| import Mathlib |
| import OAI.Analysis.Conductivity.Flux.PhysicalTensorBound |
| import OAI.Analysis.Conductivity.Flux.PhysicalChildTensor |
| import OAI.Analysis.Conductivity.Geometry.PhysicalEndRegions |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| namespace ScalarConductivity |
| open Set MeasureTheory Matrix |
| open scoped Matrix.Norms.Elementwise ENNReal |
| local instance : MeasurableSpace (Matrix (Fin 3) (Fin 3) ℝ) := borel _ |
| local instance : BorelSpace (Matrix (Fin 3) (Fin 3) ℝ) := ⟨rfl⟩ |
|
|
| lemma childAxis_pi_norm (v : Fin 3 → ℝ) : ‖fun i => v (childAxis i)‖=‖v‖ := by |
| apply le_antisymm |
| · exact (pi_norm_le_iff_of_nonneg (norm_nonneg v)).mpr |
| (fun i => norm_le_pi_norm v (childAxis i)) |
| · apply (pi_norm_le_iff_of_nonneg (norm_nonneg (fun i => v (childAxis i)))).mpr |
| intro i |
| have h := norm_le_pi_norm (fun j => v (childAxis j)) (childAxis i) |
| have he : childAxis (childAxis i)=i := by fin_cases i <;> rfl |
| simpa only [he] using h |
|
|
| lemma sourceChildTensorPush_measurable (k : Fin 2) |
| (A : (Fin 3 → ℝ) → Matrix (Fin 3) (Fin 3) ℝ) (hA : Measurable A) : |
| Measurable (sourceChildTensorPush k A) := by |
| let F : Matrix (Fin 3) (Fin 3) ℝ → Matrix (Fin 3) (Fin 3) ℝ := |
| fun M i j => sourceScale⁻¹*M (childAxis i) (childAxis j) |
| have hc : Continuous F := by |
| apply continuous_pi |
| intro i |
| apply continuous_pi |
| intro j |
| fun_prop |
| have hm : Measurable F := hc.measurable |
| exact hm.comp (hA.comp |
| (sourceChildHomeomorph (actualChildSign k)).symm.continuous.measurable) |
|
|
| lemma sourceChildTensorPush_norm_bound (k : Fin 2) |
| (A : (Fin 3 → ℝ) → Matrix (Fin 3) (Fin 3) ℝ) {B : ℝ} |
| (hB : 0 ≤ B) (hA : ∀ y,‖A y‖ ≤ B) (y : Fin 3 → ℝ) : |
| ‖sourceChildTensorPush k A y‖ ≤ |sourceScale⁻¹| * B := by |
| apply (pi_norm_le_iff_of_nonneg (mul_nonneg (abs_nonneg _) hB)).mpr |
| intro i |
| apply (pi_norm_le_iff_of_nonneg (mul_nonneg (abs_nonneg _) hB)).mpr |
| intro j |
| change ‖sourceScale⁻¹*A _ (childAxis i) (childAxis j)‖ ≤ _ |
| rw [norm_mul,Real.norm_eq_abs] |
| apply mul_le_mul_of_nonneg_left _ (abs_nonneg _) |
| exact ((norm_le_pi_norm _ (childAxis j)).trans |
| (norm_le_pi_norm _ (childAxis i))).trans (hA _) |
|
|
| def physicalEndTensor (s a : Fin 3 → ℝ) (i : Fin 3) : |
| (Fin 3 → ℝ) → Matrix (Fin 3) (Fin 3) ℝ := |
| Fin.cases (attachedCollarTensor s (a 0)) |
| (fun k => sourceChildTensorPush k (attachedCollarTensor s (a k.succ))) i |
|
|
| lemma physicalEndTensor_properties (s a : 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) |
| (ha : ∀ i,a i≠0) (i : Fin 3) : |
| Measurable (physicalEndTensor s a i) ∧ |
| (∀ y,(physicalEndTensor s a i y).IsSymm) ∧ |
| (∃ c C : ℝ,0<c ∧ c<C ∧ ∀ y (v : Fin 3 → ℝ), |
| c*‖v‖^2 ≤ v ⬝ᵥ (physicalEndTensor s a i y*ᵥv) ∧ |
| v ⬝ᵥ (physicalEndTensor s a i y*ᵥv) ≤ C*‖v‖^2) ∧ |
| (∃ B : ℝ,0<B ∧ ∀ y,‖physicalEndTensor s a i y‖ ≤ B) := by |
| refine Fin.cases ?_ (fun k => ?_) i |
| · have hp := attachedCollarTensor_properties s hs (ha 0) |
| exact ⟨hp.1,hp.2.1,hp.2.2.1,attachedFaceTensorList_bounded s (ha 0) _⟩ |
| · have hp := attachedCollarTensor_properties s hs (ha k.succ) |
| obtain ⟨c,C,hc,hcC,h⟩ := hp.2.2.1 |
| obtain ⟨B,hB,hbound⟩ := attachedFaceTensorList_bounded s (ha k.succ) |
| (Finset.univ : Finset (Fin 4 × Fin 4)).toList |
| have hspos : 0<sourceScale⁻¹ := by norm_num [sourceScale] |
| refine ⟨sourceChildTensorPush_measurable k _ hp.1, |
| sourceChildTensorPush_symmetric k _ hp.2.1, |
| ⟨sourceScale⁻¹*c,sourceScale⁻¹*C,mul_pos hspos hc, |
| mul_lt_mul_of_pos_left hcC hspos,?_⟩, |
| ⟨|sourceScale⁻¹| * B,mul_pos (abs_pos.mpr hspos.ne') hB, |
| sourceChildTensorPush_norm_bound k _ hB.le hbound⟩⟩ |
| intro y v |
| change (sourceScale⁻¹*c)*‖v‖^2 ≤ v ⬝ᵥ (sourceChildTensorPush k _ y*ᵥv) ∧ |
| v ⬝ᵥ (sourceChildTensorPush k _ y*ᵥv) ≤ (sourceScale⁻¹*C)*‖v‖^2 |
| rw [sourceChildTensorPush_quadratic] |
| have hh := h ((sourceChildHomeomorph (actualChildSign k)).symm y) |
| (fun j => v (childAxis j)) |
| rw [childAxis_pi_norm] at hh |
| exact ⟨by simpa only [mul_assoc] using mul_le_mul_of_nonneg_left hh.1 hspos.le, |
| by simpa only [mul_assoc] using mul_le_mul_of_nonneg_left hh.2 hspos.le⟩ |
| |
| def physicalBlockTensorList (s a : Fin 3 → ℝ) (is : List (Fin 3)) |
| (y : Fin 3 → ℝ) : Matrix (Fin 3) (Fin 3) ℝ := by |
| classical |
| exact is.foldr (fun i A => if y∈physicalEndRegion i then physicalEndTensor s a i y else A) 1 |
| |
| lemma physicalBlockTensorList_cons (s a : Fin 3 → ℝ) (i : Fin 3) (is : List (Fin 3)) |
| (y : Fin 3 → ℝ) : physicalBlockTensorList s a (i::is) y= |
| (by classical exact if y∈physicalEndRegion i then physicalEndTensor s a i y |
| else physicalBlockTensorList s a is y) := rfl |
| |
| lemma physicalBlockTensorList_properties (s a : 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) |
| (ha : ∀ i,a i≠0) (is : List (Fin 3)) : |
| Measurable (physicalBlockTensorList s a is) ∧ |
| (∀ y,(physicalBlockTensorList s a is y).IsSymm) ∧ |
| (∃ c C : ℝ,0<c ∧ c<C ∧ ∀ y (v : Fin 3 → ℝ), |
| c*‖v‖^2 ≤ v ⬝ᵥ (physicalBlockTensorList s a is y*ᵥv) ∧ |
| v ⬝ᵥ (physicalBlockTensorList s a is y*ᵥv) ≤ C*‖v‖^2) ∧ |
| (∃ B : ℝ,0<B ∧ ∀ y,‖physicalBlockTensorList s a is y‖ ≤ B) := by |
| classical |
| induction is with |
| | nil => |
| obtain ⟨c,C,hc,hcC,h⟩ := identity_matrix_energy_bounds |
| exact ⟨measurable_const,fun _ => Matrix.transpose_one, |
| ⟨c,C,hc,hcC,fun _ => h⟩, |
| ⟨‖(1 : Matrix (Fin 3) (Fin 3) ℝ)‖+1,by positivity,fun _ => by |
| change ‖(1 : Matrix (Fin 3) (Fin 3) ℝ)‖ ≤ _; linarith⟩⟩ |
| | cons i is ih => |
| have hi := physicalEndTensor_properties s a hs ha i |
| obtain ⟨c,C,hc,hcC,h⟩ := ih.2.2.1 |
| obtain ⟨d,D,hd,hdD,hlocal⟩ := hi.2.2.1 |
| obtain ⟨B,hB,hbound⟩ := ih.2.2.2 |
| obtain ⟨E,hE,hend⟩ := hi.2.2.2 |
| refine ⟨Measurable.ite (physicalEndRegion_compact i).measurableSet hi.1 ih.1,?_, |
| ⟨min c d,max C D,lt_min hc hd, |
| lt_of_le_of_lt (min_le_left _ _) (hcC.trans_le (le_max_left _ _)),?_⟩, |
| ⟨max B E,hB.trans_le (le_max_left _ _),?_⟩⟩ |
| · intro y |
| rw [physicalBlockTensorList_cons] |
| split_ifs |
| · exact hi.2.1 y |
| · exact ih.2.1 y |
| · intro y v |
| rw [physicalBlockTensorList_cons] |
| split_ifs with hy |
| · exact ⟨(mul_le_mul_of_nonneg_right (min_le_right _ _) (sq_nonneg _)).trans |
| (hlocal y v).1,(hlocal y v).2.trans |
| (mul_le_mul_of_nonneg_right (le_max_right _ _) (sq_nonneg _))⟩ |
| · 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 _))⟩ |
| · intro y |
| rw [physicalBlockTensorList_cons] |
| split_ifs |
| · exact (hend y).trans (le_max_right _ _) |
| · exact (hbound y).trans (le_max_left _ _) |
| |
| def physicalBlockTensor (s a : Fin 3 → ℝ) : |
| (Fin 3 → ℝ) → Matrix (Fin 3) (Fin 3) ℝ := |
| physicalBlockTensorList s a (Finset.univ : Finset (Fin 3)).toList |
| |
| lemma physicalBlockTensor_local_list (s a : Fin 3 → ℝ) (is : List (Fin 3)) |
| (i : Fin 3) (hi : i∈is) {y : Fin 3 → ℝ} (hy : y∈physicalEndRegion i) : |
| physicalBlockTensorList s a is y=physicalEndTensor s a i y := by |
| classical |
| induction is with |
| | nil => simp at hi |
| | cons j is ih => |
| rw [physicalBlockTensorList_cons] |
| by_cases he : j=i |
| · subst j; simp only [ite_eq_left hy] |
| · have hn : y∉physicalEndRegion j := fun hj => |
| (disjoint_left.mp (physicalEndRegion_pairwise_disjoint he) hj hy) |
| rw [ite_eq_right hn] |
| exact ih ((List.mem_cons.mp hi).resolve_left (Ne.symm he)) |
| |
| lemma physicalBlockTensor_local (s a : Fin 3 → ℝ) (i : Fin 3) |
| {y : Fin 3 → ℝ} (hy : y∈physicalEndRegion i) : |
| physicalBlockTensor s a y=physicalEndTensor s a i y := |
| physicalBlockTensor_local_list s a _ i (by simp) hy |
| |
| lemma physicalBlockTensorList_outside (s a : Fin 3 → ℝ) (is : List (Fin 3)) |
| {y : Fin 3 → ℝ} (hy : ∀ i∈is,y∉physicalEndRegion i) : |
| physicalBlockTensorList s a is y=1 := by |
| classical |
| induction is with |
| | nil => rfl |
| | cons i is ih => |
| rw [physicalBlockTensorList_cons,ite_eq_right (hy i List.mem_cons_self)] |
| exact ih (fun j hj => hy j (List.mem_cons_of_mem i hj)) |
| |
| theorem physicalBlockTensor_properties (s a : 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) |
| (ha : ∀ i,a i≠0) : |
| Measurable (physicalBlockTensor s a) ∧ |
| MemLp (physicalBlockTensor s a) ∞ (volume : Measure (Fin 3 → ℝ)) ∧ |
| (∀ y,(physicalBlockTensor s a y).IsSymm) ∧ |
| (∃ c C : ℝ,0<c ∧ c<C ∧ ∀ y (v : Fin 3 → ℝ), |
| c*‖v‖^2 ≤ v ⬝ᵥ (physicalBlockTensor s a y*ᵥv) ∧ |
| v ⬝ᵥ (physicalBlockTensor s a y*ᵥv) ≤ C*‖v‖^2) ∧ |
| (∀ i y,y∈physicalEndRegion i → physicalBlockTensor s a y=physicalEndTensor s a i y) ∧ |
| (∀ y,(∀ i,y∉physicalEndRegion i) → physicalBlockTensor s a y=1) := by |
| let : TopologicalSpace.PseudoMetrizableSpace (Matrix (Fin 3) (Fin 3) ℝ) := |
| inferInstanceAs (TopologicalSpace.PseudoMetrizableSpace (Fin 3 → Fin 3 → ℝ)) |
| let : SecondCountableTopology (Matrix (Fin 3) (Fin 3) ℝ) := |
| inferInstanceAs (SecondCountableTopology (Fin 3 → Fin 3 → ℝ)) |
| have h := physicalBlockTensorList_properties s a hs ha |
| (Finset.univ : Finset (Fin 3)).toList |
| obtain ⟨B,hB,hbound⟩ := h.2.2.2 |
| exact ⟨h.1,memLp_top_of_bound h.1.aestronglyMeasurable B (ae_of_all _ hbound), |
| h.2.1,h.2.2.1,fun i y hy => physicalBlockTensor_local s a i hy, |
| fun y hy => physicalBlockTensorList_outside s a _ (fun i _ => hy i)⟩ |
| |
| end ScalarConductivity |
| |
| end |
| end |
| |
| end OAI |
| |
