File size: 3,059 Bytes
9eeb9c7 | 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 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 | import OAI.Analysis.HenonEmden.IntegratedRecovery
namespace OAI
universe uE uΩ
section AngularIntegration
open MeasureTheory Set Metric
noncomputable section
namespace HenonLaneEmden.Angular
variable {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E]
[MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] [Nontrivial E]
(μ : Measure E) [μ.IsAddHaarMeasure]
open LineComponents
def rayConstant (n : ℕ) : ℝ := 2^(n+1)*(n:ℝ)
lemma long_component_measure_le {f : E → ℝ} {x : E} (hx : ‖x‖ ≤ 2)
{t l M : ℝ} (ht : 0 < t) (hl : 0 < l) (hl1 : l ≤ 1) (hM : 0 ≤ M)
(hU : IsOpen {z : sphere (0:E) 1 × ℝ | t < f (x+z.2 • z.1.1)})
(hm : (∫⁻ z in ball (0:E) 4, ENNReal.ofReal (f z) ∂μ) ≤ ENNReal.ofReal M) :
μ.toSphere.real {e | ENNReal.ofReal l < componentLength
{z : sphere (0:E) 1 × ℝ | t < f (x+z.2 • z.1.1)} e} ≤
rayConstant (Module.finrank ℝ E)*M/(t*l^(Module.finrank ℝ E)) := by
let n := Module.finrank ℝ E
have hn : 0 < (n:ℝ) := Nat.cast_pos.mpr Module.finrank_pos
have hh := (long_component_local_mass_le μ hx hl hl1 hU).trans
(mul_le_mul_right hm 2)
have hh' := ENNReal.toReal_mono (by finiteness) hh
have hf : 0 ≤ (l/2)^(Module.finrank ℝ E)/(Module.finrank ℝ E:ℝ) := by positivity
simp only [ENNReal.toReal_mul,ENNReal.toReal_ofReal ht.le,ENNReal.toReal_ofReal hf,
ENNReal.toReal_ofNat,ENNReal.toReal_ofReal hM] at hh'
rw [le_div_iff₀ (mul_pos ht (pow_pos hl _))]
unfold rayConstant
have he : t*((l/2)^n/(n:ℝ)) = (t*l^n)/(2^n*(n:ℝ)) := by
rw [div_pow]; field_simp
rw [he] at hh'
have hd : 0 < (2^n:ℝ)*(n:ℝ) := mul_pos (by positivity) hn
have hh'' := (div_le_iff₀ hd).mp (show μ.toSphere.real _*(t*l^n)/(2^n*(n:ℝ)) ≤ 2*M by
convert hh' using 1; unfold Measure.real at *; ring)
convert hh'' using 1; dsimp [n]; rw [pow_succ]; ring
end HenonLaneEmden.Angular
namespace HenonLaneEmden
lemma integral_le_constant_add_event {Ω : Type uΩ} [MeasurableSpace Ω] (ν : Measure Ω)
[IsFiniteMeasure ν] {S : Set Ω} (hS : MeasurableSet S) {F : Ω → ℝ}
(hF : Measurable F) {a H : ℝ} (ha : 0 ≤ a) (_hH : 0 ≤ H)
(hFn : ∀ z, 0 ≤ F z) (hFH : ∀ z, F z ≤ H) (hout : ∀ z ∉ S, F z ≤ a) :
(∫ z, F z ∂ν) ≤ a*ν.real univ+H*ν.real S := by
have hFi : Integrable F ν := (integrable_const H).mono' hF.aestronglyMeasurable
(ae_of_all _ (fun z => by rw [Real.norm_eq_abs,abs_of_nonneg (hFn z)]; exact hFH z))
have hIi : Integrable (S.indicator (fun _ => H)) ν := (integrable_const H).indicator hS
have hh := integral_mono hFi ((integrable_const a).add hIi) (fun z => ?_)
· simpa [integral_add (integrable_const a) hIi,integral_indicator hS,integral_const,
smul_eq_mul,mul_comm] using hh
· change F z ≤ a + S.indicator (fun _ => H) z
by_cases hz : z ∈ S
· rw [indicator_of_mem hz]; linarith [hFH z]
· rw [indicator_of_notMem hz]; simpa only [add_zero] using hout z hz
end HenonLaneEmden
end
end AngularIntegration
end OAI
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