| import Mathlib |
| import OAI.Analysis.Conductivity.Scalarization.PhysicalScalarization |
| import OAI.Analysis.Conductivity.Sources.PhysicalGradient |
| import OAI.Analysis.Conductivity.Sources.PhysicalRoot |
| import OAI.Analysis.Conductivity.Variational.PhysicalTraceMean |
|
|
| 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) |
|
|
| lemma le_of_transition_energy_zero (a : R3 → ℝ) (ha : AEStronglyMeasurable a ballMeasure) |
| (C : ℝ) (hC : 0≤C) (hb : ∀ᵐ x∂ballMeasure,|a x|≤C) |
| (hp : ∀ᵐ x∂ballMeasure,0<a x) {u : H1} (hu : u∈H10) {M : ℝ} (hM : 0≤M) |
| (he : (∫ x,a x*inner ℝ (weakGradient u x) |
| (weakGradient ((upperTransition M hM).onH1 u) x) ∂ballMeasure)=0) : |
| ∀ᵐ x∂ballMeasure,weakValue u x≤M := by |
| let F := upperTransition M hM |
| have hh0 : F.onH1 u∈H10 := F.preserves_H10 hu |
| have hg := F.onH1_gradient u |
| have hnonneg : 0 ≤ᵐ[ballMeasure] fun x => a x*inner ℝ (weakGradient u x) (weakGradient (F.onH1 u) x) := by |
| filter_upwards [hp,hg] with x ha hx |
| rw [hx,real_inner_smul_right] |
| exact mul_nonneg ha.le (mul_nonneg (upperTransition_deriv_nonneg M hM _) real_inner_self_nonneg) |
| have hzero := (integral_eq_zero_iff_of_nonneg_ae hnonneg |
| (energy_integrable a ha C hC hb u (F.onH1 u))).mp he |
| have hhg : weakGradient (F.onH1 u) =ᵐ[ballMeasure] 0 := by |
| filter_upwards [hp,hg,hzero] with x ha hx hz |
| rw [hx,real_inner_smul_right] at hz |
| have hh : deriv F (weakValue u x)*inner ℝ (weakGradient u x) (weakGradient u x)=0 := |
| (mul_eq_zero.mp hz).resolve_left ha.ne' |
| rcases mul_eq_zero.mp hh with hd | hd |
| · rw [hx,hd,zero_smul]; rfl |
| · rw [hx,inner_self_eq_zero.mp hd,smul_zero]; rfl |
| have hF0 := zero_of_weakGradient_zero hh0 hhg |
| have hval := F.onH1_value u |
| filter_upwards [hval,Lp.coeFn_zero JetFiber 2 ballMeasure] with x hx hz |
| apply (upperTransition_zero_iff M hM _).mp |
| rw [←hx,hF0] |
| change jetValue ((0:JetSpace) x)=0 |
| rw [hz]; rfl |
|
|
| theorem physical_source_le (a : R3 → ℝ) (ha : AEStronglyMeasurable a ballMeasure) |
| (C : ℝ) (hC : 0≤C) (hb : ∀ᵐ x∂ballMeasure,|a x|≤C) |
| (hp : ∀ᵐ x∂ballMeasure,0<a x) {u : H1} (hu : u∈H10) (κ : Fin 3 → ℝ) |
| (hg : ∀ h : H1,(∫ x,a x*inner ℝ (weakGradient u x) (weakGradient h x) ∂ballMeasure)= |
| constantTerminalFluxCLM κ h) {M : ℝ} (hM : 0≤M) |
| (ht : ∀ i : Fin 3,∀ᵐ θ,physicalRealTraceCLM i u θ≤M) : |
| ∀ᵐ x∂ballMeasure,weakValue u x≤M := by |
| apply le_of_transition_energy_zero a ha C hC hb hp hu hM |
| rw [hg,constantTerminalFlux_zero_of_realTrace κ _ (physicalRealTrace_transition_zero u hM ht)] |
|
|
| theorem physical_source_bounded (a : R3 → ℝ) (ha : AEStronglyMeasurable a ballMeasure) |
| (C : ℝ) (hC : 0≤C) (hb : ∀ᵐ x∂ballMeasure,|a x|≤C) |
| (hp : ∀ᵐ x∂ballMeasure,0<a x) {u : H1} (hu : u∈H10) (κ : Fin 3 → ℝ) |
| (hg : ∀ h : H1,(∫ x,a x*inner ℝ (weakGradient u x) (weakGradient h x) ∂ballMeasure)= |
| constantTerminalFluxCLM κ h) {M : ℝ} (hM : 0≤M) |
| (ht : ∀ i : Fin 3,∀ᵐ θ,|physicalRealTraceCLM i u θ|≤M) : |
| ∀ᵐ x∂ballMeasure,|weakValue u x|≤M := by |
| have hup := physical_source_le a ha C hC hb hp hu κ hg hM |
| (fun i => (ht i).mono (fun _ h => (abs_le.mp h).2)) |
| have hneg : (∫ x,a x*inner ℝ (weakGradient (-u) x) |
| (weakGradient ((upperTransition M hM).onH1 (-u)) x) ∂ballMeasure)=0 := by |
| change energy a (-u) ((upperTransition M hM).onH1 (-u))=0 |
| rw [←energyForm_apply a ha C hC hb,map_neg,neg_apply,energyForm_apply a ha C hC hb] |
| change -(∫ x,a x*inner ℝ (weakGradient u x) (weakGradient ((upperTransition M hM).onH1 (-u)) x) ∂ballMeasure)=0 |
| rw [hg] |
| have htneg : ∀ i : Fin 3,∀ᵐ θ,physicalRealTraceCLM i (-u) θ≤M := by |
| intro i |
| rw [map_neg] |
| filter_upwards [ht i,Lp.coeFn_neg (physicalRealTraceCLM i u)] with θ ht hn |
| rw [hn] |
| exact (neg_le_iff_add_nonneg).mpr (by linarith [(abs_le.mp ht).1]) |
| rw [constantTerminalFlux_zero_of_realTrace κ _ (physicalRealTrace_transition_zero (-u) hM htneg),neg_zero] |
| have hlo := le_of_transition_energy_zero a ha C hC hb hp (H10.neg_mem hu) hM hneg |
| filter_upwards [hup,hlo,weakValue_neg u] with x hx hy hz |
| rw [hz,Pi.neg_apply] at hy |
| exact abs_le.mpr ⟨by linarith,hx⟩ |
|
|
| end ScalarConductivity |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| namespace ScalarConductivity |
| open Set MeasureTheory Filter Topology Matrix UnitAddTorus |
| attribute [local instance] Classical.propDecidable |
| local instance : MeasureSpace UnitAddCircle := ⟨AddCircle.haarAddCircle⟩ |
| local instance : IsProbabilityMeasure (volume : Measure UnitAddCircle) := |
| inferInstanceAs (IsProbabilityMeasure AddCircle.haarAddCircle) |
|
|
| structure BoundedRootSources where |
| b : Coord3 → ℝ |
| measurable_b : Measurable b |
| c : ℝ |
| C : ℝ |
| K : ℝ |
| c_pos : 0<c |
| c_lt_C : c<C |
| K_pos : 0<K |
| bounds : ∀ y,b y∈Icc c C |
| W : Fin 2 → H1 |
| W_H10 : ∀ j,W j∈H10 |
| W_bound : ∀ j,∀ᵐ x∂ballMeasure,|weakValue (W j) x|≤K |
| W_energy : ∀ j,‖weakGradientL (W j)‖^2≤K |
| W_support : ∀ j,∀ᵐ x∂ballMeasure,x∉closure sourceDomain → |
| weakValue (W j) x=0 ∧ weakGradient (W j) x=0 |
| W_gradient : ∀ j,∀ᵐ x∂ballMeasure,WithLp.ofLp x∉physicalOpenBlock → weakGradient (W j) x=0 |
| green : ∀ (a : R3 → ℝ),(∀ᵐ y∂volume.restrict physicalOpenBlock,a (WithLp.toLp 2 y)=b y) → |
| ∀ j (h : H1),(∫ x,a x*inner ℝ (weakGradient (W j) x) (weakGradient h x) ∂ballMeasure)= |
| constantTerminalFluxCLM (centralBasisSlopes j) h |
|
|
| theorem boundedRootSources_nonempty : Nonempty BoundedRootSources := by |
| obtain ⟨s,hs,hres⟩ := exists_nonresonant_angular_tensor |
| let z : PhysicalFiniteEndingData s := Classical.choice (physicalFiniteEndingData_nonempty hs hres) |
| obtain ⟨σ,c,C,w,q,hσ,hσLp,hc,hcC,hσb,hq0,hwc,hwe,⟨Z,hZ,hq⟩,hgreen⟩ := z.scalar_pair_green_exists hs |
| obtain ⟨η,hη,hηc,hT⟩ := z.terminal_band_exists |
| let t := η/4 |
| have ht : 0<t := by dsimp [t]; positivity |
| have htc : 2*t<centralThickness := by dsimp [t]; linarith |
| have hT' : ∀ i : Fin 3,∀ r : ℝ,|r|≤t → (z.ending i).R<z.compression i*(r-PhysicalFiniteEndingData.terminalBase i) := by |
| intro i r hr |
| apply hT i r |
| dsimp [t] at hr |
| linarith |
| choose W hW hWv hWt using z.source_patch_trace_exists hs ht htc hT' w q hwe |
| (Z-originalPairVoltageJet physicalOpenBlock w) hZ hq |
| let b : Coord3 → ℝ := fun y => max c (min C (σ y)) |
| have hbm : Measurable b := measurable_const.max (measurable_const.min hσ) |
| have hbb (y : Coord3) : b y∈Icc c C := ⟨le_max_left _ _,max_le hcC.le (min_le_left _ _)⟩ |
| have hbe : b=ᵐ[volume.restrict physicalOpenBlock] σ := by |
| filter_upwards [hσb] with y hy |
| exact max_eq_right (by simpa only [min_eq_right hy.2] using hy.1) |>.trans (min_eq_right hy.2) |
| let a : R3 → ℝ := fun x => b (WithLp.ofLp x) |
| have ham : AEStronglyMeasurable a ballMeasure := |
| (hbm.comp (PiLp.continuous_ofLp 2 (fun _ : Fin 3 => ℝ)).measurable).aestronglyMeasurable |
| have hC : 0≤C := (hc.trans hcC).le |
| have hab : ∀ᵐ x∂ballMeasure,|a x|≤C := Eventually.of_forall (fun x => by |
| rw [abs_of_pos (hc.trans_le (hbb _).1)]; exact (hbb _).2) |
| have hap : ∀ᵐ x∂ballMeasure,0<a x := Eventually.of_forall (fun x => hc.trans_le (hbb _).1) |
| have hae : ∀ᵐ y∂volume.restrict physicalOpenBlock,a (WithLp.toLp 2 y)=σ y := hbe |
| let M : ℝ := ∑ j : Fin 2,∑ i : Fin 3,|z.terminalValue i j-z.terminalValue 0 j| |
| have hM : 0≤M := Finset.sum_nonneg (fun j _ => Finset.sum_nonneg (fun i _ => abs_nonneg _)) |
| have hMi (i : Fin 3) (j : Fin 2) : |z.terminalValue i j-z.terminalValue 0 j|≤M := by |
| have h1 : |z.terminalValue i j-z.terminalValue 0 j|≤∑ i : Fin 3,|z.terminalValue i j-z.terminalValue 0 j| := |
| Finset.single_le_sum (f:=fun i : Fin 3 => |z.terminalValue i j-z.terminalValue 0 j|) (fun k _ => abs_nonneg _) (Finset.mem_univ i) |
| have h2 : (∑ i : Fin 3,|z.terminalValue i j-z.terminalValue 0 j|)≤M := |
| Finset.single_le_sum (f:=fun j : Fin 2 => ∑ i : Fin 3,|z.terminalValue i j-z.terminalValue 0 j|) (fun k _ => Finset.sum_nonneg (fun i _ => abs_nonneg _)) (Finset.mem_univ j) |
| exact h1.trans h2 |
| have hWb (j : Fin 2) : ∀ᵐ x∂ballMeasure,|weakValue (W j) x|≤M := |
| physical_source_bounded a ham C hC hab hap (hW j) (centralBasisSlopes j) |
| (z.source_patch_green ht htc a σ (q j) (W j) j hae (hWv j) (hgreen j)) hM |
| (fun i => (hWt j i).mono (fun θ hθ => by rw [hθ]; exact hMi i j)) |
| let K := M+(∑ j : Fin 2,‖weakGradientL (W j)‖^2)+1 |
| have he0 : 0≤∑ j : Fin 2,‖weakGradientL (W j)‖^2 := Finset.sum_nonneg (fun _ _ => sq_nonneg _) |
| have hMK : M≤K := by dsimp [K]; linarith |
| have hWK (j : Fin 2) : ‖weakGradientL (W j)‖^2≤K := by |
| have hh := Finset.single_le_sum (s:=Finset.univ) (fun k _ => sq_nonneg ‖weakGradientL (W k)‖) (Finset.mem_univ j) |
| dsimp [K]; linarith |
| have hWg (j : Fin 2) := z.source_patch_gradient ht htc (q j) (W j) j (hWv j) |
| refine ⟨⟨b,hbm,c,C,K,hc,hcC,(by dsimp [K]; linarith),hbb,W,hW, |
| fun j => (hWb j).mono (fun _ h => h.trans hMK),hWK,?_,?_,?_⟩⟩ |
| · intro j |
| filter_upwards [hWv j,hWg j] with x hv hg hx |
| have ht0 : sourceCollarTime (WithLp.ofLp x)<0 := by |
| apply lt_of_not_ge |
| intro hn |
| apply hx |
| exact (sourceDomain_closure_time (WithLp.ofLp x)).mpr hn |
| have hn : WithLp.ofLp x∉physicalOpenBlock := by |
| intro hn |
| have hh := (sourceOpenBlock_time_iff _).mp hn |
| exact (not_lt.mpr ht0.le) hh.1 |
| rw [ite_eq_right hn] at hv hg |
| refine ⟨?_,hg⟩ |
| rw [hv] |
| exact z.terminalProfile_zero_outside ht htc j ht0.le |
| · intro j |
| filter_upwards [hWg j] with x hx hn |
| simpa only [ite_eq_right hn] using hx |
| · intro a ha j h |
| exact z.source_patch_green ht htc a σ (q j) (W j) j (Filter.EventuallyEq.trans ha hbe) (hWv j) (hgreen j) h |
|
|
| end ScalarConductivity |
|
|
| end |
| end |
|
|
| end OAI |
|
|
