| import Mathlib |
| import OAI.Analysis.Conductivity.Sources.CopiedEnergyAlgebra |
| import OAI.Analysis.Conductivity.Sources.WordTrace |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| namespace ScalarConductivity |
| open Set MeasureTheory Filter Topology |
|
|
| lemma sourceAveragePullIter_mean_bound (N : ℕ) (h : H1) (c K : ℝ) |
| (hm : ∀ ν : SourceWord,ν.length=N → |
| abs (sourceMeanCLM 0 (sourceWordPull ν h)-c) ≤ K) : |
| abs (sourceMeanCLM 0 (sourceAveragePullIter N h)-c) ≤ K := by |
| induction N generalizing h with |
| | zero => exact hm [] rfl |
| | succ N ih => |
| change abs (sourceMeanCLM 0 (sourceAveragePullIter N (sourceAveragePull h))-c) ≤ K |
| apply ih |
| intro ν hν |
| have h0 := abs_le.mp (hm (0::ν) (by simp only [List.length_cons,hν])) |
| have h1 := abs_le.mp (hm (1::ν) (by simp only [List.length_cons,hν])) |
| change abs (sourceMeanCLM 0 (sourceWordPull ν ((1/2:ℝ) • (childH1Pullback 0 h+childH1Pullback 1 h)))-c) ≤ K |
| simp only [map_smul,map_add,smul_eq_mul] |
| simp only [sourceWordPull,ContinuousLinearMap.comp_apply] at h0 h1 |
| apply abs_le.mpr |
| constructor <;> linarith [h0.1,h0.2,h1.1,h1.2] |
|
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| lemma sourceAveragePullIter_mean_eq (N : ℕ) (h : H1) (c : ℝ) |
| (hm : ∀ ν : SourceWord,ν.length=N → sourceMeanCLM 0 (sourceWordPull ν h)=c) : |
| sourceMeanCLM 0 (sourceAveragePullIter N h)=c := by |
| apply sub_eq_zero.mp |
| apply abs_eq_zero.mp |
| apply le_antisymm _ (abs_nonneg _) |
| apply sourceAveragePullIter_mean_bound N h c 0 |
| intro ν hν |
| rw [hm ν hν,sub_self,abs_zero] |
|
|
| lemma sourceAverageDifference_mean_bound (N : ℕ) (h : H1) (K : ℝ) |
| (hm : ∀ ν : SourceWord,ν.length=N+1 → abs (sourceMeanCLM 0 (sourceWordPull ν h)) ≤ K) : |
| abs (sourceMeanCLM 0 (sourceAveragePullIter N (sourceDifferencePull h))) ≤ 2*K := by |
| have hh := sourceAveragePullIter_mean_bound N (sourceDifferencePull h) 0 (2*K) (by |
| intro ν hν |
| change abs (sourceMeanCLM 0 (sourceWordPull ν (childH1Pullback 0 h-childH1Pullback 1 h))-0) ≤ _ |
| rw [map_sub,map_sub,sub_zero] |
| have h0 := hm (0::ν) (by simp only [List.length_cons,hν]) |
| have h1 := hm (1::ν) (by simp only [List.length_cons,hν]) |
| simp only [sourceWordPull,ContinuousLinearMap.comp_apply] at h0 h1 |
| exact (abs_sub _ _).trans (by linarith)) |
| simpa only [sub_zero] using hh |
|
|
| namespace BoundedRootSources |
| variable (r : BoundedRootSources) |
|
|
| lemma finiteSource_weighted_bound {u : H1} (hu : r.sourceHarmonic u) {M : ℝ} (hM : 0 ≤ M) |
| (hb : ∀ᵐ x∂ballMeasure,abs (weakValue u x) ≤ M) |
| {f : R3 → ℝ} (hf : ContDiff ℝ (↑(⊤:ℕ∞)) f) (hc : HasCompactSupport f) |
| {L D : ℝ} (hL : 0 ≤ L) (hD : 0 ≤ D) |
| (hlip : ∀ x y,abs (f x-f y) ≤ L*dist x y) |
| (hdiam : ∀ ν θ,dist (sourceWordBoundary ν θ) (sourceWordMap ν 0) ≤ D*sourceScale^ν.length) |
| (N : ℕ) : |
| abs (r.form (r.finiteSource N).val |
| (originalMulH1CLM hf hc (u-constantH1 (sourceMeanCLM 0 u)))) ≤ |
| (4*M*L*D)*sourceScale^N := by |
| rw [r.form_finiteSource] |
| have hh := sourceAverageDifference_mean_bound N |
| (originalMulH1CLM hf hc (u-constantH1 (sourceMeanCLM 0 u))) |
| ((2*M*L*D)*sourceScale^(N+1)) (by |
| intro ν hν |
| simpa only [hν] using sourceWord_weighted_mean_bound r hu hM hb hf hc hL hD hlip hdiam ν) |
| apply hh.trans |
| rw [pow_succ] |
| have hs : sourceScale ≤ 1 := by norm_num [sourceScale] |
| have hs0 : 0 ≤ sourceScale := by norm_num [sourceScale] |
| have hn : 0 ≤ (4*M*L*D)*sourceScale^N := by positivity |
| nlinarith |
|
|
| lemma form_weighted_test (W u : H1) (c : ℝ) {f : R3 → ℝ} |
| (hf : ContDiff ℝ (↑(⊤:ℕ∞)) f) (hc : HasCompactSupport f) : |
| r.form W (originalMulH1CLM hf hc (u-constantH1 c))= |
| ∫ x,r.coefficient x*inner ℝ (weakGradient W x) |
| ((weakValue u x-c) • gradient f x+f x • weakGradient u x) ∂ballMeasure := by |
| rw [r.form_apply] |
| apply integral_congr_ae |
| filter_upwards [originalMulH1_gradient hf hc (u-constantH1 c), |
| (H1_sub_constant_spec u c).1,(H1_sub_constant_spec u c).2] with x hx hv hg |
| rw [hx,hv,hg] |
|
|
| theorem nested_source_weighted {w : H1} |
| (hw : Tendsto (fun N => (r.finiteSource N).val) atTop (𝓝 w)) |
| {u : H1} (hu : Harmonic r.coefficient u) |
| (hb : ∃ M : ℝ,∀ᵐ x∂ballMeasure,abs (weakValue u x) ≤ M) : |
| WeightedAnnihilation r.coefficient w u (sourceMeanCLM 0 u) := by |
| obtain ⟨M,hb⟩ := hb |
| let P := max M 0 |
| have hP : 0 ≤ P := le_max_right _ _ |
| have hb' : ∀ᵐ x∂ballMeasure,abs (weakValue u x) ≤ P := hb.mono (fun x hx => hx.trans (le_max_left _ _)) |
| apply weighted_annihilation_of_geometric_error r.coefficient r.coefficient_measurable.aestronglyMeasurable |
| (max r.C 1) ((by norm_num : (0:ℝ) ≤ 1).trans (le_max_right _ _)) r.coefficient_bound hw hb' |
| (sourceMeanCLM 0 u) sourceScale (by norm_num [sourceScale]) (by norm_num [sourceScale]) |
| intro f hf hc hsupp |
| obtain ⟨L,hL,hlip⟩ := compact_smooth_lipschitz_bound f hf hc |
| obtain ⟨D,hD,hdiam⟩ := sourceWordBoundary_radius |
| refine ⟨4*P*L*D,?_⟩ |
| intro N |
| rw [←r.form_weighted_test (r.finiteSource N).val u (sourceMeanCLM 0 u) hf hc] |
| exact r.finiteSource_weighted_bound (r.harmonic_sourceHarmonic hu) hP hb' hf hc hL hD hlip hdiam N |
|
|
| end BoundedRootSources |
| end ScalarConductivity |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| namespace ScalarConductivity |
| open Set MeasureTheory Filter Topology UnitAddTorus |
| local instance iteratedAveragesMeasureSpace : MeasureSpace UnitAddCircle := ⟨AddCircle.haarAddCircle⟩ |
| local instance iteratedAveragesIsProbabilityMeasure : IsProbabilityMeasure (volume : Measure UnitAddCircle) := |
| inferInstanceAs (IsProbabilityMeasure AddCircle.haarAddCircle) |
|
|
| lemma sourceRootBoundary_mem_closure (θ : UnitAddTorus (Fin 2)) : |
| WithLp.toLp 2 (physicalOuterBoundary 0 θ)∈closure sourceDomain := by |
| rw [sourceDomain_closure_time] |
| change 0 ≤ sourceCollarTime (sourceAngularCollar 0 θ) |
| rw [sourceAngular_time (by norm_num)] |
|
|
| lemma sourceWordBoundary_mem_cell (ν : SourceWord) (θ : UnitAddTorus (Fin 2)) : |
| sourceWordBoundary ν θ∈sourceWordCell ν := |
| ⟨WithLp.toLp 2 (physicalOuterBoundary 0 θ),sourceRootBoundary_mem_closure θ,rfl⟩ |
|
|
| lemma sourceWordMean_smooth_constant (ν : SourceWord) {f : R3 → ℝ} |
| (hf : ContDiff ℝ (↑(⊤:ℕ∞)) f) {c : ℝ} |
| (hfc : ∀ θ,f (sourceWordBoundary ν θ)=c) : |
| sourceMeanCLM 0 (sourceWordPull ν (smoothH1 f hf))=c := by |
| rw [sourceMean_eq_integral] |
| change (∫ θ,sourceWordTraceCLM ν (smoothH1 f hf) θ)=c |
| rw [integral_congr_ae (sourceWordTrace_smooth ν hf)] |
| simp [hfc] |
|
|
| lemma source_first_children_separator : ∃ f : R3 → ℝ, |
| ContDiff ℝ (↑(⊤:ℕ∞)) f ∧ |
| (∀ ν θ,f (sourceWordBoundary (0::ν) θ)=1) ∧ |
| (∀ ν θ,f (sourceWordBoundary (1::ν) θ)=0) := by |
| let A := sourceChildEuclidean (actualChildSign 0) '' closure sourceDomain |
| let B := sourceChildEuclidean (actualChildSign 1) '' closure sourceDomain |
| have hA : IsClosed A := (sourceClosure_isCompact.image (sourceChildEuclidean (actualChildSign 0)).continuous).isClosed |
| have hB : IsClosed B := (sourceClosure_isCompact.image (sourceChildEuclidean (actualChildSign 1)).continuous).isClosed |
| obtain ⟨f,hf0,hf1,hfb⟩ := exists_contMDiffMap_zero_one_of_isClosed |
| (modelWithCornersSelf ℝ R3) hB hA (source_child_closed_disjoint (by decide : (1:Fin 2)≠0)) |
| (n:=↑(⊤:ℕ∞)) |
| refine ⟨f,f.contMDiff.contDiff,?_,?_⟩ |
| · intro ν θ |
| exact hf1 (sourceWordCell_cons_subset 0 ν (sourceWordBoundary_mem_cell (0::ν) θ)) |
| · intro ν θ |
| exact hf0 (sourceWordCell_cons_subset 1 ν (sourceWordBoundary_mem_cell (1::ν) θ)) |
|
|
| namespace BoundedRootSources |
| variable (r : BoundedRootSources) |
|
|
| lemma finiteSource_separator : ∃ ψ : H1,∀ N,r.form (r.finiteSource N).val ψ=1 := by |
| obtain ⟨f,hf,hf0,hf1⟩ := source_first_children_separator |
| let ψ := smoothH1 f hf |
| refine ⟨ψ,?_⟩ |
| intro N |
| rw [r.form_finiteSource] |
| apply sourceAveragePullIter_mean_eq N (sourceDifferencePull ψ) 1 |
| intro ν hν |
| change sourceMeanCLM 0 (sourceWordPull ν (childH1Pullback 0 ψ-childH1Pullback 1 ψ))=1 |
| rw [map_sub,map_sub] |
| have h0 := sourceWordMean_smooth_constant (0::ν) hf (hf0 ν) |
| have h1 := sourceWordMean_smooth_constant (1::ν) hf (hf1 ν) |
| change sourceMeanCLM 0 (sourceWordPull ν (childH1Pullback 0 ψ))=1 at h0 |
| change sourceMeanCLM 0 (sourceWordPull ν (childH1Pullback 1 ψ))=0 at h1 |
| rw [h0,h1,sub_zero] |
|
|
| lemma nested_source_nonzero {w : H1} |
| (hw : Tendsto (fun N => (r.finiteSource N).val) atTop (𝓝 w)) : |
| w≠0 ∧ 0<ballMeasure {x | weakValue w x≠0} := by |
| obtain ⟨ψ,hψ⟩ := r.finiteSource_separator |
| apply source_energy_nonzero r.coefficient r.coefficient_measurable.aestronglyMeasurable |
| ((by norm_num : (0:ℝ) ≤ 1).trans (le_max_right _ _)) r.coefficient_bound hw ψ |
| apply Eventually.of_forall |
| intro N |
| rw [←r.form_apply] |
| exact hψ N |
|
|
| end BoundedRootSources |
| end ScalarConductivity |
|
|
| end |
| end |
|
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| end OAI |
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