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