| import Mathlib |
| import OAI.Analysis.Conductivity.Sobolev.AttachedEndMatchedH1 |
|
|
| namespace OAI |
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|
| noncomputable section |
| namespace ScalarConductivity |
| open Set MeasureTheory Filter Topology UnitAddTorus |
| open scoped ENNReal |
|
|
| local instance smoothCollarTraceCircleMeasureSpace : MeasureSpace UnitAddCircle := |
| ⟨AddCircle.haarAddCircle⟩ |
| local instance smoothCollarTraceCircleIsProbabilityMeasure : |
| IsProbabilityMeasure (volume : Measure UnitAddCircle) := |
| inferInstanceAs (IsProbabilityMeasure AddCircle.haarAddCircle) |
|
|
| def smoothCollarTraceFourier (b : ℝ) {q : (Fin 3 → ℝ) → ℝ} |
| (hq : ContDiff ℝ (↑(⊤:ℕ∞)) q) : SpectralL2 TorusModes := |
| (mFourierBasis (d:=Fin 2)).repr |
| (ContinuousMap.toLp 2 volume ℂ |
| ⟨fun θ => (q (sourceAngularCollar b θ):ℂ), |
| Complex.continuous_ofReal.comp (hq.continuous.comp (continuous_sourceAngularCollar b))⟩) |
|
|
| lemma smoothCollarTraceFourier_apply (b : ℝ) {q : (Fin 3 → ℝ) → ℝ} |
| (hq : ContDiff ℝ (↑(⊤:ℕ∞)) q) (h : TorusModes) : |
| smoothCollarTraceFourier b hq h=mFourierCoeff (fun θ => (q (sourceAngularCollar b θ):ℂ)) h := by |
| rw [smoothCollarTraceFourier,mFourierBasis_repr,mFourierCoeff_toLp] |
| rfl |
|
|
| def smoothCollarTrace (s : Fin 3 → ℝ) (b : ℝ) {q : (Fin 3 → ℝ) → ℝ} |
| (hq : ContDiff ℝ (↑(⊤:ℕ∞)) q) : spectralTraceGraph (torusRate s) := |
| ((spectralTraceGraph_exists_iff (w:=torusRate s) (fun h => Real.sqrt_nonneg _) (smoothCollarTraceFourier b hq)).mpr |
| (by simpa only [smoothCollarTraceFourier_apply] using sourceAngularTrace_summable s b hq)).choose |
|
|
| lemma smoothCollarTrace_fst (s : Fin 3 → ℝ) (b : ℝ) {q : (Fin 3 → ℝ) → ℝ} |
| (hq : ContDiff ℝ (↑(⊤:ℕ∞)) q) (h : TorusModes) : |
| (smoothCollarTrace s b hq).val 0 h=mFourierCoeff (fun θ => (q (sourceAngularCollar b θ):ℂ)) h := by |
| have he := ((spectralTraceGraph_exists_iff (w:=torusRate s) (fun h => Real.sqrt_nonneg _) (smoothCollarTraceFourier b hq)).mpr |
| (by simpa only [smoothCollarTraceFourier_apply] using sourceAngularTrace_summable s b hq)).choose_spec |
| change (smoothCollarTrace s b hq).val 0=smoothCollarTraceFourier b hq at he |
| rw [he,smoothCollarTraceFourier_apply] |
|
|
| lemma smoothCollarTrace_rate_memℓp (s : Fin 3 → ℝ) (b : ℝ) {q : (Fin 3 → ℝ) → ℝ} |
| (hq : ContDiff ℝ (↑(⊤:ℕ∞)) q) : |
| Memℓp (fun h => (torusRate s h:ℂ)*(smoothCollarTrace s b hq).val 0 h) 2 := by |
| apply (memℓp_gen_iff (by norm_num : 0<(2:ℝ≥0∞).toReal)).mpr |
| simpa only [ENNReal.toReal_ofNat,Real.rpow_two,norm_mul,mul_pow, |
| Complex.norm_real,Real.norm_eq_abs,sq_abs,smoothCollarTrace_fst] using sourceAngularTrace_rate_sq_summable s b hq |
|
|
| theorem smoothCollarTrace_matched_H1 {s : Fin 3 → ℝ} |
| (hs : ∀ u v : ℝ,(1/2)*(u^2+v^2) ≤ s 0*u^2+2*s 1*u*v+s 2*v^2) |
| {q χ : (Fin 3 → ℝ) → ℝ} |
| (hq : ContDiff ℝ (↑(⊤:ℕ∞)) q) (hχ : ContDiff ℝ (↑(⊤:ℕ∞)) χ) |
| (hc : HasCompactSupport χ) (hχb : ∀ x,|χ x|≤1) |
| {a b η : ℝ} (ha : a≠0) (hη : 0<η) (hl : -(1:ℝ)/100≤b-η) (hr : b+η≤1/100) |
| (hχs : tsupport χ⊆sourceClosedCollarBand (b-η) (b+η)) : |
| let f := smoothCollarTrace s b hq |
| ∃ w : H1,w∈H10 ∧ (∀ᵐ x∂ballMeasure, |
| weakValue w x=cutMatchedValue (fun y => a*(sourceCollarTime y-b)) χ |
| (fun y => (attachedEndPoissonField s f a b 0 y).re-q y) (WithLp.ofLp x) ∧ |
| ∀ i,weakGradient w x i=cutMatchedGradient (fun y => a*(sourceCollarTime y-b)) χ |
| (fun y => (attachedEndPoissonField s f a b 0 y).re-q y) |
| (fun y => fderiv ℝ (fun y => (attachedEndPoissonField s f a b 0 y).re) y-fderiv ℝ q y) |
| i (WithLp.ofLp x)) := by |
| exact smooth_attachedEnd_matched_H1 hs _ hq hχ hc hχb ha hη hl hr hχs |
| (smoothCollarTrace_fst s b hq) |
| ⟨_,smoothCollarTrace_rate_memℓp s b hq⟩ (fun _ => rfl) |
|
|
| end ScalarConductivity |
|
|
| end |
|
|
| end OAI |
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|
