| import Mathlib |
| import OAI.Analysis.CoulombIonization.FieldAnalysis.OuterFieldMaximum |
| import OAI.Analysis.CoulombIonization.FieldAnalysis.WeakHarmonicMean |
| import OAI.Analysis.CoulombIonization.Variational.Inner |
| import OAI.Analysis.CoulombIonization.Localization.PacketComparison |
|
|
| namespace OAI |
|
|
| open MeasureTheory Filter Set Metric Laplacian |
| noncomputable section |
| namespace CoulombAtom |
| open CoulombAnalysis |
|
|
| def innerFieldTestAnnulus (h : ℝ) : Set Space := {z | h ≤ ‖z‖ ∧ ‖z‖ ≤ 3*h} |
|
|
| lemma innerFieldTestAnnulus_measurable (h : ℝ) : MeasurableSet (innerFieldTestAnnulus h) := |
| (measurableSet_le measurable_const continuous_norm.measurable).inter |
| (measurableSet_le continuous_norm.measurable measurable_const) |
|
|
| lemma innerFieldTestAnnulus_finite (h : ℝ) : volume (innerFieldTestAnnulus h) < ⊤ := |
| (measure_mono (fun z hz => show z ∈ closedBall (0 : Space) (3*h) by |
| simpa only [mem_closedBall,dist_zero_right] using hz.2)).trans_lt ((isCompact_closedBall (0 : Space) (3*h)).measure_lt_top) |
|
|
| lemma positiveCoreField_annular_integrable {N : ℕ} {psi : FormVector N} |
| (hpsi : SobolevVector psi) {Z lam h : ℝ} (hZ : 0 ≤ Z) (hlam : 0 ≤ lam) (hh : 0 < h) : |
| IntegrableOn (fun z => max (normalizedCoreField Z lam psi z) 0) (innerFieldTestAnnulus h) := by |
| let : IsFiniteMeasure (volume.restrict (innerFieldTestAnnulus h)) := |
| ⟨by rw [Measure.restrict_apply_univ]; exact innerFieldTestAnnulus_finite h⟩ |
| apply (integrable_const (Z/h)).mono' |
| ((normalizedCoreField_measurable hpsi Z lam).max measurable_const).aestronglyMeasurable |
| filter_upwards [ae_restrict_mem (innerFieldTestAnnulus_measurable h)] with z hz |
| rw [Real.norm_of_nonneg (le_max_right _ _)] |
| exact max_le ((normalizedCoreField_le_nuclear psi hZ hlam z).trans |
| (div_le_div_of_nonneg_left hZ hh hz.1)) (div_nonneg hZ hh.le) |
|
|
| lemma packet_on_inner_annulus {h : ℝ} (hh : 0 < h) {y z : Space} |
| (hy : ‖y‖ = 2*h) (hz : packetDensity y (h/4) z ≠ 0) : |
| z ∈ innerFieldTestAnnulus h := by |
| have hd := packetDensity_support y (show 0 < h/4 by positivity) hz |
| have h1 := norm_le_norm_sub_add y z |
| have h2 := norm_le_norm_sub_add z y |
| rw [norm_sub_rev y z] at h1 |
| constructor <;> linarith only [hd,h1,h2,hy,hh] |
|
|
| lemma positiveCoreField_packet_le {N : ℕ} {psi : FormVector N} |
| (hpsi : SobolevVector psi) {Z lam h : ℝ} (hZ : 0 ≤ Z) (hlam : 0 ≤ lam) (hh : 0 < h) |
| {y : Space} (hy : ‖y‖ = 2*h) : |
| Integrable (fun z => packetDensity y (h/4) z*max (normalizedCoreField Z lam psi z) 0) ∧ |
| (∫ z, packetDensity y (h/4) z*max (normalizedCoreField Z lam psi z) 0) ≤ |
| packetDensityConstant/(h/4)^3* |
| (∫ z in innerFieldTestAnnulus h, max (normalizedCoreField Z lam psi z) 0) := by |
| let A := innerFieldTestAnnulus h |
| let F := fun z => max (normalizedCoreField Z lam psi z) 0 |
| have hi := (positiveCoreField_annular_integrable hpsi hZ hlam hh).integrable_indicator |
| (innerFieldTestAnnulus_measurable h) |
| have hC : 0 ≤ packetDensityConstant/(h/4)^3 := by positivity [packetDensityConstant_pos] |
| have hb (z : Space) : packetDensity y (h/4) z*F z ≤ |
| packetDensityConstant/(h/4)^3*A.indicator F z := by |
| by_cases hz : packetDensity y (h/4) z = 0 |
| · rw [hz,zero_mul] |
| exact mul_nonneg hC (indicator_nonneg (fun z _ => le_max_right _ _) z) |
| · rw [indicator_of_mem (packet_on_inner_annulus hh hy hz)] |
| exact mul_le_mul_of_nonneg_right (packetDensity_bound y z (by positivity)) (le_max_right _ _) |
| have hp : Integrable (fun z => packetDensity y (h/4) z*F z) := by |
| apply (hi.const_mul (packetDensityConstant/(h/4)^3)).mono' |
| (((packetDensity_continuous y _).measurable.mul |
| ((normalizedCoreField_measurable hpsi Z lam).max measurable_const)).aestronglyMeasurable) |
| exact ae_of_all _ (fun z => by |
| change ‖packetDensity y (h/4) z*F z‖ ≤ _ |
| rw [Real.norm_of_nonneg (mul_nonneg (packetDensity_nonneg _ _ _) (le_max_right _ _))] |
| exact hb z) |
| refine ⟨hp,?_⟩ |
| calc |
| _ ≤ ∫ z, packetDensityConstant/(h/4)^3*A.indicator F z := integral_mono hp (hi.const_mul _) hb |
| _ = _ := by rw [integral_const_mul,integral_indicator (innerFieldTestAnnulus_measurable h)] |
|
|
| def actualInnerBoundaryCap {N : ℕ} (psi : FormVector N) (Z lam h : ℝ) : ℝ := |
| packetDensityConstant/(h/4)^3* |
| (∫ z in innerFieldTestAnnulus h, max (normalizedCoreField Z lam psi z) 0)+lam+ |
| 16*packetPotentialConstant*(∫ a, ‖a‖⁻¹ ∂(actualParticleSource psi).restrict {a | h < ‖a‖}) |
|
|
| lemma actualInnerBoundaryCap_nonneg {N : ℕ} (psi : FormVector N) (Z : ℝ) {lam h : ℝ} |
| (hlam : 0 ≤ lam) (hh : 0 < h) : 0 ≤ actualInnerBoundaryCap psi Z lam h := by |
| unfold actualInnerBoundaryCap |
| have hp := packetDensityConstant_pos |
| have hc := packetPotentialConstant_one_le |
| positivity |
|
|
| theorem actual_inner_boundary_cap {N : ℕ} {psi : FormVector N} |
| (hpsi : SobolevVector psi) (hm : formMass psi ≠ 0) {Z lam h : ℝ} |
| (hZ : 0 ≤ Z) (hlam : 0 ≤ lam) (hh : 0 < h) |
| {y : Space} (hy : ‖y‖ = 2*h) : |
| sourceField Z (actualInnerSource psi h) y ≤ actualInnerBoundaryCap psi Z lam h := by |
| let mu := actualInnerSource psi h |
| let nu := (actualParticleSource psi).restrict {a | h < ‖a‖} |
| let : IsFiniteMeasure (actualParticleSource psi) := actualParticleSource_finite hpsi |
| let : IsFiniteMeasure mu := actualInnerSource_finite hpsi h |
| have hs : ∀ᵐ a ∂mu, ‖a‖ ≤ h := actualInnerSource_support psi h |
| have hnu : ∀ᵐ a ∂nu, h < ‖a‖ := |
| ae_restrict_mem (measurableSet_lt measurable_const continuous_norm.measurable) |
| have hball : closedBall y (h/2) ⊆ {z | h < ‖z‖} := by |
| intro z hz |
| have hd : ‖z-y‖ ≤ h/2 := mem_closedBall.mp hz |
| have ht := norm_le_norm_sub_add y z |
| rw [norm_sub_rev y z] at ht |
| change h < ‖z‖ |
| linarith only [hy,ht,hd,hh] |
| have hu := (sourceField_continuousOn Z mu hh.le hs).mono hball |
| have hmean : (∫ z, packetDensity y (h/4) z*sourceField Z mu z) = sourceField Z mu y := |
| weak_harmonic_packet_mean_local hu (by positivity) (by linarith) |
| (fun g hg hcg hsg => sourceField_weak_harmonic Z mu hh.le hs hg hcg |
| (hsg.trans (ball_subset_closedBall.trans hball))) |
| have hleft : Integrable (fun z => packetDensity y (h/4) z*sourceField Z mu z) := by |
| apply Integrable.congr (continuousOn_mul_integrable_support (isCompact_closedBall y (h/2)) hu |
| (packetDensity_continuous y (h/4)) (fun z hz => ?_)) |
| · exact ae_of_all _ (fun z => mul_comm _ _) |
| · rw [mem_closedBall] |
| have hd := packetDensity_support y (show 0 < h/4 by positivity) hz |
| rw [dist_eq_norm] |
| linarith |
| obtain ⟨hpos,hposB⟩ := positiveCoreField_packet_le hpsi hZ hlam hh hy |
| have hout := packet_source_integrable nu y (show 0 < h/4 by positivity) |
| have houtB := packet_outer_source_bound nu hh hnu hy.le |
| have hconst := (packetDensity_integrable y (show 0 < h/4 by positivity)).mul_const lam |
| have hb (z : Space) : packetDensity y (h/4) z*sourceField Z mu z ≤ |
| packetDensity y (h/4) z*max (normalizedCoreField Z lam psi z) 0+ |
| packetDensity y (h/4) z*lam+packetDensity y (h/4) z*sourcePotential nu z := by |
| have he := sourceField_inner_eq hpsi hm Z lam h z |
| change sourceField Z mu z = normalizedCoreField Z lam psi z+lam+sourcePotential nu z at he |
| rw [he,mul_add,mul_add] |
| have hh := mul_le_mul_of_nonneg_left |
| (le_max_left (normalizedCoreField Z lam psi z) 0) (packetDensity_nonneg y (h/4) z) |
| linarith only [hh] |
| have hb' := integral_mono hleft ((hpos.add hconst).add hout) hb |
| simp only [Pi.add_apply] at hb' |
| have hab := integral_add (hpos.add hconst) hout |
| have hac := integral_add hpos hconst |
| simp only [Pi.add_apply] at hab hac |
| rw [hmean,hab,hac,integral_mul_const, |
| packetDensity_mass y (show 0 < h/4 by positivity),one_mul] at hb' |
| unfold actualInnerBoundaryCap |
| change sourceField Z mu y ≤ _ |
| linarith only [hb',hposB,houtB] |
|
|
| theorem actual_inner_exterior_cap {N : ℕ} {psi : FormVector N} |
| (hpsi : SobolevVector psi) (hm : formMass psi ≠ 0) {Z lam h : ℝ} |
| (hZ : 0 ≤ Z) (hlam : 0 ≤ lam) (hh : 0 < h) {y : Space} (hy : 2*h ≤ ‖y‖) : |
| sourceField Z (actualInnerSource psi h) y ≤ actualInnerBoundaryCap psi Z lam h*(2*h)/‖y‖ := by |
| let : IsFiniteMeasure (actualInnerSource psi h) := actualInnerSource_finite hpsi h |
| exact sourceField_exterior_comparison Z _ hh.le (by linarith) |
| (actualInnerBoundaryCap_nonneg psi Z hlam hh) (actualInnerSource_support psi h) |
| (fun z hz => actual_inner_boundary_cap hpsi hm hZ hlam hh hz) hy |
|
|
| end CoulombAtom |
|
|
| end |
|
|
| end OAI |
|
|
