| import Mathlib |
| import OAI.Analysis.Conductivity.Variational.NormalStretch |
| import OAI.Analysis.Conductivity.Geometry.CollarLocalInverse |
| import OAI.Analysis.Conductivity.Geometry.PhysicalCollarBand |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace ScalarConductivity |
| open Set MeasureTheory Filter Topology |
|
|
| lemma sourceCollarOpenBox_eq_interior : sourceCollarOpenBox= |
| interior (sourceExtendedBox (-(1:ℝ)/100) (1/100)) := by |
| have hbox : sourceExtendedBox (-(1:ℝ)/100) (1/100)= |
| (univ : Set (Fin 3)).pi (fun k => Icc (![(-(1:ℝ)/100),-1,-1] k) (![1/100,1,1] k)) := by |
| ext x |
| simp only [sourceExtendedBox,mem_Icc,Set.mem_pi,mem_univ,true_implies,Pi.le_def] |
| simp only [forall_and] |
| rw [hbox,interior_pi_set (finite_univ),show (fun k : Fin 3 => interior |
| (Icc (![(-(1:ℝ)/100),-1,-1] k) (![1/100,1,1] k)))= |
| (fun k => Ioo (![(-(1:ℝ)/100),-1,-1] k) (![1/100,1,1] k)) from funext (fun _ => interior_Icc)] |
| ext x |
| simp only [sourceCollarOpenBox,mem_ofPred_eq,Set.mem_pi,mem_univ,true_implies,Fin.forall_fin_succ, |
| mem_Ioo,Matrix.cons_val_zero,Matrix.cons_val_succ,Fin.forall_fin_zero, |
| abs_lt] |
| simp |
|
|
| lemma sourceCollar_frontier_null (i j : Fin 4) : |
| volume (sourceCollarPiece i j '' frontier (sourceExtendedBox (-(1:ℝ)/100) (1/100)))=0 := by |
| let B := sourceExtendedBox (-(1:ℝ)/100) (1/100) |
| have hB : IsClosed B := isClosed_Icc |
| have hs : frontier B⊆B := hB.frontier_subset |
| have hzero : volume (frontier B)=0 := (convex_Icc _ _).addHaar_frontier volume |
| have H := measure_image_le_det volume isClosed_frontier.measurableSet |
| (fun x hx => (sourceCollarPiece_contDiff i j).differentiable (by simp) x) |
| ((sourceExtendedBox_injOn i j (by rfl) (by rfl)).mono hs) |
| (C:=2) (fun x hx => by |
| rw [(sourceCollarPiece_hasFDeriv i j x).fderiv] |
| exact sourceExtended_jacobian_upper i j (by rfl) (by rfl) (hs hx)) |
| have H' : volume (sourceCollarPiece i j '' frontier B) ≤ 0 := by |
| simpa only [hzero,mul_zero] using H |
| exact le_antisymm H' bot_le |
|
|
| lemma sourceCollar_open_charts_ae : ∀ᵐ y : Fin 3 → ℝ, |
| y∈sourceClosedCollarBand (-(1:ℝ)/100) (1/100) → |
| ∃ i j : Fin 4,∃ x∈sourceCollarOpenBox,sourceCollarPiece i j x=y := by |
| let N := ⋃ i : Fin 4,⋃ j : Fin 4,sourceCollarPiece i j '' |
| frontier (sourceExtendedBox (-(1:ℝ)/100) (1/100)) |
| have hN : volume N=0 := measure_iUnion_null (fun i => |
| measure_iUnion_null (fun j => sourceCollar_frontier_null i j)) |
| have hn : ∀ᵐ y : Fin 3 → ℝ,y∉N := by |
| rw [ae_iff] |
| simpa only [not_not,Set.ofPred_mem_eq] using hN |
| filter_upwards [hn] with y hy ht |
| rw [sourceClosedCollarBand_eq _ _ (by rfl) (by rfl)] at ht |
| obtain ⟨i,hi⟩ := mem_iUnion.mp ht |
| obtain ⟨j,x,hx,he⟩ := mem_iUnion.mp hi |
| refine ⟨i,j,x,?_,he⟩ |
| rw [sourceCollarOpenBox_eq_interior] |
| by_contra hn |
| apply hy |
| exact mem_iUnion.mpr ⟨i,mem_iUnion.mpr ⟨j,⟨x, |
| ⟨subset_closure hx,hn⟩,he⟩⟩⟩ |
|
|
| end ScalarConductivity |
|
|
| end |
|
|
| end OAI |
|
|
