File size: 2,167 Bytes
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 | import Mathlib
import OAI.Analysis.CoulombIonization.FieldAnalysis.InnerCapStatistic
namespace OAI
open MeasureTheory Set Filter
noncomputable section
namespace CoulombAtom
lemma actual_inner_linear_statistic {N : ℕ} {psi : FormVector N}
(hpsi : SobolevVector psi) {Z lam h a : ℝ} (hZ : 0 ≤ Z) (hlam : 0 ≤ lam)
(hh : 0 < h) (ha : h ≤ a) {y : Space} (hy : 2*a ≤ ‖y‖) :
sourceField Z (actualInnerSource psi h) y*formMass psi ≤
(h/a)*Real.sqrt (innerCapStatistic psi Z lam h)*Real.sqrt (formMass psi) := by
have hap : 0 < a := hh.trans_le ha
have hm := formMass_nonneg psi
by_cases hm0 : formMass psi = 0
· simp only [hm0,mul_zero,Real.sqrt_zero,le_refl]
have hcap := actual_inner_exterior_cap hpsi hm0 hZ hlam hh (by linarith)
have hc := actualInnerBoundaryCap_nonneg psi Z hlam hh
have h1 : sourceField Z (actualInnerSource psi h) y ≤ actualInnerBoundaryCap psi Z lam h*(h/a) := by
calc
_ ≤ actualInnerBoundaryCap psi Z lam h*(2*h/‖y‖) := by simpa only [mul_div_assoc] using hcap
_ ≤ actualInnerBoundaryCap psi Z lam h*(2*h/(2*a)) :=
mul_le_mul_of_nonneg_left (div_le_div_of_nonneg_left (by positivity) (by positivity) hy) hc
_ = _ := by congr 1; field_simp
have hsq := actualInnerBoundaryCap_sq_le_statistic hpsi hZ hlam hh
have hstat := innerCapStatistic_nonneg psi Z lam h
have hroot : actualInnerBoundaryCap psi Z lam h*Real.sqrt (formMass psi) ≤
Real.sqrt (innerCapStatistic psi Z lam h) := by
apply (sq_le_sq₀ (mul_nonneg hc (Real.sqrt_nonneg _)) (Real.sqrt_nonneg _)).mp
simpa only [mul_pow,Real.sq_sqrt hm,Real.sq_sqrt hstat] using hsq
have h2 := mul_le_mul_of_nonneg_right hroot (Real.sqrt_nonneg (formMass psi))
rw [mul_assoc,←sq,Real.sq_sqrt hm] at h2
calc
_ ≤ actualInnerBoundaryCap psi Z lam h*(h/a)*formMass psi := mul_le_mul_of_nonneg_right h1 hm
_ = (h/a)*(actualInnerBoundaryCap psi Z lam h*formMass psi) := by ring
_ ≤ (h/a)*(Real.sqrt (innerCapStatistic psi Z lam h)*Real.sqrt (formMass psi)) :=
mul_le_mul_of_nonneg_left h2 (div_nonneg hh.le hap.le)
_ = _ := by ring
end CoulombAtom
end
end OAI
|