| 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 | |