| import Mathlib |
| import OAI.Analysis.Conductivity.Branching.ChildCentralL2 |
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|
| namespace OAI |
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| noncomputable section |
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| namespace ScalarConductivity |
|
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| section |
| open Set MeasureTheory Filter Topology |
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|
| 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] |
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| 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) |
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| 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 |
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| def childLocalJetCLM : CentralAmbient s →L[ℝ] JetSpace := |
| physicalFourJetCLM.comp (ContinuousLinearMap.pi (childLocalComponentCLM s hs ha k hχ hc)) |
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| 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) |
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| 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 |
|
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| end ScalarConductivity |
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| end |
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| end OAI |
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