| import Mathlib | |
| import OAI.Analysis.Conductivity.Flux.AssembleCompatibleTensors | |
| import OAI.Analysis.Conductivity.Flux.OpenRegularTensorTarget | |
| import OAI.Analysis.Conductivity.Geometry.RegularRegionFiniteGluing | |
| 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 | |
| structure CompactGlobalUpdate (μ : Measure Coord3) (U : Set Coord3) | |
| (u : Coord3 → Fin 2 → ℝ) (A : Coord3 → Symmetric3) where | |
| du : Coord3 → Fin 2 → ℝ | |
| dF : Coord3 → Matrix (Fin 3) (Fin 2) ℝ | |
| tensor : Coord3 → Symmetric3 | |
| smooth_du : ContDiff ℝ (↑(⊤ : ℕ∞)) du | |
| compact_du : HasCompactSupport du | |
| support_du : tsupport du ⊆ U | |
| smooth_dF : ContDiff ℝ (↑(⊤ : ℕ∞)) dF | |
| compact_dF : HasCompactSupport dF | |
| support_dF : tsupport dF ⊆ U | |
| cauchy_dF : ∀ j (ψ : Coord3 → ℝ), ContDiff ℝ (↑(⊤ : ℕ∞)) ψ → | |
| (∫ x, fderiv ℝ ψ x ((dF x).col j) ∂μ) = 0 | |
| measurable_tensor : Measurable tensor | |
| constitutive : ∀ x, conductivityFlux (fun y => u y+du y) tensor x = conductivityFlux u A x+dF x | |
| theorem global_regular_depth_reduction | |
| (μ : Measure Coord3) [μ.IsAddHaarMeasure] [Measure.InnerRegularCompactLTTop μ] | |
| (u : Coord3 → Fin 2 → ℝ) (A : Coord3 → Symmetric3) (hAm : Measurable A) | |
| {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) | |
| (hreg : μ (U \ regularRegion u A U) = 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) | |
| (hgraph : ∀ᵐ x ∂μ, x ∈ U → A x ∈ matrixFiniteLaminate a b) | |
| {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) : | |
| ∃ R : CompactGlobalUpdate μ U u A, | |
| μ (U \ regularRegion (fun x => u x+R.du x) R.tensor U) = 0 ∧ | |
| (∀ᵐ x ∂μ, x ∈ U → R.tensor x ∈ matrixFiniteLaminate a b) ∧ | |
| μ (U \ R.tensor ⁻¹' spectralExtension (depthClass T n)) ≤ μ (U \ G) + ENNReal.ofReal ε := by | |
| classical | |
| obtain ⟨N,O,P,hO,hd,hP,hPG,hloss⟩ := finite_regular_depth_reduction μ u A hUb hdiv | |
| ha hT hsub hperm hsc hGm hG hε | |
| obtain ⟨du,dF,B,hus,huc,hUs,hFs,hFc,hFS,hpair,hB,heq,hoff,hconst⟩ := | |
| assemble_compatible_tensors μ O (fun i => (hO i).1) hd u A hAm P | |
| have hOU : (⋃ i, O i) ⊆ U := iUnion_subset (fun i => subset_closure.trans (hO i).2.2.1) | |
| let R : CompactGlobalUpdate μ U u A := ⟨du,dF,B,hus,huc,hUs.trans hOU, | |
| hFs,hFc,hFS.trans hOU,hpair,hB,hconst⟩ | |
| have hnew : ∀ i, RegularPatch (fun x => u x+du x) B (O i) := by | |
| intro i | |
| exact (hP i).congr (fun x hx => by rw [(heq i).1 hx]) (heq i).2.2 | |
| have huo : EqOn (fun x => u x+du x) u (⋃ i, O i)ᶜ := by | |
| intro x hx | |
| change u x+du x = u x | |
| rw [image_eq_zero_of_notMem_tsupport (fun hs => hx (hUs hs)),add_zero] | |
| refine ⟨R,regularRegion_finite_gluing_ae μ hreg O | |
| (fun i => subset_closure.trans (hO i).2.2.1) hnew huo hoff (fun i => (hO i).2.2.2),?_,?_⟩ | |
| · filter_upwards [hgraph] with x hx | |
| intro hxU | |
| change B x ∈ matrixFiniteLaminate a b | |
| by_cases hxin : x ∈ ⋃ i, O i | |
| · obtain ⟨i,hi⟩ := mem_iUnion.mp hxin | |
| rw [(heq i).2.2 hi] | |
| exact hPG i x hi | |
| · rw [hoff hxin] | |
| exact hx hxU | |
| · let bad : Fin N → Set Coord3 := fun i => | |
| {x | x ∈ O i ∧ (P i).tensor x ∉ spectralExtension (depthClass T n)} | |
| have hset : U \ B ⁻¹' spectralExtension (depthClass T n) ⊆ | |
| (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,by rw [← (heq i).2.2 hi]; exact hx.2⟩ | |
| · exact Or.inl ⟨hxG,hxO⟩ | |
| · exact Or.inl ⟨hx.1,hxG⟩ | |
| calc | |
| μ (U \ R.tensor ⁻¹' spectralExtension (depthClass T n)) ≤ | |
| μ (U \ G) + (μ (G \ ⋃ i, O i) + μ (⋃ i, bad i)) := | |
| (measure_mono hset).trans ((measure_union_le _ _).trans | |
| (add_le_add le_rfl (measure_union_le _ _))) | |
| _ ≤ μ (U \ G) + (μ (G \ ⋃ i, O i) + ∑ i, μ (bad i)) := | |
| add_le_add le_rfl (add_le_add le_rfl (measure_iUnion_fintype_le μ bad)) | |
| _ ≤ μ (U \ G) + ENNReal.ofReal ε := add_le_add le_rfl hloss | |
| end ScalarConductivity | |
| end | |
| end OAI | |