| import Mathlib | |
| 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 | |
| lemma regularRegion_finite_gluing | |
| {I : Type*} [Fintype I] | |
| {u v : Coord3 → Fin 2 → ℝ} {A B : Coord3 → Symmetric3} {U : Set Coord3} | |
| (O : I → Set Coord3) (hOU : ∀ i, O i ⊆ U) | |
| (hnew : ∀ i, RegularPatch v B (O i)) | |
| (hu : EqOn v u (⋃ i, O i)ᶜ) (hA : EqOn B A (⋃ i, O i)ᶜ) : | |
| regularRegion u A U \ (⋃ i, frontier (O i)) ⊆ regularRegion v B U := by | |
| intro x hx | |
| by_cases hxi : x ∈ ⋃ i, O i | |
| · obtain ⟨i,hi⟩ := mem_iUnion.mp hxi | |
| exact mem_regularRegion_iff.mpr ⟨O i,hOU i,hnew i,hi⟩ | |
| · obtain ⟨V,hVU,hV,hxV⟩ := mem_regularRegion_iff.mp hx.1 | |
| let W := V \ ⋃ i, closure (O i) | |
| have hclosed : IsClosed (⋃ i, closure (O i)) := isClosed_iUnion_of_finite (fun _ => isClosed_closure) | |
| have hW : IsOpen W := hV.1.sdiff hclosed | |
| have hxW : x ∈ W := by | |
| refine ⟨hxV,?_⟩ | |
| intro hxc | |
| obtain ⟨i,hi⟩ := mem_iUnion.mp hxc | |
| apply hx.2 | |
| refine mem_iUnion.mpr ⟨i,?_⟩ | |
| rw [frontier, (hnew i).1.interior_eq] | |
| exact ⟨hi,fun hh => hxi (mem_iUnion.mpr ⟨i,hh⟩)⟩ | |
| have hWoff : W ⊆ (⋃ i, O i)ᶜ := by | |
| intro y hy hi | |
| apply hy.2 | |
| obtain ⟨i,hyi⟩ := mem_iUnion.mp hi | |
| exact 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⟩ | |
| lemma regularRegion_finite_gluing_ae | |
| (μ : Measure Coord3) {I : Type*} [Fintype I] | |
| {u v : Coord3 → Fin 2 → ℝ} {A B : Coord3 → Symmetric3} {U : Set Coord3} | |
| (hold : μ (U \ regularRegion u A U) = 0) | |
| (O : I → Set Coord3) (hOU : ∀ i, O i ⊆ U) | |
| (hnew : ∀ i, RegularPatch v B (O i)) | |
| (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, frontier (O i)) := by | |
| intro x hx | |
| by_cases hr : x ∈ regularRegion u A U | |
| · right | |
| by_contra hf | |
| exact hx.2 (regularRegion_finite_gluing O hOU hnew hu hA ⟨hr,hf⟩) | |
| · exact Or.inl ⟨hx.1,hr⟩ | |
| exact measure_mono_null hs (measure_union_null hold (measure_iUnion_null hnull)) | |
| end ScalarConductivity | |
| end | |
| end OAI | |