| import Mathlib |
| import OAI.Analysis.CoulombIonization.Variational.LocalWidthKernel |
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| namespace OAI |
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| noncomputable section |
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| open MeasureTheory Filter |
| open scoped Topology BigOperators ContDiff |
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| open MeasureTheory Filter Set Metric |
| open scoped ENNReal ContDiff |
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| namespace CoulombAtom |
| open CoulombAnalysis |
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| 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] |
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| 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 |
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| 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 |
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| 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 |
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| 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 |
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| 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 |
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| 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) |
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| 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 |
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| 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) |
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| 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] |
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| end CoulombAtom |
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| end |
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| end OAI |
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