| import Mathlib |
| import OAI.Analysis.Conductivity.Flux.CompactBoxDivergence |
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| namespace OAI |
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| noncomputable section |
| namespace ScalarConductivity |
| open Set MeasureTheory Filter Topology |
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| lemma compact_image_inner_interval {E : Type*} [TopologicalSpace E] |
| {K : Set E} (hK : IsCompact K) {f : E → ℝ} (hf : ContinuousOn f K) |
| {a b : ℝ} (hab : a<b) (hv : ∀ x∈K, a<f x ∧ f x<b) : |
| ∃ a' b' : ℝ, a<a' ∧ a'<b' ∧ b'<b ∧ ∀ x∈K, a'<f x ∧ f x<b' := by |
| by_cases hn : K.Nonempty |
| · obtain ⟨m,hm,hmin⟩ := hK.exists_isMinOn hn hf |
| obtain ⟨M,hM,hmax⟩ := hK.exists_isMaxOn hn hf |
| refine ⟨(a+f m)/2,(b+f M)/2,by linarith [(hv m hm).1],?_,by linarith [(hv M hM).2],?_⟩ |
| · have hmM : f m≤f M := hmin hM; linarith [(hv m hm).1,(hv M hM).2] |
| · intro x hx |
| have hmx : f m≤f x := hmin hx |
| have hxM : f x≤f M := hmax hx |
| constructor <;> linarith [(hv m hm).1,(hv M hM).2] |
| · refine ⟨(2*a+b)/3,(a+2*b)/3,by linarith,by linarith,by linarith,?_⟩ |
| intro x hx; exact (hn ⟨x,hx⟩).elim |
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| theorem compact_box_divergence_inside {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⊆ |
| (Ioo (a 0) (b 0) ×ˢ Ioo (a 1) (b 1)) ×ˢ Ioo (a 2) (b 2) := by |
| obtain ⟨a₀,b₀,ha₀,hab₀,hb₀,hv₀⟩ := compact_image_inner_interval hs |
| (continuous_fst.fst.continuousOn) (hab 0) (fun p hp => (hv hp).1.1) |
| obtain ⟨a₁,b₁,ha₁,hab₁,hb₁,hv₁⟩ := compact_image_inner_interval hs |
| (continuous_fst.snd.continuousOn) (hab 1) (fun p hp => (hv hp).1.2) |
| obtain ⟨a₂,b₂,ha₂,hab₂,hb₂,hv₂⟩ := compact_image_inner_interval hs |
| (continuous_snd.continuousOn) (hab 2) (fun p hp => (hv hp).2) |
| let a' : Fin 3 → ℝ := ![a₀,a₁,a₂] |
| let b' : Fin 3 → ℝ := ![b₀,b₁,b₂] |
| have hab' : ∀ i,a' i<b' i := by intro i; fin_cases i <;> assumption |
| obtain ⟨F₁,F₂,F₃,h₁,h₂,h₃,hs₁,hs₂,hs₃,hd,ht⟩ := compact_box_divergence hab' hr hs |
| (fun p hp => ⟨⟨hv₀ p hp,hv₁ p hp⟩,hv₂ p hp⟩) hz |
| refine ⟨F₁,F₂,F₃,h₁,h₂,h₃,hs₁,hs₂,hs₃,hd,ht.trans ?_⟩ |
| intro p hp |
| exact ⟨⟨⟨ha₀.trans_le hp.1.1.1,hp.1.1.2.trans_lt hb₀⟩, |
| ⟨ha₁.trans_le hp.1.2.1,hp.1.2.2.trans_lt hb₁⟩⟩, |
| ⟨ha₂.trans_le hp.2.1,hp.2.2.trans_lt hb₂⟩⟩ |
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| lemma cubePartial_tsupport {f : Box3 → ℝ} (v : Box3) : |
| tsupport (cubePartial f v)⊆tsupport f := tsupport_fderiv_apply_subset ℝ v |
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| end ScalarConductivity |
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| end |
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| end OAI |
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