(fun y => f (sourcePairCoordinates (sourceChildCoordinates (actualChildSign k) y)))
(fun y => (attachedEndPoissonField s (centralSmoothTrace s f k.succ) a (-centralThickness) 0 y).re) (WithLp.ofLp x))
(fun i => localJoinedGradient (fun y => a*(sourceCollarTime y-(-centralThickness))) χ
(fun y => f (sourcePairCoordinates (sourceChildCoordinates (actualChildSign k) y)))
(fun y => (attachedEndPoissonField s (centralSmoothTrace s f k.succ) a (-centralThickness) 0 y).re) i (WithLp.ofLp x)))) := by
have h₀ := ae_restrict_of_ae (s:=ball) ((PiLp.volume_preserving_ofLp (Fin 3)).quasiMeasurePreserving.ae_eq_comp
(childLocalComponentCLM_smooth_value s hs ha k hχ hc hχs f))
have h₁ (i : Fin 3) := ae_restrict_of_ae (s:=ball) ((PiLp.volume_preserving_ofLp (Fin 3)).quasiMeasurePreserving.ae_eq_comp
(childLocalComponentCLM_smooth_gradient s hs ha k hχ hc hχs f i))
apply (physicalFourJetCLM_ae (fun i => childLocalComponentCLM s hs ha k hχ hc i (centralEmbedL s f))).trans
filter_upwards [h₀,ae_all_iff.mpr h₁] with x hv hg
ext i
refine Fin.cases ?_ (fun j => ?_) i
· exact hv
· exact hg j
end ScalarConductivity
end
end OAI
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