import Mathlib import OAI.Analysis.Conductivity.Branching.ParentCompletedEndJet import OAI.Analysis.Conductivity.Variational.EndPartitionJet import OAI.Analysis.Conductivity.Walls.CentralSeamCutoff namespace OAI noncomputable section namespace ScalarConductivity open Set MeasureTheory Filter Topology theorem centralParent_complement_partition : ∃ (r : ℝ) (η : (Fin 3 → ℝ) → ℝ),0 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 ⟨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 end ScalarConductivity namespace ScalarConductivity open Set MeasureTheory Filter Topology 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 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)