openAI-math-dataset / lean /OAI /Analysis /CoulombIonization /FieldAnalysis /FineLocalPotential.lean
| import Mathlib | |
| import OAI.Analysis.CoulombIonization.FieldAnalysis.FineSmearingPotential | |
| import OAI.Analysis.CoulombIonization.RadialBounds.TranslatedShortRange | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory Filter | |
| open scoped Topology BigOperators ContDiff | |
| open MeasureTheory Filter Set Metric | |
| open scoped BigOperators ENNReal | |
| namespace CoulombNeumann | |
| open CoulombAtom CoulombAnalysis | |
| lemma varthetaScaled_local_potential_le {b q : ℝ} (hb : 0 < b) | |
| (y a : Space) (ha : a ≠ y) : | |
| (∫ z in ball y q, varthetaScaled b (z-a)/‖y-z‖) ≤ | |
| (ball y (q+Real.sqrt 3*b)).indicator (fun z => 1/‖y-z‖) a := by | |
| by_cases hnear : a ∈ ball y (q+Real.sqrt 3*b) | |
| · rw [indicator_of_mem hnear] | |
| calc | |
| _ ≤ ∫ z : Space, varthetaScaled b (z-a)/‖y-z‖ := | |
| setIntegral_le_integral | |
| (translated_pole_integrable (varthetaScaled_integrable hb) (varthetaScaled_memLp hb) y a) | |
| (Eventually.of_forall fun z => div_nonneg (varthetaScaled_nonneg hb _) (norm_nonneg _)) | |
| _ = tfPotential (varthetaScaled b) (y-a) := integral_translated_pole _ _ _ | |
| _ ≤ _ := varthetaScaled_potential_le hb (sub_ne_zero.mpr ha.symm) | |
| · rw [indicator_of_notMem hnear] | |
| apply le_of_eq | |
| apply integral_eq_zero_of_ae | |
| filter_upwards [ae_restrict_mem measurableSet_ball] with z hz | |
| have hzero : varthetaScaled b (z-a) = 0 := by | |
| by_contra hne | |
| have hs := varthetaScaled_support_bound hb (z-a) hne | |
| have hd := dist_triangle a z y | |
| have hfar : q+Real.sqrt 3*b ≤ dist a y := le_of_not_gt hnear | |
| have hclose : dist z y < q := hz | |
| rw [←dist_eq_norm,dist_comm z a] at hs | |
| linarith | |
| simp only [hzero,zero_div,Pi.zero_apply] | |
| lemma retainedSmear_local_potential_le {N : ℕ} {b q : ℝ} (hb : 0 < b) | |
| (S : Finset (Fin N)) (x : Configuration N) (y : Space) | |
| (hxy : ∀ i ∈ S, x i ≠ y) : | |
| (∫ z in ball y q, retainedSmear b S x z/‖y-z‖) ≤ | |
| ∑ i ∈ S, (ball y (q+Real.sqrt 3*b)).indicator (fun z => 1/‖y-z‖) (x i) := by | |
| simp only [retainedSmear,Finset.sum_div] | |
| rw [integral_finsetSum _ (fun i _ => | |
| (translated_pole_integrable (varthetaScaled_integrable hb) (varthetaScaled_memLp hb) y (x i)).restrict)] | |
| exact Finset.sum_le_sum (fun i hi => varthetaScaled_local_potential_le hb y (x i) (hxy i hi)) | |
| theorem retainedPatchLp_local_potential_le {N : ℕ} {b q R : ℝ} (hb : 0 < b) (hqR : q ≤ R) | |
| (S : Finset (Fin N)) (x : Configuration N) (y : Space) | |
| (hxy : ∀ i ∈ S, x i ≠ y) : | |
| (∫ z in ball 0 q, ‖retainedPatchLp hb S x y R z‖/‖z‖) ≤ | |
| ∑ i ∈ S, (ball y (q+Real.sqrt 3*b)).indicator (fun z => 1/‖y-z‖) (x i) := by | |
| have he := ae_mono (Measure.restrict_mono (ball_subset_ball hqR) le_rfl) | |
| (retainedPatchLp_ae hb S x y R) | |
| have hid : (∫ z in ball 0 q, ‖retainedPatchLp hb S x y R z‖/‖z‖) = | |
| ∫ z in ball 0 q, retainedSmear b S x (y+z)/‖z‖ := by | |
| apply integral_congr_ae | |
| filter_upwards [he] with z hz | |
| rw [hz,Real.norm_of_nonneg (retainedSmear_nonneg hb S x _)] | |
| rw [hid] | |
| have ht := setIntegral_translate_ball y q (fun z => retainedSmear b S x z/‖y-z‖) | |
| simp only [sub_add_cancel_left,norm_neg] at ht | |
| rw [ht] | |
| exact retainedSmear_local_potential_le hb S x y hxy | |
| end CoulombNeumann | |
| end | |
| end OAI | |