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 end OAI