| import Mathlib |
| import OAI.Analysis.Conductivity.Variational.UniformC1Bounds |
|
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| namespace OAI |
|
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| section |
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| noncomputable section |
| namespace ScalarConductivity |
| open Set MeasureTheory Filter Topology |
|
|
| theorem compact_box_divergence_C1 {a b : Fin 3 → ℝ} (hab : ∀ i,a i<b i) : |
| ∃ C : ℝ,0<C ∧ ∀ (r : Box3 → ℝ),ContDiff ℝ (↑(⊤ : ℕ∞)) r → |
| HasCompactSupport r → |
| tsupport r⊆(Ioo (a 0) (b 0) ×ˢ Ioo (a 1) (b 1)) ×ˢ Ioo (a 2) (b 2) → |
| (∫ p,r p)=0 → ∀ M : ℝ,0≤M → UniformC1Bound r M → |
| ∃ F₁ F₂ F₃ : Box3 → ℝ, |
| ContDiff ℝ (↑(⊤ : ℕ∞)) F₁ ∧ ContDiff ℝ (↑(⊤ : ℕ∞)) F₂ ∧ |
| ContDiff ℝ (↑(⊤ : ℕ∞)) F₃ ∧ |
| HasCompactSupport F₁ ∧ HasCompactSupport F₂ ∧ HasCompactSupport F₃ ∧ |
| (∀ p,cubePartial F₁ ((1,0),0) p+cubePartial F₂ ((0,1),0) p+ |
| cubePartial F₃ ((0,0),1) p=r p) ∧ |
| tsupport F₁∪tsupport F₂∪tsupport F₃⊆ |
| (Icc (a 0) (b 0) ×ˢ Icc (a 1) (b 1)) ×ˢ Icc (a 2) (b 2) ∧ |
| UniformC1Bound F₁ (C*M) ∧ UniformC1Bound F₂ (C*M) ∧ UniformC1Bound F₃ (C*M) := by |
| obtain ⟨η₃,hη₃,hsη₃,hvη₃,hiη₃⟩ := exists_unit_interval_bump (hab 2) |
| obtain ⟨η₂,hη₂,hsη₂,hvη₂,hiη₂⟩ := exists_unit_interval_bump (hab 1) |
| obtain ⟨K₃,hK₃,hbη₃⟩ := exists_uniformC1Bound hη₃ hsη₃ |
| obtain ⟨K₂,hK₂,hbη₂⟩ := exists_uniformC1Bound hη₂ hsη₂ |
| have hl₀ : 0<b 0-a 0 := sub_pos.mpr (hab 0) |
| have hl₁ : 0<b 1-a 1 := sub_pos.mpr (hab 1) |
| have hl₂ : 0<b 2-a 2 := sub_pos.mpr (hab 2) |
| let c₁ := 4*K₃*K₂*(b 0-a 0+1)*(b 1-a 1)*(b 2-a 2) |
| let c₂ := 2*K₃*(b 1-a 1+1)*(1+K₂*(b 1-a 1))*(b 2-a 2) |
| let c₃ := (b 2-a 2+1)*(1+K₃*(b 2-a 2)) |
| have hc₁ : 0≤c₁ := by dsimp [c₁] |
| have hc₂ : 0≤c₂ := by dsimp [c₂] |
| have hc₃ : 0≤c₃ := by dsimp [c₃] |
| let C := c₁+c₂+c₃+1 |
| refine ⟨C,by dsimp [C] |
| intro r hr hs hv hz M hM hbound |
| let R := compactFiberMarginal (a 2) (b 2) r |
| have hR := compactFiberMarginal_smooth (hab 2).le hr |
| have hsR := (compactFiberMarginal_support hs (a := a 2) (b := b 2)).1 |
| have hvR : tsupport R⊆Ioo (a 0) (b 0) ×ˢ Ioo (a 1) (b 1) := by |
| intro x hx |
| obtain ⟨p,hp,rfl⟩ := (compactFiberMarginal_support hs).2 hx |
| exact (hv hp).1 |
| have hbR : UniformC1Bound R ((b 2-a 2)*M) := compactFiberMarginal_C1Bound (hab 2).le hr hM hbound |
| obtain ⟨F₃,hF₃,hsF₃,hd₃,ht₃,hb₃,hdB₃⟩ := compact_fiber_decomposition_C1 (hab 2) hr hs |
| (fun p hp => (hv hp).2) hη₃ hsη₃ hvη₃ hiη₃ hM hK₃.le |
| (fun p => (hbound p).1) (fun p => (hbound p).2) (fun t => (hbη₃ t).1) |
| let S := compactFiberMarginal (a 1) (b 1) R |
| have hS := compactFiberMarginal_smooth (hab 1).le hR |
| have hsS := (compactFiberMarginal_support hsR (a := a 1) (b := b 1)).1 |
| have hvS : tsupport S⊆Ioo (a 0) (b 0) := by |
| intro x hx |
| obtain ⟨p,hp,rfl⟩ := (compactFiberMarginal_support hsR).2 hx |
| exact (hvR hp).1 |
| have hbS : UniformC1Bound S ((b 1-a 1)*((b 2-a 2)*M)) := |
| compactFiberMarginal_C1Bound (hab 1).le hR (by positivity) hbR |
| obtain ⟨G,hG,hsG,hdG,htG,hbG₀,hdbG⟩ := compact_fiber_decomposition_C1 (hab 1) hR hsR |
| (fun p hp => (hvR hp).2) hη₂ hsη₂ hvη₂ hiη₂ (by positivity) hK₂.le |
| (fun p => (hbR p).1) (fun p => (hbR p).2) (fun t => (hbη₂ t).1) |
| have hbG : UniformC1Bound G ((b 1-a 1+1)*(1+K₂*(b 1-a 1))*((b 2-a 2)*M)) := by |
| intro p |
| refine ⟨(hbG₀ p).trans ?_,hdbG p⟩ |
| nlinarith [mul_nonneg (by positivity : 0≤1+K₂*(b 1-a 1)) (by positivity : 0≤(b 2-a 2)*M)] |
| have hzR : (∫ x,R x)=0 := by |
| rw [compactFiberMarginal_integral (hab 2).le |
| (hr.continuous.integrable_of_hasCompactSupport hs) |
| (fun p hp => (hv (subset_tsupport r hp)).2)] |
| exact hz |
| have hzS : (∫ x,S x)=0 := by |
| rw [compactFiberMarginal_integral (hab 1).le |
| (hR.continuous.integrable_of_hasCompactSupport hsR) |
| (fun p hp => (hvR (subset_tsupport R hp)).2)] |
| exact hzR |
| obtain ⟨P,hP,hsP,hdP,htP,hbP⟩ := compact_scalar_primitive_bounded (hab 0) hS |
| (fun x hx => hvS (subset_tsupport S hx)) hzS (by positivity) (fun t => (hbS t).1) |
| let F₁ : Box3 → ℝ := fun p => η₃ p.2*η₂ p.1.2*P p.1.1 |
| let F₂ : Box3 → ℝ := fun p => η₃ p.2*G p.1 |
| have hF₁ : ContDiff ℝ (↑(⊤ : ℕ∞)) F₁ := |
| ((hη₃.comp contDiff_snd).mul (hη₂.comp (contDiff_snd.comp contDiff_fst))).mul |
| (hP.comp (contDiff_fst.comp contDiff_fst)) |
| have hF₂ : ContDiff ℝ (↑(⊤ : ℕ∞)) F₂ := |
| (hη₃.comp contDiff_snd).mul (hG.comp contDiff_fst) |
| have hbf : UniformC1Bound (fun p : Box3 => η₃ p.2) K₃ := |
| hbη₃.comp_snd (hη₃.differentiable (by simp)) hK₃.le |
| have hbg : UniformC1Bound (fun p : Box3 => η₂ p.1.2) K₂ := |
| (hbη₂.comp_snd (E := ℝ) (hη₂.differentiable (by simp)) hK₂.le).comp_fst |
| ((hη₂.comp contDiff_snd).differentiable (by simp)) hK₂.le |
| have hbp : UniformC1Bound (fun p : Box3 => P p.1.1) |
| ((b 0-a 0+1)*((b 1-a 1)*((b 2-a 2)*M))) := |
| (hbP.comp_fst (P := ℝ) (hP.differentiable (by simp)) (by positivity)).comp_fst |
| ((hP.comp contDiff_fst).differentiable (by simp)) (by positivity) |
| have hbf₁ : UniformC1Bound F₁ (c₁*M) := by |
| have hh := (hbf.mul ((hη₃.comp contDiff_snd).differentiable (by simp)) |
| ((hη₂.comp (contDiff_snd.comp contDiff_fst)).differentiable (by simp)) hK₃.le hK₂.le hbg).mul |
| (((hη₃.comp contDiff_snd).mul (hη₂.comp (contDiff_snd.comp contDiff_fst))).differentiable (by simp)) |
| ((hP.comp (contDiff_fst.comp contDiff_fst)).differentiable (by simp)) (by positivity) (by positivity) hbp |
| convert hh using 1 < |
| ring |
| have hbf₂ : UniformC1Bound F₂ (c₂*M) := by |
| have hh := hbf.mul ((hη₃.comp contDiff_snd).differentiable (by simp)) |
| ((hG.comp contDiff_fst).differentiable (by simp)) hK₃.le (by positivity) |
| (hbG.comp_fst (hG.differentiable (by simp)) (by positivity)) |
| convert hh using 1 < |
| ring |
| have hbf₃ : UniformC1Bound F₃ (c₃*M) := by |
| intro p |
| refine ⟨(hb₃ p).trans ?_,hdB₃ p⟩ |
| dsimp [c₃] |
| nlinarith [mul_nonneg (by positivity : 0≤1+K₃*(b 2-a 2)) hM] |
| let K := (Icc (a 0) (b 0) ×ˢ Icc (a 1) (b 1)) ×ˢ Icc (a 2) (b 2) |
| have hK : IsCompact K := (isCompact_Icc.prod isCompact_Icc).prod isCompact_Icc |
| have ht₁ : tsupport F₁⊆K := by |
| apply closure_minimal _ hK.isClosed |
| intro p hp |
| obtain ⟨h₁,h₂⟩ := mul_ne_zero_iff.mp hp |
| obtain ⟨h₃,h₄⟩ := mul_ne_zero_iff.mp h₁ |
| exact ⟨⟨htP (subset_tsupport P h₂),Ioo_subset_Icc_self (hvη₂ (subset_tsupport η₂ h₄))⟩, |
| Ioo_subset_Icc_self (hvη₃ (subset_tsupport η₃ h₃))⟩ |
| have ht₂ : tsupport F₂⊆K := by |
| apply closure_minimal _ hK.isClosed |
| intro p hp |
| obtain ⟨h₁,h₂⟩ := mul_ne_zero_iff.mp hp |
| have hh := htG (subset_tsupport G h₂) |
| obtain ⟨q,hq,hqx⟩ := hh.1 |
| exact ⟨⟨Ioo_subset_Icc_self (hqx ▸ (hvR hq).1),hh.2⟩, |
| Ioo_subset_Icc_self (hvη₃ (subset_tsupport η₃ h₁))⟩ |
| have ht₃' : tsupport F₃⊆K := by |
| intro p hp |
| have hh := ht₃ hp |
| obtain ⟨q,hq,hqp⟩ := hh.1 |
| have hxy := (hv hq).1 |
| change q.1∈_ at hxy |
| rw [hqp] at hxy |
| exact ⟨⟨Ioo_subset_Icc_self hxy.1,Ioo_subset_Icc_self hxy.2⟩,hh.2⟩ |
| refine ⟨F₁,F₂,F₃,hF₁,hF₂,hF₃, |
| HasCompactSupport.of_support_subset_isCompact hK (subset_tsupport F₁ |>.trans ht₁), |
| HasCompactSupport.of_support_subset_isCompact hK (subset_tsupport F₂ |>.trans ht₂),hsF₃,?_, |
| union_subset (union_subset ht₁ ht₂) ht₃', |
| hbf₁.mono (mul_le_mul_of_nonneg_right (by dsimp [C] |
| hbf₂.mono (mul_le_mul_of_nonneg_right (by dsimp [C] |
| hbf₃.mono (mul_le_mul_of_nonneg_right (by dsimp [C] |
| intro p |
| have hd₁ : cubePartial F₁ ((1,0),0) p=η₃ p.2*η₂ p.1.2*S p.1.1 := by |
| have hline := (hasDerivAt_id p.1.1).prodMk (hasDerivAt_const p.1.1 p.1.2) |
| have he := ((hF₁.differentiable (by simp)) p).hasFDerivAt.comp_hasDerivAt p.1.1 |
| (hline.prodMk (hasDerivAt_const p.1.1 p.2)) |
| have he' : HasDerivAt (fun t => F₁ ((t,p.1.2),p.2)) |
| (cubePartial F₁ ((1,0),0) p) p.1.1 := he |
| have hp := ((hP.differentiable (by simp)) p.1.1).hasDerivAt.const_mul |
| (η₃ p.2*η₂ p.1.2) |
| rw [hdP] at hp |
| exact he'.unique hp |
| have hd₂ : cubePartial F₂ ((0,1),0) p=η₃ p.2*wallDerivative G p.1 := by |
| have hline := (hasDerivAt_const p.1.2 p.1.1).prodMk (hasDerivAt_id p.1.2) |
| have he := ((hF₂.differentiable (by simp)) p).hasFDerivAt.comp_hasDerivAt p.1.2 |
| (hline.prodMk (hasDerivAt_const p.1.2 p.2)) |
| have he' : HasDerivAt (fun t => F₂ ((p.1.1,t),p.2)) |
| (cubePartial F₂ ((0,1),0) p) p.1.2 := he |
| have hg := (((hG.differentiable (by simp)) p.1).hasFDerivAt.comp_hasDerivAt |
| p.1.2 hline).const_mul (η₃ p.2) |
| exact he'.unique hg |
| rw [hd₁,hd₂,cube_partial_fiber] |
| have h₁ := hdG p.1 |
| have h₂ := hd₃ p |
| dsimp [S,R] at * |
| linear_combination (η₃ p.2)*h₁+h₂ |
|
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| end ScalarConductivity |
| |
| end |
| end |
|
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| end OAI |
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