| import Mathlib |
| import OAI.Analysis.Conductivity.Sobolev.OriginalSmoothMultiplier |
| import OAI.Analysis.Conductivity.Sources.PhysicalGradient |
| import OAI.Analysis.Conductivity.Variational.PhysicalRealTrace |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| namespace ScalarConductivity |
| open Set MeasureTheory Filter Topology UnitAddTorus |
| open scoped ENNReal NNReal |
| local instance sourcePhysicalTraceMeasureSpace : MeasureSpace UnitAddCircle := ⟨AddCircle.haarAddCircle⟩ |
| local instance sourcePhysicalTraceIsProbabilityMeasure : IsProbabilityMeasure (volume : Measure UnitAddCircle) := |
| inferInstanceAs (IsProbabilityMeasure AddCircle.haarAddCircle) |
|
|
| lemma physicalOuterTrace_compact_mul {χ : R3 → ℝ} |
| (hχ : ContDiff ℝ (↑(⊤:ℕ∞)) χ) (hc : HasCompactSupport χ) |
| (hs : tsupport χ⊆(WithLp.ofLp) ⁻¹' physicalOpenBlock) (u v : H1) |
| (hv : weakValue v=ᵐ[ballMeasure] fun x => χ x*weakValue u x) (i : Fin 3) : |
| physicalOuterTraceCLM i v=0 := by |
| let P := H1Space.orthogonalProjectionOnto |
| let T : H1 →L[ℝ] SpectralL2 TorusModes := |
| (physicalOuterTraceCLM i).comp (P.comp (originalMulJetCLM hχ hc)) |
| have hz : T u=0 := by |
| apply (closure_minimal (s:={u : H1 | u.val∈smoothJets}) ?_ |
| (isClosed_eq T.continuous continuous_const)) (smoothH1_dense u) |
| rintro w ⟨f,hf,hm,he⟩ |
| have hw : w=smoothH1 f hf := Subtype.ext he |
| change T w=0 |
| rw [hw] |
| change physicalOuterTraceCLM i (P (originalMulJetCLM hχ hc (smoothH1 f hf)))=0 |
| rw [originalMulJetCLM_smooth] |
| simp only [P,H1Space.orthogonalProjectionOnto_mem_subspace_eq_self] |
| let E := (PiLp.continuousLinearEquiv 2 ℝ (fun _ : Fin 3 => ℝ)).symm |
| have hd : ContDiff ℝ (↑(⊤:ℕ∞)) ((χ*f) ∘ E) := (hχ.mul hf).comp E.contDiff |
| have hev : piSmoothH1 hd=smoothH1 (χ*f) (hχ.mul hf) := by |
| unfold piSmoothH1 |
| congr 1 |
| rw [←hev,physicalOuterTrace_smooth i hd (fun _ => 1), |
| physicalSmoothOuterTrace_compact_zero (fun _ => 1) hd ?_] |
| rfl |
| have hss : tsupport ((χ*f) ∘ E)⊆E ⁻¹' tsupport χ := |
| closure_minimal (fun x hx => tsupport_mul_subset_left (subset_tsupport (χ*f) hx)) |
| ((isClosed_tsupport χ).preimage E.continuous) |
| intro y hy |
| exact hs (hss hy) |
| have hm := originalMulJetCLM_mem_zero hχ hc (fun x hx => physicalOpenBlock_subset_ball (hs hx)) u |
| let a : H1 := ⟨originalMulJetCLM hχ hc u,zeroTraceAmbient_le_H1Space hm⟩ |
| have ha : a=v := by |
| apply H1_eq_of_value_ae |
| filter_upwards [originalMulJetCLM_ae hχ hc u,hv] with x hx hv |
| exact (congrArg (fun z : JetFiber => z 0) hx).trans hv.symm |
| change physicalOuterTraceCLM i (P a.val)=0 at hz |
| rw [H1Space.orthogonalProjectionOnto_mem_subspace_eq_self,ha] at hz |
| exact hz |
|
|
| lemma physicalOuterTrace_compact_lipschitz {C : ℝ≥0} {f : R3 → ℝ} |
| (hf : LipschitzWith C f) {R : ℝ} (hR : R<3) (hs : ∀ x,R<‖x‖ → f x=0) |
| (u : H1) (hu : weakValue u=ᵐ[ballMeasure] f) (i : Fin 3) : |
| physicalOuterTraceCLM i u=continuousOuterFourier i hf.continuous := by |
| have hm : MemLp f 2 volume := hf.continuous.memLp_of_hasCompactSupport (bounded_support_compact hs) |
| have hd (j : Fin 3) := lipschitz_lineDeriv_memLp hf hs (EuclideanSpace.single j 1) |
| let F : Fin 4 → WholeL2 := Fin.cases (hm.toLp f) |
| (fun j => (hd j).toLp (fun x => lineDeriv ℝ f x (EuclideanSpace.single j 1))) |
| have hv : (F 0 : R3 → ℝ)=ᵐ[volume] f := hm.coeFn_toLp |
| have hw (j : Fin 3) : WeakL2Direction (F 0) (F j.succ) (EuclideanSpace.single j 1) := |
| (lipschitz_weakDirection hf _).congr hv ((hd j).coeFn_toLp) |
| have hz := compact_weakJet_mem_zeroTrace F hw hR (by |
| filter_upwards [hv] with x hx hn; exact hx.trans (hs x hn)) |
| let v : H1 := ⟨ballJet F,zeroTraceAmbient_le_H1Space hz⟩ |
| have he : u=v := by |
| apply H1_eq_of_value_ae |
| filter_upwards [hu,ae_restrict_of_ae (s:=ball) hv,ballJet_ae F] with x hu hx hb |
| exact hu.trans (hx.symm.trans (hb 0).symm) |
| rw [he] |
| exact physicalOuterTrace_lipschitz hf F hv hw v rfl i |
|
|
| lemma physicalRealTrace_compact_lipschitz {C : ℝ≥0} {f : R3 → ℝ} |
| (hf : LipschitzWith C f) {R : ℝ} (hR : R<3) (hs : ∀ x,R<‖x‖ → f x=0) |
| (u : H1) (hu : weakValue u=ᵐ[ballMeasure] f) (i : Fin 3) : |
| physicalRealTraceCLM i u=ᵐ[volume] fun θ => f (WithLp.toLp 2 (physicalOuterBoundary i θ)) := by |
| have he : physicalComplexTraceCLM i u=ContinuousMap.toLp 2 volume ℂ (continuousOuterMap i hf.continuous) := by |
| change (mFourierBasis (d:=Fin 2)).repr.symm (physicalOuterTraceCLM i u)=_ |
| rw [physicalOuterTrace_compact_lipschitz hf hR hs u hu] |
| exact (mFourierBasis (d:=Fin 2)).repr.symm_apply_apply _ |
| have hh := Complex.reCLM.coeFn_compLpL (p:=2) (μ:=volume) (physicalComplexTraceCLM i u) |
| change physicalRealTraceCLM i u=ᵐ[volume] _ at hh |
| rw [he] at hh |
| filter_upwards [hh,ContinuousMap.coeFn_toLp (continuousOuterMap i hf.continuous) (p:=2) (μ:=volume) (𝕜:=ℂ)] |
| with θ hθ hfv |
| rw [hθ,hfv] |
| rfl |
|
|
| namespace PhysicalFiniteEndingData |
| variable {s : Fin 3 → ℝ} (z : PhysicalFiniteEndingData s) |
|
|
| lemma terminalTime_outer (i : Fin 3) (θ : UnitAddTorus (Fin 2)) : |
| terminalTime i (physicalOuterBoundary i θ)=0 := by |
| refine Fin.cases ?_ (fun k => ?_) i |
| · change sourceCollarTime (sourceAngularCollar 0 θ)=0 |
| exact sourceAngular_time (by norm_num) θ |
| · change sourceCollarTime ((sourceChildHomeomorph (actualChildSign k)).symm |
| ((sourceChildHomeomorph (actualChildSign k)) (sourceAngularCollar 0 θ)))=0 |
| rw [Homeomorph.symm_apply_apply,sourceAngular_time (by norm_num)] |
|
|
| lemma terminalProfile_boundary {η : ℝ} (hη : 0<η) (hηc : 2*η<centralThickness) |
| (j : Fin 2) (i : Fin 3) (θ : UnitAddTorus (Fin 2)) : |
| z.terminalProfile η j (physicalOuterBoundary i θ)=z.terminalValue i j-z.terminalValue 0 j := by |
| refine Fin.cases ?_ (fun k => ?_) i |
| · rw [z.terminalProfile_zero_outside hη hηc j (terminalTime_outer 0 θ).le,sub_self] |
| · rw [z.terminalProfile_child_band hη hηc j k (by rw [terminalTime_outer]; linarith),terminalTime_outer] |
| simp |
|
|
| lemma terminalProfile_trace {η : ℝ} (hη : 0<η) (hηc : 2*η<centralThickness) |
| (j : Fin 2) (f : H1) |
| (hf : weakValue f=ᵐ[ballMeasure] fun x => z.terminalProfile η j (WithLp.ofLp x)) (i : Fin 3) : |
| physicalRealTraceCLM i f=ᵐ[volume] fun _ => z.terminalValue i j-z.terminalValue 0 j := by |
| let E : R3 ≃L[ℝ] Coord3 := PiLp.continuousLinearEquiv 2 ℝ (fun _ : Fin 3 => ℝ) |
| have hh (x : R3) : LocalLipAt (z.terminalProfile η j ∘ E) x := |
| (z.terminalProfile_localLip η j (E x)).comp |
| (E.toContinuousLinearMap.lipschitzWith.locallyLipschitz x) |
| have hz (x : R3) (hx : 5/2<‖x‖) : (z.terminalProfile η j ∘ E) x=0 := by |
| apply z.terminalProfile_zero_outside hη hηc j |
| by_contra hn |
| have hd := (sourceCollarTime_pos_iff_domain (E x)).mp (lt_of_not_ge hn) |
| have hb := sourceClosure_subset_ball (subset_closure hd) |
| change x∈Metric.ball 0 (5/2) at hb |
| simp only [Metric.mem_ball,dist_zero_right] at hb |
| linarith |
| obtain ⟨K,hK⟩ := compact_locallyLipschitz hh (bounded_support_compact hz) |
| have ht := physicalRealTrace_compact_lipschitz hK (by norm_num : (5/2:ℝ)<3) hz f hf i |
| exact ht.mono (fun θ h => h.trans (z.terminalProfile_boundary hη hηc j i θ)) |
|
|
| end PhysicalFiniteEndingData |
| end ScalarConductivity |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| namespace ScalarConductivity |
| open Set MeasureTheory Filter Topology Matrix |
| attribute [local instance] Classical.propDecidable |
|
|
| lemma patch_energy_eq (a : R3 → ℝ) (σ : Coord3 → ℝ) (q W h : H1) |
| (ha : ∀ᵐ y∂volume.restrict physicalOpenBlock,a (WithLp.toLp 2 y)=σ y) |
| (hW : ∀ᵐ x∂ballMeasure,weakGradient W x= |
| if WithLp.ofLp x∈physicalOpenBlock then weakGradient q x else 0) : |
| (∫ x,a x*inner ℝ (weakGradient W x) (weakGradient h x) ∂ballMeasure)= |
| ∫ y in physicalOpenBlock,σ y*(originalPiGradient q y ⬝ᵥ originalPiGradient h y) := by |
| let E : R3 ≃L[ℝ] Coord3 := PiLp.continuousLinearEquiv 2 ℝ (fun _ : Fin 3 => ℝ) |
| let U := E ⁻¹' physicalOpenBlock |
| let f : R3 → ℝ := fun x => a x*inner ℝ (weakGradient q x) (weakGradient h x) |
| have hU : MeasurableSet U := physicalOpenBlock_open.measurableSet.preimage E.continuous.measurable |
| have hUb : U⊆ball := fun _ hx => physicalOpenBlock_subset_ball hx |
| have hi : (∫ x,a x*inner ℝ (weakGradient W x) (weakGradient h x) ∂ballMeasure)= |
| ∫ x,f x ∂volume.restrict U := by |
| calc |
| _ = ∫ x,U.indicator f x ∂ballMeasure := integral_congr_ae (by |
| filter_upwards [hW] with x hx |
| rw [hx] |
| by_cases hb : WithLp.ofLp x∈physicalOpenBlock |
| · rw [ite_eq_left hb,Set.indicator_of_mem (show x∈U from hb)] |
| · rw [ite_eq_right hb,Set.indicator_of_notMem (show x∉U from hb),inner_zero_left,mul_zero]) |
| _ = ∫ x in U,f x ∂ballMeasure := integral_indicator hU |
| _ = _ := by rw [ballMeasure,Measure.restrict_restrict hU,inter_eq_left.mpr hUb] |
| rw [hi] |
| have ht := (PiLp.volume_preserving_ofLp (Fin 3)).setIntegral_preimage_emb |
| E.toHomeomorph.measurableEmbedding (fun y => f (WithLp.toLp 2 y)) physicalOpenBlock |
| change (∫ x in U,f x)=_ at ht |
| rw [ht] |
| apply integral_congr_ae |
| filter_upwards [ha] with y hy |
| change a (WithLp.toLp 2 y)*inner ℝ (weakGradient q (WithLp.toLp 2 y)) |
| (weakGradient h (WithLp.toLp 2 y))=_ |
| rw [hy] |
| congr 1 |
| simp only [EuclideanSpace.inner_eq_star_dotProduct,star_trivial] |
| exact dotProduct_comm _ _ |
|
|
| namespace PhysicalFiniteEndingData |
| variable {s : Fin 3 → ℝ} (z : PhysicalFiniteEndingData s) |
|
|
| theorem source_patch_green {η : ℝ} (hη : 0<η) (hηc : 2*η<centralThickness) |
| (a : R3 → ℝ) (σ : Coord3 → ℝ) (q W : H1) (j : Fin 2) |
| (ha : ∀ᵐ y∂volume.restrict physicalOpenBlock,a (WithLp.toLp 2 y)=σ y) |
| (hv : ∀ᵐ x∂ballMeasure,weakValue W x= |
| if WithLp.ofLp x∈physicalOpenBlock then weakValue q x-z.terminalValue 0 j |
| else z.terminalProfile η j (WithLp.ofLp x)) |
| (hg : ∀ h : H1,(∫ y in physicalOpenBlock,σ y*(originalPiGradient q y ⬝ᵥ |
| originalPiGradient h y))=constantTerminalFluxCLM (centralBasisSlopes j) h) : |
| ∀ h : H1,(∫ x,a x*inner ℝ (weakGradient W x) (weakGradient h x) ∂ballMeasure)= |
| constantTerminalFluxCLM (centralBasisSlopes j) h := by |
| intro h |
| exact (patch_energy_eq a σ q W h ha (z.source_patch_gradient hη hηc q W j hv)).trans (hg h) |
|
|
| end PhysicalFiniteEndingData |
| end ScalarConductivity |
|
|
| end |
| end |
|
|
| end OAI |
|
|
