| import Mathlib |
| import OAI.Analysis.Conductivity.Sobolev.CentralNeighborhoodCompletion |
| import OAI.Analysis.Conductivity.Variational.WholeBallJetFormula |
| import OAI.Analysis.Conductivity.Sobolev.PhysicalProductJet |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace ScalarConductivity |
|
|
| section |
| 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) |
|
|
| lemma centralInteriorCompletion_smooth_ae (f : centralSmoothFunctions) : |
| (centralInteriorCompletion s (centralJoinPartition_smooth 0) |
| centralJoinPartition_interior_compact centralJoinPartition_interior_support |
| (fun _ hy => centralPhysical_subset_ball (centralJoinPartition_interior_support hy)) |
| (centralSmoothElement s f)).val=ᵐ[ballMeasure] |
| (fun x => piSmoothJet ((centralJoinPartition 0)*(fun y => f (sourcePairCoordinates y))) (WithLp.ofLp x)) := by |
| have h₀ := ae_restrict_of_ae (s:=ball) ((PiLp.volume_preserving_ofLp (Fin 3)).quasiMeasurePreserving.ae_eq_comp |
| (centralInteriorComponentCLM_smooth_value s (centralJoinPartition_smooth 0) |
| centralJoinPartition_interior_compact centralJoinPartition_interior_support f)) |
| have h₁ (i : Fin 3) := ae_restrict_of_ae (s:=ball) ((PiLp.volume_preserving_ofLp (Fin 3)).quasiMeasurePreserving.ae_eq_comp |
| (centralInteriorComponentCLM_smooth_gradient s (centralJoinPartition_smooth 0) |
| centralJoinPartition_interior_compact centralJoinPartition_interior_support f i)) |
| filter_upwards [physicalFourJetCLM_ae (fun i => centralInteriorComponentCLM s |
| (centralJoinPartition_smooth 0) centralJoinPartition_interior_compact i (centralEmbedL s f)),h₀,ae_all_iff.mpr h₁] with x hj hv hg |
| change physicalFourJetCLM (fun i => centralInteriorComponentCLM s |
| (centralJoinPartition_smooth 0) centralJoinPartition_interior_compact i (centralEmbedL s f)) x=_ |
| rw [hj] |
| ext i |
| refine Fin.cases ?_ (fun j => ?_) i |
| · exact hv |
| · exact hg j |
|
|
| lemma parentCompletionJoin_smooth_central {a : ℝ} (ha : a<0) (f : centralSmoothFunctions) : |
| ∀ᵐ x∂ballMeasure,WithLp.ofLp x∈centralPhysical → |
| (parentCompletionJoin s hs ha (centralJoinPartition_smooth 1) |
| centralJoinPartition_parent_compact (centralJoinPartition_bound 1) |
| centralJoinPartition_parent_support (centralSmoothElement s f)).val x= |
| piSmoothJet ((centralJoinPartition 1)*(fun y => f (sourcePairCoordinates y))) (WithLp.ofLp x) := by |
| filter_upwards [parentLocalJetCLM_smooth_ae s hs ha (centralJoinPartition_smooth 1) |
| centralJoinPartition_parent_compact centralJoinPartition_parent_support f] with x hx |
| intro hy |
| change parentLocalJetCLM s hs ha _ _ (centralEmbedL s f) x=_ |
| rw [hx] |
| apply localJoinedJet_eq_piSmoothJet |
| · exact mul_nonpos_of_nonpos_of_nonneg ha.le (sub_nonneg.mpr ((centralPhysical_time_iff _).mp hy).1) |
| · exact (centralJoinPartition_smooth 1).differentiable (by simp) _ |
| · exact ((central_smooth f).comp sourcePairCLE.contDiff).differentiable (by simp) _ |
|
|
| lemma physicalChildCompletionJoin_smooth_central {a : ℝ} (ha : 0<a) (k : Fin 2) |
| (f : centralSmoothFunctions) : |
| ∀ᵐ x∂ballMeasure,WithLp.ofLp x∈centralPhysical → |
| (physicalChildCompletionJoin s hs ha k (centralJoinChildCutoff_smooth k) |
| (centralJoinChildCutoff_compact k) (centralJoinChildCutoff_bound k) |
| (centralJoinChildCutoff_support k) (centralSmoothElement s f)).val x= |
| piSmoothJet ((centralJoinPartition (centralChildPatch k))*(fun y => f (sourcePairCoordinates y))) (WithLp.ofLp x) := by |
| have hB : ∀ y∈tsupport (centralJoinChildCutoff k),WithLp.toLp 2 y∈ball := by |
| intro y hy |
| have hb := centralJoinChildCutoff_support k hy |
| exact sourceBand_mem_ball ⟨le_trans (by norm_num [centralThickness]) hb.1, |
| le_trans hb.2 (by norm_num)⟩ |
| have hw := ballWholePiJetCLM_localJoined_ae hB |
| (childLocalJetCLM_smooth_ae s hs ha k (centralJoinChildCutoff_smooth k) |
| (centralJoinChildCutoff_compact k) (centralJoinChildCutoff_support k) f) |
| filter_upwards [childTransportJetCLM_ae_of_ae k _ _ hw] with x hx |
| intro hy |
| change childTransportJetCLM k (childLocalJetCLM s hs ha k |
| (centralJoinChildCutoff_smooth k) (centralJoinChildCutoff_compact k) (centralEmbedL s f)) x=_ |
| rw [hx] |
| let y := (sourceChildHomeomorph (actualChildSign k)).symm (WithLp.ofLp x) |
| have ht : a*(sourceCollarTime y-(-centralThickness)) ≤ 0 := |
| mul_nonpos_of_nonneg_of_nonpos ha.le (sub_nonpos.mpr (((centralPhysical_time_iff _).mp hy).2 k)) |
| have hq := ((central_smooth f).comp sourcePairCLE.contDiff).comp (sourceChildCoordinates_contDiff (actualChildSign k)) |
| have he := localJoinedJet_eq_piSmoothJet |
| (τ:=fun y => a*(sourceCollarTime y-(-centralThickness))) |
| (χ:=centralJoinChildCutoff k) |
| (q:=fun y => f (sourcePairCoordinates (sourceChildCoordinates (actualChildSign k) y))) |
| (p:=fun y => |
| (attachedEndPoissonField s (centralSmoothTrace s f k.succ) a (-centralThickness) 0 y).re) |
| ht ((centralJoinChildCutoff_smooth k).differentiable (by simp) y) (hq.differentiable (by simp) y) |
| have he' := congrArg (fun z : JetFiber => (WithLp.toLp 2 |
| (Fin.cases (z 0) (fun j => sourceScale⁻¹*z (childAxis j).succ)) : JetFiber)) he |
| exact he'.trans (sourceChild_piSmoothJet_cancel |
| ((centralJoinPartition (centralChildPatch k))*(fun y => f (sourcePairCoordinates y))) |
| ((centralJoinPartition_smooth _).mul ((central_smooth f).comp sourcePairCLE.contDiff)) |
| (actualChildSign k) (WithLp.ofLp x)) |
|
|
| end |
|
|
| open Set MeasureTheory Filter Topology |
|
|
| lemma centralPhysical_ae_interior : |
| ∀ᵐ y : Fin 3 → ℝ,y∈centralPhysical → y∈interior centralPhysical := by |
| have hp := sourceColevel_ae_ne |
| (show centralThickness∈Icc (-(1:ℝ)/100) (1/100) by norm_num [centralThickness]) |
| have hc (k : Fin 2) := (sourceChildInverse_quasi (actualChildSign k)).ae |
| (sourceColevel_ae_ne (show -centralThickness∈Icc (-(1:ℝ)/100) (1/100) by norm_num [centralThickness])) |
| filter_upwards [hp,ae_all_iff.mpr hc] with y hp hc |
| intro hy |
| have ht := (centralPhysical_time_iff y).mp hy |
| exact centralPhysical_mem_interior_of_strict (lt_of_le_of_ne ht.1 hp.symm) |
| (fun k => lt_of_le_of_ne (ht.2 k) (hc k)) |
|
|
| lemma centralPhysical_ball_ae_interior : |
| ∀ᵐ x∂ballMeasure,WithLp.ofLp x∈centralPhysical → WithLp.ofLp x∈interior centralPhysical := |
| ae_restrict_of_ae ((PiLp.volume_preserving_ofLp (Fin 3)).quasiMeasurePreserving.ae centralPhysical_ae_interior) |
|
|
| 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 : Fin 3 → ℝ} (ha₀ : a 0<0) (ha : ∀ k : Fin 2,0<a k.succ) |
|
|
| def centralNeighborhoodRawCLM : CentralAmbient s →L[ℝ] JetSpace := |
| centralInteriorJetCLM s (centralJoinPartition_smooth 0) centralJoinPartition_interior_compact+ |
| parentLocalJetCLM s hs ha₀ (centralJoinPartition_smooth 1) centralJoinPartition_parent_compact+ |
| ((childTransportJetCLM 0).comp (childLocalJetCLM s hs (ha 0) 0 |
| (centralJoinChildCutoff_smooth 0) (centralJoinChildCutoff_compact 0))+ |
| (childTransportJetCLM 1).comp (childLocalJetCLM s hs (ha 1) 1 |
| (centralJoinChildCutoff_smooth 1) (centralJoinChildCutoff_compact 1))) |
|
|
| lemma centralNeighborhoodCompletion_val (u : centralEnergySpace s) : |
| (centralNeighborhoodCompletion s hs ha₀ ha u).val=centralNeighborhoodRawCLM s hs ha₀ ha u.val := by |
| simp only [centralNeighborhoodCompletion,Fin.sum_univ_two,centralNeighborhoodRawCLM, |
| add_apply,ContinuousLinearMap.comp_apply,Submodule.coe_add] |
| rfl |
|
|
| def centralFullJetCLM : CentralAmbient s →L[ℝ] JetSpace := |
| physicalFourJetCLM.comp (ContinuousLinearMap.pi (fun i => |
| centralPhysicalWholeCLM.comp (centralAmbientComponent s i))) |
|
|
| lemma centralFullJetCLM_smooth_ae (f : centralSmoothFunctions) : |
| ∀ᵐ x∂ballMeasure,WithLp.ofLp x∈centralPhysical → |
| centralFullJetCLM s (centralEmbedL s f) x= |
| piSmoothJet (fun y => f (sourcePairCoordinates y)) (WithLp.ofLp x) := by |
| have hi (i : Fin 4) := ae_restrict_of_ae (s:=ball) |
| ((PiLp.volume_preserving_ofLp (Fin 3)).quasiMeasurePreserving.ae_eq_comp |
| (centralPhysicalWholeCLM_smooth_ae f i)) |
| filter_upwards [physicalFourJetCLM_ae (fun i => centralPhysicalWholeCLM (centralSmoothJet f i)), |
| ae_all_iff.mpr hi] with x hx hi |
| intro hy |
| simp only [Function.comp_apply] at hi |
| change physicalFourJetCLM (fun i => centralPhysicalWholeCLM (centralSmoothJet f i)) x=_ |
| rw [hx] |
| ext i |
| refine Fin.cases ?_ (fun j => ?_) i |
| · simpa only [indicator_of_mem hy,piSmoothJet,PiLp.toLp_apply,Fin.cases_zero, |
| centralJetField,Matrix.cons_val_zero] using hi 0 |
| · change centralPhysicalWholeCLM (centralSmoothJet f j.succ) (WithLp.ofLp x)=_ |
| rw [hi j.succ,indicator_of_mem hy,central_parent_partial] |
| rfl |
|
|
| lemma centralNeighborhoodRawCLM_smooth_ae (f : centralSmoothFunctions) : |
| ∀ᵐ x∂ballMeasure,WithLp.ofLp x∈centralPhysical → |
| centralNeighborhoodRawCLM s hs ha₀ ha (centralEmbedL s f) x= |
| piSmoothJet (fun y => f (sourcePairCoordinates y)) (WithLp.ofLp x) := by |
| let zi := centralInteriorJetCLM s (centralJoinPartition_smooth 0) |
| centralJoinPartition_interior_compact (centralEmbedL s f) |
| let zp := parentLocalJetCLM s hs ha₀ (centralJoinPartition_smooth 1) |
| centralJoinPartition_parent_compact (centralEmbedL s f) |
| let zc (k : Fin 2) := childTransportJetCLM k (childLocalJetCLM s hs (ha k) k |
| (centralJoinChildCutoff_smooth k) (centralJoinChildCutoff_compact k) (centralEmbedL s f)) |
| filter_upwards [Lp.coeFn_add (zi+zp) (zc 0+zc 1),Lp.coeFn_add zi zp,Lp.coeFn_add (zc 0) (zc 1), |
| centralInteriorCompletion_smooth_ae s f,parentCompletionJoin_smooth_central s hs ha₀ f, |
| physicalChildCompletionJoin_smooth_central s hs (ha 0) 0 f, |
| physicalChildCompletionJoin_smooth_central s hs (ha 1) 1 f, |
| centralPhysical_ball_ae_interior] with x hz hip hcc hi hp hc₀ hc₁ hinter |
| intro hy |
| change (zi+zp+(zc 0+zc 1)) x=_ |
| rw [hz] |
| change (zi+zp) x+(zc 0+zc 1) x=_ |
| rw [hip,hcc] |
| change zi x+zp x+(zc 0 x+zc 1 x)=_ |
| change zi x=_ at hi |
| have hp' : zp x=piSmoothJet ((centralJoinPartition 1)*(fun y => f (sourcePairCoordinates y))) (WithLp.ofLp x) := hp hy |
| have h₀ : zc 0 x=piSmoothJet ((centralJoinPartition 2)*(fun y => f (sourcePairCoordinates y))) (WithLp.ofLp x) := hc₀ hy |
| have h₁ : zc 1 x=piSmoothJet ((centralJoinPartition 3)*(fun y => f (sourcePairCoordinates y))) (WithLp.ofLp x) := hc₁ hy |
| rw [hi,hp',h₀,h₁] |
| have he := piSmoothJet_partition (q:=fun y => f (sourcePairCoordinates y)) centralJoinPartition_smooth |
| ((central_smooth f).comp sourcePairCLE.contDiff) (WithLp.ofLp x) |
| (centralJoinPartition_sum hy) (centralJoinPartition_derivative_sum (hinter hy)) |
| simpa only [Fin.sum_univ_four,add_assoc] using he |
|
|
| end ScalarConductivity |
|
|
| end |
|
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| end OAI |
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|
