| import Mathlib |
| import OAI.Analysis.Conductivity.Sobolev.CentralPoincare |
| import OAI.Analysis.Conductivity.Sobolev.CentralPhysicalTrace |
| import OAI.Analysis.Conductivity.Variational.CentralVariational |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace ScalarConductivity |
|
|
| section |
| open Set MeasureTheory UnitAddTorus |
| open scoped ENNReal |
|
|
| def normalizedTorusVolume : Measure (UnitAddTorus (Fin 2)) := |
| Measure.pi (fun _ => AddCircle.haarAddCircle) |
|
|
| lemma unitTorus_volume_eq_normalized : (volume : Measure (UnitAddTorus (Fin 2)))=normalizedTorusVolume := by |
| change Measure.pi (fun _ : Fin 2 => (volume : Measure UnitAddCircle))=_ |
| simp only [unitCircle_volume_eq_haar] |
| rfl |
|
|
| lemma central_physical_trace_normalized {f : Box3 → ℝ} |
| (hf : ContDiff ℝ (↑(⊤ : ℕ∞)) f) (i : Fin 3) : |
| (∫ x : UnitAddTorus (Fin 2),(f (centralBoundary i x))^2 ∂normalizedTorusVolume)≤ |
| centralTraceConstant*(∫ z in centralClosed,192*cubeEnergyDensity f z+2*(f z)^2) := by |
| rw [←unitTorus_volume_eq_normalized] |
| exact central_physical_trace hf i |
|
|
| local instance : MeasureSpace UnitAddCircle := ⟨AddCircle.haarAddCircle⟩ |
| local instance : IsProbabilityMeasure (volume : Measure UnitAddCircle) := |
| inferInstanceAs (IsProbabilityMeasure AddCircle.haarAddCircle) |
|
|
| abbrev TorusModes := Fin 2 → ℤ |
|
|
| def centralTraceFunction (f : centralSmoothFunctions) (i : Fin 3) : |
| C(UnitAddTorus (Fin 2),ℂ) := |
| ⟨fun x => (f (centralBoundary i x):ℂ), |
| Complex.continuous_ofReal.comp ((central_smooth f).continuous.comp (continuous_centralBoundary i))⟩ |
|
|
| def centralTraceFourier (f : centralSmoothFunctions) (i : Fin 3) : SpectralL2 TorusModes := |
| ((mFourierBasis (d:=Fin 2)).repr.restrictScalars ℝ) (ContinuousMap.toLp 2 volume ℝ (centralTraceFunction f i)) |
|
|
| lemma centralTraceFourier_apply (f : centralSmoothFunctions) (i : Fin 3) (h : TorusModes) : |
| centralTraceFourier f i h=mFourierCoeff (fun x => (f (centralBoundary i x):ℂ)) h := by |
| change (mFourierBasis (d:=Fin 2)).repr (ContinuousMap.toLp 2 volume ℂ (centralTraceFunction f i)) h=_ |
| rw [mFourierBasis_repr,mFourierCoeff_toLp] |
| rfl |
|
|
| lemma centralTrace_summable (s : Fin 3 → ℝ) (f : centralSmoothFunctions) (i : Fin 3) : |
| Summable (fun h => torusRate s h*‖centralTraceFourier f i h‖^2) := by |
| simp only [centralTraceFourier_apply] |
| have hf : ContDiff ℝ (↑(⊤ : ℕ∞)) (fun y : Fin 3 → ℝ => f (sourcePairCoordinates y)) := |
| (central_smooth f).comp sourcePairCLE.contDiff |
| fin_cases i |
| · simpa [centralBoundary] using |
| (sourceAngularTrace_summable s centralThickness hf) |
| · simpa [centralBoundary,Function.comp_def] using |
| (sourceAngularTrace_summable s (-centralThickness) (hf.comp (sourceChildCoordinates_contDiff 1))) |
| · simpa [centralBoundary,Function.comp_def] using |
| (sourceAngularTrace_summable s (-centralThickness) (hf.comp (sourceChildCoordinates_contDiff (-1)))) |
|
|
| def centralSmoothTrace (s : Fin 3 → ℝ) (f : centralSmoothFunctions) (i : Fin 3) : |
| spectralTraceGraph (torusRate s) := |
| ((spectralTraceGraph_exists_iff (fun frequency => Real.sqrt_nonneg (torusQuadratic s frequency)) (centralTraceFourier f i)).mpr |
| (centralTrace_summable s f i)).choose |
|
|
| lemma centralSmoothTrace_fst (s : Fin 3 → ℝ) (f : centralSmoothFunctions) (i : Fin 3) : |
| (centralSmoothTrace s f i).val 0=centralTraceFourier f i := |
| ((spectralTraceGraph_exists_iff (fun frequency => Real.sqrt_nonneg (torusQuadratic s frequency)) (centralTraceFourier f i)).mpr |
| (centralTrace_summable s f i)).choose_spec |
|
|
| lemma centralTraceFunction_add (f g : centralSmoothFunctions) (i : Fin 3) : |
| centralTraceFunction (f+g) i=centralTraceFunction f i+centralTraceFunction g i := by |
| ext x |
| exact Complex.ofReal_add _ _ |
|
|
| lemma centralTraceFunction_smul (c : ℝ) (f : centralSmoothFunctions) (i : Fin 3) : |
| centralTraceFunction (c•f) i=c•centralTraceFunction f i := by |
| ext x |
| exact Complex.ofReal_mul _ _ |
|
|
| def centralSmoothTraceL (s : Fin 3 → ℝ) (i : Fin 3) : |
| centralSmoothFunctions →ₗ[ℝ] spectralTraceGraph (torusRate s) where |
| toFun := fun f => centralSmoothTrace s f i |
| map_add' := by |
| intro f g |
| apply spectralTraceGraph_fst_injective |
| change (centralSmoothTrace s (f+g) i).val 0=(centralSmoothTrace s f i).val 0+(centralSmoothTrace s g i).val 0 |
| rw [centralSmoothTrace_fst,centralSmoothTrace_fst,centralSmoothTrace_fst] |
| simp only [centralTraceFourier,centralTraceFunction_add,map_add] |
| map_smul' := by |
| intro c f |
| apply spectralTraceGraph_fst_injective |
| change (centralSmoothTrace s (c•f) i).val 0=c•(centralSmoothTrace s f i).val 0 |
| rw [centralSmoothTrace_fst,centralSmoothTrace_fst] |
| simp only [centralTraceFourier,centralTraceFunction_smul,map_smul] |
|
|
| lemma centralTraceFourier_norm (f : centralSmoothFunctions) (i : Fin 3) : |
| ‖centralTraceFourier f i‖^2=∫ x : UnitAddTorus (Fin 2),(f (centralBoundary i x))^2 := by |
| change ‖(mFourierBasis (d:=Fin 2)).repr (ContinuousMap.toLp 2 volume ℂ (centralTraceFunction f i))‖^2=_ |
| rw [LinearIsometryEquiv.norm_map,←real_inner_self_eq_norm_sq] |
| rw [L2.inner_def] |
| apply integral_congr_ae |
| filter_upwards [ContinuousMap.coeFn_toLp (𝕜:=ℂ) (p:=2) volume (centralTraceFunction f i)] with x hx |
| rw [hx,real_inner_self_eq_norm_sq] |
| simp [centralTraceFunction,Complex.norm_real,Real.norm_eq_abs,sq_abs] |
|
|
| lemma centralSmoothTrace_norm (s : Fin 3 → ℝ) (f : centralSmoothFunctions) (i : Fin 3) : |
| ‖centralSmoothTrace s f i‖^2=‖centralTraceFourier f i‖^2+ |
| ‖spectralGraphWeight (torusRate s) (centralSmoothTrace s f i)‖^2 := by |
| rw [←Submodule.norm_coe,PiLp.norm_sq_eq_of_L2,Fin.sum_univ_two,centralSmoothTrace_fst] |
| rfl |
|
|
| end |
|
|
| open Set MeasureTheory UnitAddTorus |
| open scoped ENNReal |
|
|
| local instance : MeasureSpace UnitAddCircle := ⟨AddCircle.haarAddCircle⟩ |
| local instance : IsProbabilityMeasure (volume : Measure UnitAddCircle) := |
| inferInstanceAs (IsProbabilityMeasure AddCircle.haarAddCircle) |
|
|
| def centralSmoothOne : centralSmoothFunctions := ⟨fun _ => 1,by change ContDiff ℝ (↑(⊤ : ℕ∞)) (fun _ : Box3 => (1 : ℝ)); exact contDiff_const⟩ |
|
|
| lemma centralOne_gradient : centralJetD (centralSmoothJet centralSmoothOne)=0 := by |
| apply PiLp.ext |
| intro i |
| apply Lp.ext |
| filter_upwards [centralSmoothJet_ae centralSmoothOne i.succ,Lp.coeFn_zero ℝ 2 centralMeasure] with z hz h0 |
| change centralSmoothJet centralSmoothOne i.succ z=(0 : CentralL2) z |
| rw [hz,h0] |
| fin_cases i <;> simp [centralJetField,centralSmoothOne,cubePartial] |
|
|
| lemma centralOne_traceFunction (i : Fin 3) : |
| centralTraceFunction centralSmoothOne i=mFourier (0 : TorusModes) := by |
| ext x |
| simp [centralTraceFunction,centralSmoothOne,mFourier_zero] |
|
|
| lemma centralOne_fourier (i : Fin 3) : |
| centralTraceFourier centralSmoothOne i=lp.single 2 (0 : TorusModes) 1 := by |
| change (mFourierBasis (d:=Fin 2)).repr (ContinuousMap.toLp 2 volume ℂ |
| (centralTraceFunction centralSmoothOne i))=_ |
| rw [centralOne_traceFunction] |
| simpa only [coe_mFourierBasis] using (mFourierBasis (d:=Fin 2)).repr_self 0 |
|
|
| lemma centralOne_weight (s : Fin 3 → ℝ) (i : Fin 3) : |
| spectralGraphWeight (torusRate s) (centralSmoothTrace s centralSmoothOne i)=0 := by |
| ext h |
| change ((centralSmoothTrace s centralSmoothOne i).val 1) h=0 |
| rw [(centralSmoothTrace s centralSmoothOne i).property h,centralSmoothTrace_fst,centralOne_fourier] |
| by_cases hh : h=0 |
| · subst h |
| simp [torusRate,torusQuadratic] |
| · simp [lp.single_apply,Ne.symm hh] |
|
|
| lemma centralOne_mean (s : Fin 3 → ℝ) (i : Fin 3) : |
| spectralGraphMean (torusRate s) (centralSmoothTrace s centralSmoothOne i)=1 := by |
| change (((centralSmoothTrace s centralSmoothOne i).val 0) 0).re=1 |
| rw [centralSmoothTrace_fst,centralOne_fourier] |
| simp [lp.single_apply] |
|
|
| lemma centralTrace_anchor_bound (f : centralSmoothFunctions) (i : Fin 3) : |
| ‖centralTraceFourier f i‖^2≤ |
| centralTraceConstant*(192*‖centralJetD (centralSmoothJet f)‖^2+ |
| 2*(∫ z in centralClosed,(f z)^2)) := by |
| rw [centralTraceFourier_norm] |
| have hh := central_physical_trace_normalized (central_smooth f) i |
| have hE : IntegrableOn (cubeEnergyDensity f) centralClosed := (continuous_cubeEnergyDensity (central_smooth f)).continuousOn.integrableOn_compact centralClosed_compact |
| have hf : IntegrableOn (fun z => (f z)^2) centralClosed := ((central_smooth f).continuous.pow 2).continuousOn.integrableOn_compact centralClosed_compact |
| rw [integral_add (hE.const_mul 192) (hf.const_mul 2),integral_const_mul,integral_const_mul] at hh |
| rw [centralSmoothJet_gradient_norm] |
| exact hh |
|
|
| end ScalarConductivity |
|
|
| end |
|
|
| end OAI |
|
|
