| import Mathlib | |
| import OAI.Analysis.Conductivity.Variational.Unchanged | |
| 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 TwoFieldRankRegular.congr {u v : Coord3 → Fin 2 → ℝ} {O : Set Coord3} | |
| (hO : IsOpen O) (h : TwoFieldRankRegular u O) (he : EqOn v u O) : | |
| TwoFieldRankRegular v O := by | |
| rcases h with h | h | ⟨κ,hκ⟩ | |
| · left | |
| intro x hx | |
| rw [fderiv_eq_of_eqOn_open hO he hx] | |
| exact h x hx | |
| · right; left | |
| obtain ⟨w,l,κ,hw,hl,hwd,hrel⟩ := h | |
| exact ⟨w,l,κ,hw,hl,hwd,fun x hx j => by rw [he hx]; exact hrel x hx j⟩ | |
| · exact Or.inr (Or.inr ⟨κ,fun x hx => (he hx).trans (hκ x hx)⟩) | |
| def RegularPatch (u : Coord3 → Fin 2 → ℝ) (A : Coord3 → Symmetric3) | |
| (O : Set Coord3) : Prop := | |
| IsOpen O ∧ ContDiffOn ℝ (↑(⊤ : ℕ∞)) u O ∧ | |
| ContDiffOn ℝ (↑(⊤ : ℕ∞)) (fun x => (A x).val) O ∧ TwoFieldRankRegular u O | |
| lemma RegularPatch.mono {u : Coord3 → Fin 2 → ℝ} {A : Coord3 → Symmetric3} | |
| {O W : Set Coord3} (h : RegularPatch u A O) (hW : IsOpen W) (hWO : W ⊆ O) : | |
| RegularPatch u A W := | |
| ⟨hW,h.2.1.mono hWO,h.2.2.1.mono hWO,h.2.2.2.mono hWO⟩ | |
| lemma RegularPatch.congr {u v : Coord3 → Fin 2 → ℝ} {A B : Coord3 → Symmetric3} | |
| {O : Set Coord3} (h : RegularPatch u A O) (hu : EqOn v u O) (hA : EqOn B A O) : | |
| RegularPatch v B O := by | |
| exact ⟨h.1,h.2.1.congr hu,h.2.2.1.congr (fun x hx => congrArg Subtype.val (hA hx)), | |
| h.2.2.2.congr h.1 hu⟩ | |
| def regularRegion (u : Coord3 → Fin 2 → ℝ) (A : Coord3 → Symmetric3) | |
| (U : Set Coord3) : Set Coord3 := | |
| ⋃ (O : Set Coord3) (_ : O ⊆ U) (_ : RegularPatch u A O), O | |
| lemma isOpen_regularRegion (u : Coord3 → Fin 2 → ℝ) (A : Coord3 → Symmetric3) | |
| (U : Set Coord3) : IsOpen (regularRegion u A U) := by | |
| exact isOpen_iUnion (fun O => isOpen_iUnion (fun _ => isOpen_iUnion (fun h => h.1))) | |
| lemma regularRegion_subset (u : Coord3 → Fin 2 → ℝ) (A : Coord3 → Symmetric3) | |
| (U : Set Coord3) : regularRegion u A U ⊆ U := by | |
| intro x hx | |
| obtain ⟨O,hx⟩ := mem_iUnion.mp hx | |
| obtain ⟨hOU,hx⟩ := mem_iUnion.mp hx | |
| obtain ⟨_,hx⟩ := mem_iUnion.mp hx | |
| exact hOU hx | |
| lemma mem_regularRegion_iff {u : Coord3 → Fin 2 → ℝ} {A : Coord3 → Symmetric3} | |
| {U : Set Coord3} {p : Coord3} : p ∈ regularRegion u A U ↔ | |
| ∃ O : Set Coord3, O ⊆ U ∧ RegularPatch u A O ∧ p ∈ O := by | |
| simp only [regularRegion, mem_iUnion, exists_prop] | |
| lemma exists_precompact_regular_patch {u : Coord3 → Fin 2 → ℝ} | |
| {A : Coord3 → Symmetric3} {U : Set Coord3} {p : Coord3} | |
| (hp : p ∈ regularRegion u A U) : | |
| ∃ W : Set Coord3, p ∈ W ∧ closure W ⊆ U ∧ IsCompact (closure W) ∧ RegularPatch u A W := by | |
| obtain ⟨O,hOU,hO,hpO⟩ := mem_regularRegion_iff.mp hp | |
| obtain ⟨W,hW,hpW,hWO,hWc⟩ := exists_precompact_open_inside hO.1 hpO | |
| exact ⟨W,hpW,hWO.trans hOU,hWc,hO.mono hW (subset_closure.trans hWO)⟩ | |
| end ScalarConductivity | |
| end | |
| end OAI | |