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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 | import Mathlib
import OAI.Analysis.Conductivity.Variational.CompactRegularCorrection
import OAI.Analysis.Conductivity.Flux.CompactInteriorDivergence
import OAI.Analysis.Conductivity.Variational.FiniteMomentTransfer
namespace OAI
noncomputable section
namespace ScalarConductivity
open Set MeasureTheory
lemma exists_unit_global_interval_bump {a b : ℝ} (hab : a<b) :
∃ η : ℝ → ℝ, ContDiff ℝ (↑(⊤ : ℕ∞)) η ∧ HasCompactSupport η ∧
tsupport η⊆Ioo a b ∧ (∫ t,η t)=1 := by
obtain ⟨η,hη,hs,hv,hi⟩ := exists_unit_interval_bump hab
refine ⟨η,hη,hs,hv,?_⟩
rw [intervalIntegral.integral_of_le hab.le] at hi
have he : (∫ t in Ioc a b,η t)=(∫ t,η t) := by
apply setIntegral_eq_integral_of_forall_compl_eq_zero
intro t ht
exact image_eq_zero_of_notMem_tsupport (fun hh => ht ⟨(hv hh).1,(hv hh).2.le⟩)
rwa [he] at hi
lemma exists_unit_box_bump {a b : Fin 3 → ℝ} (hab : ∀ i,a i<b i) :
∃ K : Box3 → ℝ,ContDiff ℝ (↑(⊤ : ℕ∞)) K ∧ HasCompactSupport K ∧
tsupport K⊆(Ioo (a 0) (b 0) ×ˢ Ioo (a 1) (b 1)) ×ˢ Ioo (a 2) (b 2) ∧
(∫ p,K p)=1 := by
obtain ⟨η₁,h₁,hs₁,hv₁,hi₁⟩ := exists_unit_global_interval_bump (hab 0)
obtain ⟨η₂,h₂,hs₂,hv₂,hi₂⟩ := exists_unit_global_interval_bump (hab 1)
obtain ⟨η₃,h₃,hs₃,hv₃,hi₃⟩ := exists_unit_global_interval_bump (hab 2)
let K : Box3 → ℝ := fun p => (η₁ p.1.1*η₂ p.1.2)*η₃ p.2
have hK : ContDiff ℝ (↑(⊤ : ℕ∞)) K :=
((h₁.comp (contDiff_fst.comp contDiff_fst)).mul
(h₂.comp (contDiff_snd.comp contDiff_fst))).mul (h₃.comp contDiff_snd)
have hsK : HasCompactSupport K := by
apply HasCompactSupport.of_support_subset_isCompact ((hs₁.prod hs₂).prod hs₃)
intro p hp
have hh := mul_ne_zero_iff.mp hp
have hh' := mul_ne_zero_iff.mp hh.1
exact ⟨⟨subset_tsupport η₁ hh'.1,subset_tsupport η₂ hh'.2⟩,subset_tsupport η₃ hh.2⟩
have htK : tsupport K⊆(tsupport η₁ ×ˢ tsupport η₂) ×ˢ tsupport η₃ := by
apply closure_minimal _ ((hs₁.isClosed.prod hs₂.isClosed).prod hs₃.isClosed)
intro p hp
have hh := mul_ne_zero_iff.mp hp
have hh' := mul_ne_zero_iff.mp hh.1
exact ⟨⟨subset_tsupport η₁ hh'.1,subset_tsupport η₂ hh'.2⟩,subset_tsupport η₃ hh.2⟩
refine ⟨K,hK,hsK,htK.trans (prod_mono (prod_mono hv₁ hv₂) hv₃),?_⟩
change (∫ p : (ℝ×ℝ)×ℝ,(η₁ p.1.1*η₂ p.1.2)*η₃ p.2)=1
rw [Measure.volume_eq_prod,integral_prod_mul (fun p : ℝ×ℝ => η₁ p.1*η₂ p.2) η₃,
Measure.volume_eq_prod,integral_prod_mul η₁ η₂,hi₁,hi₂,hi₃]
norm_num
lemma cubePartial_integral_zero {K : Box3 → ℝ}
(hK : ContDiff ℝ (↑(⊤ : ℕ∞)) K) (hs : HasCompactSupport K) (v : Box3) :
(∫ p,cubePartial K v p)=0 := by
have hh := cube_integration_by_parts (f := fun _ => (1:ℝ)) contDiff_const hK hs v
simpa [cubePartial] using hh
theorem box_moment_right_inverse {a b : Fin 3 → ℝ} (hab : ∀ i,a i<b i)
(m : Fin 3 → ℝ) :
∃ 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)=m 0 ∧ (∫ p,r₂ p)=m 1 ∧
(∫ p,p.1.2*r₁ p-p.1.1*r₂ p)=m 2 := by
obtain ⟨K,hK,hs,hv,hi⟩ := exists_unit_box_bump hab
let D := cubePartial K ((0,1),0)
let c := m 0*(∫ p : Box3,p.1.2*K p)-m 1*(∫ p : Box3,p.1.1*K p)-m 2
let r₁ := fun p => m 0*K p+c*D p
let r₂ := fun p => m 1*K p
have hD : ContDiff ℝ (↑(⊤ : ℕ∞)) D := cubePartial_smooth hK _
have hsD : HasCompactSupport D := cubePartial_compact hs _
have hvD : tsupport D⊆(Ioo (a 0) (b 0) ×ˢ Ioo (a 1) (b 1)) ×ˢ Ioo (a 2) (b 2) :=
(cubePartial_tsupport _).trans hv
have iK : Integrable K := hK.continuous.integrable_of_hasCompactSupport hs
have iD : Integrable D := hD.continuous.integrable_of_hasCompactSupport hsD
have hDz : (∫ p,D p)=0 := cubePartial_integral_zero hK hs _
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 hDm : (∫ p : Box3,p.1.2*D p)= -1 := by
have hh := cube_integration_by_parts hf₂ hK hs ((0,1),0)
simpa only [cubePartial_snd_fst,one_mul,hi] using hh
have iK₁ : Integrable (fun p : Box3 => p.1.1*K p) :=
(hf₁.continuous.mul hK.continuous).integrable_of_hasCompactSupport hs.mul_left
have iK₂ : Integrable (fun p : Box3 => p.1.2*K p) :=
(hf₂.continuous.mul hK.continuous).integrable_of_hasCompactSupport hs.mul_left
have iD₂ : Integrable (fun p : Box3 => p.1.2*D p) :=
(hf₂.continuous.mul hD.continuous).integrable_of_hasCompactSupport hsD.mul_left
refine ⟨r₁,r₂,(contDiff_const.mul hK).add (contDiff_const.mul hD),
contDiff_const.mul hK,(hs.mul_left).add hsD.mul_left,hs.mul_left,?_,?_,?_,?_,?_⟩
· exact (tsupport_add _ _).trans (union_subset
(tsupport_mul_subset_right.trans hv) (tsupport_mul_subset_right.trans hvD))
· exact tsupport_mul_subset_right.trans hv
· dsimp only [r₁]
rw [integral_add (iK.const_mul _) (iD.const_mul _),integral_const_mul,integral_const_mul,hi,hDz]
ring
· dsimp only [r₂]
rw [integral_const_mul,hi,mul_one]
· have he : (fun p : Box3 => p.1.2*r₁ p-p.1.1*r₂ p)=
(fun p => m 0*(p.1.2*K p)+c*(p.1.2*D p)-m 1*(p.1.1*K p)) := by
funext p; dsimp only [r₁,r₂]; ring
have iKD : Integrable (fun p : Box3 => m 0*(p.1.2*K p)+c*(p.1.2*D p)) :=
(iK₂.const_mul _).add (iD₂.const_mul _)
rw [he,integral_sub iKD (iK₁.const_mul _),
integral_add (iK₂.const_mul _) (iD₂.const_mul _),integral_const_mul,integral_const_mul,
integral_const_mul,hDm]
dsimp only [c]
ring
end ScalarConductivity
namespace ScalarConductivity
open Set MeasureTheory
theorem box_moment_bounded_right_inverse {a b : Fin 3 → ℝ} (hab : ∀ i,a i<b i) :
∃ g : Fin 3 → Fin 2 → Box3 → ℝ,
(∀ i j,ContDiff ℝ (↑(⊤ : ℕ∞)) (g i j)) ∧
(∀ i j,HasCompactSupport (g i j)) ∧
(∀ i j,tsupport (g i j)⊆(Ioo (a 0) (b 0) ×ˢ Ioo (a 1) (b 1)) ×ˢ Ioo (a 2) (b 2)) ∧
(∀ m : Fin 3 → ℝ,
(∫ p,∑ i,m i*g i 0 p)=m 0 ∧
(∫ p,∑ i,m i*g i 1 p)=m 1 ∧
(∫ p,p.1.2*(∑ i,m i*g i 0 p)-p.1.1*(∑ i,m i*g i 1 p))=m 2) ∧
(∀ j n,∃ B : ℝ,0<B ∧ ∀ (m : Fin 3 → ℝ) (p : Box3),
‖iteratedFDeriv ℝ n (fun p => ∑ i,m i*g i j p) p‖≤B*‖m‖) := by
choose g₀ g₁ hg₀ hg₁ hs₀ hs₁ hv₀ hv₁ hi₀ hi₁ hit using
fun i : Fin 3 => box_moment_right_inverse hab (Pi.single i 1)
let g : Fin 3 → Fin 2 → Box3 → ℝ := fun i => ![g₀ i,g₁ i]
have hg (i : Fin 3) (j : Fin 2) : ContDiff ℝ (↑(⊤ : ℕ∞)) (g i j) := by
fin_cases j
· exact hg₀ i
· exact hg₁ i
have hs (i : Fin 3) (j : Fin 2) : HasCompactSupport (g i j) := by
fin_cases j
· exact hs₀ i
· exact hs₁ i
have hv (i : Fin 3) (j : Fin 2) : tsupport (g i j)⊆
(Ioo (a 0) (b 0) ×ˢ Ioo (a 1) (b 1)) ×ˢ Ioo (a 2) (b 2) := by
fin_cases j
· exact hv₀ i
· exact hv₁ i
have hm (i : Fin 3) (j : Fin 2) : (∫ p,g i j p)=(Pi.single i (1 : ℝ) : Fin 3 → ℝ) (j.castLE (by decide : 2≤3)) := by
fin_cases j
· exact hi₀ i
· exact hi₁ i
have hmt (i : Fin 3) : (∫ p : Box3,p.1.2*g i 0 p-p.1.1*g i 1 p)= (Pi.single i (1 : ℝ) : Fin 3 → ℝ) 2 := hit i
have hI (i : Fin 3) (j : Fin 2) : Integrable (g i j) :=
(hg i j).continuous.integrable_of_hasCompactSupport (hs i j)
refine ⟨g,hg,hs,hv,?_,?_⟩
· intro m
have hh (j : Fin 2) : (∫ p,∑ i,m i*g i j p)=m (j.castLE (by decide : 2≤3)) := by
rw [integral_finsetSum _ (fun i _ => (hI i j).const_mul (m i))]
simp_rw [integral_const_mul,hm]
simp [Pi.single_apply]
refine ⟨hh 0,hh 1,?_⟩
have hU : ContDiff ℝ (↑(⊤ : ℕ∞)) (fun p : Box3 => p.1.1) := contDiff_fst.comp contDiff_fst
have hV : ContDiff ℝ (↑(⊤ : ℕ∞)) (fun p : Box3 => p.1.2) := contDiff_snd.comp contDiff_fst
have htI (i : Fin 3) : Integrable (fun p : Box3 => p.1.2*g i 0 p-p.1.1*g i 1 p) :=
((hV.continuous.mul (hg i 0).continuous).integrable_of_hasCompactSupport (hs i 0).mul_left).sub
((hU.continuous.mul (hg i 1).continuous).integrable_of_hasCompactSupport (hs i 1).mul_left)
have he : (fun p : Box3 => p.1.2*(∑ i,m i*g i 0 p)-p.1.1*(∑ i,m i*g i 1 p))=
(fun p => ∑ i,m i*(p.1.2*g i 0 p-p.1.1*g i 1 p)) := by
funext p
simp only [Finset.mul_sum,←Finset.sum_sub_distrib]
apply Finset.sum_congr rfl
intro i _
ring
rw [he,integral_finsetSum _ (fun i _ => (htI i).const_mul (m i))]
simp_rw [integral_const_mul,hmt]
simp [Pi.single_apply]
· intro j n
obtain ⟨B,hB,hbound⟩ := fixed_smooth_tests_bound (fun i => g i j)
(fun i => hg i j) (fun i => hs i j) n
refine ⟨B,hB,?_⟩
intro m p
simpa only [smul_eq_mul] using hbound m ‖m‖ (norm_nonneg _) (norm_le_pi_norm m) p
end ScalarConductivity
end
end OAI
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