| import Mathlib |
| import OAI.Analysis.Conductivity.Variational.CompactRegularCorrection |
| import OAI.Analysis.Conductivity.Flux.CompactInteriorDivergence |
| import OAI.Analysis.Conductivity.Flux.CompactBoxDivergenceC1 |
|
|
| namespace OAI |
|
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| section |
|
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| noncomputable section |
| namespace ScalarConductivity |
| open Set MeasureTheory Filter Topology |
|
|
| lemma cubePartial_bound_of_C1 {f : Box3 → ℝ} {M : ℝ} |
| (hb : UniformC1Bound f M) {v : Box3} (hv : ‖v‖≤1) (p : Box3) : |
| |cubePartial f v p|≤M := by |
| have hM : 0≤M := (norm_nonneg _).trans (hb p).2 |
| exact ((fderiv ℝ f p).le_opNorm v).trans |
| ((mul_le_mul (hb p).2 hv (norm_nonneg _) hM).trans_eq (mul_one M)) |
|
|
| theorem compact_regular_box_correction_C0 {a b c d : Fin 3 → ℝ} |
| (hab : ∀ i,a i<b i) (hca : ∀ i,c i<a i) (hbd : ∀ i,b i<d i) : |
| ∃ L : ℝ,0<L ∧ ∀ (r₁ r₂ : Box3 → ℝ), |
| ContDiff ℝ (↑(⊤ : ℕ∞)) r₁ → ContDiff ℝ (↑(⊤ : ℕ∞)) r₂ → |
| HasCompactSupport r₁ → HasCompactSupport r₂ → |
| tsupport r₁⊆(Ioo (a 0) (b 0) ×ˢ Ioo (a 1) (b 1)) ×ˢ Ioo (a 2) (b 2) → |
| tsupport r₂⊆(Ioo (a 0) (b 0) ×ˢ Ioo (a 1) (b 1)) ×ˢ Ioo (a 2) (b 2) → |
| (∫ p,r₁ p)=0 → (∫ p,r₂ p)=0 → |
| (∫ p,p.1.2*r₁ p-p.1.1*r₂ p)=0 → ∀ M : ℝ,0≤M → |
| UniformC1Bound r₁ M → UniformC1Bound r₂ M → |
| ∃ 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)⊆(Icc (c 0) (d 0) ×ˢ Icc (c 1) (d 1)) ×ˢ Icc (c 2) (d 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) ∧ |
| (∀ i j p,|H i j p|≤L*M) := by |
| have hcd : ∀ i,c i<d i := fun i => (hca i).trans ((hab i).trans (hbd i)) |
| obtain ⟨D₀,hD₀,solve₀⟩ := compact_box_divergence_C1 hab |
| obtain ⟨D₁,hD₁,solve₁⟩ := compact_box_divergence_C1 hcd |
| let L := D₀+2*D₁*D₀ |
| refine ⟨L,by dsimp [L] |
| intro r₁ r₂ hr₁ hr₂ hs₁ hs₂ hv₁ hv₂ hz₁ hz₂ ht M hM hb₁ hb₂ |
| obtain ⟨F₁,F₂,F₃,hF₁,hF₂,hF₃,hsF₁,hsF₂,hsF₃,hdF,htF,hbF₁,hbF₂,hbF₃⟩ := solve₀ r₁ hr₁ hs₁ hv₁ hz₁ M hM hb₁ |
| obtain ⟨G₁,G₂,G₃,hG₁,hG₂,hG₃,hsG₁,hsG₂,hsG₃,hdG,htG,hbG₁,hbG₂,hbG₃⟩ := solve₀ r₂ hr₂ hs₂ hv₂ hz₂ M hM hb₂ |
| 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 := (Icc (c 0) (d 0) ×ˢ Icc (c 1) (d 1)) ×ˢ Icc (c 2) (d 2) |
| let V := (Ioo (c 0) (d 0) ×ˢ Ioo (c 1) (d 1)) ×ˢ Ioo (c 2) (d 2) |
| have hinner : (Icc (a 0) (b 0) ×ˢ Icc (a 1) (b 1)) ×ˢ Icc (a 2) (b 2) ⊆ V := by |
| intro p hp |
| exact ⟨⟨⟨(hca 0).trans_le hp.1.1.1,hp.1.1.2.trans_lt (hbd 0)⟩, |
| ⟨(hca 1).trans_le hp.1.2.1,hp.1.2.2.trans_lt (hbd 1)⟩⟩, |
| ⟨(hca 2).trans_le hp.2.1,hp.2.2.trans_lt (hbd 2)⟩⟩ |
| have hVU : V⊆U := fun _ hp => |
| ⟨⟨Ioo_subset_Icc_self hp.1.1,Ioo_subset_Icc_self hp.1.2⟩,Ioo_subset_Icc_self hp.2⟩ |
| have htFV := htF.trans hinner |
| have htGV := htG.trans hinner |
| have htF₁ : tsupport F₁⊆U := ((subset_union_left.trans subset_union_left).trans htFV).trans hVU |
| have htF₂ : tsupport F₂⊆U := ((subset_union_right.trans subset_union_left).trans htFV).trans hVU |
| have htF₃ : tsupport F₃⊆U := (subset_union_right.trans htFV).trans hVU |
| have htG₁ : tsupport G₁⊆U := ((subset_union_left.trans subset_union_left).trans htGV).trans hVU |
| have htG₂ : tsupport G₂⊆U := ((subset_union_right.trans subset_union_left).trans htGV).trans hVU |
| have htG₃ : tsupport G₃⊆U := (subset_union_right.trans htGV).trans hVU |
| have htq : tsupport (fun p => F₂ p-G₁ p)⊆V := |
| (tsupport_sub _ _).trans (union_subset |
| ((subset_union_right.trans subset_union_left).trans htFV) |
| ((subset_union_left.trans subset_union_left).trans htGV)) |
| have hbq : UniformC1Bound (fun p => F₂ p-G₁ p) (2*D₀*M) := by |
| convert hbF₂.sub (hF₂.differentiable (by simp)) (hG₁.differentiable (by simp)) hbG₁ using 1 |
| ring |
| obtain ⟨h₁,h₂,h₃,hh₁,hh₂,hh₃,hsh₁,hsh₂,hsh₃,hdh,hth,hbh₁,hbh₂,hbh₃⟩ := |
| solve₁ (fun p => F₂ p-G₁ p) (hF₂.sub hG₁) (hsF₂.sub hsG₁) htq hzq |
| (2*D₀*M) (by positivity) hbq |
| 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] |
|
|
| · have hcomb (f g : Box3 → ℝ) (hf : UniformC1Bound f (D₀*M)) |
| (hg : UniformC1Bound g (D₁*(2*D₀*M))) (v : Box3) (hv : ‖v‖≤1) (p : Box3) : |
| |f p+cubePartial g v p|≤L*M ∧ |f p-cubePartial g v p|≤L*M := by |
| have hd := cubePartial_bound_of_C1 hg hv p |
| have he : D₀*M+D₁*(2*D₀*M)=L*M := by dsimp [L] |
| exact ⟨(abs_add_le _ _).trans ((add_le_add (hf p).1 hd).trans_eq he), |
| (abs_sub _ _).trans ((add_le_add (hf p).1 hd).trans_eq he)⟩ |
| have hA₀ (p : Box3) : |A p|≤L*M := (hcomb F₁ h₁ hbF₁ hbh₁ ((0,1),0) (by simp [Prod.norm_def]) p).1 |
| have hB₀ (p : Box3) : |B p|≤L*M := (hcomb G₁ h₂ hbG₁ hbh₂ ((0,1),0) (by simp [Prod.norm_def]) p).1 |
| have hC₀ (p : Box3) : |C p|≤L*M := (hcomb F₃ h₃ hbF₃ hbh₃ ((0,1),0) (by simp [Prod.norm_def]) p).1 |
| have hD₀' (p : Box3) : |D p|≤L*M := (hcomb G₂ h₂ hbG₂ hbh₂ ((1,0),0) (by simp [Prod.norm_def]) p).2 |
| have hG₀ (p : Box3) : |G₃ p|≤L*M := (hbG₃ p).1.trans |
| (mul_le_mul_of_nonneg_right (by dsimp [L]; exact le_add_of_nonneg_right (by positivity)) hM) |
| intro i j p |
| fin_cases i <;> fin_cases j |
| all_goals first | exact hA₀ p | exact hB₀ p | exact hC₀ p | exact hD₀' p | exact hG₀ p | skip |
| change |(0:ℝ)|≤L*M |
| rw [abs_zero] |
| dsimp [L] |
| positivity |
|
|
| end ScalarConductivity |
| |
| end |
| end |
|
|
| end OAI |
|
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