| import Mathlib |
| import OAI.Analysis.Conductivity.Branching.AttachedFlatTransport |
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| namespace OAI |
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| noncomputable section |
| namespace ScalarConductivity |
| open Set MeasureTheory Matrix |
| open scoped Matrix.Norms.Elementwise |
| local instance physicalTensorPatchMeasurableSpace : |
| MeasurableSpace (Matrix (Fin 3) (Fin 3) ℝ) := borel _ |
| local instance physicalTensorPatchBorelSpace : BorelSpace (Matrix (Fin 3) (Fin 3) ℝ) := ⟨rfl⟩ |
|
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| def sourceFlatPatch (i j : Fin 4) : Set (Fin 3 → ℝ) := |
| sourceCollarPiece i j '' sourceExtendedBox (-(1:ℝ)/100) (1/100) |
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| lemma isCompact_sourceFlatPatch (i j : Fin 4) : IsCompact (sourceFlatPatch i j) := |
| isCompact_Icc.image (sourceCollarPiece_contDiff i j).continuous |
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|
| lemma continuousAt_sourceCollarInverse_patch (i j : Fin 4) {y : Fin 3 → ℝ} |
| (hy : y∈sourceFlatPatch i j) : ContinuousAt (sourceCollarInverse i j) y := by |
| obtain ⟨x,hx,rfl⟩ := hy |
| obtain ⟨ht,hi,hj⟩ := mem_sourceExtendedBox.mp hx |
| have hr : 0<sourceCollarRadius j (x 0) (x 2) := |
| (by norm_num : (0:ℝ)<1/50).trans_le (sourceCollarRadius_extended j ht hj).1 |
| have hrad := sourceCollarPiece_radial_pos i j hr.le hi |
| have htime := sourceCollarPiece_time_pos i j hr.le (by linarith [ht.2] : x 0≤1) hi hj |
| have hc : Continuous sourceCollarTime := by |
| unfold sourceCollarTime sourceCrossCoordinates sourceRadial |
| fun_prop |
| have hcR : Continuous sourceRadial := by unfold sourceRadial; fun_prop |
| have hcX : Continuous sourceCrossCoordinates := by |
| unfold sourceCrossCoordinates sourceRadial |
| fun_prop |
| unfold sourceCollarInverse |
| apply continuousAt_pi.mpr |
| intro k |
| fin_cases k |
| · exact hc.continuousAt |
| · apply ContinuousAt.div |
| · unfold squareFaceParameter |
| fun_prop |
| · exact hcR.continuousAt |
| · rw [hrad] |
| exact hr.ne' |
| · apply ContinuousAt.div |
| · exact ((show Continuous (squareFaceParameter j) by |
| unfold squareFaceParameter; fun_prop).comp hcX).continuousAt |
| · exact continuousAt_const.sub hc.continuousAt |
| · rw [htime] |
| linarith [ht.2] |
|
|
| lemma continuousOn_attachedPhysicalFlatTensor (s : Fin 3 → ℝ) {a : ℝ} (ha : a≠0) |
| (i j : Fin 4) : ContinuousOn (attachedPhysicalFlatTensor s a i j) (sourceFlatPatch i j) := by |
| apply (continuousOn_attachedFlatTensor s ha i j).comp |
| (fun y hy => (continuousAt_sourceCollarInverse_patch i j hy).continuousWithinAt) |
| rintro _ ⟨x,hx,rfl⟩ |
| simpa only [sourceCollarInverse_extended i j (le_refl _) (le_refl _) hx] using hx |
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| def attachedPatchTensor (s : Fin 3 → ℝ) (a : ℝ) (i j : Fin 4) |
| (y : Fin 3 → ℝ) : Matrix (Fin 3) (Fin 3) ℝ := |
| by classical exact if y∈sourceFlatPatch i j then attachedPhysicalFlatTensor s a i j y else 1 |
|
|
| lemma measurable_attachedPatchTensor (s : Fin 3 → ℝ) {a : ℝ} (ha : a≠0) |
| (i j : Fin 4) : Measurable (attachedPatchTensor s a i j) := by |
| classical |
| exact (continuousOn_attachedPhysicalFlatTensor s ha i j).measurable_piecewise |
| continuousOn_const (isCompact_sourceFlatPatch i j).isClosed.measurableSet |
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| lemma attachedPatchTensor_symmetric (s : Fin 3 → ℝ) (a : ℝ) (i j : Fin 4) |
| (y : Fin 3 → ℝ) : (attachedPatchTensor s a i j y)ᵀ=attachedPatchTensor s a i j y := by |
| classical |
| unfold attachedPatchTensor |
| split_ifs |
| · exact attachedPhysicalFlatTensor_symmetric s a i j y |
| · exact Matrix.transpose_one |
|
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| end ScalarConductivity |
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| end |
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|
| noncomputable section |
| namespace ScalarConductivity |
| open Set MeasureTheory Matrix |
| open scoped Matrix.Norms.Elementwise |
| local instance attachedFaceTensorListMeasurableSpace : |
| MeasurableSpace (Matrix (Fin 3) (Fin 3) ℝ) := borel _ |
| local instance attachedFaceTensorListBorelSpace : BorelSpace (Matrix (Fin 3) (Fin 3) ℝ) := ⟨rfl⟩ |
|
|
| def attachedFaceTensorList (s : Fin 3 → ℝ) (a : ℝ) (ks : List (Fin 4 × Fin 4)) |
| (y : Fin 3 → ℝ) : Matrix (Fin 3) (Fin 3) ℝ := by |
| classical |
| exact ks.foldr (fun k A => if y∈sourceFlatPatch k.1 k.2 then |
| attachedPatchTensor s a k.1 k.2 y else A) 1 |
|
|
| lemma attachedFaceTensorList_cons (s : Fin 3 → ℝ) (a : ℝ) |
| (k : Fin 4 × Fin 4) (ks : List (Fin 4 × Fin 4)) (y : Fin 3 → ℝ) : |
| attachedFaceTensorList s a (k::ks) y= |
| (by classical exact if y∈sourceFlatPatch k.1 k.2 then attachedPatchTensor s a k.1 k.2 y |
| else attachedFaceTensorList s a ks y) := by |
| classical |
| rfl |
|
|
| lemma attachedFaceTensorList_measurable (s : Fin 3 → ℝ) {a : ℝ} (ha : a≠0) |
| (ks : List (Fin 4 × Fin 4)) : Measurable (attachedFaceTensorList s a ks) := by |
| classical |
| induction ks with |
| | nil => exact measurable_const |
| | cons k ks ih => |
| exact Measurable.ite (isCompact_sourceFlatPatch k.1 k.2).isClosed.measurableSet |
| (measurable_attachedPatchTensor s ha k.1 k.2) ih |
|
|
| lemma attachedFaceTensorList_symmetric (s : Fin 3 → ℝ) (a : ℝ) |
| (ks : List (Fin 4 × Fin 4)) (y : Fin 3 → ℝ) : |
| (attachedFaceTensorList s a ks y)ᵀ=attachedFaceTensorList s a ks y := by |
| classical |
| induction ks with |
| | nil => exact Matrix.transpose_one |
| | cons k ks ih => |
| simp only [attachedFaceTensorList_cons] |
| split_ifs |
| · exact attachedPatchTensor_symmetric s a k.1 k.2 y |
| · exact ih |
|
|
| lemma identity_matrix_energy_bounds : ∃ c C : ℝ,0<c ∧ c<C ∧ ∀ v : Fin 3 → ℝ, |
| c*‖v‖^2≤v ⬝ᵥ ((1 : Matrix (Fin 3) (Fin 3) ℝ)*ᵥv) ∧ |
| v ⬝ᵥ ((1 : Matrix (Fin 3) (Fin 3) ℝ)*ᵥv)≤C*‖v‖^2 := by |
| have hp (u : Unit) (_ : u∈({()} : Set Unit)) (v : Fin 3 → ℝ) (hv : v≠0) : |
| 0<v ⬝ᵥ ((1 : Matrix (Fin 3) (Fin 3) ℝ)*ᵥv) := by |
| rw [Matrix.one_mulVec] |
| exact lt_of_le_of_ne (Finset.sum_nonneg (fun k _ => mul_self_nonneg (v k))) |
| (Ne.symm (mt dotProduct_self_eq_zero.mp hv)) |
| obtain ⟨c,C,hc,hcC,h⟩ := compact_matrix_energy_bounds isCompact_singleton |
| (fun _ : Unit => (1 : Matrix (Fin 3) (Fin 3) ℝ)) continuousOn_const hp |
| exact ⟨c,C,hc,hcC,h () (mem_singleton ())⟩ |
|
|
| lemma attachedFaceTensorList_elliptic (s : Fin 3 → ℝ) |
| (hs : ∀ x y : ℝ,(1/2)*(x^2+y^2) ≤ s 0*x^2+2*s 1*x*y+s 2*y^2) |
| {a : ℝ} (ha : a≠0) (ks : List (Fin 4 × Fin 4)) : |
| ∃ c C : ℝ,0<c ∧ c<C ∧ ∀ y (v : Fin 3 → ℝ), |
| c*‖v‖^2≤v ⬝ᵥ (attachedFaceTensorList s a ks y*ᵥv) ∧ |
| v ⬝ᵥ (attachedFaceTensorList s a ks y*ᵥv)≤C*‖v‖^2 := by |
| classical |
| induction ks with |
| | nil => |
| obtain ⟨c,C,hc,hcC,h⟩ := identity_matrix_energy_bounds |
| exact ⟨c,C,hc,hcC,fun _ => h⟩ |
| | cons k ks ih => |
| obtain ⟨c,C,hc,hcC,h⟩ := ih |
| obtain ⟨d,D,hd,hdD,hk⟩ := attachedPhysicalFlatTensor_elliptic s hs ha k.1 k.2 |
| refine ⟨min c d,max C D,lt_min hc hd, |
| lt_of_le_of_lt (min_le_left _ _) (hcC.trans_le (le_max_left _ _)),?_⟩ |
| intro y v |
| simp only [attachedFaceTensorList_cons] |
| by_cases hy : y∈sourceFlatPatch k.1 k.2 |
| · simp only [ite_eq_left hy,attachedPatchTensor] |
| exact ⟨(mul_le_mul_of_nonneg_right (min_le_right _ _) (sq_nonneg _)).trans |
| (hk y hy v).1, |
| (hk y hy v).2.trans (mul_le_mul_of_nonneg_right (le_max_right _ _) (sq_nonneg _))⟩ |
| · simp only [ite_eq_right hy] |
| exact ⟨(mul_le_mul_of_nonneg_right (min_le_left _ _) (sq_nonneg _)).trans |
| (h y v).1, |
| (h y v).2.trans (mul_le_mul_of_nonneg_right (le_max_left _ _) (sq_nonneg _))⟩ |
|
|
| lemma attachedFaceTensorList_outside (s : Fin 3 → ℝ) (a : ℝ) |
| (ks : List (Fin 4 × Fin 4)) (y : Fin 3 → ℝ) |
| (hy : ∀ k∈ks,y∉sourceFlatPatch k.1 k.2) : attachedFaceTensorList s a ks y=1 := by |
| classical |
| induction ks with |
| | nil => rfl |
| | cons k ks ih => |
| simpa only [attachedFaceTensorList,List.foldr_cons,ite_eq_right (hy k (List.mem_cons_self))] |
| using ih (fun l hl => hy l (List.mem_cons_of_mem k hl)) |
|
|
| lemma attachedFaceTensorList_local (s : Fin 3 → ℝ) (a : ℝ) |
| (ks : List (Fin 4 × Fin 4)) (y : Fin 3 → ℝ) |
| (hy : ∃ k∈ks,y∈sourceFlatPatch k.1 k.2) : |
| ∃ k∈ks,y∈sourceFlatPatch k.1 k.2 ∧ |
| attachedFaceTensorList s a ks y=attachedPhysicalFlatTensor s a k.1 k.2 y := by |
| classical |
| induction ks with |
| | nil => simp at hy |
| | cons k ks ih => |
| by_cases hk : y∈sourceFlatPatch k.1 k.2 |
| · refine ⟨k,List.mem_cons_self,hk,?_⟩ |
| simp only [attachedFaceTensorList,List.foldr_cons,ite_eq_left hk,attachedPatchTensor] |
| · obtain ⟨l,hl,hy⟩ := hy |
| obtain ⟨l,hl,hly,he⟩ := ih ⟨l,(List.mem_cons.mp hl).resolve_left |
| (fun e => hk (e ▸ hy)),hy⟩ |
| refine ⟨l,List.mem_cons_of_mem k hl,hly,?_⟩ |
| simpa only [attachedFaceTensorList,List.foldr_cons,ite_eq_right hk] using he |
|
|
| def attachedCollarTensor (s : Fin 3 → ℝ) (a : ℝ) : |
| (Fin 3 → ℝ) → Matrix (Fin 3) (Fin 3) ℝ := |
| attachedFaceTensorList s a (Finset.univ : Finset (Fin 4 × Fin 4)).toList |
|
|
| theorem attachedCollarTensor_properties (s : Fin 3 → ℝ) |
| (hs : ∀ x y : ℝ,(1/2)*(x^2+y^2) ≤ s 0*x^2+2*s 1*x*y+s 2*y^2) |
| {a : ℝ} (ha : a≠0) : |
| Measurable (attachedCollarTensor s a) ∧ |
| (∀ y,(attachedCollarTensor s a y)ᵀ=attachedCollarTensor s a y) ∧ |
| (∃ c C : ℝ,0<c ∧ c<C ∧ ∀ y (v : Fin 3 → ℝ), |
| c*‖v‖^2≤v ⬝ᵥ (attachedCollarTensor s a y*ᵥv) ∧ |
| v ⬝ᵥ (attachedCollarTensor s a y*ᵥv)≤C*‖v‖^2) ∧ |
| (∀ y,(∀ i j,y∉sourceFlatPatch i j) → attachedCollarTensor s a y=1) ∧ |
| (∀ y,(∃ i j,y∈sourceFlatPatch i j) → ∃ i j,y∈sourceFlatPatch i j ∧ |
| attachedCollarTensor s a y=attachedPhysicalFlatTensor s a i j y) := by |
| refine ⟨attachedFaceTensorList_measurable s ha _, |
| attachedFaceTensorList_symmetric s a _,attachedFaceTensorList_elliptic s hs ha _,?_,?_⟩ |
| · intro y hy |
| exact attachedFaceTensorList_outside s a _ y (fun k _ => hy k.1 k.2) |
| · rintro y ⟨i,j,hy⟩ |
| obtain ⟨k,_,hk,he⟩ := attachedFaceTensorList_local s a |
| (Finset.univ : Finset (Fin 4 × Fin 4)).toList y ⟨(i,j),by simp,hy⟩ |
| exact ⟨k.1,k.2,hk,he⟩ |
|
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| end ScalarConductivity |
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| end |
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| end OAI |
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