| import Mathlib |
| import OAI.Analysis.Conductivity.Geometry.RegularPatch |
| import OAI.Analysis.Conductivity.Variational.FiniteEndPhysicalRegularity |
| import OAI.Analysis.Conductivity.Flux.FiniteEndPhysicalSmoothTensor |
| import OAI.Analysis.Conductivity.Flux.VariableCollarTensor |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| namespace ScalarConductivity |
| open Set Filter Topology MeasureTheory Matrix |
| open scoped Matrix.Norms.Elementwise |
| |
| def variableFaceField (u : Coord3 → Fin 2 → ℝ) (a b : ℝ) (i j : Fin 4) : Coord3 → Fin 2 → ℝ := |
| fun y => u (flatEndCoordinates a b (sourceFaceAngles i j (sourceCollarInverse i j y))) |
|
|
| def variableCollarField (u : Coord3 → Fin 2 → ℝ) (a b : ℝ) : Coord3 → Fin 2 → ℝ := |
| sourceFacePaste (variableFaceField u a b) 0 |
|
|
| lemma variableCollarField_open (u : Coord3 → Fin 2 → ℝ) (a b : ℝ) |
| (i j : Fin 4) {x : Coord3} (hx : x∈sourceCollarOpenBox) : |
| variableCollarField u a b (sourceCollarPiece i j x)= |
| u (flatEndCoordinates a b (sourceFaceAngles i j x)) := by |
| rw [variableCollarField,sourceFacePaste_open _ _ i j hx] |
| simp only [variableFaceField,sourceCollarInverse_point i j hx] |
|
|
| lemma flatEndCoordinates_fderiv_surjective {a : ℝ} (ha : a≠0) (b : ℝ) (x : Coord3) : |
| Function.Surjective (fderiv ℝ (flatEndCoordinates a b) x) := by |
| change Function.Surjective (fderiv ℝ (endAxialAffine a b) x) |
| rw [(endAxialAffine_hasFDeriv a b x).fderiv] |
| exact (LinearMap.equivOfDetNeZero (Matrix.toLin' (endAxialMatrix a)) (by |
| simpa [LinearMap.det_toLin',endAxialMatrix,Matrix.det_diagonal,Fin.prod_univ_succ] using ha)).surjective |
|
|
| lemma sourceFaceAngles_fderiv_surjective (i j : Fin 4) {x : Coord3} |
| (hx : x∈sourceCollarOpenBox) : Function.Surjective (fderiv ℝ (sourceFaceAngles i j) x) := by |
| rw [(sourceFaceAngles_hasFDeriv i j x).fderiv] |
| exact (LinearMap.equivOfDetNeZero (Matrix.toLin' (sourceFaceAngleMatrix i j x)) (by |
| rw [LinearMap.det_toLin'] |
|
|
| lemma variableCollarTensor_local_eq (K : Coord3 → Mat3) (a b : ℝ) |
| (i j : Fin 4) {x : Coord3} (hx : x∈sourceCollarOpenBox) : |
| variableCollarTensor K a b =ᶠ[𝓝 (sourceCollarPiece i j x)] |
| (attachedVariableTensor (K ∘ flatEndCoordinates a b ∘ sourceFaceAngles i j) a i j ∘ sourceCollarInverse i j) := by |
| change ∀ᶠ y in 𝓝 (sourceCollarPiece i j x), _ = _ |
| rw [←sourceCollarPiece_map_nhds i j hx,eventually_map] |
| filter_upwards [isOpen_sourceCollarOpenBox.mem_nhds hx] with y hy |
| simpa only [Function.comp_def,sourceCollarInverse_point i j hy] using |
| variableCollarTensor_open K a b i j hy |
|
|
| lemma variableCollarTensor_smooth_at {K : Coord3 → Mat3} {a : ℝ} (ha : a≠0) (b : ℝ) |
| (i j : Fin 4) {x : Coord3} (hx : x∈sourceCollarOpenBox) |
| (hK : ContDiffAt ℝ (↑(⊤:ℕ∞)) K (flatEndCoordinates a b (sourceFaceAngles i j x))) : |
| ContDiffAt ℝ (↑(⊤:ℕ∞)) (variableCollarTensor K a b) (sourceCollarPiece i j x) := by |
| have hk : ContDiffAt ℝ (↑(⊤:ℕ∞)) (K ∘ flatEndCoordinates a b ∘ sourceFaceAngles i j) x := |
| hK.comp x ((contDiff_flatEndCoordinates a b).contDiffAt.comp x |
| (contDiff_sourceFaceAngles i j).contDiffAt) |
| have ht : ContDiffAt ℝ (↑(⊤:ℕ∞)) |
| (attachedVariableTensor (K ∘ flatEndCoordinates a b ∘ sourceFaceAngles i j) a i j) |
| (sourceCollarInverse i j (sourceCollarPiece i j x)) := by |
| rw [sourceCollarInverse_point i j hx] |
| exact contDiffAt_attachedVariableTensor hk ha i j hx |
| exact (ht.comp _ (sourceCollarInverse_contDiffAt i j hx)).congr_of_eventuallyEq |
| (variableCollarTensor_local_eq K a b i j hx) |
|
|
| theorem variableCollar_regular_at {u : Coord3 → Fin 2 → ℝ} {K : Coord3 → Mat3} |
| (hS : ∀ x,(K x).IsSymm) {a : ℝ} (ha : a≠0) (b : ℝ) |
| (i j : Fin 4) {x : Coord3} (hx : x∈sourceCollarOpenBox) {U : Set Coord3} |
| (hr : flatEndCoordinates a b (sourceFaceAngles i j x)∈ |
| regularRegion u (fun y => ⟨K y,hS y⟩) U) : |
| sourceCollarPiece i j x∈regularRegion (variableCollarField u a b) |
| (fun y => ⟨variableCollarTensor K a b y,variableCollarTensor_symm hS a b y⟩) |
| (sourceCollarPiece i j '' sourceCollarOpenBox) := by |
| obtain ⟨O,hOU,hO,hxO⟩ := mem_regularRegion_iff.mp hr |
| let V := sourceCollarPiece i j '' sourceCollarOpenBox |
| let g : Coord3 → Coord3 := fun y => flatEndCoordinates a b (sourceFaceAngles i j (sourceCollarInverse i j y)) |
| have hV : IsOpen V := isOpen_sourceCollarOpenImage i j |
| have hg : ContDiffOn ℝ (↑(⊤:ℕ∞)) g V := |
| (contDiff_flatEndCoordinates a b).comp_contDiffOn |
| ((contDiff_sourceFaceAngles i j).comp_contDiffOn (sourceCollarInverse_contDiffOn i j)) |
| let W := V∩g ⁻¹' O |
| have hW : IsOpen W := hg.continuousOn.isOpen_inter_preimage hV hO.1 |
| have hm : MapsTo g W O := fun _ hy => hy.2 |
| have hd : ∀ y∈W,Function.Surjective (fderiv ℝ g y) := by |
| rintro y ⟨⟨z,hz,rfl⟩,_⟩ |
| have hinv := (sourceCollarInverse_contDiffAt i j hz).differentiableAt (by simp) |
| have hang := (contDiff_sourceFaceAngles i j).differentiable (by simp) |
| have hflat := (contDiff_flatEndCoordinates a b).differentiable (by simp) |
| change Function.Surjective (fderiv ℝ (flatEndCoordinates a b ∘ (sourceFaceAngles i j ∘ sourceCollarInverse i j)) _) |
| rw [fderiv_comp _ (hflat _) ((hang _).comp _ hinv),fderiv_comp _ (hang _) hinv] |
| have hsur : Function.Surjective (fderiv ℝ (sourceFaceAngles i j) |
| (sourceCollarInverse i j (sourceCollarPiece i j z))) := by |
| simpa only [sourceCollarInverse_point i j hz] using sourceFaceAngles_fderiv_surjective i j hz |
| exact (flatEndCoordinates_fderiv_surjective ha b _).comp |
| (hsur.comp (sourceCollarInverse_fderiv_surjective i j ⟨z,hz,rfl⟩)) |
| have hB : ContDiffOn ℝ (↑(⊤:ℕ∞)) (variableCollarTensor K a b) W := by |
| rintro y hy |
| obtain ⟨z,hz,rfl⟩ := hy.1 |
| apply (variableCollarTensor_smooth_at ha b i j hz ?_).contDiffWithinAt |
| have hgz : g (sourceCollarPiece i j z)=flatEndCoordinates a b (sourceFaceAngles i j z) := by |
| simp only [g,sourceCollarInverse_point i j hz] |
| simpa only [hgz] using (hO.2.2.1 _ hy.2).contDiffAt (hO.1.mem_nhds hy.2) |
| have hp : RegularPatch (fun y => u (g y)) |
| (fun y => ⟨variableCollarTensor K a b y,variableCollarTensor_symm hS a b y⟩) W := |
| ⟨hW,hO.2.1.comp (hg.mono inter_subset_left) hm,hB, |
| hO.2.2.2.smoothCoordinates hO.1 hW hO.2.1 (hg.mono inter_subset_left) hm hd⟩ |
| have hc : EqOn (variableCollarField u a b) (fun y => u (g y)) W := by |
| rintro y ⟨⟨z,hz,rfl⟩,_⟩ |
| simpa only [g,sourceCollarInverse_point i j hz] using variableCollarField_open u a b i j hz |
| refine mem_regularRegion_iff.mpr ⟨W,inter_subset_left,hp.congr hc (fun _ _ => rfl),?_⟩ |
| refine ⟨⟨x,hx,rfl⟩,?_⟩ |
| change g (sourceCollarPiece i j x)∈O |
| simpa only [g,sourceCollarInverse_point i j hx] using hxO |
|
|
| end ScalarConductivity |
| |
| end |
| end |
|
|
| end OAI |
|
|
