| import Mathlib |
| import OAI.Analysis.Conductivity.Variational.CompactDepthReductionRegular |
| import OAI.Analysis.Conductivity.Geometry.RegularPatch |
|
|
| 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 isOpen_regular_tensor_target (u : Coord3 → Fin 2 → ℝ) |
| (A : Coord3 → Symmetric3) (U : Set Coord3) {T : Set Symmetric3} (hT : IsOpen T) : |
| IsOpen (regularRegion u A U ∩ A ⁻¹' T) := by |
| rw [isOpen_iff_mem_nhds] |
| intro x hx |
| obtain ⟨O,hOU,hO,hxO⟩ := mem_regularRegion_iff.mp hx.1 |
| have hAc : ContinuousAt A x := Topology.IsInducing.subtypeVal.continuousAt_iff.mpr |
| (hO.2.2.1.continuousOn.continuousAt (hO.1.mem_nhds hxO)) |
| exact Filter.inter_mem ((isOpen_regularRegion u A U).mem_nhds hx.1) |
| (hAc.preimage_mem_nhds (hT.mem_nhds hx.2)) |
|
|
| theorem finite_regular_depth_reduction |
| (μ : Measure Coord3) [μ.IsAddHaarMeasure] [Measure.InnerRegularCompactLTTop μ] |
| (u : Coord3 → Fin 2 → ℝ) (A : Coord3 → Symmetric3) |
| {U : Set Coord3} (hUb : Bornology.IsBounded U) |
| (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) |
| {a b : ℝ} (ha : 0 < a) {T : Set DiagonalTriple} |
| (hT : IsOpen T) (hsub : ∀ d ∈ T, IsFiniteLaminate a b d) |
| (hperm : ∀ d ∈ T, ∀ e : Equiv.Perm (Fin 3), d ∘ e ∈ T) |
| {n : ℕ} (hsc : (spectralExtension (depthClass T n)).Nonempty) |
| {G : Set Coord3} (hGm : MeasurableSet G) |
| (hG : G ⊆ regularRegion u A U ∩ A ⁻¹' spectralExtension (depthClass T (n+1))) |
| {ε : ℝ} (hε : 0 < ε) : |
| ∃ (N : ℕ) (O : Fin N → Set Coord3) (R : ∀ i, CompactCompatibleReplacement μ (O i) u A), |
| (∀ i, IsOpen (O i) ∧ IsCompact (closure (O i)) ∧ closure (O i) ⊆ U ∧ μ (frontier (O i)) = 0) ∧ |
| Pairwise (fun i j => Disjoint (O i) (O j)) ∧ |
| (∀ i, RegularPatch (fun x => u x+(R i).du x) (R i).tensor (O i)) ∧ |
| (∀ i x, x ∈ O i → (R i).tensor x ∈ matrixFiniteLaminate a b) ∧ |
| μ (G \ ⋃ i, O i) + |
| ∑ i, μ {x | x ∈ O i ∧ (R i).tensor x ∉ spectralExtension (depthClass T n)} |
| ≤ ENNReal.ofReal ε := by |
| classical |
| have hlocal : ∀ p : G, ∃ W : Set Coord3, |
| IsOpen W ∧ p.val ∈ W ∧ closure W ⊆ U ∧ IsCompact (closure W) ∧ |
| ∀ O : Set Coord3, IsOpen O → O ⊆ W → ∀ δ : ℝ, 0 < δ → |
| ∃ R : CompactCompatibleReplacement μ O u A, |
| RegularPatch (fun x => u x+R.du x) R.tensor O ∧ |
| (∀ x ∈ O, R.tensor x ∈ matrixFiniteLaminate a b) ∧ |
| μ {x | x ∈ O ∧ R.tensor x ∉ spectralExtension (depthClass T n)} ≤ ENNReal.ofReal δ := by |
| intro p |
| obtain ⟨V,hVU,hV,hpV⟩ := mem_regularRegion_iff.mp (hG p.property).1 |
| obtain ⟨W,hW,hpW,hWV,hWc,hop⟩ := exists_compact_depth_reduction_regular μ u A |
| hV.1 (hUb.subset hVU) hV.2.1 hV.2.2.1 |
| (fun j ψ hs hc ht => hdiv j ψ hs hc (ht.trans hVU)) hV.2.2.2 |
| ha hT hsub hperm hpV (hG p.property).2 hsc |
| refine ⟨W,hW,hpW,hWV.trans hVU,hWc,?_⟩ |
| intro O hO hOW δ hδ |
| obtain ⟨R,hr,hRG,hbad⟩ := hop O hO hOW δ hδ |
| exact ⟨R,⟨hO,(hV.2.1.mono (hOW.trans (subset_closure.trans hWV))).add R.smooth_du.contDiffOn, |
| R.smooth_tensor,hr⟩,hRG,hbad⟩ |
| choose W hWo hpW hWU hWc hop using hlocal |
| have hμG : μ G ≠ ∞ := (hUb.subset (hG.trans (inter_subset_left.trans (regularRegion_subset u A U)))).measure_lt_top.ne |
| have hcover : G ⊆ ⋃ p : G, W p := fun x hx => mem_iUnion.mpr ⟨⟨x,hx⟩,hpW ⟨x,hx⟩⟩ |
| obtain ⟨N,O,tag,hO,hd,hmiss⟩ := finite_disjoint_localization_null μ hGm hμG W hWo hcover |
| (ENNReal.ofReal_pos.mpr (by linarith : 0 < ε/2)).ne' |
| let δ : ℝ := (ε/2) / (N+1) |
| have hδ : 0 < δ := div_pos (by linarith) (by positivity) |
| choose R hR hRG hbad using fun i => hop (tag i) (O i) (hO i).1 |
| (subset_closure.trans (hO i).2.2.1) δ hδ |
| refine ⟨N,O,R,fun i => ⟨(hO i).1,(hO i).2.1,(hO i).2.2.1.trans (subset_closure.trans (hWU (tag i))), |
| (hO i).2.2.2⟩,hd,hR,hRG,?_⟩ |
| have hsum : (∑ i : Fin N, μ {x | x ∈ O i ∧ (R i).tensor x ∉ spectralExtension (depthClass T n)}) |
| ≤ ENNReal.ofReal (ε/2) := by |
| calc |
| _ ≤ ∑ _i : Fin N, ENNReal.ofReal δ := Finset.sum_le_sum (fun i _ => hbad i) |
| _ = ENNReal.ofReal ((N : ℝ) * δ) := by simp [ENNReal.ofReal_mul] |
| _ ≤ ENNReal.ofReal (ε/2) := ENNReal.ofReal_le_ofReal (by |
| dsimp [δ] |
| rw [← mul_div_assoc] |
| apply (div_le_iff₀ (by positivity : (0:ℝ) < N+1)).mpr |
| nlinarith) |
| calc |
| _ ≤ ENNReal.ofReal (ε/2) + ENNReal.ofReal (ε/2) := 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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|
