| import OAI.Analysis.HenonEmden.ShortComponents |
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| namespace OAI |
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| universe uE |
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| section Angular |
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| open MeasureTheory MeasureTheory.Measure Set Metric |
| open scoped ENNReal Pointwise |
| 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] |
|
|
| /-- Polar lower bound contributed by a set of directions whose full radial |
| segment has source at least `t`. Surface measure here is not normalized. -/ |
| lemma ray_mass_le {f : E → ℝ≥0∞} {S : Set (sphere (0:E) 1)} |
| (hS : MeasurableSet S) {l : ℝ} (hl : 0 < l) (t : ℝ≥0∞) |
| (h : ∀ e ∈ S, ∀ r ∈ Ioo 0 l, t ≤ f (r • e.1)) : |
| t * μ.toSphere S * ENNReal.ofReal (l ^ Module.finrank ℝ E / Module.finrank ℝ E) |
| ≤ ∫⁻ z, f z ∂μ := by |
| let L : Ioi (0:ℝ) := ⟨l,hl⟩ |
| let T := S ×ˢ Iio L |
| have hm : MeasurableSet T := hS.prod measurableSet_Iio |
| rw [Polar.lintegral_polar_prod μ] |
| have hb : ∀ z : sphere (0:E) 1 × Ioi (0:ℝ), T.indicator (fun _ => t) z ≤ f (z.2.1 • z.1.1) := by |
| intro z |
| by_cases hz : z ∈ T |
| · rw [indicator_of_mem hz] |
| exact h z.1 hz.1 z.2.1 ⟨z.2.2,hz.2⟩ |
| · simp only [indicator_of_notMem hz,zero_le] |
| have he : ∫⁻ z, T.indicator (fun _ => t) z |
| ∂μ.toSphere.prod (volumeIoiPow (Module.finrank ℝ E-1)) = |
| t * μ.toSphere S * ENNReal.ofReal (l ^ Module.finrank ℝ E / Module.finrank ℝ E) := by |
| rw [lintegral_indicator hm,lintegral_const,Measure.restrict_apply_univ, |
| Measure.prod_prod,volumeIoiPow_apply_Iio] |
| have hn : Module.finrank ℝ E-1+1 = Module.finrank ℝ E := |
| Nat.sub_add_cancel (Module.finrank_pos : 0 < Module.finrank ℝ E) |
| have hnr : ((Module.finrank ℝ E-1 : ℕ) : ℝ)+1 = Module.finrank ℝ E := by |
| exact_mod_cast hn |
| simp only [hn,hnr,L,mul_assoc] |
| exact he ▸ lintegral_mono hb |
|
|
| open LineComponents |
|
|
| /-- The long-component alternative on a line uses either forward or backward |
| rays. In particular it requires no independence of any cap weight and layer. -/ |
| lemma long_component_mass_le {f : E → ℝ≥0∞} |
| {U : Set (sphere (0:E) 1 × ℝ)} (hU : IsOpen U) |
| {l : ℝ} (hl : 0 < l) (t : ℝ≥0∞) |
| (h : ∀ e r, |r| ≤ l/2 → (e,r) ∈ U → t ≤ f (r • e.1)) : |
| t * ENNReal.ofReal ((l/2)^Module.finrank ℝ E / Module.finrank ℝ E) * |
| μ.toSphere {e | ENNReal.ofReal l < componentLength U e} ≤ |
| 2 * ∫⁻ z, f z ∂μ := by |
| let F := {e | ENNReal.ofReal (l/2) < forwardExit U e} |
| let G := {e | ENNReal.ofReal (l/2) < backwardExit U e} |
| have hF : MeasurableSet F := measurableSet_lt measurable_const (measurable_forwardExit hU) |
| have hG : MeasurableSet G := measurableSet_lt measurable_const (measurable_backwardExit hU) |
| have hs : {e | ENNReal.ofReal l < componentLength U e} ⊆ F ∪ G := by |
| intro e he |
| by_contra hn |
| have hne : ¬ e ∈ F ∧ ¬ e ∈ G := not_or.mp hn |
| have hh : componentLength U e ≤ ENNReal.ofReal l := by |
| calc |
| componentLength U e ≤ ENNReal.ofReal (l/2) + ENNReal.ofReal (l/2) := |
| add_le_add (le_of_not_gt hne.1) (le_of_not_gt hne.2) |
| _ = ENNReal.ofReal l := by rw [← ENNReal.ofReal_add (by positivity) (by positivity)]; congr 1; ring |
| exact (not_lt_of_ge hh) he |
| have hforward := ray_mass_le μ hF (half_pos hl) t (f := f) (by |
| intro e he r hr |
| have hin : SegmentInside U e (l/2) := (lt_forwardExit_iff hU (by positivity)).mp he |
| exact h e r (by rw [abs_of_pos hr.1]; exact hr.2.le) (hin r (by rw [uIcc_of_le (by positivity)]; exact ⟨hr.1.le,hr.2.le⟩))) |
| have hbackward := ray_mass_le μ hG (half_pos hl) t (f := fun z => f (-z)) (by |
| intro e he r hr |
| have hin : SegmentInside U e (-(l/2)) := (lt_backwardExit_iff hU (by positivity)).mp he |
| have hm := hin (-r) (by rw [uIcc_of_ge (by linarith)]; constructor <;> linarith [hr.1,hr.2]) |
| simpa only [neg_smul] using h e (-r) (by rw [abs_neg,abs_of_pos hr.1]; exact hr.2.le) hm) |
| rw [lintegral_neg_eq_self] at hbackward |
| have hmeasure : μ.toSphere {e | ENNReal.ofReal l < componentLength U e} ≤ |
| μ.toSphere F + μ.toSphere G := (measure_mono hs).trans (measure_union_le F G) |
| calc |
| _ ≤ t * ENNReal.ofReal ((l/2)^Module.finrank ℝ E / Module.finrank ℝ E) * |
| (μ.toSphere F + μ.toSphere G) := by gcongr |
| _ = t * μ.toSphere F * ENNReal.ofReal ((l/2)^Module.finrank ℝ E / Module.finrank ℝ E) + |
| t * μ.toSphere G * ENNReal.ofReal ((l/2)^Module.finrank ℝ E / Module.finrank ℝ E) := by ring |
| _ ≤ (∫⁻ z, f z ∂μ) + ∫⁻ z, f z ∂μ := add_le_add hforward hbackward |
| _ = _ := (two_mul _).symm |
|
|
| /-- Translation of the ray estimate to a local source; the only quantitative |
| input is its mass on the fixed ball `B₄`. -/ |
| lemma long_component_local_mass_le {f : E → ℝ} {x : E} (hx : ‖x‖ ≤ 2) |
| {t l : ℝ} (hl : 0 < l) (hl1 : l ≤ 1) |
| (hU : IsOpen {z : sphere (0:E) 1 × ℝ | t < f (x + z.2 • z.1.1)}) : |
| ENNReal.ofReal t * ENNReal.ofReal ((l/2)^Module.finrank ℝ E / Module.finrank ℝ E) * |
| μ.toSphere {e | ENNReal.ofReal l < componentLength |
| {z : sphere (0:E) 1 × ℝ | t < f (x + z.2 • z.1.1)} e} ≤ |
| 2 * ∫⁻ z in ball (0:E) 4, ENNReal.ofReal (f z) ∂μ := by |
| let F : E → ℝ≥0∞ := (ball (0:E) 4).indicator (fun z => ENNReal.ofReal (f z)) |
| have hh := long_component_mass_le μ hU hl (ENNReal.ofReal t) |
| (f := fun z => F (x+z)) (by |
| intro e r hr ht |
| have hn : ‖(e:E)‖ = 1 := by simpa only [mem_sphere,dist_zero_right] using e.2 |
| have hb : x+r • (e:E) ∈ ball (0:E) 4 := by |
| rw [mem_ball,dist_zero_right] |
| have hs := norm_add_le x (r • (e:E)) |
| rw [norm_smul,Real.norm_eq_abs,hn,mul_one] at hs |
| linarith |
| change ENNReal.ofReal t ≤ (ball (0:E) 4).indicator (fun z => ENNReal.ofReal (f z)) (x+r • (e:E)) |
| rw [indicator_of_mem hb] |
| exact ENNReal.ofReal_le_ofReal ht.le) |
| rwa [lintegral_add_left_eq_self,lintegral_indicator measurableSet_ball] at hh |
|
|
| end HenonLaneEmden.Angular |
|
|
| namespace HenonLaneEmden |
| /-- With value zero assigned at the origin, the positive source subgraph is |
| open jointly in spatial location and positive level. -/ |
| lemma isOpen_source_subgraph {n : ℕ} {u : Space n → ℝ} |
| (hu : ContDiffOn ℝ 2 u ({0}ᶜ : Set (Space n))) (hup : ∀ x, 0 < u x) (q B : ℝ) : |
| IsOpen {z : Space n × ℝ | 0 < z.2 ∧ z.2 < source u q B z.1} := by |
| rw [isOpen_iff_mem_nhds] |
| intro z hz |
| have hx : z.1 ≠ 0 := by |
| intro he |
| simpa [he,source_zero] using hz.1.trans hz.2 |
| have hc : ContinuousAt (source u q B) z.1 := |
| (regular_source hu hup q B).continuousOn.continuousAt (isClosed_singleton.isOpen_compl.mem_nhds hx) |
| exact (continuousAt_const.eventually_lt continuousAt_snd hz.1).and |
| (continuousAt_snd.eventually_lt (hc.comp continuousAt_fst) hz.2) |
|
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| lemma isOpen_source_lineLayer {n : ℕ} {u : Space n → ℝ} |
| (hu : ContDiffOn ℝ 2 u ({0}ᶜ : Set (Space n))) (hup : ∀ x, 0 < u x) (q B : ℝ) |
| (x : Space n) {t : ℝ} (ht : 0 < t) : |
| IsOpen {z : sphere (0:Space n) 1 × ℝ | t < source u q B (x + z.2 • z.1.1)} := by |
| have hh := (isOpen_source_subgraph hu hup q B).preimage |
| (show Continuous (fun z : sphere (0:Space n) 1 × ℝ => (x+z.2 • z.1.1,t)) from |
| (continuous_const.add (continuous_snd.smul continuous_fst.subtype_val)).prodMk continuous_const) |
| simpa only [Set.mem_ofPred_eq,true_and,ht,Set.preimage_ofPred_eq] using hh |
|
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| end HenonLaneEmden |
| end |
| end Angular |
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| end OAI |
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