| import Mathlib |
| import OAI.Analysis.Conductivity.Branching.AttachedEndL2 |
| import OAI.Analysis.Conductivity.Flux.AttachedEndDifferential |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace ScalarConductivity |
| open Set MeasureTheory Filter Topology UnitAddTorus Matrix |
| open scoped ENNReal |
|
|
| lemma sourceAngularCollarMeasure_self (b : ℝ) : sourceAngularCollarMeasure b b=0 := by |
| simp [sourceAngularCollarMeasure] |
|
|
| lemma sourceCollarTime_level_null {b : ℝ} (hb : b∈Icc (-(1:ℝ)/100) (1/100)) : |
| volume {y : Fin 3 → ℝ | sourceCollarTime y=b}=0 := by |
| have hle := (sourceClosedCollarBand_measure_le hb.1 hb.2).trans |
| (sourcePhysicalCollarMeasure_le (le_refl b) hb.1 hb.2) |
| rw [sourceAngularCollarMeasure_self,smul_zero] at hle |
| have he := Measure.le_iff.mp hle univ MeasurableSet.univ |
| have hs : sourceClosedCollarBand b b={y : Fin 3 → ℝ | sourceCollarTime y=b} := by |
| ext y; simp [sourceClosedCollarBand] |
| simp only [Measure.restrict_apply_univ,Measure.coe_zero,Pi.zero_apply,hs] at he |
| exact le_antisymm he bot_le |
|
|
| lemma sourceColevel_ae_ne {b : ℝ} (hb : b∈Icc (-(1:ℝ)/100) (1/100)) : |
| ∀ᵐ y : Fin 3 → ℝ,sourceCollarTime y≠b := by |
| rw [ae_iff] |
| simpa only [not_not] using sourceCollarTime_level_null hb |
|
|
| lemma attachedEndPoisson_grad_bound (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) |
| (f : spectralTraceGraph (torusRate s)) (a b : ℝ) : |
| ∃ C : ℝ,0<C ∧ ∀ k : Fin 3,∀ᵐ y : Fin 3 → ℝ, |
| y∈sourceClosedCollarBand (-(1:ℝ)/100) (1/100) → 0<a*(sourceCollarTime y-b) → |
| |fderiv ℝ (fun y => (attachedEndPoissonField s f a b 0 y).re) y (Pi.single k 1)|≤ |
| C*∑ d : Fin 3,‖attachedEndPoissonField s f a b d.succ y‖ := by |
| obtain ⟨C,hC,hbound⟩ := sourceCartesianGradient_uniform_bound |
| refine ⟨C*(|a|+1),mul_pos hC (by positivity),?_⟩ |
| intro k |
| filter_upwards [sourceCollar_open_charts_ae] with y hy hb ht |
| obtain ⟨i,j,x,hx,rfl⟩ := hy hb |
| rw [sourceCollarPiece_time_open hx] at ht |
| rw [attachedEndPoisson_real_fderiv s hs f a b i j hx ht,dotProduct_single,mul_one] |
| rw [Matrix.mulVec] |
| apply (Finset.abs_sum_le_sum_abs _ _).trans |
| rw [Finset.mul_sum] |
| apply Finset.sum_le_sum |
| intro d _ |
| rw [abs_mul,endAxialMatrix,Matrix.mulVec_diagonal,abs_mul] |
| have hd : |![a,1,1] d|≤|a|+1 := by |
| fin_cases d |
| · change |a|≤|a|+1; linarith |
| · change |(1:ℝ)|≤|a|+1; rw [abs_one]; linarith [abs_nonneg a] |
| · change |(1:ℝ)|≤|a|+1; rw [abs_one]; linarith [abs_nonneg a] |
| have he := Complex.abs_re_le_norm (attachedEndPoissonField s f a b d.succ (sourceCollarPiece i j x)) |
| calc |
| _ ≤ C*((|a|+1)*‖attachedEndPoissonField s f a b d.succ (sourceCollarPiece i j x)‖) := |
| mul_le_mul (hbound i j x (sourceCollarOpenBox_subset hx) k d) |
| (mul_le_mul hd he (abs_nonneg _) (by positivity)) (by positivity) hC.le |
| _ = _ := by ring |
|
|
| lemma attachedEndPoisson_grad_memLp (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) |
| (f : spectralTraceGraph (torusRate s)) {a b l r R : ℝ} |
| (ha : a≠0) (hR : 0≤R) (hb : b∈Icc (-(1:ℝ)/100) (1/100)) |
| (hlr : l≤r) (hl : -(1:ℝ)/100≤l) (hr : r≤1/100) |
| (hT : ∀ t∈Icc l r,affineEndTime a b t∈Icc 0 R) (k : Fin 3) : |
| MemLp (fun y => fderiv ℝ (fun y => (attachedEndPoissonField s f a b 0 y).re) y (Pi.single k 1)) |
| 2 (volume.restrict (sourceClosedCollarBand l r)) := by |
| obtain ⟨C,hC,hbound⟩ := attachedEndPoisson_grad_bound s hs f a b |
| have hjet (d : Fin 3) := (attachedEndPoissonField_memLp s hs f ha hR hlr hl hr hT d.succ).norm |
| have hsum : MemLp (fun y => C*∑ d : Fin 3,‖attachedEndPoissonField s f a b d.succ y‖) 2 |
| (volume.restrict (sourceClosedCollarBand l r)) := |
| (memLp_finsetSum Finset.univ (fun d _ => hjet d)).const_mul C |
| apply hsum.mono' (measurable_fderiv_apply_const ℝ _ _).aestronglyMeasurable |
| have hband : MeasurableSet (sourceClosedCollarBand l r) := |
| measurableSet_Icc.preimage measurable_sourcePhysicalCoordinates.fst |
| filter_upwards [ae_restrict_mem hband,ae_restrict_of_ae (hbound k), |
| ae_restrict_of_ae (sourceColevel_ae_ne hb)] with y hy hbd hne |
| rw [Real.norm_eq_abs] |
| exact hbd ⟨hl.trans hy.1,hy.2.trans hr⟩ |
| (lt_of_le_of_ne (hT _ hy).1 (Ne.symm (mul_ne_zero ha (sub_ne_zero.mpr hne)))) |
| |
| end ScalarConductivity |
| |
| end |
| |
| end OAI |
| |
