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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 | import Mathlib
import OAI.Analysis.Conductivity.Variational.CompactFiberDecomposition
namespace OAI
noncomputable section
namespace ScalarConductivity
open Set MeasureTheory Filter Topology
lemma interval_integral_eq_integral_of_support {a b : ℝ} (hab : a≤b)
{f : ℝ → ℝ} (hv : ∀ t, f t≠0 → t∈Ioo a b) :
(∫ t in a..b, f t)=∫ t, f t := by
rw [intervalIntegral.integral_of_le hab]
apply setIntegral_eq_integral_of_forall_compl_eq_zero
intro x hx
by_contra hn
exact hx ⟨(hv x hn).1,(hv x hn).2.le⟩
lemma compactFiberMarginal_integral {E : Type} [NormedAddCommGroup E] [NormedSpace ℝ E]
[MeasureSpace E] [BorelSpace E] [SFinite (volume : Measure E)]
{a b : ℝ} (hab : a≤b) {f : E×ℝ → ℝ} (hf : Integrable f)
(hv : ∀ p, f p≠0 → p.2∈Ioo a b) :
(∫ x, compactFiberMarginal a b f x)=∫ p, f p := by
have he (x : E) : compactFiberMarginal a b f x=∫ t, f (x,t) :=
interval_integral_eq_integral_of_support hab (fun t ht => hv (x,t) ht)
simp_rw [he]
exact (integral_prod f hf).symm
lemma compact_scalar_primitive {a b : ℝ} (hab : a<b) {r : ℝ → ℝ}
(hr : ContDiff ℝ (↑(⊤ : ℕ∞)) r)
(hv : ∀ t, r t≠0 → t∈Ioo a b) (hz : (∫ t, r t)=0) :
∃ P : ℝ → ℝ, ContDiff ℝ (↑(⊤ : ℕ∞)) P ∧ HasCompactSupport P ∧
(∀ t, deriv P t=r t) ∧ tsupport P⊆Icc a b := by
let f : ℝ×ℝ → ℝ := fun p => r p.2
have hf : ContDiff ℝ (↑(⊤ : ℕ∞)) f := hr.comp contDiff_snd
let P : ℝ → ℝ := fun t => fiberPrimitive a f (0,t)
have hP : ContDiff ℝ (↑(⊤ : ℕ∞)) P :=
(fiberPrimitive_smooth a hf).comp (contDiff_const.prodMk contDiff_id)
have hvf : ∀ p, f p≠0 → p.2∈Ioo a b := fun p hp => hv p.2 hp
have hzf : ∀ x, (∫ t in a..b, f (x,t))=0 := by
intro x
exact (interval_integral_eq_integral_of_support hab.le hv).trans hz
have hs : Function.support P⊆Icc a b := by
intro t ht
refine ⟨?_,?_⟩
· by_contra hn
exact ht (fiberPrimitive_below hf.continuous hvf (0,t) (le_of_not_ge hn))
· by_contra hn
exact ht (fiberPrimitive_above hf.continuous hvf hzf (0,t) (le_of_not_ge hn))
refine ⟨P,hP,HasCompactSupport.of_support_subset_isCompact isCompact_Icc hs,?_,
closure_minimal hs isClosed_Icc⟩
intro t
have hd := (((fiberPrimitive_smooth a hf).differentiable (by simp)) (0,t)).hasFDerivAt.comp_hasDerivAt t
((hasDerivAt_const t (0:ℝ)).prodMk (hasDerivAt_id t))
exact hd.deriv.trans (fiberPrimitive_derivative a hf (0,t))
lemma cube_partial_fiber (f : Box3 → ℝ) (p : Box3) :
cubePartial f ((0,0),1) p=wallDerivative f p := rfl
theorem compact_box_divergence {a b : Fin 3 → ℝ} (hab : ∀ i,a i<b i)
{r : Box3 → ℝ} (hr : ContDiff ℝ (↑(⊤ : ℕ∞)) r) (hs : HasCompactSupport r)
(hv : tsupport r⊆(Ioo (a 0) (b 0) ×ˢ Ioo (a 1) (b 1)) ×ˢ Ioo (a 2) (b 2))
(hz : (∫ p, r p)=0) :
∃ F₁ F₂ F₃ : Box3 → ℝ,
ContDiff ℝ (↑(⊤ : ℕ∞)) F₁ ∧ ContDiff ℝ (↑(⊤ : ℕ∞)) F₂ ∧
ContDiff ℝ (↑(⊤ : ℕ∞)) F₃ ∧
HasCompactSupport F₁ ∧ HasCompactSupport F₂ ∧ HasCompactSupport F₃ ∧
(∀ p, cubePartial F₁ ((1,0),0) p+cubePartial F₂ ((0,1),0) p+
cubePartial F₃ ((0,0),1) p=r p) ∧
tsupport F₁∪tsupport F₂∪tsupport F₃⊆
(Icc (a 0) (b 0) ×ˢ Icc (a 1) (b 1)) ×ˢ Icc (a 2) (b 2) := by
obtain ⟨η₃,hη₃,hsη₃,hvη₃,hiη₃⟩ := exists_unit_interval_bump (hab 2)
obtain ⟨η₂,hη₂,hsη₂,hvη₂,hiη₂⟩ := exists_unit_interval_bump (hab 1)
let R := compactFiberMarginal (a 2) (b 2) r
have hR := compactFiberMarginal_smooth (hab 2).le hr
have hsR := (compactFiberMarginal_support hs (a := a 2) (b := b 2)).1
have hvR : tsupport R⊆Ioo (a 0) (b 0) ×ˢ Ioo (a 1) (b 1) := by
intro x hx
obtain ⟨p,hp,rfl⟩ := (compactFiberMarginal_support hs).2 hx
exact (hv hp).1
obtain ⟨F₃,hF₃,hsF₃,hd₃,ht₃⟩ := compact_fiber_decomposition (hab 2) hr hs
(fun p hp => (hv (subset_tsupport r hp)).2) hη₃ hsη₃ hvη₃ hiη₃
let S := compactFiberMarginal (a 1) (b 1) R
have hS := compactFiberMarginal_smooth (hab 1).le hR
have hsS := (compactFiberMarginal_support hsR (a := a 1) (b := b 1)).1
have hvS : tsupport S⊆Ioo (a 0) (b 0) := by
intro x hx
obtain ⟨p,hp,rfl⟩ := (compactFiberMarginal_support hsR).2 hx
exact (hvR hp).1
obtain ⟨G,hG,hsG,hdG,htG⟩ := compact_fiber_decomposition (hab 1) hR hsR
(fun p hp => (hvR (subset_tsupport R hp)).2) hη₂ hsη₂ hvη₂ hiη₂
have hzR : (∫ x,R x)=0 := by
rw [compactFiberMarginal_integral (hab 2).le
(hr.continuous.integrable_of_hasCompactSupport hs)
(fun p hp => (hv (subset_tsupport r hp)).2)]
exact hz
have hzS : (∫ x,S x)=0 := by
rw [compactFiberMarginal_integral (hab 1).le
(hR.continuous.integrable_of_hasCompactSupport hsR)
(fun p hp => (hvR (subset_tsupport R hp)).2)]
exact hzR
obtain ⟨P,hP,hsP,hdP,htP⟩ := compact_scalar_primitive (hab 0) hS
(fun x hx => hvS (subset_tsupport S hx)) hzS
let F₁ : Box3 → ℝ := fun p => η₃ p.2*η₂ p.1.2*P p.1.1
let F₂ : Box3 → ℝ := fun p => η₃ p.2*G p.1
have hF₁ : ContDiff ℝ (↑(⊤ : ℕ∞)) F₁ :=
((hη₃.comp contDiff_snd).mul (hη₂.comp (contDiff_snd.comp contDiff_fst))).mul
(hP.comp (contDiff_fst.comp contDiff_fst))
have hF₂ : ContDiff ℝ (↑(⊤ : ℕ∞)) F₂ :=
(hη₃.comp contDiff_snd).mul (hG.comp contDiff_fst)
let K := (Icc (a 0) (b 0) ×ˢ Icc (a 1) (b 1)) ×ˢ Icc (a 2) (b 2)
have hK : IsCompact K := (isCompact_Icc.prod isCompact_Icc).prod isCompact_Icc
have ht₁ : tsupport F₁⊆K := by
apply closure_minimal _ hK.isClosed
intro p hp
obtain ⟨h₁,h₂⟩ := mul_ne_zero_iff.mp hp
obtain ⟨h₃,h₄⟩ := mul_ne_zero_iff.mp h₁
exact ⟨⟨htP (subset_tsupport P h₂),Ioo_subset_Icc_self (hvη₂ (subset_tsupport η₂ h₄))⟩,
Ioo_subset_Icc_self (hvη₃ (subset_tsupport η₃ h₃))⟩
have ht₂ : tsupport F₂⊆K := by
apply closure_minimal _ hK.isClosed
intro p hp
obtain ⟨h₁,h₂⟩ := mul_ne_zero_iff.mp hp
have hh := htG (subset_tsupport G h₂)
obtain ⟨q,hq,hqx⟩ := hh.1
exact ⟨⟨Ioo_subset_Icc_self (hqx ▸ (hvR hq).1),hh.2⟩,
Ioo_subset_Icc_self (hvη₃ (subset_tsupport η₃ h₁))⟩
have ht₃' : tsupport F₃⊆K := by
intro p hp
have hh := ht₃ hp
obtain ⟨q,hq,hqp⟩ := hh.1
have hxy := (hv hq).1
change q.1∈_ at hxy
rw [hqp] at hxy
exact ⟨⟨Ioo_subset_Icc_self hxy.1,Ioo_subset_Icc_self hxy.2⟩,hh.2⟩
refine ⟨F₁,F₂,F₃,hF₁,hF₂,hF₃,
HasCompactSupport.of_support_subset_isCompact hK (subset_tsupport F₁ |>.trans ht₁),
HasCompactSupport.of_support_subset_isCompact hK (subset_tsupport F₂ |>.trans ht₂),hsF₃,?_,
union_subset (union_subset ht₁ ht₂) ht₃'⟩
intro p
have hd₁ : cubePartial F₁ ((1,0),0) p=η₃ p.2*η₂ p.1.2*S p.1.1 := by
have hline := (hasDerivAt_id p.1.1).prodMk (hasDerivAt_const p.1.1 p.1.2)
have he := ((hF₁.differentiable (by simp)) p).hasFDerivAt.comp_hasDerivAt p.1.1
(hline.prodMk (hasDerivAt_const p.1.1 p.2))
have he' : HasDerivAt (fun t => F₁ ((t,p.1.2),p.2))
(cubePartial F₁ ((1,0),0) p) p.1.1 := he
have hp := ((hP.differentiable (by simp)) p.1.1).hasDerivAt.const_mul
(η₃ p.2*η₂ p.1.2)
rw [hdP] at hp
exact he'.unique hp
have hd₂ : cubePartial F₂ ((0,1),0) p=η₃ p.2*wallDerivative G p.1 := by
have hline := (hasDerivAt_const p.1.2 p.1.1).prodMk (hasDerivAt_id p.1.2)
have he := ((hF₂.differentiable (by simp)) p).hasFDerivAt.comp_hasDerivAt p.1.2
(hline.prodMk (hasDerivAt_const p.1.2 p.2))
have he' : HasDerivAt (fun t => F₂ ((p.1.1,t),p.2))
(cubePartial F₂ ((0,1),0) p) p.1.2 := he
have hg := (((hG.differentiable (by simp)) p.1).hasFDerivAt.comp_hasDerivAt
p.1.2 hline).const_mul (η₃ p.2)
exact he'.unique hg
rw [hd₁,hd₂,cube_partial_fiber]
have h₁ := hdG p.1
have h₂ := hd₃ p
dsimp [S,R] at *
linear_combination (η₃ p.2)*h₁+h₂
end ScalarConductivity
end
end OAI
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