| import Mathlib |
| import OAI.Analysis.CoulombIonization.Variational.CoulombTestCharge |
| import OAI.Analysis.CoulombIonization.Variational.BoundedDensityPotential |
|
|
| namespace OAI |
|
|
| open MeasureTheory Set Metric Laplacian |
| open scoped Topology |
| noncomputable section |
| namespace CoulombAtom |
| open CoulombAnalysis |
|
|
| def sourcePotential (mu : Measure Space) (y : Space) : ℝ := |
| ∫ a, (‖y-a‖)⁻¹ ∂mu |
|
|
| lemma inner_source_separation {h d : ℝ} {a y : Space} |
| (ha : ‖a‖ ≤ h) (hy : h+d ≤ ‖y‖) : d ≤ ‖y-a‖ := by |
| have := norm_le_norm_sub_add y a |
| linarith |
|
|
| lemma source_kernel_integrable (mu : Measure Space) [IsFiniteMeasure mu] |
| {h d : ℝ} (hs : ∀ᵐ a ∂mu, ‖a‖ ≤ h) (hd : 0 < d) {y : Space} |
| (hy : h+d ≤ ‖y‖) : Integrable (fun a => (‖y-a‖)⁻¹) mu := by |
| apply (integrable_const (d⁻¹)).mono' (by fun_prop) |
| filter_upwards [hs] with a ha |
| rw [Real.norm_of_nonneg (inv_nonneg.mpr (norm_nonneg _))] |
| exact inv_anti₀ hd (inner_source_separation ha hy) |
|
|
| lemma sourcePotential_nonneg (mu : Measure Space) (y : Space) : |
| 0 ≤ sourcePotential mu y := integral_nonneg (fun _ => inv_nonneg.mpr (norm_nonneg _)) |
|
|
| lemma sourcePotential_le (mu : Measure Space) [IsFiniteMeasure mu] |
| {h d : ℝ} (hs : ∀ᵐ a ∂mu, ‖a‖ ≤ h) (hd : 0 < d) {y : Space} |
| (hy : h+d ≤ ‖y‖) : sourcePotential mu y ≤ (mu Set.univ).toReal/d := by |
| calc |
| _ ≤ ∫ _a, d⁻¹ ∂mu := integral_mono_ae (source_kernel_integrable mu hs hd hy) |
| (integrable_const _) (hs.mono (fun a ha => inv_anti₀ hd (inner_source_separation ha hy))) |
| _ = _ := by simp only [integral_const,smul_eq_mul,Measure.real,div_eq_mul_inv] |
|
|
| lemma sourcePotential_difference (mu : Measure Space) [IsFiniteMeasure mu] |
| {h d : ℝ} (hs : ∀ᵐ a ∂mu, ‖a‖ ≤ h) (hd : 0 < d) {y z : Space} |
| (hy : h+d ≤ ‖y‖) (hz : h+d ≤ ‖z‖) : |
| |sourcePotential mu y-sourcePotential mu z| ≤ |
| ((mu Set.univ).toReal/d^2)*‖y-z‖ := by |
| have hi := source_kernel_integrable mu hs hd hy |
| have hj := source_kernel_integrable mu hs hd hz |
| have hb : ∀ᵐ a ∂mu, ‖(‖y-a‖)⁻¹-(‖z-a‖)⁻¹‖ ≤ ‖y-z‖/d^2 := by |
| filter_upwards [hs] with a ha |
| have hya := inner_source_separation ha hy |
| have hza := inner_source_separation ha hz |
| have hn1 : a ≠ y := by intro he; subst a; simp only [sub_self,norm_zero] at hya; linarith |
| have hn2 : a ≠ z := by intro he; subst a; simp only [sub_self,norm_zero] at hza; linarith |
| have hiy := pow_le_pow_left₀ (inv_nonneg.mpr (norm_nonneg (y-a))) (inv_anti₀ hd hya) 2 |
| have hiz := pow_le_pow_left₀ (inv_nonneg.mpr (norm_nonneg (z-a))) (inv_anti₀ hd hza) 2 |
| rw [Real.norm_eq_abs] |
| calc _ ≤ ‖y-z‖*((‖y-a‖)⁻¹^2+(‖z-a‖)⁻¹^2)/2 := coulomb_inv_difference hn1 hn2 |
| _ ≤ ‖y-z‖*(d⁻¹^2+d⁻¹^2)/2 := by gcongr |
| _ = _ := by ring |
| have hh := norm_integral_le_of_norm_le (integrable_const (‖y-z‖/d^2) : |
| Integrable (fun _ : Space => ‖y-z‖/d^2) mu) hb |
| rw [integral_sub hi hj,Real.norm_eq_abs] at hh |
| calc |
| _ ≤ ∫ _a : Space, ‖y-z‖/d^2 ∂mu := hh |
| _ = _ := by simp only [integral_const,Measure.real,smul_eq_mul]; ring |
|
|
| lemma sourcePotential_lipschitzOn (mu : Measure Space) [IsFiniteMeasure mu] |
| {h d : ℝ} (hs : ∀ᵐ a ∂mu, ‖a‖ ≤ h) (hd : 0 < d) : |
| LipschitzOnWith ⟨(mu Set.univ).toReal/d^2,by positivity⟩ (sourcePotential mu) |
| {y | h+d ≤ ‖y‖} := by |
| apply LipschitzOnWith.of_dist_le_mul |
| intro y hy z hz |
| change dist (sourcePotential mu y) (sourcePotential mu z) ≤ |
| ((mu univ).toReal/d^2)*dist y z |
| simpa only [dist_eq_norm,Real.norm_eq_abs] using |
| sourcePotential_difference mu hs hd hy hz |
|
|
| lemma sourcePotential_continuousOn_exterior (mu : Measure Space) [IsFiniteMeasure mu] |
| {h : ℝ} (hs : ∀ᵐ a ∂mu, ‖a‖ ≤ h) : |
| ContinuousOn (sourcePotential mu) {y | h < ‖y‖} := by |
| intro y hy |
| change h < ‖y‖ at hy |
| have hd : 0 < (‖y‖-h)/2 := by linarith [hy] |
| have hnb : {z : Space | h+(‖y‖-h)/2 ≤ ‖z‖} ∈ 𝓝 y := |
| Filter.mem_of_superset |
| (continuous_norm.continuousAt.preimage_mem_nhds |
| (Ioi_mem_nhds (show h+(‖y‖-h)/2 < ‖y‖ by linarith [hy]))) |
| (fun z hz => show h+(‖y‖-h)/2 ≤ ‖z‖ from hz.le) |
| exact ((sourcePotential_lipschitzOn mu hs hd).continuousOn.continuousAt hnb).continuousWithinAt |
|
|
| lemma sourcePotential_pairing_integrable (mu : Measure Space) [IsFiniteMeasure mu] |
| {h d : ℝ} (hs : ∀ᵐ a ∂mu, ‖a‖ ≤ h) (hd : 0 < d) |
| {g : Space → ℝ} (hg : ContDiff ℝ 2 g) (hcg : HasCompactSupport g) |
| (hsg : tsupport g ⊆ {y | h+d ≤ ‖y‖}) : |
| Integrable (fun p : Space × Space => (‖p.2-p.1‖)⁻¹*Δ g p.2) |
| (mu.prod volume) := by |
| have hi : Integrable (Δ g) := (tfLaplacian_continuous hg).integrable_of_hasCompactSupport (tfLaplacian_compact hg hcg) |
| apply ((integrable_const (d⁻¹) : Integrable (fun _ : Space => d⁻¹) mu).mul_prod hi.norm).mono' |
| (by have hc := tfLaplacian_continuous hg; fun_prop) |
| apply (Measure.ae_prod_iff_ae_ae (measurableSet_le (by have hc := tfLaplacian_continuous hg; fun_prop) |
| (by have hc := tfLaplacian_continuous hg; fun_prop))).mpr |
| filter_upwards [hs] with a ha |
| apply ae_of_all |
| intro y |
| rw [norm_mul,Real.norm_of_nonneg (inv_nonneg.mpr (norm_nonneg _))] |
| by_cases hz : Δ g y = 0 |
| · simp only [hz,norm_zero,mul_zero,le_refl] |
| · exact mul_le_mul_of_nonneg_right |
| (inv_anti₀ hd (inner_source_separation ha (hsg (tfLaplacian_support hg hz)))) (norm_nonneg _) |
|
|
| lemma sourcePotential_weak_harmonic_separated (mu : Measure Space) [IsFiniteMeasure mu] |
| {h d : ℝ} (hs : ∀ᵐ a ∂mu, ‖a‖ ≤ h) (hd : 0 < d) |
| {g : Space → ℝ} (hg : ContDiff ℝ 2 g) (hcg : HasCompactSupport g) |
| (hsg : tsupport g ⊆ {y | h+d ≤ ‖y‖}) : |
| (∫ y, sourcePotential mu y*Δ g y) = 0 := by |
| have hf := sourcePotential_pairing_integrable mu hs hd hg hcg hsg |
| have he : (∫ y, sourcePotential mu y*Δ g y) = |
| ∫ a, ∫ y, (‖y-a‖)⁻¹*Δ g y ∂volume ∂mu := by |
| simp only [sourcePotential,← integral_mul_const] |
| exact (integral_integral_swap hf).symm |
| rw [he] |
| apply integral_eq_zero_of_ae |
| filter_upwards [hs] with a ha |
| have hga : g a = 0 := by |
| apply image_eq_zero_of_notMem_tsupport |
| intro hm |
| have hh : h+d ≤ ‖a‖ := hsg hm |
| linarith |
| have hh := tfPotential_laplacian hg hcg a |
| change (∫ y, Δ g y/‖a-y‖) = _ at hh |
| simpa [norm_sub_rev,div_eq_mul_inv,mul_comm,hga] using hh |
|
|
| lemma compact_exterior_separated {K : Set Space} (hK : IsCompact K) {h : ℝ} |
| (hs : K ⊆ {y | h < ‖y‖}) : ∃ d > 0, K ⊆ {y | h+d ≤ ‖y‖} := by |
| rcases K.eq_empty_or_nonempty with he | hn |
| · exact ⟨1,by norm_num,by simp [he]⟩ |
| obtain ⟨y,hy,hmin⟩ := hK.exists_isMinOn hn continuous_norm.continuousOn |
| refine ⟨‖y‖-h,sub_pos.mpr (hs hy),?_⟩ |
| intro z hz |
| change h+(‖y‖-h) ≤ ‖z‖ |
| simp only [add_sub_cancel] |
| exact hmin hz |
|
|
| theorem sourcePotential_weak_harmonic_exterior (mu : Measure Space) [IsFiniteMeasure mu] |
| {h : ℝ} (hs : ∀ᵐ a ∂mu, ‖a‖ ≤ h) |
| {g : Space → ℝ} (hg : ContDiff ℝ 2 g) (hcg : HasCompactSupport g) |
| (hsg : tsupport g ⊆ {y | h < ‖y‖}) : |
| (∫ y, sourcePotential mu y*Δ g y) = 0 := by |
| obtain ⟨d,hd,hdg⟩ := compact_exterior_separated hcg hsg |
| exact sourcePotential_weak_harmonic_separated mu hs hd hg hcg hdg |
|
|
| end CoulombAtom |
|
|
| end |
|
|
| end OAI |
|
|
