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a458723 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 | import Mathlib
import OAI.Analysis.Conductivity.Branching.ParentFullEndEnergy
import OAI.Analysis.Conductivity.Walls.ChildSeamCutoff
namespace OAI
noncomputable section
namespace ScalarConductivity
open Set MeasureTheory Filter Topology
theorem centralChild_complement_partition (k : Fin 2) :
∃ (l : ℝ) (η : (Fin 3 → ℝ) → ℝ),-centralThickness<l ∧ l<0 ∧
ContDiff ℝ (↑(⊤:ℕ∞)) η ∧ HasCompactSupport η ∧
(∀ y,|η y|≤1) ∧ tsupport η⊆sourceClosedCollarBand l (centralThickness/2) ∧
(∀ y,sourceCollarTime y≤-centralThickness → η=ᶠ[𝓝 y] 0) ∧
(∀ y∈sourceClosedCollarBand (-centralThickness) 0,
(fun z => centralJoinChildCutoff k z+η z)=ᶠ[𝓝 y] (fun _ => 1)) := by
let χ : (Fin 3 → ℝ) → ℝ := centralJoinChildCutoff k
let q := fun y => 1-χ y
let K := sourceClosedCollarBand (-centralThickness) 0 ∩ tsupport q
have hK : IsCompact K :=
(isCompact_sourceClosedCollarBand (show -(1:ℝ)/100≤-centralThickness by norm_num [centralThickness])
(show (0:ℝ)≤1/100 by norm_num)).inter_right (isClosed_tsupport q)
have hKt (y) (hy : y∈K) : -centralThickness<sourceCollarTime y := by
apply lt_of_le_of_ne hy.1.1
intro he
have hn : y∉tsupport q := notMem_tsupport_iff_eventuallyEq.mpr (by
filter_upwards [centralJoinChildCutoff_one k he.symm] with z hz
change 1-χ z=0
change χ z=1 at hz
rw [hz]; ring)
exact hn hy.2
have hl : ∃ l : ℝ,-centralThickness<l ∧ l<0 ∧ ∀ y∈K,l<sourceCollarTime y := by
by_cases hn : K.Nonempty
· obtain ⟨y,hy,hmin⟩ := hK.exists_isMinOn hn locallyLipschitz_sourceCollarTime.continuous.continuousOn
refine ⟨(sourceCollarTime y-centralThickness)/2,?_,?_,?_⟩
· linarith [hKt y hy]
· have hp := hy.1.2
have : (0:ℝ)<centralThickness := by norm_num [centralThickness]
linarith
· intro z hz
have hh : sourceCollarTime y ≤ sourceCollarTime z := hmin 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 ⟨l,hl,hl₀,hlK⟩ := hl
have hl' : -(1:ℝ)/100≤l := (show -(1:ℝ)/100≤-centralThickness by norm_num [centralThickness]).trans hl.le
have hbig := isCompact_sourceClosedCollarBand hl'
(show centralThickness/2≤1/100 by norm_num [centralThickness])
have hi : K⊆interior (sourceClosedCollarBand l (centralThickness/2)) := 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 l (centralThickness/2) from
⟨hlK y hy,by have := hy.1.2; dsimp [centralThickness] at *; linarith⟩))
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 l (centralThickness/2) := 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 l (centralThickness/2) :=
tsupport_mul_subset_right.trans hψs
refine ⟨l,η,hl,hl₀,(contDiff_const.sub (centralJoinChildCutoff_smooth k)).mul ψ.contMDiff.contDiff,
hψc.mul_left,?_,hηs,?_,?_⟩
· intro y
have h₀ := centralJoinPartition.nonneg (centralChildPatch k) (sourceChildCoordinates (actualChildSign k) y)
have h₁ := centralJoinPartition.le_one (centralChildPatch k) (sourceChildCoordinates (actualChildSign k) y)
change 0≤χ y at h₀
change χ y≤1 at h₁
have h₂ := (hψb y).1
have h₃ := (hψb y).2
dsimp only [η,q]
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).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 centralJoinChildCutoff k z+((1-centralJoinChildCutoff k 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 childAttachedCorrection_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 : 0<a) (k : Fin 2) (κ : ℝ) (p : centralEnergySpace s) :
∃ d : H1,d∈H10 ∧
(∀ᵐ x∂ballMeasure,sourceCollarTime (WithLp.ofLp x)≤-centralThickness →
weakValue d x=0 ∧ ∀ i,weakGradient d x i=0) ∧
(∀ᵐ x∂ballMeasure,WithLp.ofLp x∈sourceClosedCollarBand (-centralThickness) 0 →
weakValue (childCompletionJoin s hs ha k (centralJoinChildCutoff_smooth k)
(centralJoinChildCutoff_compact k) (centralJoinChildCutoff_bound k)
(centralJoinChildCutoff_support k) p+d) x=
fullAttachedEndValue s (centralT s k.succ p) a (-centralThickness) κ (WithLp.ofLp x) ∧
∀ i,weakGradient (childCompletionJoin s hs ha k (centralJoinChildCutoff_smooth k)
(centralJoinChildCutoff_compact k) (centralJoinChildCutoff_bound k)
(centralJoinChildCutoff_support k) p+d) x i=
fullAttachedEndGradient s (centralT s k.succ p) a (-centralThickness) κ (WithLp.ofLp x) i) := by
obtain ⟨l,η,hl,hl₀,hη,hηc,hηb,hηs,hηzero,hpart⟩ := centralChild_complement_partition k
obtain ⟨v,hv,hve⟩ := compact_collar_slope_value_gradient (centralJoinChildCutoff_smooth k)
(centralJoinChildCutoff_compact k) (centralJoinChildCutoff_bound k) ha.ne'
(show -centralThickness∈Icc (-(1:ℝ)/100) (1/100) by norm_num [centralThickness])
κ (-2*centralThickness) 0 (by norm_num [centralThickness]) (by norm_num) (centralJoinChildCutoff_support k)
have hR : 0≤a*(3*centralThickness/2) :=
mul_nonneg ha.le (by norm_num [centralThickness])
have hT : ∀ t∈Icc l (centralThickness/2),
affineEndTime a (-centralThickness) t∈Icc 0 (a*(3*centralThickness/2)) := by
intro t ht
dsimp only [affineEndTime]
constructor
· exact mul_nonneg ha.le (sub_nonneg.mpr (hl.le.trans ht.1))
· have hm := mul_nonneg ha.le (sub_nonneg.mpr ht.2)
nlinarith
have hpos : ∀ t∈Icc l (centralThickness/2),0<a*(t-(-centralThickness)) :=
fun t ht => mul_pos ha (sub_pos.mpr (hl.trans_le ht.1))
obtain ⟨z,hz,hze⟩ := cut_fullAttachedEnd_H10 s hs (centralT s k.succ p) κ ha.ne' hR
(show -centralThickness∈Icc (-(1:ℝ)/100) (1/100) by norm_num [centralThickness])
(show l≤centralThickness/2 by
have ht : (0:ℝ)<centralThickness := by norm_num [centralThickness]
linarith)
((show -(1:ℝ)/100≤-centralThickness by norm_num [centralThickness]).trans hl.le)
(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 hn : a*(sourceCollarTime (WithLp.ofLp x)-(-centralThickness))≤0 :=
mul_nonpos_of_nonneg_of_nonpos ha.le (sub_nonpos.mpr hcentral)
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,_root_.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 (-centralThickness) 0 →
0<a*(sourceCollarTime (WithLp.ofLp x)-(-centralThickness)) := by
filter_upwards [hne] with x hx hb
exact mul_pos ha (sub_pos.mpr (lt_of_le_of_ne hb.1 (Ne.symm hx)))
have hh := fullEnd_partition_jet s (centralT s k.succ p) a (-centralThickness) κ
(centralJoinChildCutoff_smooth k) hη hpart (v:=v) (z:=z)
(u:=childCompletionJoin s hs ha k (centralJoinChildCutoff_smooth k)
(centralJoinChildCutoff_compact k) (centralJoinChildCutoff_bound k)
(centralJoinChildCutoff_support k) p) hp
(childCompletionJoin_end_ae s hs ha k (centralJoinChildCutoff_smooth k)
(centralJoinChildCutoff_compact k) (centralJoinChildCutoff_bound k)
(centralJoinChildCutoff_support k) p) hve hze
simpa only [add_assoc] using hh
end ScalarConductivity
end
end OAI
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