| import Mathlib |
| import OAI.Analysis.Conductivity.Sources.NestedWeightedLimit |
| import OAI.Analysis.Conductivity.Branching.ChildJetScaling |
| import OAI.Analysis.Conductivity.Sources.WordGeometry |
| import OAI.Analysis.Conductivity.Sources.PhysicalMaximum |
| import OAI.Analysis.Conductivity.Sources.CopiedEnergyAlgebra |
| import OAI.Analysis.Conductivity.Branching.PhysicalTraceTransport |
| import OAI.Analysis.Conductivity.Sobolev.OriginalMultiplierOperator |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| namespace ScalarConductivity |
| open Set MeasureTheory Filter Topology UnitAddTorus |
| local instance : MeasureSpace UnitAddCircle := ⟨AddCircle.haarAddCircle⟩ |
| local instance : IsProbabilityMeasure (volume : Measure UnitAddCircle) := |
| inferInstanceAs (IsProbabilityMeasure AddCircle.haarAddCircle) |
|
|
| def sourceWordPull : SourceWord → H1 →L[ℝ] H1 |
| | [] => ContinuousLinearMap.id ℝ H1 |
| | k::ν => (sourceWordPull ν).comp (childH1Pullback k) |
|
|
| lemma sourceWordMap_contDiff (ν : SourceWord) : |
| ContDiff ℝ (↑(⊤:ℕ∞)) (sourceWordMap ν : R3 → R3) := by |
| induction ν with |
| | nil => exact contDiff_id |
| | cons k ν ih => exact (sourceChildEuclidean_contDiff _).comp ih |
|
|
| lemma sourceWordPull_smooth (ν : SourceWord) {f : R3 → ℝ} |
| (hf : ContDiff ℝ (↑(⊤:ℕ∞)) f) : |
| sourceWordPull ν (smoothH1 f hf)=smoothH1 (f ∘ sourceWordMap ν) (hf.comp (sourceWordMap_contDiff ν)) := by |
| induction ν generalizing f with |
| | nil => rfl |
| | cons k ν ih => |
| change sourceWordPull ν (childH1Pullback k (smoothH1 f hf))=_ |
| rw [childH1Pullback_smooth,ih] |
| rfl |
|
|
| lemma childH1Pullback_bound (k : Fin 2) (u : H1) {M : ℝ} |
| (hb : ∀ᵐ x∂ballMeasure,abs (weakValue u x) ≤ M) : |
| ∀ᵐ x∂ballMeasure,abs (weakValue (childH1Pullback k u) x) ≤ M := by |
| filter_upwards [childH1Pullback_value k u,(sourceChildEuclidean_ball_quasi k).ae hb] with x hx hy |
| rw [hx]; exact hy |
|
|
| lemma sourceWordPull_bound (ν : SourceWord) (u : H1) {M : ℝ} |
| (hb : ∀ᵐ x∂ballMeasure,abs (weakValue u x) ≤ M) : |
| ∀ᵐ x∂ballMeasure,abs (weakValue (sourceWordPull ν u) x) ≤ M := by |
| induction ν generalizing u with |
| | nil => exact hb |
| | cons k ν ih => exact ih _ (childH1Pullback_bound k u hb) |
|
|
| def sourceWordTraceCLM (ν : SourceWord) : H1 →L[ℝ] Lp ℝ 2 (volume : Measure (UnitAddTorus (Fin 2))) := |
| (physicalRealTraceCLM 0).comp (sourceWordPull ν) |
|
|
| def sourceWordBoundary (ν : SourceWord) (θ : UnitAddTorus (Fin 2)) : R3 := |
| sourceWordMap ν (WithLp.toLp 2 (physicalOuterBoundary 0 θ)) |
|
|
| lemma sourceWordBoundary_continuous (ν : SourceWord) : Continuous (sourceWordBoundary ν) := |
| (sourceWordMap ν).continuous.comp ((PiLp.continuous_toLp 2 (fun _ : Fin 3 => ℝ)).comp |
| (physicalOuterBoundary_continuous 0)) |
|
|
| lemma sourceWordTrace_smooth (ν : SourceWord) {f : R3 → ℝ} |
| (hf : ContDiff ℝ (↑(⊤:ℕ∞)) f) : |
| sourceWordTraceCLM ν (smoothH1 f hf)=ᵐ[volume] (fun θ => f (sourceWordBoundary ν θ)) := by |
| change physicalRealTraceCLM 0 (sourceWordPull ν (smoothH1 f hf))=ᵐ[volume] _ |
| rw [sourceWordPull_smooth] |
| exact physicalRealTrace_smooth_ae 0 (hf.comp (sourceWordMap_contDiff ν)) |
|
|
| lemma sourceWordTrace_mul (ν : SourceWord) {f : R3 → ℝ} |
| (hf : ContDiff ℝ (↑(⊤:ℕ∞)) f) (hc : HasCompactSupport f) (u : H1) : |
| sourceWordTraceCLM ν (originalMulH1CLM hf hc u)=ᵐ[volume] |
| (fun θ => f (sourceWordBoundary ν θ)*sourceWordTraceCLM ν u θ) := by |
| let χ : UnitAddTorus (Fin 2) → ℝ := f ∘ sourceWordBoundary ν |
| have hχ : Continuous χ := hf.continuous.comp (sourceWordBoundary_continuous ν) |
| have hχc : HasCompactSupport χ := HasCompactSupport.of_compactSpace _ |
| have he : sourceWordTraceCLM ν (originalMulH1CLM hf hc u)= |
| compactMultiplierCLM hχ hχc (sourceWordTraceCLM ν u) := by |
| have hclosed : IsClosed {u : H1 | sourceWordTraceCLM ν (originalMulH1CLM hf hc u)= |
| compactMultiplierCLM hχ hχc (sourceWordTraceCLM ν u)} := |
| isClosed_eq ((sourceWordTraceCLM ν).continuous.comp (originalMulH1CLM hf hc).continuous) |
| ((compactMultiplierCLM hχ hχc).continuous.comp (sourceWordTraceCLM ν).continuous) |
| apply (closure_minimal (s:={u : H1 | u.val∈smoothJets}) ?_ hclosed) (smoothH1_dense u) |
| rintro v ⟨g,hg,_,hv⟩ |
| have he : v=smoothH1 g hg := Subtype.ext hv |
| change sourceWordTraceCLM ν (originalMulH1CLM hf hc v)=_ |
| rw [he,originalMulH1_smooth] |
| apply Lp.ext |
| filter_upwards [sourceWordTrace_smooth ν (hf.mul hg),sourceWordTrace_smooth ν hg, |
| compactMultiplierCLM_ae hχ hχc (sourceWordTraceCLM ν (smoothH1 g hg))] with θ h0 h1 h2 |
| exact h0.trans (by rw [h2,h1]; rfl) |
| rw [he] |
| exact compactMultiplierCLM_ae hχ hχc (sourceWordTraceCLM ν u) |
|
|
| end ScalarConductivity |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| namespace ScalarConductivity |
| open Set MeasureTheory Filter Topology UnitAddTorus |
| local instance : MeasureSpace UnitAddCircle := ⟨AddCircle.haarAddCircle⟩ |
| local instance : IsProbabilityMeasure (volume : Measure UnitAddCircle) := |
| inferInstanceAs (IsProbabilityMeasure AddCircle.haarAddCircle) |
|
|
| lemma constantH1_eq_smooth (c : ℝ) : constantH1 c=smoothH1 (fun _ => c) contDiff_const := by |
| apply H1_eq_of_value_ae |
| exact (constantH1_value c).trans (smoothH1_value _ contDiff_const).symm |
|
|
| lemma sourceWordTrace_constant (ν : SourceWord) (c : ℝ) : |
| sourceWordTraceCLM ν (constantH1 c)=ᵐ[volume] (fun _ => c) := by |
| rw [constantH1_eq_smooth] |
| exact sourceWordTrace_smooth ν contDiff_const |
|
|
| lemma sourceWordMean_constant (ν : SourceWord) (c : ℝ) : |
| sourceMeanCLM 0 (sourceWordPull ν (constantH1 c))=c := by |
| rw [sourceMean_eq_integral] |
| change (∫ θ,sourceWordTraceCLM ν (constantH1 c) θ)=c |
| rw [integral_congr_ae (sourceWordTrace_constant ν c)] |
| simp |
|
|
| namespace BoundedRootSources |
| variable (r : BoundedRootSources) |
| lemma sourceHarmonic_word {u : H1} (hu : r.sourceHarmonic u) (ν : SourceWord) : |
| r.sourceHarmonic (sourceWordPull ν u) := by |
| induction ν generalizing u with |
| | nil => exact hu |
| | cons k ν ih => exact ih (r.sourceHarmonic_child hu k) |
|
|
| lemma sourceHarmonic_word_mean {u : H1} (hu : r.sourceHarmonic u) (ν : SourceWord) : |
| sourceMeanCLM 0 (sourceWordPull ν u)=sourceMeanCLM 0 u := by |
| induction ν generalizing u with |
| | nil => rfl |
| | cons k ν ih => |
| change sourceMeanCLM 0 (sourceWordPull ν (childH1Pullback k u))=_ |
| rw [ih (r.sourceHarmonic_child hu k),r.sourceHarmonic_mean hu k] |
|
|
| lemma sourceHarmonic_trace_centered_mean {u : H1} (hu : r.sourceHarmonic u) (ν : SourceWord) : |
| (∫ θ,sourceWordTraceCLM ν (u-constantH1 (sourceMeanCLM 0 u)) θ)=0 := by |
| change (∫ θ,physicalRealTraceCLM 0 (sourceWordPull ν (u-constantH1 (sourceMeanCLM 0 u))) θ)=0 |
| rw [←sourceMean_eq_integral,map_sub,map_sub,r.sourceHarmonic_word_mean hu ν,sourceWordMean_constant,sub_self] |
| end BoundedRootSources |
|
|
| lemma sourceWordBoundary_radius : ∃ D : ℝ,0 ≤ D ∧ ∀ ν θ, |
| dist (sourceWordBoundary ν θ) (sourceWordMap ν 0) ≤ D*sourceScale^ν.length := by |
| let f : UnitAddTorus (Fin 2) → ℝ := fun θ => dist (sourceWordBoundary [] θ) 0 |
| have hf : Continuous f := (sourceWordBoundary_continuous []).dist continuous_const |
| obtain ⟨D,hD⟩ := (HasCompactSupport.of_compactSpace f).exists_bound_of_continuous hf |
| have hD0 : 0 ≤ D := (norm_nonneg (f 0)).trans (hD 0) |
| refine ⟨D,hD0,?_⟩ |
| intro ν θ |
| change dist (sourceWordMap ν (WithLp.toLp 2 (physicalOuterBoundary 0 θ))) (sourceWordMap ν 0) ≤ _ |
| rw [sourceWordMap_dist] |
| have hh : dist (WithLp.toLp 2 (physicalOuterBoundary 0 θ)) 0 ≤ D := by |
| simpa only [f,sourceWordBoundary,sourceWordMap_nil,Real.norm_eq_abs,abs_of_nonneg dist_nonneg] using hD θ |
| nlinarith [pow_nonneg (show 0 ≤ sourceScale by norm_num [sourceScale]) ν.length] |
|
|
| lemma sourceWord_weighted_mean_bound (r : BoundedRootSources) {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) |
| (ν : SourceWord) : |
| abs (sourceMeanCLM 0 (sourceWordPull ν |
| (originalMulH1CLM hf hc (u-constantH1 (sourceMeanCLM 0 u))))) ≤ |
| (2*M*L*D)*sourceScale^ν.length := by |
| let v := u-constantH1 (sourceMeanCLM 0 u) |
| have hv : ∀ᵐ x∂ballMeasure,abs (weakValue v x) ≤ 2*M := by |
| filter_upwards [hb,(H1_sub_constant_spec u (sourceMeanCLM 0 u)).1] with x hx he |
| change abs (weakValue (u-constantH1 (sourceMeanCLM 0 u)) x) ≤ _ |
| rw [he] |
| have hm := sourceMean_bound 0 u hM hb |
| exact (abs_sub _ _).trans (by linarith) |
| let t := sourceWordTraceCLM ν v |
| have ht : Integrable t := (Lp.memLp t).integrable (by norm_num) |
| have htb : ∀ᵐ θ,abs (t θ) ≤ 2*M := physicalRealTrace_bound 0 (sourceWordPull ν v) |
| (by positivity) (sourceWordPull_bound ν v hv) |
| have hz : (∫ θ,t θ)=0 := r.sourceHarmonic_trace_centered_mean hu ν |
| obtain ⟨A,hA⟩ := hc.exists_bound_of_continuous hf.continuous |
| have htf : Integrable (fun θ => t θ*f (sourceWordBoundary ν θ)) := |
| ht.mul_bdd (hf.continuous.comp (sourceWordBoundary_continuous ν)).aestronglyMeasurable |
| (Eventually.of_forall (fun θ => hA (sourceWordBoundary ν θ))) |
| have ho : ∀ᵐ θ,abs (f (sourceWordBoundary ν θ)-f (sourceWordMap ν 0)) ≤ L*(D*sourceScale^ν.length) := |
| Eventually.of_forall (fun θ => (hlip _ _).trans (mul_le_mul_of_nonneg_left (hdiam ν θ) hL)) |
| have hh := surface_cancellation_bound volume t (f ∘ sourceWordBoundary ν) |
| (f (sourceWordMap ν 0)) (2*M) (L*(D*sourceScale^ν.length)) (by positivity) |
| (mul_nonneg hL (mul_nonneg hD (pow_nonneg (by norm_num [sourceScale]) _))) ht htf hz htb ho |
| rw [sourceMean_eq_integral] |
| change abs (∫ θ,sourceWordTraceCLM ν (originalMulH1CLM hf hc v) θ) ≤ _ |
| rw [integral_congr_ae (sourceWordTrace_mul ν hf hc v)] |
| have he : (∫ θ,f (sourceWordBoundary ν θ)*t θ)=(∫ θ,t θ*f (sourceWordBoundary ν θ)) := by |
| congr 1; funext θ; ring |
| rw [he] |
| exact hh.trans_eq (by ring) |
|
|
| end ScalarConductivity |
|
|
| end |
| end |
|
|
| end OAI |
|
|
