| import Mathlib | |
| import OAI.Analysis.Conductivity.Variational.CompactFiberDecomposition | |
| namespace OAI | |
| noncomputable section | |
| namespace ScalarConductivity | |
| open Set MeasureTheory Filter Topology | |
| lemma interval_integral_eq_integral_of_support {a b : ℝ} (hab : a≤b) | |
| {f : ℝ → ℝ} (hv : ∀ t, f t≠0 → t∈Ioo a b) : | |
| (∫ t in a..b, f t)=∫ t, f t := by | |
| rw [intervalIntegral.integral_of_le hab] | |
| apply setIntegral_eq_integral_of_forall_compl_eq_zero | |
| intro x hx | |
| by_contra hn | |
| exact hx ⟨(hv x hn).1,(hv x hn).2.le⟩ | |
| lemma compactFiberMarginal_integral {E : Type} [NormedAddCommGroup E] [NormedSpace ℝ E] | |
| [MeasureSpace E] [BorelSpace E] [SFinite (volume : Measure E)] | |
| {a b : ℝ} (hab : a≤b) {f : E×ℝ → ℝ} (hf : Integrable f) | |
| (hv : ∀ p, f p≠0 → p.2∈Ioo a b) : | |
| (∫ x, compactFiberMarginal a b f x)=∫ p, f p := by | |
| have he (x : E) : compactFiberMarginal a b f x=∫ t, f (x,t) := | |
| interval_integral_eq_integral_of_support hab (fun t ht => hv (x,t) ht) | |
| simp_rw [he] | |
| exact (integral_prod f hf).symm | |
| lemma compact_scalar_primitive {a b : ℝ} (hab : a<b) {r : ℝ → ℝ} | |
| (hr : ContDiff ℝ (↑(⊤ : ℕ∞)) r) | |
| (hv : ∀ t, r t≠0 → t∈Ioo a b) (hz : (∫ t, r t)=0) : | |
| ∃ P : ℝ → ℝ, ContDiff ℝ (↑(⊤ : ℕ∞)) P ∧ HasCompactSupport P ∧ | |
| (∀ t, deriv P t=r t) ∧ tsupport P⊆Icc a b := by | |
| let f : ℝ×ℝ → ℝ := fun p => r p.2 | |
| have hf : ContDiff ℝ (↑(⊤ : ℕ∞)) f := hr.comp contDiff_snd | |
| let P : ℝ → ℝ := fun t => fiberPrimitive a f (0,t) | |
| have hP : ContDiff ℝ (↑(⊤ : ℕ∞)) P := | |
| (fiberPrimitive_smooth a hf).comp (contDiff_const.prodMk contDiff_id) | |
| have hvf : ∀ p, f p≠0 → p.2∈Ioo a b := fun p hp => hv p.2 hp | |
| have hzf : ∀ x, (∫ t in a..b, f (x,t))=0 := by | |
| intro x | |
| exact (interval_integral_eq_integral_of_support hab.le hv).trans hz | |
| have hs : Function.support P⊆Icc a b := by | |
| intro t ht | |
| refine ⟨?_,?_⟩ | |
| · by_contra hn | |
| exact ht (fiberPrimitive_below hf.continuous hvf (0,t) (le_of_not_ge hn)) | |
| · by_contra hn | |
| exact ht (fiberPrimitive_above hf.continuous hvf hzf (0,t) (le_of_not_ge hn)) | |
| refine ⟨P,hP,HasCompactSupport.of_support_subset_isCompact isCompact_Icc hs,?_, | |
| closure_minimal hs isClosed_Icc⟩ | |
| intro t | |
| have hd := (((fiberPrimitive_smooth a hf).differentiable (by simp)) (0,t)).hasFDerivAt.comp_hasDerivAt t | |
| ((hasDerivAt_const t (0:ℝ)).prodMk (hasDerivAt_id t)) | |
| exact hd.deriv.trans (fiberPrimitive_derivative a hf (0,t)) | |
| lemma cube_partial_fiber (f : Box3 → ℝ) (p : Box3) : | |
| cubePartial f ((0,0),1) p=wallDerivative f p := rfl | |
| theorem compact_box_divergence {a b : Fin 3 → ℝ} (hab : ∀ i,a i<b i) | |
| {r : Box3 → ℝ} (hr : ContDiff ℝ (↑(⊤ : ℕ∞)) r) (hs : HasCompactSupport r) | |
| (hv : tsupport r⊆(Ioo (a 0) (b 0) ×ˢ Ioo (a 1) (b 1)) ×ˢ Ioo (a 2) (b 2)) | |
| (hz : (∫ p, r p)=0) : | |
| ∃ 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) := 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) | |
| 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 | |
| obtain ⟨F₃,hF₃,hsF₃,hd₃,ht₃⟩ := compact_fiber_decomposition (hab 2) hr hs | |
| (fun p hp => (hv (subset_tsupport r hp)).2) hη₃ hsη₃ hvη₃ hiη₃ | |
| 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 | |
| obtain ⟨G,hG,hsG,hdG,htG⟩ := compact_fiber_decomposition (hab 1) hR hsR | |
| (fun p hp => (hvR (subset_tsupport R hp)).2) hη₂ hsη₂ hvη₂ hiη₂ | |
| 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⟩ := compact_scalar_primitive (hab 0) hS | |
| (fun x hx => hvS (subset_tsupport S hx)) hzS | |
| 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) | |
| 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'⟩ | |
| 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 | |
| end ScalarConductivity | |
| end | |
| end OAI | |