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159357f | 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 | import Mathlib
import OAI.Analysis.Conductivity.Branching.PhysicalJoin
import OAI.Analysis.Conductivity.Branching.PhysicalAttachedAssembly
namespace OAI
section
noncomputable section
namespace ScalarConductivity
open Set MeasureTheory Filter Topology
def attachedPhysicalJet (s a κ : Fin 3 → ℝ) (p : centralEnergySpace s)
(i : Fin 3) (x : R3) : JetFiber :=
Fin.cases
(physicalJet (fullAttachedEndValue s (centralT s 0 p) (a 0) centralThickness
(κ 0) (WithLp.ofLp x))
(fullAttachedEndGradient s (centralT s 0 p) (a 0) centralThickness
(κ 0) (WithLp.ofLp x)))
(fun k => physicalJet
(fullAttachedEndValue s (centralT s k.succ p) (a k.succ) (-centralThickness)
(κ k.succ) ((sourceChildHomeomorph (actualChildSign k)).symm (WithLp.ofLp x)))
(fun j => sourceScale⁻¹*fullAttachedEndGradient s (centralT s k.succ p)
(a k.succ) (-centralThickness) (κ k.succ)
((sourceChildHomeomorph (actualChildSign k)).symm (WithLp.ofLp x)) (childAxis j))) i
theorem complete_physical_attachment (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 κ : Fin 3 → ℝ) (ha₀ : a 0<0) (ha : ∀ k : Fin 2,0<a k.succ)
(p : centralEnergySpace s) :
∃ w : H1,w∈H10 ∧
H1JetOn w centralPhysical (centralFullJetCLM s p.val) ∧
∀ i : Fin 3,H1JetOn w (physicalEndRegion i) (attachedPhysicalJet s a κ p i) := by
let zi := centralInteriorCompletion s (centralJoinPartition_smooth 0)
centralJoinPartition_interior_compact centralJoinPartition_interior_support
(fun _ hy => centralPhysical_subset_ball (centralJoinPartition_interior_support hy)) p
let zp := parentCompletionJoin s hs ha₀ (centralJoinPartition_smooth 1)
centralJoinPartition_parent_compact (centralJoinPartition_bound 1)
centralJoinPartition_parent_support p
let zc (k : Fin 2) := physicalChildCompletionJoin s hs (ha k) k (centralJoinChildCutoff_smooth k)
(centralJoinChildCutoff_compact k) (centralJoinChildCutoff_bound k)
(centralJoinChildCutoff_support k) p
obtain ⟨dp,hdp,hds,hdpZ,hdpE⟩ := parentAttachedCorrection_supported s hs ha₀ (κ 0) p
have hdcEx (k : Fin 2) := physicalChildAttachedCorrection_supported s hs (ha k) k (κ k.succ) p
choose dc hdc hdcZ hdcE using hdcEx
let w : H1 := zi+(zp+dp)+((zc 0+dc 0)+(zc 1+dc 1))
have hw : w∈H10 := H10.add_mem
(H10.add_mem (centralInteriorCompletion_mem_H10 s _ _ _ _ p)
(H10.add_mem (parentCompletionJoin_mem_H10 s hs ha₀ _ _ _ _ p) hdp))
(H10.add_mem
(H10.add_mem (physicalChildCompletionJoin_mem_H10 s hs (ha 0) 0 _ _ _ _ p) (hdc 0))
(H10.add_mem (physicalChildCompletionJoin_mem_H10 s hs (ha 1) 1 _ _ _ _ p) (hdc 1)))
have hpC : H1JetOn dp centralPhysical (fun _ => 0) :=
(H1JetOn.zero_of_weak (D:={y | centralThickness≤ sourceCollarTime y}) hdpZ).mono
(fun y hy => ((centralPhysical_time_iff y).mp hy).1)
have hcC (k : Fin 2) : H1JetOn (dc k) centralPhysical (fun _ => 0) :=
(H1JetOn.zero_of_weak (D:={y | sourceCollarTime
((sourceChildHomeomorph (actualChildSign k)).symm y)≤-centralThickness}) (hdcZ k)).mono
(fun y hy => ((centralPhysical_time_iff y).mp hy).2 k)
have hpO (k : Fin 2) : H1JetOn (zp+dp) (physicalEndRegion k.succ) (fun _ => 0) := by
apply H1JetOn.add_zero
· exact H1JetOn.zero_of_weak (parent_join_zero_on_children s hs ha₀ p k)
· apply (H1JetOn.zero_of_weak (D:={y | centralThickness≤ sourceCollarTime y}) hdpZ).mono
intro y hy
exact (show centralThickness≤2*centralThickness by norm_num [centralThickness]).trans
(physicalChildEnd_parent_time k hy)
have hcO (k : Fin 2) (i : Fin 3) (hik : i≠k.succ) :
H1JetOn (zc k+dc k) (physicalEndRegion i) (fun _ => 0) := by
apply H1JetOn.add_zero
· exact H1JetOn.zero_of_weak (physical_child_join_zero_on_other_ends s hs (ha k) p k i hik)
· apply (H1JetOn.zero_of_weak (D:={y | sourceCollarTime
((sourceChildHomeomorph (actualChildSign k)).symm y)≤-centralThickness}) (hdcZ k)).mono
intro y hy
exact (physicalOtherEnd_child_time k i hik hy).le.trans
(by norm_num [centralThickness])
have hiO (i : Fin 3) : H1JetOn zi (physicalEndRegion i) (fun _ => 0) :=
H1JetOn.zero_of_weak (central_interior_join_zero_on_ends s p i)
have hpE : H1JetOn (zp+dp) (physicalEndRegion 0) (attachedPhysicalJet s a κ p 0) :=
H1JetOn.of_weak hdpE
have hcE (k : Fin 2) : H1JetOn (zc k+dc k) (physicalEndRegion k.succ)
(attachedPhysicalJet s a κ p k.succ) := H1JetOn.of_weak (hdcE k)
refine ⟨w,hw,?_,?_⟩
· have he : w=centralNeighborhoodCompletion s hs ha₀ ha p+(dp+(dc 0+dc 1)) := by
simp only [w,centralNeighborhoodCompletion,Fin.sum_univ_two,add_apply]
change zi+(zp+dp)+((zc 0+dc 0)+(zc 1+dc 1))=zi+zp+(zc 0+zc 1)+(dp+(dc 0+dc 1))
abel
rw [he]
have hc : H1JetOn (centralNeighborhoodCompletion s hs ha₀ ha p) centralPhysical
(centralFullJetCLM s p.val) := centralNeighborhoodCompletion_central_ae s hs ha₀ ha p
exact hc.add_zero (hpC.add_zero ((hcC 0).add_zero (hcC 1)))
· intro i
refine Fin.cases ?_ (fun k => ?_) i
· exact ((hiO 0).zero_add hpE).add_zero
((hcO 0 0 (by decide)).add_zero (hcO 1 0 (by decide)))
· fin_cases k
· exact ((hiO 1).add_zero (hpO 0)).zero_add
((hcE 0).add_zero (hcO 1 1 (by decide)))
· exact ((hiO 2).add_zero (hpO 1)).zero_add
((hcO 0 2 (by decide)).zero_add (hcE 1))
end ScalarConductivity
end
end
end OAI
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