| import Mathlib |
| import OAI.Analysis.CoulombIonization.FieldAnalysis.PuncturedGreenBarrier |
| import OAI.Analysis.CoulombIonization.Variational.InnerPotential |
|
|
| namespace OAI |
|
|
| open MeasureTheory Filter Set Metric Laplacian |
| noncomputable section |
| namespace CoulombAtom |
| open CoulombAnalysis |
|
|
| def packetPotentialConstant : ℝ := 1+9*packetDensityConstant*unitCoulombIntegral |
|
|
| lemma packetPotentialConstant_one_le : 1 ≤ packetPotentialConstant := by |
| unfold packetPotentialConstant |
| nlinarith only [mul_nonneg packetDensityConstant_pos.le unitCoulombIntegral_nonneg] |
|
|
| lemma packet_potential_uniform (y a : Space) {r : ℝ} (hr : 0 < r) : |
| tfPotential (packetDensity y r) a ≤ packetPotentialConstant/r := by |
| by_cases ha : 2*r ≤ ‖a-y‖ |
| · change (∫ z, packetDensity y r z/‖a-z‖) ≤ _ |
| rw [packetDensity_potential y hr (by linarith)] |
| exact (div_le_div_of_nonneg_left (by norm_num) hr (by linarith)).trans |
| (div_le_div_of_nonneg_right packetPotentialConstant_one_le hr.le) |
| · have ha' : ‖a-y‖ < 2*r := lt_of_not_ge ha |
| have hk : Integrable (fun z : Space => truncatedCoulomb (3*r) (a-z)) := |
| (integrable_comp_sub_left _ a).mpr (truncatedCoulomb_integrable _) |
| calc |
| _ ≤ ∫ z : Space, (packetDensityConstant/r^3)*truncatedCoulomb (3*r) (a-z) := by |
| apply integral_mono (tfPotential_integrable (packetDensity_integrable y hr) |
| (packetDensity_memLp y hr) a) (hk.const_mul _) |
| intro z |
| change packetDensity y r z / ‖a-z‖ ≤ (packetDensityConstant/r^3)*truncatedCoulomb (3*r) (a-z) |
| by_cases hz : packetDensity y r z = 0 |
| · simp only [hz,zero_div] |
| exact mul_nonneg (by positivity [packetDensityConstant_pos]) (truncatedCoulomb_nonneg _ _) |
| · have hs := packetDensity_support y hr hz |
| have hd : ‖a-z‖ < 3*r := by |
| have ht := norm_sub_le a y |
| have ht' := norm_sub_le (a-y) (z-y) |
| have he : (a-y)-(z-y) = a-z := by abel |
| rw [he] at ht' |
| linarith only [ht',hs,ha'] |
| rw [truncatedCoulomb,indicator_of_mem (mem_ball_zero_iff.mpr hd),mul_one_div] |
| exact div_le_div_of_nonneg_right (packetDensity_bound y z hr) (norm_nonneg _) |
| _ = 9*packetDensityConstant*unitCoulombIntegral/r := by |
| rw [integral_const_mul,integral_sub_left_eq_self,truncatedCoulomb_integral (by positivity)] |
| field_simp |
| ring |
| _ ≤ _ := by |
| apply div_le_div_of_nonneg_right _ hr.le |
| unfold packetPotentialConstant |
| linarith |
|
|
| lemma packet_source_pair_integrable (mu : Measure Space) [IsFiniteMeasure mu] |
| (y : Space) {r : ℝ} (hr : 0 < r) : |
| Integrable (fun p : Space × Space => packetDensity y r p.2/‖p.1-p.2‖) |
| (mu.prod volume) := by |
| have hm : Measurable (fun p : Space × Space => packetDensity y r p.2/‖p.1-p.2‖) := |
| ((packetDensity_continuous y r).measurable.comp measurable_snd).div |
| (measurable_fst.sub measurable_snd).norm |
| apply (integrable_prod_iff hm.aestronglyMeasurable).mpr |
| refine ⟨ae_of_all _ (fun a => tfPotential_integrable (packetDensity_integrable y hr) |
| (packetDensity_memLp y hr) a),?_⟩ |
| have hnorm (a : Space) : (∫ z, ‖packetDensity y r z/‖a-z‖‖) = |
| tfPotential (packetDensity y r) a := by |
| simp only [Real.norm_of_nonneg (div_nonneg (packetDensity_nonneg y r _) (norm_nonneg _)),tfPotential] |
| simp only [hnorm] |
| apply (integrable_const (packetPotentialConstant/r)).mono' |
| (hm.stronglyMeasurable.integral_prod_right.aestronglyMeasurable) |
| apply ae_of_all |
| intro a |
| change ‖(∫ z, packetDensity y r z/‖a-z‖)‖ ≤ _ |
| rw [Real.norm_of_nonneg (integral_nonneg (fun z => div_nonneg (packetDensity_nonneg y r z) (norm_nonneg _)))] |
| exact packet_potential_uniform y a hr |
|
|
| lemma packet_source_pairing (mu : Measure Space) [IsFiniteMeasure mu] |
| (y : Space) {r : ℝ} (hr : 0 < r) : |
| (∫ z, packetDensity y r z*sourcePotential mu z) = |
| ∫ a, tfPotential (packetDensity y r) a ∂mu := by |
| have hi := packet_source_pair_integrable mu y hr |
| simp only [sourcePotential,← integral_const_mul,tfPotential] |
| calc |
| (∫ z, ∫ a, packetDensity y r z*‖z-a‖⁻¹ ∂mu) = |
| ∫ z, ∫ a, packetDensity y r z/‖a-z‖ ∂mu := by |
| simp only [div_eq_mul_inv,norm_sub_rev] |
| _ = _ := (integral_integral_swap hi).symm |
|
|
| lemma packet_source_integrable (mu : Measure Space) [IsFiniteMeasure mu] |
| (y : Space) {r : ℝ} (hr : 0 < r) : |
| Integrable (fun z => packetDensity y r z*sourcePotential mu z) := by |
| have hi := (packet_source_pair_integrable mu y hr).integral_prod_right |
| simpa only [sourcePotential,← integral_const_mul,div_eq_mul_inv,norm_sub_rev] using hi |
|
|
| lemma packet_outer_potential (y a : Space) {h : ℝ} (hh : 0 < h) |
| (hy : ‖y‖ ≤ 2*h) (ha : h < ‖a‖) : |
| tfPotential (packetDensity y (h/4)) a ≤ 16*packetPotentialConstant/‖a‖ := by |
| have hap : 0 < ‖a‖ := hh.trans ha |
| by_cases hfar : 4*h ≤ ‖a‖ |
| · have hd : ‖a‖/2 ≤ ‖a-y‖ := by |
| have ht := norm_le_norm_sub_add a y |
| linarith |
| change (∫ z, packetDensity y (h/4) z/‖a-z‖) ≤ _ |
| rw [packetDensity_potential y (by positivity) (by linarith)] |
| calc |
| _ ≤ 1/(‖a‖/2) := div_le_div_of_nonneg_left (by norm_num) (by positivity) hd |
| _ = 2/‖a‖ := by ring |
| _ ≤ _ := div_le_div_of_nonneg_right (by nlinarith [packetPotentialConstant_one_le]) hap.le |
| · have hnear : ‖a‖ < 4*h := lt_of_not_ge hfar |
| calc |
| _ ≤ packetPotentialConstant/(h/4) := packet_potential_uniform y a (by positivity) |
| _ ≤ 16*packetPotentialConstant/‖a‖ := by |
| apply (div_le_div_iff₀ (by positivity : 0 < h/4) hap).mpr |
| have hc := packetPotentialConstant_one_le |
| nlinarith only [mul_le_mul_of_nonneg_left hnear.le (zero_le_one.trans hc)] |
|
|
| theorem packet_outer_source_bound (mu : Measure Space) [IsFiniteMeasure mu] |
| {h : ℝ} (hh : 0 < h) (hs : ∀ᵐ a ∂mu, h < ‖a‖) |
| {y : Space} (hy : ‖y‖ ≤ 2*h) : |
| (∫ z, packetDensity y (h/4) z*sourcePotential mu z) ≤ |
| 16*packetPotentialConstant*(∫ a, ‖a‖⁻¹ ∂mu) := by |
| have hi : Integrable (fun a : Space => ‖a‖⁻¹) mu := by |
| apply (integrable_const (h⁻¹)).mono' (by fun_prop) |
| filter_upwards [hs] with a ha |
| rw [Real.norm_of_nonneg (inv_nonneg.mpr (norm_nonneg _))] |
| exact inv_anti₀ hh ha.le |
| rw [packet_source_pairing mu y (by positivity),←integral_const_mul] |
| apply integral_mono_ae (packet_source_pair_integrable mu y (by positivity)).integral_prod_left |
| (hi.const_mul _) |
| filter_upwards [hs] with a ha |
| simpa only [tfPotential,div_eq_mul_inv] using packet_outer_potential y a hh hy ha |
|
|
| end CoulombAtom |
|
|
| end |
|
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| end OAI |
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