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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 | import Mathlib
import OAI.Analysis.CoulombIonization.Variational.LocalWidthKernel
namespace OAI
noncomputable section
open MeasureTheory Filter
open scoped Topology BigOperators ContDiff
open MeasureTheory Filter Set Metric
open scoped ENNReal ContDiff
namespace CoulombAtom
open CoulombAnalysis
lemma scaled_packet_density_radial (b : ℝ) {g : Space → ℝ} (hr : IsRadial g) :
IsRadial (fun x => (scaledRealPacket b g x)^2) := by
intro x y hxy
dsimp only
rw [scaledRealPacket_radial b hr x y hxy]
lemma scaled_packet_density_support {b : ℝ} (hb : 0 < b) {g : Space → ℝ}
(hs : tsupport g ⊆ ball 0 1) (x : Space) (hx : (scaledRealPacket b g x)^2 ≠ 0) :
‖x‖ ≤ b := (scaledRealPacket_support hb hs (by intro hz; exact hx (by rw [hz]; norm_num))).le
lemma scaled_packet_potential_le {b : ℝ} (hb : 0 < b)
{g : Space → ℝ} (hg : ContDiff ℝ ∞ g) (hcg : HasCompactSupport g)
(hgn : ∫ x, (g x)^2 = 1) (hr : IsRadial g) (hs : tsupport g ⊆ ball 0 1)
{x : Space} (hx : x ≠ 0) :
tfPotential (fun z => (scaledRealPacket b g z)^2) x ≤ 1/‖x‖ := by
have hc : Continuous (fun z => (scaledRealPacket b g z)^2) := by
exact (scaledRealPacket_smooth b hg).continuous.pow 2
have hcomp : HasCompactSupport (fun z => (scaledRealPacket b g z)^2) := by
convert! (scaledRealPacket_compact hb hcg).mul_right (f' := scaledRealPacket b g) using 1
funext z
exact pow_two _
have hh := radial_potential_le_point (hc.integrable_of_hasCompactSupport hcomp)
(hc.memLp_of_hasCompactSupport hcomp) (fun z => sq_nonneg _)
(scaled_packet_density_radial b hr) hb.le (scaled_packet_density_support hb hs) hx
simpa only [scaledRealPacket_mass hb,hgn] using hh
lemma scaled_packet_potential_eq {b : ℝ} (hb : 0 < b)
{g : Space → ℝ} (hg : ContDiff ℝ ∞ g) (hcg : HasCompactSupport g)
(hgn : ∫ x, (g x)^2 = 1) (hr : IsRadial g) (hs : tsupport g ⊆ ball 0 1)
{x : Space} (hx : b ≤ ‖x‖) :
tfPotential (fun z => (scaledRealPacket b g z)^2) x = 1/‖x‖ := by
have hc : Continuous (fun z => (scaledRealPacket b g z)^2) := by
exact (scaledRealPacket_smooth b hg).continuous.pow 2
have hcomp : HasCompactSupport (fun z => (scaledRealPacket b g z)^2) := by
convert! (scaledRealPacket_compact hb hcg).mul_right (f' := scaledRealPacket b g) using 1
funext z
exact pow_two _
have hh := tfPotential_newton_exterior_closed (hc.integrable_of_hasCompactSupport hcomp)
(hc.memLp_of_hasCompactSupport hcomp) (scaled_packet_density_radial b hr) hb
(scaled_packet_density_support hb hs) hx
simpa only [scaledRealPacket_mass hb,hgn] using hh
lemma masterKernel_potential_centered (c₁ r₀ s : ℝ) (g : Space → ℝ) (x y : Space) :
tfPotential (masterKernel c₁ r₀ s g x) y =
tfPotential (fun z => (scaledRealPacket (masterWidth c₁ r₀ s x) g z)^2) (y-x) :=
integral_translated_pole (fun z => (scaledRealPacket (masterWidth c₁ r₀ s x) g z)^2) y x
lemma masterKernel_potential_nonneg (c₁ r₀ s : ℝ) (g : Space → ℝ) (x y : Space) :
0 ≤ tfPotential (masterKernel c₁ r₀ s g x) y :=
tfPotential_nonneg (ae_of_all _ (masterKernel_nonneg c₁ r₀ s g x)) y
lemma masterKernel_potential_le {c₁ r₀ s : ℝ} (hc : 0 < c₁) (hr : 0 < r₀) (hs : 0 < s)
{g : Space → ℝ} (hg : ContDiff ℝ ∞ g) (hcg : HasCompactSupport g)
(hgn : ∫ z, (g z)^2 = 1) (hrad : IsRadial g) (hgs : tsupport g ⊆ ball 0 1)
{x y : Space} (hxy : x ≠ y) :
tfPotential (masterKernel c₁ r₀ s g x) y ≤ 1/‖y-x‖ := by
rw [masterKernel_potential_centered]
exact scaled_packet_potential_le (masterWidth_pos hc hr hs x) hg hcg hgn hrad hgs
(sub_ne_zero.mpr hxy.symm)
lemma masterWidth_le_distance {c₁ r₀ s : ℝ}
(hc : 0 < c₁) (hcL : c₁ < (10*(100000:ℝ))⁻¹)
(hr : 0 < r₀) (hs : 0 < s) (hs1 : s ≤ 1) {x y : Space}
(hxy : 2*masterWidth c₁ r₀ s y ≤ ‖y-x‖) :
masterWidth c₁ r₀ s x ≤ ‖y-x‖ := by
have hh := (masterWidth_small_lipschitz hc hcL hr hs hs1).dist_le_mul x y
simp only [dist_eq_norm,Real.norm_eq_abs,NNReal.coe_div,NNReal.coe_one,NNReal.coe_ofNat] at hh
rw [norm_sub_rev x y] at hh
have h := (le_abs_self (masterWidth c₁ r₀ s x-masterWidth c₁ r₀ s y)).trans hh
linarith
lemma masterKernel_potential_eq_far {c₁ r₀ s : ℝ}
(hc : 0 < c₁) (hcL : c₁ < (10*(100000:ℝ))⁻¹)
(hr : 0 < r₀) (hs : 0 < s) (hs1 : s ≤ 1)
{g : Space → ℝ} (hg : ContDiff ℝ ∞ g) (hcg : HasCompactSupport g)
(hgn : ∫ z, (g z)^2 = 1) (hrad : IsRadial g) (hgs : tsupport g ⊆ ball 0 1)
{x y : Space} (hxy : 2*masterWidth c₁ r₀ s y ≤ ‖y-x‖) :
tfPotential (masterKernel c₁ r₀ s g x) y = 1/‖y-x‖ := by
rw [masterKernel_potential_centered]
exact scaled_packet_potential_eq (masterWidth_pos hc hr hs x) hg hcg hgn hrad hgs
(masterWidth_le_distance hc hcL hr hs hs1 hxy)
theorem masterKernel_potential_loss {c₁ r₀ s : ℝ}
(hc : 0 < c₁) (hcL : c₁ < (10*(100000:ℝ))⁻¹)
(hr : 0 < r₀) (hs : 0 < s) (hs1 : s ≤ 1)
{g : Space → ℝ} (hg : ContDiff ℝ ∞ g) (hcg : HasCompactSupport g)
(hgn : ∫ z, (g z)^2 = 1) (hrad : IsRadial g) (hgs : tsupport g ⊆ ball 0 1)
{x y : Space} (hxy : x ≠ y) :
0 ≤ 1/‖y-x‖-tfPotential (masterKernel c₁ r₀ s g x) y ∧
1/‖y-x‖-tfPotential (masterKernel c₁ r₀ s g x) y ≤
(ball y (2*masterWidth c₁ r₀ s y)).indicator (fun x => 1/‖y-x‖) x := by
refine ⟨sub_nonneg.mpr (masterKernel_potential_le hc hr hs hg hcg hgn hrad hgs hxy),?_⟩
by_cases hx : x ∈ ball y (2*masterWidth c₁ r₀ s y)
· rw [indicator_of_mem hx]
linarith [masterKernel_potential_nonneg c₁ r₀ s g x y]
· rw [indicator_of_notMem hx]
have hfar : 2*masterWidth c₁ r₀ s y ≤ ‖y-x‖ := by
simpa only [mem_ball,dist_eq_norm,norm_sub_rev,not_lt] using hx
rw [masterKernel_potential_eq_far hc hcL hr hs hs1 hg hcg hgn hrad hgs hfar,sub_self]
end CoulombAtom
end
end OAI
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