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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 | import Mathlib
import OAI.Analysis.Conductivity.Variational.CompactRegularCorrection
import OAI.Analysis.Conductivity.Flux.CompactInteriorDivergence
import OAI.Analysis.Conductivity.Flux.CompactBoxDivergenceC1
namespace OAI
section
noncomputable section
namespace ScalarConductivity
open Set MeasureTheory Filter Topology
lemma cubePartial_bound_of_C1 {f : Box3 → ℝ} {M : ℝ}
(hb : UniformC1Bound f M) {v : Box3} (hv : ‖v‖≤1) (p : Box3) :
|cubePartial f v p|≤M := by
have hM : 0≤M := (norm_nonneg _).trans (hb p).2
exact ((fderiv ℝ f p).le_opNorm v).trans
((mul_le_mul (hb p).2 hv (norm_nonneg _) hM).trans_eq (mul_one M))
theorem compact_regular_box_correction_C0 {a b c d : Fin 3 → ℝ}
(hab : ∀ i,a i<b i) (hca : ∀ i,c i<a i) (hbd : ∀ i,b i<d i) :
∃ L : ℝ,0<L ∧ ∀ (r₁ r₂ : Box3 → ℝ),
ContDiff ℝ (↑(⊤ : ℕ∞)) r₁ → ContDiff ℝ (↑(⊤ : ℕ∞)) r₂ →
HasCompactSupport r₁ → HasCompactSupport r₂ →
tsupport r₁⊆(Ioo (a 0) (b 0) ×ˢ Ioo (a 1) (b 1)) ×ˢ Ioo (a 2) (b 2) →
tsupport r₂⊆(Ioo (a 0) (b 0) ×ˢ Ioo (a 1) (b 1)) ×ˢ Ioo (a 2) (b 2) →
(∫ p,r₁ p)=0 → (∫ p,r₂ p)=0 →
(∫ p,p.1.2*r₁ p-p.1.1*r₂ p)=0 → ∀ M : ℝ,0≤M →
UniformC1Bound r₁ M → UniformC1Bound r₂ M →
∃ H : Fin 3 → Fin 3 → Box3 → ℝ,
(∀ i j, ContDiff ℝ (↑(⊤ : ℕ∞)) (H i j)) ∧
(∀ i j, HasCompactSupport (H i j)) ∧
(∀ i j, tsupport (H i j)⊆(Icc (c 0) (d 0) ×ˢ Icc (c 1) (d 1)) ×ˢ Icc (c 2) (d 2)) ∧
(∀ i j p,H i j p=H j i p) ∧
(∀ p,cubePartial (H 0 0) ((1,0),0) p+cubePartial (H 1 0) ((0,1),0) p+
cubePartial (H 2 0) ((0,0),1) p=r₁ p) ∧
(∀ p,cubePartial (H 0 1) ((1,0),0) p+cubePartial (H 1 1) ((0,1),0) p+
cubePartial (H 2 1) ((0,0),1) p=r₂ p) ∧
(∀ i j p,|H i j p|≤L*M) := by
have hcd : ∀ i,c i<d i := fun i => (hca i).trans ((hab i).trans (hbd i))
obtain ⟨D₀,hD₀,solve₀⟩ := compact_box_divergence_C1 hab
obtain ⟨D₁,hD₁,solve₁⟩ := compact_box_divergence_C1 hcd
let L := D₀+2*D₁*D₀
refine ⟨L,by dsimp [L]; positivity,?_⟩
intro r₁ r₂ hr₁ hr₂ hs₁ hs₂ hv₁ hv₂ hz₁ hz₂ ht M hM hb₁ hb₂
obtain ⟨F₁,F₂,F₃,hF₁,hF₂,hF₃,hsF₁,hsF₂,hsF₃,hdF,htF,hbF₁,hbF₂,hbF₃⟩ := solve₀ r₁ hr₁ hs₁ hv₁ hz₁ M hM hb₁
obtain ⟨G₁,G₂,G₃,hG₁,hG₂,hG₃,hsG₁,hsG₂,hsG₃,hdG,htG,hbG₁,hbG₂,hbG₃⟩ := solve₀ r₂ hr₂ hs₂ hv₂ hz₂ M hM hb₂
have hf₁ : ContDiff ℝ (↑(⊤ : ℕ∞)) (fun p : Box3 => p.1.1) := contDiff_fst.comp contDiff_fst
have hf₂ : ContDiff ℝ (↑(⊤ : ℕ∞)) (fun p : Box3 => p.1.2) := contDiff_snd.comp contDiff_fst
have hIF := cube_coordinate_divergence_moment hF₁ hF₂ hF₃ hsF₁ hsF₂ hsF₃ hdF hf₂
have hIG := cube_coordinate_divergence_moment hG₁ hG₂ hG₃ hsG₁ hsG₂ hsG₃ hdG hf₁
simp only [cubePartial_fst_fst,cubePartial_snd_fst,zero_mul,one_mul,integral_zero,neg_zero,
sub_zero,zero_sub] at hIF hIG
have hi₁ : Integrable (fun p : Box3 => p.1.2*r₁ p) := (hf₂.continuous.mul hr₁.continuous).integrable_of_hasCompactSupport (μ := volume) hs₁.mul_left
have hi₂ : Integrable (fun p : Box3 => p.1.1*r₂ p) := (hf₁.continuous.mul hr₂.continuous).integrable_of_hasCompactSupport (μ := volume) hs₂.mul_left
have hzq : (∫ p,F₂ p-G₁ p)=0 := by
rw [integral_sub (hF₂.continuous.integrable_of_hasCompactSupport hsF₂)
(hG₁.continuous.integrable_of_hasCompactSupport hsG₁)]
rw [integral_sub hi₁ hi₂,hIF,hIG] at ht
linarith
let U := (Icc (c 0) (d 0) ×ˢ Icc (c 1) (d 1)) ×ˢ Icc (c 2) (d 2)
let V := (Ioo (c 0) (d 0) ×ˢ Ioo (c 1) (d 1)) ×ˢ Ioo (c 2) (d 2)
have hinner : (Icc (a 0) (b 0) ×ˢ Icc (a 1) (b 1)) ×ˢ Icc (a 2) (b 2) ⊆ V := by
intro p hp
exact ⟨⟨⟨(hca 0).trans_le hp.1.1.1,hp.1.1.2.trans_lt (hbd 0)⟩,
⟨(hca 1).trans_le hp.1.2.1,hp.1.2.2.trans_lt (hbd 1)⟩⟩,
⟨(hca 2).trans_le hp.2.1,hp.2.2.trans_lt (hbd 2)⟩⟩
have hVU : V⊆U := fun _ hp =>
⟨⟨Ioo_subset_Icc_self hp.1.1,Ioo_subset_Icc_self hp.1.2⟩,Ioo_subset_Icc_self hp.2⟩
have htFV := htF.trans hinner
have htGV := htG.trans hinner
have htF₁ : tsupport F₁⊆U := ((subset_union_left.trans subset_union_left).trans htFV).trans hVU
have htF₂ : tsupport F₂⊆U := ((subset_union_right.trans subset_union_left).trans htFV).trans hVU
have htF₃ : tsupport F₃⊆U := (subset_union_right.trans htFV).trans hVU
have htG₁ : tsupport G₁⊆U := ((subset_union_left.trans subset_union_left).trans htGV).trans hVU
have htG₂ : tsupport G₂⊆U := ((subset_union_right.trans subset_union_left).trans htGV).trans hVU
have htG₃ : tsupport G₃⊆U := (subset_union_right.trans htGV).trans hVU
have htq : tsupport (fun p => F₂ p-G₁ p)⊆V :=
(tsupport_sub _ _).trans (union_subset
((subset_union_right.trans subset_union_left).trans htFV)
((subset_union_left.trans subset_union_left).trans htGV))
have hbq : UniformC1Bound (fun p => F₂ p-G₁ p) (2*D₀*M) := by
convert hbF₂.sub (hF₂.differentiable (by simp)) (hG₁.differentiable (by simp)) hbG₁ using 1
ring
obtain ⟨h₁,h₂,h₃,hh₁,hh₂,hh₃,hsh₁,hsh₂,hsh₃,hdh,hth,hbh₁,hbh₂,hbh₃⟩ :=
solve₁ (fun p => F₂ p-G₁ p) (hF₂.sub hG₁) (hsF₂.sub hsG₁) htq hzq
(2*D₀*M) (by positivity) hbq
have hth₁ : tsupport h₁⊆U := (subset_union_left.trans subset_union_left).trans hth
have hth₂ : tsupport h₂⊆U := (subset_union_right.trans subset_union_left).trans hth
have hth₃ : tsupport h₃⊆U := subset_union_right.trans hth
let A := fun p => F₁ p+cubePartial h₁ ((0,1),0) p
let B := fun p => G₁ p+cubePartial h₂ ((0,1),0) p
let C := fun p => F₃ p+cubePartial h₃ ((0,1),0) p
let D := fun p => G₂ p-cubePartial h₂ ((1,0),0) p
let H : Fin 3 → Fin 3 → Box3 → ℝ := ![![A,B,C],![B,D,G₃],![C,G₃,fun _ => 0]]
have hA := hF₁.add (cubePartial_smooth hh₁ ((0,1),0))
have hB := hG₁.add (cubePartial_smooth hh₂ ((0,1),0))
have hC := hF₃.add (cubePartial_smooth hh₃ ((0,1),0))
have hD := hG₂.sub (cubePartial_smooth hh₂ ((1,0),0))
have hsA := hsF₁.add (cubePartial_compact hsh₁ ((0,1),0))
have hsB := hsG₁.add (cubePartial_compact hsh₂ ((0,1),0))
have hsC := hsF₃.add (cubePartial_compact hsh₃ ((0,1),0))
have hsD := hsG₂.sub (cubePartial_compact hsh₂ ((1,0),0))
have hB' : B=(fun p => F₂ p-cubePartial h₁ ((1,0),0) p-cubePartial h₃ ((0,0),1) p) := by
funext p
change G₁ p+cubePartial h₂ ((0,1),0) p=_
linarith [hdh p]
have htA : tsupport A⊆U := (tsupport_add _ _).trans
(union_subset htF₁ ((cubePartial_tsupport _).trans hth₁))
have htB : tsupport B⊆U := (tsupport_add _ _).trans
(union_subset htG₁ ((cubePartial_tsupport _).trans hth₂))
have htC : tsupport C⊆U := (tsupport_add _ _).trans
(union_subset htF₃ ((cubePartial_tsupport _).trans hth₃))
have htD : tsupport D⊆U := (tsupport_sub _ _).trans
(union_subset htG₂ ((cubePartial_tsupport _).trans hth₂))
have ht0 : tsupport (fun _ : Box3 => (0:ℝ))⊆U := by simp
refine ⟨H,?_,?_,?_,?_,?_,?_,?_⟩
· intro i j; fin_cases i <;> fin_cases j
all_goals first | exact hA | exact hB | exact hC | exact hD | exact hG₃ | exact contDiff_const
· intro i j; fin_cases i <;> fin_cases j
all_goals first | exact hsA | exact hsB | exact hsC | exact hsD | exact hsG₃ | exact HasCompactSupport.zero
· intro i j; fin_cases i <;> fin_cases j
all_goals first | exact htA | exact htB | exact htC | exact htD | exact htG₃ | exact ht0
· intro i j p; fin_cases i <;> fin_cases j <;> rfl
· intro p
change cubePartial A ((1,0),0) p+cubePartial B ((0,1),0) p+cubePartial C ((0,0),1) p=r₁ p
rw [hB']
change cubePartial (fun p => F₁ p+cubePartial h₁ ((0,1),0) p) ((1,0),0) p+
cubePartial (fun p => F₂ p-cubePartial h₁ ((1,0),0) p-cubePartial h₃ ((0,0),1) p) ((0,1),0) p+
cubePartial (fun p => F₃ p+cubePartial h₃ ((0,1),0) p) ((0,0),1) p=r₁ p
rw [cubePartial_add (hF₁.differentiable (by simp)) ((cubePartial_smooth hh₁ _).differentiable (by simp)),
cubePartial_sub ((hF₂.sub (cubePartial_smooth hh₁ _)).differentiable (by simp))
((cubePartial_smooth hh₃ _).differentiable (by simp)),
cubePartial_sub (hF₂.differentiable (by simp)) ((cubePartial_smooth hh₁ _).differentiable (by simp)),
cubePartial_add (hF₃.differentiable (by simp)) ((cubePartial_smooth hh₃ _).differentiable (by simp)),
cubePartial_commute hh₁ ((0,1),0) ((1,0),0),cubePartial_commute hh₃ ((0,1),0) ((0,0),1)]
linarith [hdF p]
· intro p
change cubePartial B ((1,0),0) p+cubePartial D ((0,1),0) p+cubePartial G₃ ((0,0),1) p=r₂ p
change cubePartial (fun p => G₁ p+cubePartial h₂ ((0,1),0) p) ((1,0),0) p+
cubePartial (fun p => G₂ p-cubePartial h₂ ((1,0),0) p) ((0,1),0) p+
cubePartial G₃ ((0,0),1) p=r₂ p
rw [cubePartial_add (hG₁.differentiable (by simp)) ((cubePartial_smooth hh₂ _).differentiable (by simp)),
cubePartial_sub (hG₂.differentiable (by simp)) ((cubePartial_smooth hh₂ _).differentiable (by simp)),
cubePartial_commute hh₂ ((0,1),0) ((1,0),0)]
linarith [hdG p]
· have hcomb (f g : Box3 → ℝ) (hf : UniformC1Bound f (D₀*M))
(hg : UniformC1Bound g (D₁*(2*D₀*M))) (v : Box3) (hv : ‖v‖≤1) (p : Box3) :
|f p+cubePartial g v p|≤L*M ∧ |f p-cubePartial g v p|≤L*M := by
have hd := cubePartial_bound_of_C1 hg hv p
have he : D₀*M+D₁*(2*D₀*M)=L*M := by dsimp [L]; ring
exact ⟨(abs_add_le _ _).trans ((add_le_add (hf p).1 hd).trans_eq he),
(abs_sub _ _).trans ((add_le_add (hf p).1 hd).trans_eq he)⟩
have hA₀ (p : Box3) : |A p|≤L*M := (hcomb F₁ h₁ hbF₁ hbh₁ ((0,1),0) (by simp [Prod.norm_def]) p).1
have hB₀ (p : Box3) : |B p|≤L*M := (hcomb G₁ h₂ hbG₁ hbh₂ ((0,1),0) (by simp [Prod.norm_def]) p).1
have hC₀ (p : Box3) : |C p|≤L*M := (hcomb F₃ h₃ hbF₃ hbh₃ ((0,1),0) (by simp [Prod.norm_def]) p).1
have hD₀' (p : Box3) : |D p|≤L*M := (hcomb G₂ h₂ hbG₂ hbh₂ ((1,0),0) (by simp [Prod.norm_def]) p).2
have hG₀ (p : Box3) : |G₃ p|≤L*M := (hbG₃ p).1.trans
(mul_le_mul_of_nonneg_right (by dsimp [L]; exact le_add_of_nonneg_right (by positivity)) hM)
intro i j p
fin_cases i <;> fin_cases j
all_goals first | exact hA₀ p | exact hB₀ p | exact hC₀ p | exact hD₀' p | exact hG₀ p | skip
change |(0:ℝ)|≤L*M
rw [abs_zero]
dsimp [L]
positivity
end ScalarConductivity
end
end
end OAI
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