| import Mathlib |
| import OAI.Analysis.Conductivity.Branching.ParentCompletedEndJet |
| import OAI.Analysis.Conductivity.Variational.EndPartitionJet |
| import OAI.Analysis.Conductivity.Walls.CentralSeamCutoff |
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| namespace OAI |
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| noncomputable section |
| namespace ScalarConductivity |
| open Set MeasureTheory Filter Topology |
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| theorem centralParent_complement_partition : |
| ∃ (r : ℝ) (η : (Fin 3 → ℝ) → ℝ),0<r ∧ r<centralThickness ∧ |
| ContDiff ℝ (↑(⊤:ℕ∞)) η ∧ HasCompactSupport η ∧ |
| (∀ y,|η y|≤1) ∧ tsupport η⊆sourceClosedCollarBand (-centralThickness/2) r ∧ |
| (∀ y∈centralPhysical,η=ᶠ[𝓝 y] 0) ∧ |
| (∀ y∈sourceClosedCollarBand 0 centralThickness, |
| (fun z => centralJoinPartition 1 z+η z)=ᶠ[𝓝 y] (fun _ => 1)) := by |
| let χ : (Fin 3 → ℝ) → ℝ := centralJoinPartition 1 |
| let q := fun y => 1-χ y |
| let K := sourceClosedCollarBand 0 centralThickness ∩ tsupport q |
| have hK : IsCompact K := |
| (isCompact_sourceClosedCollarBand (show -(1:ℝ)/100≤0 by norm_num) |
| (show centralThickness≤1/100 by norm_num [centralThickness])).inter_right (isClosed_tsupport q) |
| have hKt (y) (hy : y∈K) : sourceCollarTime y<centralThickness := by |
| apply lt_of_le_of_ne hy.1.2 |
| intro he |
| have hn : y∉tsupport q := notMem_tsupport_iff_eventuallyEq.mpr (by |
| filter_upwards [centralJoinPartition_parent_one he] with z hz |
| change 1-χ z=0 |
| change χ z=1 at hz |
| rw [hz]; ring) |
| exact hn hy.2 |
| have hr : ∃ r : ℝ,0<r ∧ r<centralThickness ∧ ∀ y∈K,sourceCollarTime y<r := by |
| by_cases hn : K.Nonempty |
| · obtain ⟨y,hy,hmax⟩ := hK.exists_isMaxOn hn locallyLipschitz_sourceCollarTime.continuous.continuousOn |
| refine ⟨(sourceCollarTime y+centralThickness)/2,?_,?_,?_⟩ |
| · have hp := hy.1.1 |
| have : (0:ℝ)<centralThickness := by norm_num [centralThickness] |
| linarith |
| · linarith [hKt y hy] |
| · intro z hz |
| have hh : sourceCollarTime z ≤ sourceCollarTime y := hmax hz |
| linarith [hKt y hy] |
| · refine ⟨centralThickness/2,by norm_num [centralThickness],by norm_num [centralThickness],?_⟩ |
| intro y hy |
| exact (hn ⟨y,hy⟩).elim |
| obtain ⟨r,hr₀,hr,hrK⟩ := hr |
| have hr' : r≤1/100 := hr.le.trans (by norm_num [centralThickness]) |
| have hbig := isCompact_sourceClosedCollarBand |
| (show -(1:ℝ)/100≤-centralThickness/2 by norm_num [centralThickness]) hr' |
| have hi : K⊆interior (sourceClosedCollarBand (-centralThickness/2) r) := by |
| intro y hy |
| apply mem_interior_iff_mem_nhds.mpr |
| have hm := locallyLipschitz_sourceCollarTime.continuous.continuousAt |
| (isOpen_Ioo.mem_nhds (show sourceCollarTime y∈Ioo (-centralThickness/2) r from |
| ⟨by have := hy.1.1; dsimp [centralThickness] at *; linarith,hrK y hy⟩)) |
| exact mem_of_superset hm (fun z hz => ⟨hz.1.le,hz.2.le⟩) |
| obtain ⟨ψ,hψ1,hψ0,hψb⟩ := exists_contMDiffMap_one_nhds_of_subset_interior |
| (modelWithCornersSelf ℝ (Fin 3 → ℝ)) hK.isClosed hi (n:=↑(⊤:ℕ∞)) |
| have hψc : HasCompactSupport ψ := HasCompactSupport.intro hbig hψ0 |
| have hψs : tsupport ψ⊆sourceClosedCollarBand (-centralThickness/2) r := by |
| apply closure_minimal _ hbig.isClosed |
| intro y hy |
| by_contra hn |
| exact hy (hψ0 y hn) |
| let η := fun y => q y*ψ y |
| have hηs : tsupport η⊆sourceClosedCollarBand (-centralThickness/2) r := |
| tsupport_mul_subset_right.trans hψs |
| refine ⟨r,η,hr₀,hr,(contDiff_const.sub (centralJoinPartition_smooth 1)).mul ψ.contMDiff.contDiff, |
| hψc.mul_left,?_,hηs,?_,?_⟩ |
| · intro y |
| have h₀ := centralJoinPartition.nonneg 1 y |
| have h₁ := centralJoinPartition.le_one 1 y |
| have h₂ := (hψb y).1 |
| have h₃ := (hψb y).2 |
| dsimp only [η,q,χ] |
| rw [abs_of_nonneg (mul_nonneg (sub_nonneg.mpr h₁) h₂)] |
| nlinarith [mul_nonneg h₀ h₂] |
| · intro y hy |
| apply notMem_tsupport_iff_eventuallyEq.mp |
| intro hn |
| have hle := (hηs hn).2 |
| have hge := ((centralPhysical_time_iff y).mp hy).1 |
| linarith |
| · intro y hy |
| by_cases hk : y∈K |
| · have hψ := hψ1.filter_mono (nhds_le_nhdsSet hk) |
| filter_upwards [hψ] with z hz |
| change centralJoinPartition 1 z+((1-centralJoinPartition 1 z)*ψ z)=1 |
| rw [hz] |
| ring |
| · have hq : y∉tsupport q := fun hn => hk ⟨hy,hn⟩ |
| filter_upwards [notMem_tsupport_iff_eventuallyEq.mp hq] with z hz |
| change q z=0 at hz |
| change χ z+q z*ψ z=1 |
| rw [hz,zero_mul,add_zero] |
| dsimp only [q] at hz |
| linarith |
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| end ScalarConductivity |
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| namespace ScalarConductivity |
| open Set MeasureTheory Filter Topology |
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| theorem parentAttachedCorrection_exists (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 : a<0) (κ : ℝ) (p : centralEnergySpace s) : |
| ∃ d : H1,d∈H10 ∧ |
| (∀ᵐ x∂ballMeasure,WithLp.ofLp x∈centralPhysical → |
| weakValue d x=0 ∧ ∀ i,weakGradient d x i=0) ∧ |
| (∀ᵐ x∂ballMeasure,WithLp.ofLp x∈sourceClosedCollarBand 0 centralThickness → |
| weakValue (parentCompletionJoin s hs ha (centralJoinPartition_smooth 1) |
| centralJoinPartition_parent_compact (centralJoinPartition_bound 1) |
| centralJoinPartition_parent_support p+d) x= |
| fullAttachedEndValue s (centralT s 0 p) a centralThickness κ (WithLp.ofLp x) ∧ |
| ∀ i,weakGradient (parentCompletionJoin s hs ha (centralJoinPartition_smooth 1) |
| centralJoinPartition_parent_compact (centralJoinPartition_bound 1) |
| centralJoinPartition_parent_support p+d) x i= |
| fullAttachedEndGradient s (centralT s 0 p) a centralThickness κ (WithLp.ofLp x) i) := by |
| obtain ⟨r,η,hr₀,hr,hη,hηc,hηb,hηs,hηzero,hpart⟩ := centralParent_complement_partition |
| obtain ⟨v,hv,hve⟩ := compact_collar_slope_value_gradient (centralJoinPartition_smooth 1) |
| centralJoinPartition_parent_compact (centralJoinPartition_bound 1) ha.ne |
| (show centralThickness∈Icc (-(1:ℝ)/100) (1/100) by norm_num [centralThickness]) |
| κ 0 (2*centralThickness) (by norm_num) (by norm_num [centralThickness]) centralJoinPartition_parent_support |
| have hR : 0≤-a*(3*centralThickness/2) := |
| mul_nonneg (neg_nonneg.mpr ha.le) (by norm_num [centralThickness]) |
| have hT : ∀ t∈Icc (-centralThickness/2) r, |
| affineEndTime a centralThickness t∈Icc 0 (-a*(3*centralThickness/2)) := by |
| intro t ht |
| dsimp only [affineEndTime] |
| constructor |
| · exact mul_nonneg_of_nonpos_of_nonpos ha.le (sub_nonpos.mpr (ht.2.trans hr.le)) |
| · have hm := mul_nonpos_of_nonpos_of_nonneg ha.le (sub_nonneg.mpr ht.1) |
| nlinarith |
| have hpos : ∀ t∈Icc (-centralThickness/2) r,0<a*(t-centralThickness) := |
| fun t ht => mul_pos_of_neg_of_neg ha (sub_neg.mpr (ht.2.trans_lt hr)) |
| obtain ⟨z,hz,hze⟩ := cut_fullAttachedEnd_H10 s hs (centralT s 0 p) κ ha.ne hR |
| (show centralThickness∈Icc (-(1:ℝ)/100) (1/100) by norm_num [centralThickness]) |
| (show -centralThickness/2≤r by |
| have ht : (0:ℝ)<centralThickness := by norm_num [centralThickness] |
| linarith) |
| (by norm_num [centralThickness]) (hr.le.trans (by norm_num [centralThickness])) |
| hT hpos hη hηc hηb hηs |
| refine ⟨v+z,H10.add_mem hv hz,?_,?_⟩ |
| · filter_upwards [hve,hze,weakValue_add v z,weakGradient_add v z] with x hx hy hval hgrad hcentral |
| have ht := ((centralPhysical_time_iff (WithLp.ofLp x)).mp hcentral).1 |
| have hn : a*(sourceCollarTime (WithLp.ofLp x)-centralThickness)≤0 := |
| mul_nonpos_of_nonpos_of_nonneg ha.le (sub_nonneg.mpr ht) |
| have he : η (WithLp.ofLp x)=0 := (hηzero _ hcentral).self_of_nhds |
| have hd : fderiv ℝ η (WithLp.ofLp x)=0 := by |
| rw [(hηzero _ hcentral).fderiv_eq] |
| exact fderiv_const_apply (0:ℝ) |
| constructor |
| · rw [hval] |
| simp only [Pi.add_apply,hx.1,hy.1,max_eq_left hn,he,mul_zero,zero_mul,add_zero] |
| · intro i |
| rw [hgrad] |
| simp only [WithLp.ofLp_add,Pi.add_apply,hx.2 i,hy.2 i,max_eq_left hn, |
| ite_eq_right (not_lt_of_ge hn),he,hd,zero_apply,mul_zero,zero_mul,add_zero] |
| · have hne := ae_restrict_of_ae (s:=ball) |
| ((PiLp.volume_preserving_ofLp (Fin 3)).quasiMeasurePreserving.ae |
| (sourceColevel_ae_ne (show centralThickness∈Icc (-(1:ℝ)/100) (1/100) by |
| norm_num [centralThickness]))) |
| have hp : ∀ᵐ x∂ballMeasure,WithLp.ofLp x∈sourceClosedCollarBand 0 centralThickness → |
| 0<a*(sourceCollarTime (WithLp.ofLp x)-centralThickness) := by |
| filter_upwards [hne] with x hx hb |
| exact mul_pos_of_neg_of_neg ha (sub_neg.mpr (lt_of_le_of_ne hb.2 hx)) |
| have hh := fullEnd_partition_jet s (centralT s 0 p) a centralThickness κ |
| (centralJoinPartition_smooth 1) hη hpart (v:=v) (z:=z) |
| (u:=parentCompletionJoin s hs ha (centralJoinPartition_smooth 1) |
| centralJoinPartition_parent_compact (centralJoinPartition_bound 1) |
| centralJoinPartition_parent_support p) hp |
| (parentCompletionJoin_end_ae s hs ha (centralJoinPartition_smooth 1) |
| centralJoinPartition_parent_compact (centralJoinPartition_bound 1) |
| centralJoinPartition_parent_support p) hve hze |
| simpa only [add_assoc] using hh |
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| end ScalarConductivity |
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| end |
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| end OAI |
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