| import Mathlib |
| import OAI.Analysis.Conductivity.Variational.CompactRegularCorrection |
| import OAI.Analysis.Conductivity.Flux.CompactInteriorDivergence |
|
|
| namespace OAI |
|
|
|
|
| noncomputable section |
| namespace ScalarConductivity |
| open Set MeasureTheory Filter Topology |
|
|
| theorem compact_regular_box_correction {a b : Fin 3 → ℝ} (hab : ∀ i,a i<b i) |
| {r₁ r₂ : Box3 → ℝ} |
| (hr₁ : ContDiff ℝ (↑(⊤ : ℕ∞)) r₁) (hr₂ : ContDiff ℝ (↑(⊤ : ℕ∞)) r₂) |
| (hs₁ : HasCompactSupport r₁) (hs₂ : HasCompactSupport r₂) |
| (hv₁ : tsupport r₁⊆(Ioo (a 0) (b 0) ×ˢ Ioo (a 1) (b 1)) ×ˢ Ioo (a 2) (b 2)) |
| (hv₂ : tsupport r₂⊆(Ioo (a 0) (b 0) ×ˢ Ioo (a 1) (b 1)) ×ˢ Ioo (a 2) (b 2)) |
| (hz₁ : (∫ p,r₁ p)=0) (hz₂ : (∫ p,r₂ p)=0) |
| (ht : (∫ p,p.1.2*r₁ p-p.1.1*r₂ p)=0) : |
| ∃ H : Fin 3 → Fin 3 → Box3 → ℝ, |
| (∀ i j, ContDiff ℝ (↑(⊤ : ℕ∞)) (H i j)) ∧ |
| (∀ i j, HasCompactSupport (H i j)) ∧ |
| (∀ i j, tsupport (H i j)⊆(Ioo (a 0) (b 0) ×ˢ Ioo (a 1) (b 1)) ×ˢ Ioo (a 2) (b 2)) ∧ |
| (∀ i j p,H i j p=H j i p) ∧ |
| (∀ p,cubePartial (H 0 0) ((1,0),0) p+cubePartial (H 1 0) ((0,1),0) p+ |
| cubePartial (H 2 0) ((0,0),1) p=r₁ p) ∧ |
| (∀ p,cubePartial (H 0 1) ((1,0),0) p+cubePartial (H 1 1) ((0,1),0) p+ |
| cubePartial (H 2 1) ((0,0),1) p=r₂ p) := by |
| obtain ⟨F₁,F₂,F₃,hF₁,hF₂,hF₃,hsF₁,hsF₂,hsF₃,hdF,htF⟩ := compact_box_divergence_inside hab hr₁ hs₁ hv₁ hz₁ |
| obtain ⟨G₁,G₂,G₃,hG₁,hG₂,hG₃,hsG₁,hsG₂,hsG₃,hdG,htG⟩ := compact_box_divergence_inside hab hr₂ hs₂ hv₂ hz₂ |
| have hf₁ : ContDiff ℝ (↑(⊤ : ℕ∞)) (fun p : Box3 => p.1.1) := contDiff_fst.comp contDiff_fst |
| have hf₂ : ContDiff ℝ (↑(⊤ : ℕ∞)) (fun p : Box3 => p.1.2) := contDiff_snd.comp contDiff_fst |
| have hIF := cube_coordinate_divergence_moment hF₁ hF₂ hF₃ hsF₁ hsF₂ hsF₃ hdF hf₂ |
| have hIG := cube_coordinate_divergence_moment hG₁ hG₂ hG₃ hsG₁ hsG₂ hsG₃ hdG hf₁ |
| simp only [cubePartial_fst_fst,cubePartial_snd_fst,zero_mul,one_mul,integral_zero,neg_zero, |
| sub_zero,zero_sub] at hIF hIG |
| have hi₁ : Integrable (fun p : Box3 => p.1.2*r₁ p) := (hf₂.continuous.mul hr₁.continuous).integrable_of_hasCompactSupport (μ := volume) hs₁.mul_left |
| have hi₂ : Integrable (fun p : Box3 => p.1.1*r₂ p) := (hf₁.continuous.mul hr₂.continuous).integrable_of_hasCompactSupport (μ := volume) hs₂.mul_left |
| have hzq : (∫ p,F₂ p-G₁ p)=0 := by |
| rw [integral_sub (hF₂.continuous.integrable_of_hasCompactSupport hsF₂) |
| (hG₁.continuous.integrable_of_hasCompactSupport hsG₁)] |
| rw [integral_sub hi₁ hi₂,hIF,hIG] at ht |
| linarith |
| let U := (Ioo (a 0) (b 0) ×ˢ Ioo (a 1) (b 1)) ×ˢ Ioo (a 2) (b 2) |
| have htF₁ : tsupport F₁⊆U := (subset_union_left.trans subset_union_left).trans htF |
| have htF₂ : tsupport F₂⊆U := (subset_union_right.trans subset_union_left).trans htF |
| have htF₃ : tsupport F₃⊆U := subset_union_right.trans htF |
| have htG₁ : tsupport G₁⊆U := (subset_union_left.trans subset_union_left).trans htG |
| have htG₂ : tsupport G₂⊆U := (subset_union_right.trans subset_union_left).trans htG |
| have htG₃ : tsupport G₃⊆U := subset_union_right.trans htG |
| have htq : tsupport (fun p => F₂ p-G₁ p)⊆U := |
| (tsupport_sub _ _).trans (union_subset htF₂ htG₁) |
| obtain ⟨h₁,h₂,h₃,hh₁,hh₂,hh₃,hsh₁,hsh₂,hsh₃,hdh,hth⟩ := |
| compact_box_divergence_inside hab (hF₂.sub hG₁) (hsF₂.sub hsG₁) htq hzq |
| have hth₁ : tsupport h₁⊆U := (subset_union_left.trans subset_union_left).trans hth |
| have hth₂ : tsupport h₂⊆U := (subset_union_right.trans subset_union_left).trans hth |
| have hth₃ : tsupport h₃⊆U := subset_union_right.trans hth |
| let A := fun p => F₁ p+cubePartial h₁ ((0,1),0) p |
| let B := fun p => G₁ p+cubePartial h₂ ((0,1),0) p |
| let C := fun p => F₃ p+cubePartial h₃ ((0,1),0) p |
| let D := fun p => G₂ p-cubePartial h₂ ((1,0),0) p |
| let H : Fin 3 → Fin 3 → Box3 → ℝ := ![![A,B,C],![B,D,G₃],![C,G₃,fun _ => 0]] |
| have hA := hF₁.add (cubePartial_smooth hh₁ ((0,1),0)) |
| have hB := hG₁.add (cubePartial_smooth hh₂ ((0,1),0)) |
| have hC := hF₃.add (cubePartial_smooth hh₃ ((0,1),0)) |
| have hD := hG₂.sub (cubePartial_smooth hh₂ ((1,0),0)) |
| have hsA := hsF₁.add (cubePartial_compact hsh₁ ((0,1),0)) |
| have hsB := hsG₁.add (cubePartial_compact hsh₂ ((0,1),0)) |
| have hsC := hsF₃.add (cubePartial_compact hsh₃ ((0,1),0)) |
| have hsD := hsG₂.sub (cubePartial_compact hsh₂ ((1,0),0)) |
| have hB' : B=(fun p => F₂ p-cubePartial h₁ ((1,0),0) p-cubePartial h₃ ((0,0),1) p) := by |
| funext p |
| change G₁ p+cubePartial h₂ ((0,1),0) p=_ |
| linarith [hdh p] |
| have htA : tsupport A⊆U := (tsupport_add _ _).trans |
| (union_subset htF₁ ((cubePartial_tsupport _).trans hth₁)) |
| have htB : tsupport B⊆U := (tsupport_add _ _).trans |
| (union_subset htG₁ ((cubePartial_tsupport _).trans hth₂)) |
| have htC : tsupport C⊆U := (tsupport_add _ _).trans |
| (union_subset htF₃ ((cubePartial_tsupport _).trans hth₃)) |
| have htD : tsupport D⊆U := (tsupport_sub _ _).trans |
| (union_subset htG₂ ((cubePartial_tsupport _).trans hth₂)) |
| have ht0 : tsupport (fun _ : Box3 => (0:ℝ))⊆U := by simp |
| refine ⟨H,?_,?_,?_,?_,?_,?_⟩ |
| · intro i j; fin_cases i <;> fin_cases j |
| all_goals first | exact hA | exact hB | exact hC | exact hD | exact hG₃ | exact contDiff_const |
| · intro i j; fin_cases i <;> fin_cases j |
| all_goals first | exact hsA | exact hsB | exact hsC | exact hsD | exact hsG₃ | exact HasCompactSupport.zero |
| · intro i j; fin_cases i <;> fin_cases j |
| all_goals first | exact htA | exact htB | exact htC | exact htD | exact htG₃ | exact ht0 |
| · intro i j p; fin_cases i <;> fin_cases j <;> rfl |
| · intro p |
| change cubePartial A ((1,0),0) p+cubePartial B ((0,1),0) p+cubePartial C ((0,0),1) p=r₁ p |
| rw [hB'] |
| change cubePartial (fun p => F₁ p+cubePartial h₁ ((0,1),0) p) ((1,0),0) p+ |
| cubePartial (fun p => F₂ p-cubePartial h₁ ((1,0),0) p-cubePartial h₃ ((0,0),1) p) ((0,1),0) p+ |
| cubePartial (fun p => F₃ p+cubePartial h₃ ((0,1),0) p) ((0,0),1) p=r₁ p |
| rw [cubePartial_add (hF₁.differentiable (by simp)) ((cubePartial_smooth hh₁ _).differentiable (by simp)), |
| cubePartial_sub ((hF₂.sub (cubePartial_smooth hh₁ _)).differentiable (by simp)) |
| ((cubePartial_smooth hh₃ _).differentiable (by simp)), |
| cubePartial_sub (hF₂.differentiable (by simp)) ((cubePartial_smooth hh₁ _).differentiable (by simp)), |
| cubePartial_add (hF₃.differentiable (by simp)) ((cubePartial_smooth hh₃ _).differentiable (by simp)), |
| cubePartial_commute hh₁ ((0,1),0) ((1,0),0),cubePartial_commute hh₃ ((0,1),0) ((0,0),1)] |
| linarith [hdF p] |
| · intro p |
| change cubePartial B ((1,0),0) p+cubePartial D ((0,1),0) p+cubePartial G₃ ((0,0),1) p=r₂ p |
| change cubePartial (fun p => G₁ p+cubePartial h₂ ((0,1),0) p) ((1,0),0) p+ |
| cubePartial (fun p => G₂ p-cubePartial h₂ ((1,0),0) p) ((0,1),0) p+ |
| cubePartial G₃ ((0,0),1) p=r₂ p |
| rw [cubePartial_add (hG₁.differentiable (by simp)) ((cubePartial_smooth hh₂ _).differentiable (by simp)), |
| cubePartial_sub (hG₂.differentiable (by simp)) ((cubePartial_smooth hh₂ _).differentiable (by simp)), |
| cubePartial_commute hh₂ ((0,1),0) ((1,0),0)] |
| linarith [hdG p] |
|
|
| end ScalarConductivity |
| |
| end |
|
|
| end OAI |
|
|
