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159357f | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 | 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
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