| import Mathlib |
| import OAI.Analysis.Conductivity.Variational.CompactRegularCorrection |
| import OAI.Analysis.Conductivity.Flux.CompactInteriorDivergence |
| import OAI.Analysis.Conductivity.Variational.FiniteMomentTransfer |
|
|
| namespace OAI |
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| noncomputable section |
| namespace ScalarConductivity |
| open Set MeasureTheory |
|
|
| lemma exists_unit_global_interval_bump {a b : ℝ} (hab : a<b) : |
| ∃ η : ℝ → ℝ, ContDiff ℝ (↑(⊤ : ℕ∞)) η ∧ HasCompactSupport η ∧ |
| tsupport η⊆Ioo a b ∧ (∫ t,η t)=1 := by |
| obtain ⟨η,hη,hs,hv,hi⟩ := exists_unit_interval_bump hab |
| refine ⟨η,hη,hs,hv,?_⟩ |
| rw [intervalIntegral.integral_of_le hab.le] at hi |
| have he : (∫ t in Ioc a b,η t)=(∫ t,η t) := by |
| apply setIntegral_eq_integral_of_forall_compl_eq_zero |
| intro t ht |
| exact image_eq_zero_of_notMem_tsupport (fun hh => ht ⟨(hv hh).1,(hv hh).2.le⟩) |
| rwa [he] at hi |
|
|
| lemma exists_unit_box_bump {a b : Fin 3 → ℝ} (hab : ∀ i,a i<b i) : |
| ∃ K : Box3 → ℝ,ContDiff ℝ (↑(⊤ : ℕ∞)) K ∧ HasCompactSupport K ∧ |
| tsupport K⊆(Ioo (a 0) (b 0) ×ˢ Ioo (a 1) (b 1)) ×ˢ Ioo (a 2) (b 2) ∧ |
| (∫ p,K p)=1 := by |
| obtain ⟨η₁,h₁,hs₁,hv₁,hi₁⟩ := exists_unit_global_interval_bump (hab 0) |
| obtain ⟨η₂,h₂,hs₂,hv₂,hi₂⟩ := exists_unit_global_interval_bump (hab 1) |
| obtain ⟨η₃,h₃,hs₃,hv₃,hi₃⟩ := exists_unit_global_interval_bump (hab 2) |
| let K : Box3 → ℝ := fun p => (η₁ p.1.1*η₂ p.1.2)*η₃ p.2 |
| have hK : ContDiff ℝ (↑(⊤ : ℕ∞)) K := |
| ((h₁.comp (contDiff_fst.comp contDiff_fst)).mul |
| (h₂.comp (contDiff_snd.comp contDiff_fst))).mul (h₃.comp contDiff_snd) |
| have hsK : HasCompactSupport K := by |
| apply HasCompactSupport.of_support_subset_isCompact ((hs₁.prod hs₂).prod hs₃) |
| intro p hp |
| have hh := mul_ne_zero_iff.mp hp |
| have hh' := mul_ne_zero_iff.mp hh.1 |
| exact ⟨⟨subset_tsupport η₁ hh'.1,subset_tsupport η₂ hh'.2⟩,subset_tsupport η₃ hh.2⟩ |
| have htK : tsupport K⊆(tsupport η₁ ×ˢ tsupport η₂) ×ˢ tsupport η₃ := by |
| apply closure_minimal _ ((hs₁.isClosed.prod hs₂.isClosed).prod hs₃.isClosed) |
| intro p hp |
| have hh := mul_ne_zero_iff.mp hp |
| have hh' := mul_ne_zero_iff.mp hh.1 |
| exact ⟨⟨subset_tsupport η₁ hh'.1,subset_tsupport η₂ hh'.2⟩,subset_tsupport η₃ hh.2⟩ |
| refine ⟨K,hK,hsK,htK.trans (prod_mono (prod_mono hv₁ hv₂) hv₃),?_⟩ |
| change (∫ p : (ℝ×ℝ)×ℝ,(η₁ p.1.1*η₂ p.1.2)*η₃ p.2)=1 |
| rw [Measure.volume_eq_prod,integral_prod_mul (fun p : ℝ×ℝ => η₁ p.1*η₂ p.2) η₃, |
| Measure.volume_eq_prod,integral_prod_mul η₁ η₂,hi₁,hi₂,hi₃] |
| norm_num |
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|
| lemma cubePartial_integral_zero {K : Box3 → ℝ} |
| (hK : ContDiff ℝ (↑(⊤ : ℕ∞)) K) (hs : HasCompactSupport K) (v : Box3) : |
| (∫ p,cubePartial K v p)=0 := by |
| have hh := cube_integration_by_parts (f := fun _ => (1:ℝ)) contDiff_const hK hs v |
| simpa [cubePartial] using hh |
|
|
| theorem box_moment_right_inverse {a b : Fin 3 → ℝ} (hab : ∀ i,a i<b i) |
| (m : Fin 3 → ℝ) : |
| ∃ 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)=m 0 ∧ (∫ p,r₂ p)=m 1 ∧ |
| (∫ p,p.1.2*r₁ p-p.1.1*r₂ p)=m 2 := by |
| obtain ⟨K,hK,hs,hv,hi⟩ := exists_unit_box_bump hab |
| let D := cubePartial K ((0,1),0) |
| let c := m 0*(∫ p : Box3,p.1.2*K p)-m 1*(∫ p : Box3,p.1.1*K p)-m 2 |
| let r₁ := fun p => m 0*K p+c*D p |
| let r₂ := fun p => m 1*K p |
| have hD : ContDiff ℝ (↑(⊤ : ℕ∞)) D := cubePartial_smooth hK _ |
| have hsD : HasCompactSupport D := cubePartial_compact hs _ |
| have hvD : tsupport D⊆(Ioo (a 0) (b 0) ×ˢ Ioo (a 1) (b 1)) ×ˢ Ioo (a 2) (b 2) := |
| (cubePartial_tsupport _).trans hv |
| have iK : Integrable K := hK.continuous.integrable_of_hasCompactSupport hs |
| have iD : Integrable D := hD.continuous.integrable_of_hasCompactSupport hsD |
| have hDz : (∫ p,D p)=0 := cubePartial_integral_zero hK hs _ |
| 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 hDm : (∫ p : Box3,p.1.2*D p)= -1 := by |
| have hh := cube_integration_by_parts hf₂ hK hs ((0,1),0) |
| simpa only [cubePartial_snd_fst,one_mul,hi] using hh |
| have iK₁ : Integrable (fun p : Box3 => p.1.1*K p) := |
| (hf₁.continuous.mul hK.continuous).integrable_of_hasCompactSupport hs.mul_left |
| have iK₂ : Integrable (fun p : Box3 => p.1.2*K p) := |
| (hf₂.continuous.mul hK.continuous).integrable_of_hasCompactSupport hs.mul_left |
| have iD₂ : Integrable (fun p : Box3 => p.1.2*D p) := |
| (hf₂.continuous.mul hD.continuous).integrable_of_hasCompactSupport hsD.mul_left |
| refine ⟨r₁,r₂,(contDiff_const.mul hK).add (contDiff_const.mul hD), |
| contDiff_const.mul hK,(hs.mul_left).add hsD.mul_left,hs.mul_left,?_,?_,?_,?_,?_⟩ |
| · exact (tsupport_add _ _).trans (union_subset |
| (tsupport_mul_subset_right.trans hv) (tsupport_mul_subset_right.trans hvD)) |
| · exact tsupport_mul_subset_right.trans hv |
| · dsimp only [r₁] |
| rw [integral_add (iK.const_mul _) (iD.const_mul _),integral_const_mul,integral_const_mul,hi,hDz] |
| ring |
| · dsimp only [r₂] |
| rw [integral_const_mul,hi,mul_one] |
| · have he : (fun p : Box3 => p.1.2*r₁ p-p.1.1*r₂ p)= |
| (fun p => m 0*(p.1.2*K p)+c*(p.1.2*D p)-m 1*(p.1.1*K p)) := by |
| funext p; dsimp only [r₁,r₂]; ring |
| have iKD : Integrable (fun p : Box3 => m 0*(p.1.2*K p)+c*(p.1.2*D p)) := |
| (iK₂.const_mul _).add (iD₂.const_mul _) |
| rw [he,integral_sub iKD (iK₁.const_mul _), |
| integral_add (iK₂.const_mul _) (iD₂.const_mul _),integral_const_mul,integral_const_mul, |
| integral_const_mul,hDm] |
| dsimp only [c] |
| ring |
|
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| end ScalarConductivity |
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|
| namespace ScalarConductivity |
| open Set MeasureTheory |
|
|
| theorem box_moment_bounded_right_inverse {a b : Fin 3 → ℝ} (hab : ∀ i,a i<b i) : |
| ∃ g : Fin 3 → Fin 2 → Box3 → ℝ, |
| (∀ i j,ContDiff ℝ (↑(⊤ : ℕ∞)) (g i j)) ∧ |
| (∀ i j,HasCompactSupport (g i j)) ∧ |
| (∀ i j,tsupport (g i j)⊆(Ioo (a 0) (b 0) ×ˢ Ioo (a 1) (b 1)) ×ˢ Ioo (a 2) (b 2)) ∧ |
| (∀ m : Fin 3 → ℝ, |
| (∫ p,∑ i,m i*g i 0 p)=m 0 ∧ |
| (∫ p,∑ i,m i*g i 1 p)=m 1 ∧ |
| (∫ p,p.1.2*(∑ i,m i*g i 0 p)-p.1.1*(∑ i,m i*g i 1 p))=m 2) ∧ |
| (∀ j n,∃ B : ℝ,0<B ∧ ∀ (m : Fin 3 → ℝ) (p : Box3), |
| ‖iteratedFDeriv ℝ n (fun p => ∑ i,m i*g i j p) p‖≤B*‖m‖) := by |
| choose g₀ g₁ hg₀ hg₁ hs₀ hs₁ hv₀ hv₁ hi₀ hi₁ hit using |
| fun i : Fin 3 => box_moment_right_inverse hab (Pi.single i 1) |
| let g : Fin 3 → Fin 2 → Box3 → ℝ := fun i => ![g₀ i,g₁ i] |
| have hg (i : Fin 3) (j : Fin 2) : ContDiff ℝ (↑(⊤ : ℕ∞)) (g i j) := by |
| fin_cases j |
| · exact hg₀ i |
| · exact hg₁ i |
| have hs (i : Fin 3) (j : Fin 2) : HasCompactSupport (g i j) := by |
| fin_cases j |
| · exact hs₀ i |
| · exact hs₁ i |
| have hv (i : Fin 3) (j : Fin 2) : tsupport (g i j)⊆ |
| (Ioo (a 0) (b 0) ×ˢ Ioo (a 1) (b 1)) ×ˢ Ioo (a 2) (b 2) := by |
| fin_cases j |
| · exact hv₀ i |
| · exact hv₁ i |
| have hm (i : Fin 3) (j : Fin 2) : (∫ p,g i j p)=(Pi.single i (1 : ℝ) : Fin 3 → ℝ) (j.castLE (by decide : 2≤3)) := by |
| fin_cases j |
| · exact hi₀ i |
| · exact hi₁ i |
| have hmt (i : Fin 3) : (∫ p : Box3,p.1.2*g i 0 p-p.1.1*g i 1 p)= (Pi.single i (1 : ℝ) : Fin 3 → ℝ) 2 := hit i |
| have hI (i : Fin 3) (j : Fin 2) : Integrable (g i j) := |
| (hg i j).continuous.integrable_of_hasCompactSupport (hs i j) |
| refine ⟨g,hg,hs,hv,?_,?_⟩ |
| · intro m |
| have hh (j : Fin 2) : (∫ p,∑ i,m i*g i j p)=m (j.castLE (by decide : 2≤3)) := by |
| rw [integral_finsetSum _ (fun i _ => (hI i j).const_mul (m i))] |
| simp_rw [integral_const_mul,hm] |
| simp [Pi.single_apply] |
| refine ⟨hh 0,hh 1,?_⟩ |
| have hU : ContDiff ℝ (↑(⊤ : ℕ∞)) (fun p : Box3 => p.1.1) := contDiff_fst.comp contDiff_fst |
| have hV : ContDiff ℝ (↑(⊤ : ℕ∞)) (fun p : Box3 => p.1.2) := contDiff_snd.comp contDiff_fst |
| have htI (i : Fin 3) : Integrable (fun p : Box3 => p.1.2*g i 0 p-p.1.1*g i 1 p) := |
| ((hV.continuous.mul (hg i 0).continuous).integrable_of_hasCompactSupport (hs i 0).mul_left).sub |
| ((hU.continuous.mul (hg i 1).continuous).integrable_of_hasCompactSupport (hs i 1).mul_left) |
| have he : (fun p : Box3 => p.1.2*(∑ i,m i*g i 0 p)-p.1.1*(∑ i,m i*g i 1 p))= |
| (fun p => ∑ i,m i*(p.1.2*g i 0 p-p.1.1*g i 1 p)) := by |
| funext p |
| simp only [Finset.mul_sum,←Finset.sum_sub_distrib] |
| apply Finset.sum_congr rfl |
| intro i _ |
| ring |
| rw [he,integral_finsetSum _ (fun i _ => (htI i).const_mul (m i))] |
| simp_rw [integral_const_mul,hmt] |
| simp [Pi.single_apply] |
| · intro j n |
| obtain ⟨B,hB,hbound⟩ := fixed_smooth_tests_bound (fun i => g i j) |
| (fun i => hg i j) (fun i => hs i j) n |
| refine ⟨B,hB,?_⟩ |
| intro m p |
| simpa only [smul_eq_mul] using hbound m ‖m‖ (norm_nonneg _) (norm_le_pi_norm m) p |
|
|
| end ScalarConductivity |
|
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| end |
|
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| end OAI |
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