| import Mathlib |
| import OAI.Analysis.Conductivity.Sources.LocalPhysicalCorrection |
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| namespace OAI |
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| noncomputable section |
| namespace ScalarConductivity |
| open Set MeasureTheory Filter Topology Matrix |
| open scoped Matrix.Norms.Elementwise |
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| def potentialCompletion (D : Coord3 →L[ℝ] (Fin 2 → ℝ)) : Mat3 := |
| ![(gradientColumns D).col 0,(gradientColumns D).col 1, |
| crossProduct ((gradientColumns D).col 0) ((gradientColumns D).col 1)] |
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| lemma potentialCompletion_det_ne_zero (D : Coord3 →L[ℝ] (Fin 2 → ℝ)) |
| (hD : Function.Surjective D) : (potentialCompletion D).det≠0 := by |
| have hc : crossProduct ((gradientColumns D).col 0) ((gradientColumns D).col 1)≠0 := by |
| apply crossProduct_ne_zero_iff_linearIndependent.mpr |
| have he : ![(gradientColumns D).col 0,(gradientColumns D).col 1] = |
| (gradientColumns D).col := by |
| ext j i |
| fin_cases j <;> rfl |
| rw [he] |
| exact gradientColumns_rank D hD |
| rw [potentialCompletion,←triple_product_eq_det, |
| triple_product_permutation ((gradientColumns D).col 0), |
| triple_product_permutation ((gradientColumns D).col 1)] |
| exact mt dotProduct_self_eq_zero.mp hc |
|
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| lemma potentialCompletion_pairing (D : Coord3 →L[ℝ] (Fin 2 → ℝ)) (x : Coord3) : |
| coordinatePair (potentialCompletion D*ᵥx)=D x := by |
| rw [←gradientColumns_apply D x] |
| ext j |
| fin_cases j <;> rfl |
|
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| def potentialCompletionEquiv (D : Coord3 →L[ℝ] (Fin 2 → ℝ)) |
| (hD : Function.Surjective D) : Coord3 ≃L[ℝ] Coord3 := |
| (LinearMap.equivOfDetNeZero (Matrix.toLin' (potentialCompletion D)) |
| (by rw [LinearMap.det_toLin']; exact potentialCompletion_det_ne_zero D hD)).toContinuousLinearEquiv |
|
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| lemma potentialCompletionEquiv_pairing (D : Coord3 →L[ℝ] (Fin 2 → ℝ)) |
| (hD : Function.Surjective D) (x : Coord3) : |
| coordinatePair (potentialCompletionEquiv D hD x)=D x := |
| potentialCompletion_pairing D x |
|
|
| theorem exists_regular_potential_coordinates |
| {u : Coord3 → Fin 2 → ℝ} (hu : ContDiff ℝ (↑(⊤:ℕ∞)) u) |
| (a : Coord3) (hD : Function.Surjective (fderiv ℝ u a)) |
| {U : Set Coord3} (hU : IsOpen U) (ha : a∈U) : |
| ∃ X : OpenPartialHomeomorph Coord3 Coord3, |
| a∈X.target ∧ X.target⊆U ∧ |
| ContDiffOn ℝ (↑(⊤:ℕ∞)) X X.source ∧ |
| ContDiff ℝ (↑(⊤:ℕ∞)) X.symm ∧ |
| u=coordinatePair∘X.symm := by |
| obtain ⟨Y,haY,hYU,hY,hYi,he,hder⟩ := exists_potential_chart hu a hD hU ha |
| let L : Coord3 ≃L[ℝ] Coord3 := potentialCompletionEquiv (fderiv ℝ u a) hD |
| let Z := Y.transHomeomorph L.toHomeomorph |
| refine ⟨Z.symm,haY,hYU,?_,?_,?_⟩ |
| · change ContDiffOn ℝ (↑(⊤:ℕ∞)) (fun z => Y.symm (L.symm z)) {z | L.symm z∈Y.target} |
| exact hYi.comp L.symm.contDiff.contDiffOn (fun _ hx => hx) |
| · change ContDiff ℝ (↑(⊤:ℕ∞)) (fun z => L (Y z)) |
| exact L.contDiff.comp hY |
| · funext x |
| exact (he x).symm.trans (potentialCompletionEquiv_pairing _ hD (Y x)).symm |
|
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| end ScalarConductivity |
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| end |
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|
| noncomputable section |
| namespace ScalarConductivity |
| open Set MeasureTheory Filter Topology Matrix |
| open scoped Matrix.Norms.Elementwise |
|
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| lemma mem_correctionBox {a b x : Coord3} : x∈correctionBox a b ↔ ∀ i,a i<x i ∧ x i<b i := by |
| constructor |
| · rintro ⟨⟨h0,h1⟩,h2⟩ i |
| fin_cases i |
| · exact h0 |
| · exact h1 |
| · exact h2 |
| · intro h |
| exact ⟨⟨h 0,h 1⟩,h 2⟩ |
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| lemma correctionBox_isOpen (a b : Coord3) : IsOpen (correctionBox a b) := |
| ((isOpen_Ioo.prod isOpen_Ioo).prod isOpen_Ioo).preimage boxCoordinates.continuous |
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| lemma exists_correctionBox_inside {U : Set Coord3} (hU : IsOpen U) {x : Coord3} (hx : x∈U) : |
| ∃ a b : Coord3,(∀ i,a i<b i) ∧ x∈correctionBox a b ∧ correctionBox a b⊆U := by |
| obtain ⟨ε,hε,hball⟩ := Metric.isOpen_iff.mp hU x hx |
| refine ⟨fun i => x i-ε,fun i => x i+ε,fun i => by linarith, |
| mem_correctionBox.mpr (fun i => by constructor <;> linarith),?_⟩ |
| intro y hy |
| apply hball |
| rw [Metric.mem_ball,dist_pi_lt_iff hε] |
| intro i |
| rw [Real.dist_eq,abs_lt] |
| have hi := mem_correctionBox.mp hy i |
| constructor <;> linarith |
|
|
| structure RegularCorrectionBox (u : Coord3 → Fin 2 → ℝ) (U : Set Coord3) where |
| chart : OpenPartialHomeomorph Coord3 Coord3 |
| lower : Coord3 |
| upper : Coord3 |
| widths : ∀ i,lower i<upper i |
| subset_source : correctionBox lower upper⊆chart.source |
| smooth : ContDiffOn ℝ (↑(⊤:ℕ∞)) chart chart.source |
| inverse_smooth : ContDiffOn ℝ (↑(⊤:ℕ∞)) chart.symm chart.target |
| potentials : EqOn u (coordinatePair∘chart.symm) chart.target |
| target_subset : chart.target⊆U |
|
|
| namespace RegularCorrectionBox |
| variable {u : Coord3 → Fin 2 → ℝ} {U : Set Coord3} |
| def region (B : RegularCorrectionBox u U) : Set Coord3 := |
| B.chart '' correctionBox B.lower B.upper |
|
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| lemma region_open (B : RegularCorrectionBox u U) : IsOpen B.region := |
| B.chart.isOpen_image_of_subset_source (correctionBox_isOpen _ _) B.subset_source |
|
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| lemma region_target (B : RegularCorrectionBox u U) : B.region⊆B.chart.target := by |
| rintro y ⟨x,hx,rfl⟩ |
| exact B.chart.map_source (B.subset_source hx) |
|
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| lemma region_subset (B : RegularCorrectionBox u U) : B.region⊆U := |
| B.region_target.trans B.target_subset |
|
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| lemma region_nonempty (B : RegularCorrectionBox u U) : B.region.Nonempty := by |
| refine ⟨B.chart (fun i => (B.lower i+B.upper i)/2), |
| (fun i => (B.lower i+B.upper i)/2),?_,rfl⟩ |
| exact mem_correctionBox.mpr (fun i => by have := B.widths i; constructor <;> linarith) |
|
|
| end RegularCorrectionBox |
|
|
| theorem exists_regular_correction_box |
| {u : Coord3 → Fin 2 → ℝ} (hu : ContDiff ℝ (↑(⊤:ℕ∞)) u) |
| {U : Set Coord3} (hU : IsOpen U) {p : Coord3} (hp : p∈U) |
| (hD : Function.Surjective (fderiv ℝ u p)) : |
| ∃ B : RegularCorrectionBox u U,p∈B.region := by |
| obtain ⟨X,hpX,hXU,hX,hXi,he⟩ := exists_regular_potential_coordinates hu p hD hU hp |
| obtain ⟨a,b,hab,hx,hs⟩ := exists_correctionBox_inside X.open_source (X.map_target hpX) |
| let B : RegularCorrectionBox u U := ⟨X,a,b,hab,hs,hX,hXi.contDiffOn,fun _ _ => congrFun he _,hXU⟩ |
| exact ⟨B,X.symm p,hx,X.right_inv hpX⟩ |
|
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| end ScalarConductivity |
|
|
| end |
|
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| end OAI |
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