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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 | import Mathlib
import OAI.Analysis.Conductivity.Fourier.SpectralExtensionNonemptyNonempty
import OAI.Analysis.Conductivity.Branching.MeasurableDisjointGluingMeasurable
namespace OAI
noncomputable section
open MeasureTheory
open scoped ENNReal
open Matrix Filter Topology
open Set MeasureTheory Filter Topology
open scoped BigOperators
open Set MeasureTheory Filter Topology
open scoped Manifold
open Set Filter
open scoped Topology
open Set Filter MeasureTheory
open scoped Topology Manifold ENNReal
open Set
namespace ScalarConductivity
open Matrix Set MeasureTheory Filter Topology
open scoped Matrix.Norms.Elementwise ENNReal
lemma regularRegion_congr_subset {u v : Coord3 → Fin 2 → ℝ}
{A B : Coord3 → Symmetric3} {O U : Set Coord3} (hOU : O ⊆ U)
(hu : EqOn v u O) (hA : EqOn B A O) :
regularRegion u A O ⊆ regularRegion v B U := by
intro x hx
obtain ⟨V,hVO,hV,hxV⟩ := mem_regularRegion_iff.mp hx
exact mem_regularRegion_iff.mpr ⟨V,hVO.trans hOU,hV.congr (hu.mono hVO) (hA.mono hVO),hxV⟩
lemma regularRegion_piecewise_ae
(μ : Measure Coord3) {I : Type*} [Fintype I]
{u v : Coord3 → Fin 2 → ℝ} {A B : Coord3 → Symmetric3} {U : Set Coord3}
(O : I → Set Coord3) (hO : ∀ i, IsOpen (O i)) (hOU : ∀ i, O i ⊆ U)
(old : μ (U \ regularRegion u A U) = 0)
(localgood : ∀ i, μ (O i \ regularRegion v B (O i)) = 0)
(hu : EqOn v u (⋃ i, O i)ᶜ) (hA : EqOn B A (⋃ i, O i)ᶜ)
(hnull : ∀ i, μ (frontier (O i)) = 0) :
μ (U \ regularRegion v B U) = 0 := by
have hs : U \ regularRegion v B U ⊆ (U \ regularRegion u A U) ∪
(⋃ i, O i \ regularRegion v B (O i)) ∪ (⋃ i, frontier (O i)) := by
intro x hx
by_cases hr : x ∈ regularRegion u A U
· by_cases hi : x ∈ ⋃ i, O i
· obtain ⟨i,hi⟩ := mem_iUnion.mp hi
exact Or.inl (Or.inr (mem_iUnion.mpr ⟨i,hi,fun hh => hx.2
(regularRegion_congr_subset (hOU i) (fun _ _ => rfl) (fun _ _ => rfl) hh)⟩))
· by_cases hf : x ∈ ⋃ i, frontier (O i)
· exact Or.inr hf
· exfalso
apply hx.2
obtain ⟨V,hVU,hV,hxV⟩ := mem_regularRegion_iff.mp hr
let W := V \ ⋃ i, closure (O i)
have hW : IsOpen W := hV.1.sdiff (isClosed_iUnion_of_finite (fun _ => isClosed_closure))
have hxW : x ∈ W := by
refine ⟨hxV,?_⟩
intro hh
obtain ⟨i,hi'⟩ := mem_iUnion.mp hh
apply hf
refine mem_iUnion.mpr ⟨i,?_⟩
rw [frontier,(hO i).interior_eq]
exact ⟨hi',fun hh => hi (mem_iUnion.mpr ⟨i,hh⟩)⟩
have hWoff : W ⊆ (⋃ i, O i)ᶜ := by
intro y hy hh
obtain ⟨i,hyi⟩ := mem_iUnion.mp hh
exact hy.2 (mem_iUnion.mpr ⟨i,subset_closure hyi⟩)
exact mem_regularRegion_iff.mpr ⟨W,sdiff_subset.trans hVU,
(hV.mono hW sdiff_subset).congr (hu.mono hWoff) (hA.mono hWoff),hxW⟩
· exact Or.inl (Or.inl ⟨hx.1,hr⟩)
exact measure_mono_null hs (measure_union_null (measure_union_null old
(measure_iUnion_null localgood)) (measure_iUnion_null hnull))
end ScalarConductivity
namespace ScalarConductivity
open Matrix Set MeasureTheory Filter Topology
open scoped Matrix.Norms.Elementwise ENNReal
lemma regularRegion_restrict_open {u : Coord3 → Fin 2 → ℝ} {A : Coord3 → Symmetric3}
{U O : Set Coord3} (hO : IsOpen O) :
O ∩ regularRegion u A U ⊆ regularRegion u A O := by
intro x hx
obtain ⟨V,hVU,hV,hxV⟩ := mem_regularRegion_iff.mp hx.2
exact mem_regularRegion_iff.mpr ⟨V ∩ O,inter_subset_right,
hV.mono (hV.1.inter hO) inter_subset_left,hxV,hx.1⟩
theorem fine_global_scalar_tensor_approximation
(μ : Measure Coord3) [μ.IsAddHaarMeasure] [Measure.InnerRegularCompactLTTop μ]
{a b : ℝ} (ha : 0 < a) (hab : a < b)
{U : Set Coord3} (hUb : Bornology.IsBounded U)
(u : Coord3 → Fin 2 → ℝ) (A : Coord3 → Symmetric3) (hAm : Measurable A)
(hint : SmoothFluxIntegrable μ U (conductivityFlux u A))
(hdiv : ∀ j (ψ : Coord3 → ℝ), ContDiff ℝ (↑(⊤ : ℕ∞)) ψ → HasCompactSupport ψ →
tsupport ψ ⊆ U → (∫ x, fderiv ℝ ψ x ((conductivityFlux u A x).col j) ∂μ) = 0)
(hreg : μ (U \ regularRegion u A U) = 0)
(hgraph : ∀ᵐ x ∂μ, x ∈ U → A x ∈ matrixFiniteLaminate a b)
{η ε δ : ℝ} (hη : 0 < η) (hε : 0 < ε) (hδ : 0 < δ) :
∃ (N : ℕ) (O : Fin N → Set Coord3) (center : Fin N → Coord3)
(R : ∀ i, CompactGlobalUpdate μ (O i) u A) (S : CompactGlobalUpdate μ U u A),
(∀ i, IsOpen (O i) ∧ closure (O i) ⊆ U ∧ closure (O i) ⊆ Metric.ball (center i) δ) ∧
Pairwise (fun i j => Disjoint (O i) (O j)) ∧
S.du = (fun x => ∑ i, (R i).du x) ∧ S.dF = (fun x => ∑ i, (R i).dF x) ∧
μ (U \ regularRegion (fun x => u x+S.du x) S.tensor U) = 0 ∧
(∀ᵐ x ∂μ, x ∈ U → S.tensor x ∈ matrixFiniteLaminate a b) ∧
μ (U \ S.tensor ⁻¹' spectralExtension (scalarNeighbourhood a b η)) ≤ ENNReal.ofReal ε := by
classical
let G := regularRegion u A U
have hGo : IsOpen G := isOpen_regularRegion u A U
have hGU : G ⊆ U := regularRegion_subset u A U
obtain ⟨N,O,tag,center,hO,hd,hmiss⟩ := finite_disjoint_localization_fine μ hGo.measurableSet
(hUb.subset hGU).measure_lt_top.ne (fun _ : Unit => G) (fun _ => hGo)
(fun x hx => mem_iUnion.mpr ⟨(),hx⟩)
(ENNReal.ofReal_pos.mpr (by linarith : 0 < ε/2)).ne' hδ
have hOU : ∀ i, O i ⊆ U := fun i => subset_closure.trans ((hO i).2.2.1.trans hGU)
have hOr : ∀ i, μ (O i \ regularRegion u A (O i)) = 0 := by
intro i
have hs : O i ⊆ regularRegion u A (O i) := fun x hx =>
regularRegion_restrict_open (hO i).1 ⟨hx,(hO i).2.2.1 (subset_closure hx)⟩
simp only [sdiff_eq_empty.mpr hs,measure_empty]
let ρ : ℝ := (ε/2)/(N+1)
have hρ : 0 < ρ := div_pos (by linarith) (by positivity)
have hlocal : ∀ i, ∃ R : CompactGlobalUpdate μ (O i) u A,
μ (O i \ regularRegion (fun x => u x+R.du x) R.tensor (O i)) = 0 ∧
(∀ᵐ x ∂μ, x ∈ O i → R.tensor x ∈ matrixFiniteLaminate a b) ∧
μ (O i \ R.tensor ⁻¹' spectralExtension (scalarNeighbourhood a b η)) ≤ ENNReal.ofReal ρ := by
intro i
apply global_scalar_tensor_approximation μ ha hab (hO i).1.measurableSet (hUb.subset (hOU i)) u A hAm
(fun j ψ hs hc ht => hint j ψ hs hc (ht.trans (hOU i)))
(fun j ψ hs hc ht => hdiv j ψ hs hc (ht.trans (hOU i))) (hOr i)
· filter_upwards [hgraph] with x hx
exact fun hi => hx (hOU i hi)
· exact hη
· exact hρ
choose R hRreg hRG hRbad using hlocal
obtain ⟨S,hsu,hsF,hS,hSoff⟩ := assemble_global_updates μ O (fun i => (hO i).1) hOU hd u A hAm R
have hsuoff : EqOn (fun x => u x+S.du x) u (⋃ i, O i)ᶜ := by
intro x hx
change u x+S.du x = u x
rw [hsu]
change u x+(∑ i, (R i).du x) = u x
rw [supported_sum_eq_zero_off O _ (fun i => (R i).support_du) hx,add_zero]
have hregS : μ (U \ regularRegion (fun x => u x+S.du x) S.tensor U) = 0 := by
apply regularRegion_piecewise_ae μ O (fun i => (hO i).1) hOU hreg _ hsuoff hSoff
(fun i => (hO i).2.2.2.1)
intro i
apply measure_mono_null _ (hRreg i)
exact sdiff_subset_sdiff_right (regularRegion_congr_subset Subset.rfl
(fun x hx => by rw [(hS i).1 hx]) (hS i).2.2)
refine ⟨N,O,center,R,S,fun i => ⟨(hO i).1,(hO i).2.2.1.trans hGU,(hO i).2.2.2.2⟩,
hd,hsu,hsF,hregS,?_,?_⟩
· have hRGall : ∀ᵐ x ∂μ, ∀ i, x ∈ O i → (R i).tensor x ∈ matrixFiniteLaminate a b :=
(ae_all_iff).mpr hRG
filter_upwards [hgraph,hRGall] with x hx hRx
intro hxU
by_cases hi : x ∈ ⋃ i, O i
· obtain ⟨i,hi⟩ := mem_iUnion.mp hi
rw [(hS i).2.2 hi]
exact hRx i hi
· rw [hSoff hi]
exact hx hxU
· let T := spectralExtension (scalarNeighbourhood a b η)
let bad : Fin N → Set Coord3 := fun i => O i \ (R i).tensor ⁻¹' T
have hs : U \ S.tensor ⁻¹' T ⊆ (U \ G) ∪ ((G \ ⋃ i, O i) ∪ ⋃ i, bad i) := by
intro x hx
by_cases hxG : x ∈ G
· right
by_cases hxO : x ∈ ⋃ i, O i
· right
obtain ⟨i,hi⟩ := mem_iUnion.mp hxO
exact mem_iUnion.mpr ⟨i,hi,fun ht => hx.2 (by change S.tensor x ∈ T; rw [(hS i).2.2 hi]; exact ht)⟩
· exact Or.inl ⟨hxG,hxO⟩
· exact Or.inl ⟨hx.1,hxG⟩
have hsum : (∑ i : Fin N, μ (bad i)) ≤ ENNReal.ofReal (ε/2) := by
calc
_ ≤ ∑ _i : Fin N, ENNReal.ofReal ρ := Finset.sum_le_sum (fun i _ => hRbad i)
_ = ENNReal.ofReal ((N : ℝ)*ρ) := by simp [ENNReal.ofReal_mul]
_ ≤ ENNReal.ofReal (ε/2) := ENNReal.ofReal_le_ofReal (by
dsimp [ρ]
rw [← mul_div_assoc]
apply (div_le_iff₀ (by positivity : (0:ℝ) < N+1)).mpr
nlinarith)
calc
_ ≤ μ (U \ G) + (μ (G \ ⋃ i, O i) + ∑ i, μ (bad i)) :=
(measure_mono (μ := μ) hs).trans ((measure_union_le _ _).trans
(add_le_add le_rfl ((measure_union_le _ _).trans (add_le_add le_rfl (measure_iUnion_fintype_le μ bad)))))
_ ≤ ENNReal.ofReal (ε/2)+ENNReal.ofReal (ε/2) := by
rw [hreg,zero_add]
exact add_le_add hmiss.le hsum
_ = ENNReal.ofReal ε := by
rw [← ENNReal.ofReal_add (by linarith) (by linarith)]
congr 1
ring
end ScalarConductivity
end
end OAI
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