| import OAI.Analysis.HenonEmden.PotentialContinuity |
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| namespace OAI |
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| universe uE |
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| section CapGeometry |
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| open MeasureTheory MeasureTheory.Measure Set Metric |
| open scoped ENNReal Pointwise |
| noncomputable section |
| namespace HenonLaneEmden.CapGeometry |
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| /-- Radial bands in the positive-radius factor of polar coordinates. -/ |
| def band (a b : ℝ) : Set (Ioi (0:ℝ)) := |
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| lemma volumeIoiPow_band_bounds (k : ℕ) (ha : 0 ≤ a) (hab : a ≤ b) : |
| ENNReal.ofReal (a^k*(b-a)) ≤ volumeIoiPow k (band a b) ∧ |
| volumeIoiPow k (band a b) ≤ ENNReal.ofReal (b^k*(b-a)) := by |
| have hband : MeasurableSet (band a b) := (measurable_subtype_coe measurableSet_Ioo) |
| have he : Subtype.val '' band a b = Ioo a b := by |
| ext x |
| constructor |
| · rintro ⟨r,hr,rfl⟩; exact hr |
| · intro hx; exact ⟨⟨x,ha.trans_lt hx.1⟩,hx,rfl⟩ |
| rw [volumeIoiPow,withDensity_apply _ hband, |
| setLIntegral_subtype measurableSet_Ioi _ (fun r : ℝ => ENNReal.ofReal (r^k)),he] |
| have hconst (c : ℝ) (hc : 0 ≤ c) : |
| (∫⁻ _ in Ioo a b, ENNReal.ofReal c) = ENNReal.ofReal (c*(b-a)) := by |
| rw [lintegral_const,Measure.restrict_apply_univ,Real.volume_Ioo,← ENNReal.ofReal_mul hc] |
| constructor |
| · rw [← hconst (a^k) (pow_nonneg ha k)] |
| apply setLIntegral_mono' measurableSet_Ioo |
| intro x hx |
| exact ENNReal.ofReal_le_ofReal (pow_le_pow_left₀ ha hx.1.le k) |
| · rw [← hconst (b^k) (pow_nonneg (ha.trans hab) k)] |
| apply setLIntegral_mono' measurableSet_Ioo |
| intro x hx |
| exact ENNReal.ofReal_le_ofReal (pow_le_pow_left₀ (ha.trans hx.1.le) hx.2.le k) |
| |
| variable [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] [Nontrivial E] |
| (μ : Measure E) [μ.IsAddHaarMeasure] |
| |
| lemma polar_rectangle_le |
| (hS : MeasurableSet S) (hI : MeasurableSet I) (hA : MeasurableSet A) |
| (h : ∀ e ∈ S, ∀ r ∈ I, r.1 • e.1 ∈ A) : |
| μ.toSphere S * volumeIoiPow (Module.finrank ℝ E-1) I ≤ μ A := by |
| rw [← Measure.prod_prod] |
| have he : μ A = ∫⁻ z : sphere (0:E) 1 × Ioi (0:ℝ), A.indicator (fun _ => (1:ℝ≥0∞)) |
| (z.2.1 • z.1.1) ∂μ.toSphere.prod (volumeIoiPow (Module.finrank ℝ E-1)) := by |
| rw [← Polar.lintegral_polar_prod μ,lintegral_indicator hA,lintegral_one,Measure.restrict_apply_univ] |
| have hr : (μ.toSphere.prod (volumeIoiPow (Module.finrank ℝ E-1))) (S ×ˢ I) = |
| ∫⁻ z, (S ×ˢ I).indicator (fun _ => (1:ℝ≥0∞)) z |
| ∂μ.toSphere.prod (volumeIoiPow (Module.finrank ℝ E-1)) := by |
| rw [lintegral_indicator (hS.prod hI),lintegral_one,Measure.restrict_apply_univ] |
| rw [he,hr] |
| apply lintegral_mono |
| intro z |
| by_cases hz : z ∈ S ×ˢ I |
| · simp only [indicator_of_mem hz,indicator_of_mem (h z.1 hz.1 z.2 hz.2),le_refl] |
| · simp only [indicator_of_notMem hz,zero_le] |
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| lemma measure_le_polar_rectangle |
| (hS : MeasurableSet S) (hI : MeasurableSet I) (hA : MeasurableSet A) |
| (h : ∀ e r, r.1 • e.1 ∈ A → e ∈ S ∧ r ∈ I) : |
| μ A ≤ μ.toSphere S * volumeIoiPow (Module.finrank ℝ E-1) I := by |
| rw [← Measure.prod_prod] |
| have he : μ A = ∫⁻ z : sphere (0:E) 1 × Ioi (0:ℝ), A.indicator (fun _ => (1:ℝ≥0∞)) |
| (z.2.1 • z.1.1) ∂μ.toSphere.prod (volumeIoiPow (Module.finrank ℝ E-1)) := by |
| rw [← Polar.lintegral_polar_prod μ,lintegral_indicator hA,lintegral_one,Measure.restrict_apply_univ] |
| have hr : (μ.toSphere.prod (volumeIoiPow (Module.finrank ℝ E-1))) (S ×ˢ I) = |
| ∫⁻ z, (S ×ˢ I).indicator (fun _ => (1:ℝ≥0∞)) z |
| ∂μ.toSphere.prod (volumeIoiPow (Module.finrank ℝ E-1)) := by |
| rw [lintegral_indicator (hS.prod hI),lintegral_one,Measure.restrict_apply_univ] |
| rw [he,hr] |
| apply lintegral_mono |
| intro z |
| by_cases hz : z.2.1 • z.1.1 ∈ A |
| · have hz' : z ∈ S ×ˢ I := h z.1 z.2 hz |
| simp only [indicator_of_mem hz,indicator_of_mem hz',le_refl] |
| · simp only [indicator_of_notMem hz,zero_le] |
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| omit [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] [Nontrivial E] in |
| lemma norm_unit (e : sphere (0:E) 1) : ‖(e:E)‖ = 1 := by |
| simpa only [mem_sphere,dist_zero_right] using e.2 |
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| omit [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] [Nontrivial E] in |
| lemma polar_unit_distance (e θ : sphere (0:E) 1) (hr : 0 ≤ r) : |
| |r-1| ≤ ‖r • (e:E) - (θ:E)‖ ∧ |
| dist e θ ≤ 2 * ‖r • (e:E) - (θ:E)‖ := by |
| have hn : ‖r • (e:E)‖ = r := by rw [norm_smul,Real.norm_eq_abs,abs_of_nonneg hr,norm_unit,mul_one] |
| have hab : |r-1| ≤ ‖r • (e:E) - (θ:E)‖ := by |
| simpa only [hn,norm_unit] using abs_norm_sub_norm_le (r • (e:E)) (θ:E) |
| refine ⟨hab,?_⟩ |
| have heq : ‖(e:E)-r • (e:E)‖ = |r-1| := by |
| have hev : (e:E)-r • (e:E) = (1-r) • (e:E) := by rw [sub_smul,one_smul] |
| rw [hev,norm_smul,Real.norm_eq_abs,norm_unit,mul_one,abs_sub_comm] |
| have hh := dist_triangle (e:E) (r • (e:E)) (θ:E) |
| simp only [dist_eq_norm] at hh |
| rw [heq] at hh |
| rw [Subtype.dist_eq,dist_eq_norm] |
| linarith |
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| lemma cap_lower_product (θ : sphere (0:E) 1) (hl : 0 < l) (hl1 : l ≤ 1) : |
| μ (ball (θ:E) (l/4)) ≤ μ.toSphere (ball θ l) * |
| ENNReal.ofReal (2^(Module.finrank ℝ E-1)*l) := by |
| have ha : 0 ≤ 1-l/4 := by linarith |
| have hb : 1-l/4 ≤ 1+l/4 := by linarith |
| have hgeom := measure_le_polar_rectangle μ (S := ball θ l) |
| (I := band (1-l/4) (1+l/4)) measurableSet_ball |
| (measurable_subtype_coe measurableSet_Ioo) measurableSet_ball (A := ball (θ:E) (l/4)) (by |
| intro e r hr |
| have hd : ‖r.1 • (e:E)-(θ:E)‖ < l/4 := by simpa only [mem_ball,dist_eq_norm] using hr |
| have hh := polar_unit_distance e θ r.2.le |
| constructor |
| · change dist e θ < l |
| linarith [hh.2] |
| · change 1-l/4 < r.1 ∧ r.1 < 1+l/4 |
| have har : |r.1-1| < l/4 := hh.1.trans_lt hd |
| obtain ⟨hlo,hhi⟩ := abs_lt.mp har |
| constructor <;> linarith) |
| apply hgeom.trans |
| gcongr |
| apply (volumeIoiPow_band_bounds _ ha hb).2.trans |
| apply ENNReal.ofReal_le_ofReal |
| have hp : (1+l/4)^(Module.finrank ℝ E-1) ≤ 2^(Module.finrank ℝ E-1) := |
| pow_le_pow_left₀ (by linarith) (by linarith) _ |
| calc |
| (1+l/4)^(Module.finrank ℝ E-1)*((1+l/4)-(1-l/4)) = |
| (1+l/4)^(Module.finrank ℝ E-1)*(l/2) := by ring |
| _ ≤ 2^(Module.finrank ℝ E-1)*(l/2) := by gcongr |
| _ ≤ _ := by gcongr; linarith |
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| lemma cap_upper_product (θ : sphere (0:E) 1) (hl : 0 < l) (hl1 : l ≤ 1) : |
| μ.toSphere (ball θ l) * ENNReal.ofReal ((1/2:ℝ)^(Module.finrank ℝ E-1)*(l/2)) ≤ |
| μ (ball (θ:E) (2*l)) := by |
| have ha : 0 ≤ 1-l/2 := by linarith |
| have hb : 1-l/2 ≤ 1 := by linarith |
| have hgeom := polar_rectangle_le μ (S := ball θ l) |
| (I := band (1-l/2) 1) measurableSet_ball |
| (measurable_subtype_coe measurableSet_Ioo) measurableSet_ball (A := ball (θ:E) (2*l)) (by |
| intro e he r hr |
| have hd : ‖(e:E)-(θ:E)‖ < l := by simpa only [mem_ball,Subtype.dist_eq,dist_eq_norm] using he |
| have hr1 : r.1 < 1 := hr.2 |
| have hrl : 1-l/2 < r.1 := hr.1 |
| have heq : ‖r.1 • (e:E)-(e:E)‖ = 1-r.1 := by |
| have hev : r.1 • (e:E)-(e:E) = (r.1-1) • (e:E) := by rw [sub_smul,one_smul] |
| rw [hev,norm_smul,Real.norm_eq_abs,norm_unit,mul_one,abs_of_neg (by linarith)] |
| ring |
| have hh := dist_triangle (r.1 • (e:E)) (e:E) (θ:E) |
| simp only [dist_eq_norm] at hh |
| rw [heq] at hh |
| rw [mem_ball,dist_eq_norm] |
| linarith) |
| apply le_trans _ hgeom |
| gcongr |
| apply le_trans _ (volumeIoiPow_band_bounds _ ha hb).1 |
| apply ENNReal.ofReal_le_ofReal |
| have hp : (1/2:ℝ)^(Module.finrank ℝ E-1) ≤ (1-l/2)^(Module.finrank ℝ E-1) := |
| pow_le_pow_left₀ (by norm_num) (by linarith) _ |
| have heq : 1-(1-l/2) = l/2 := by ring |
| rw [heq] |
| gcongr |
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| lemma cap_real_bounds (θ : sphere (0:E) 1) (hl : 0 < l) (hl1 : l ≤ 1) : |
| (μ (ball (0:E) 1)).toReal / (4^Module.finrank ℝ E * 2^(Module.finrank ℝ E-1)) * |
| l^(Module.finrank ℝ E-1) ≤ (μ.toSphere (ball θ l)).toReal ∧ |
| (μ.toSphere (ball θ l)).toReal ≤ |
| 4^Module.finrank ℝ E * (μ (ball (0:E) 1)).toReal * l^(Module.finrank ℝ E-1) := by |
| 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 hbf : μ (ball (0:E) 1) ≠ ⊤ := (isBounded_ball.measure_lt_top (μ := μ)).ne |
| have hcf : μ.toSphere (ball θ l) ≠ ⊤ := measure_ne_top _ _ |
| have hlo := cap_lower_product μ θ hl hl1 |
| rw [μ.addHaar_ball _ (by positivity)] at hlo |
| have hlor := ENNReal.toReal_mono (ENNReal.mul_ne_top hcf ENNReal.ofReal_ne_top) hlo |
| simp only [ENNReal.toReal_mul,ENNReal.toReal_ofReal (show 0 ≤ (l/4)^Module.finrank ℝ E by positivity), |
| ENNReal.toReal_ofReal (show 0 ≤ (2:ℝ)^(Module.finrank ℝ E-1)*l by positivity)] at hlor |
| have hhi := cap_upper_product μ θ hl hl1 |
| rw [μ.addHaar_ball _ (by positivity)] at hhi |
| have hhir := ENNReal.toReal_mono (ENNReal.mul_ne_top ENNReal.ofReal_ne_top hbf) hhi |
| simp only [ENNReal.toReal_mul, |
| ENNReal.toReal_ofReal (show 0 ≤ (2*l)^Module.finrank ℝ E by positivity), |
| ENNReal.toReal_ofReal (show 0 ≤ (1/2:ℝ)^(Module.finrank ℝ E-1)*(l/2) by positivity)] at hhir |
| have hlpow : l^Module.finrank ℝ E = l^(Module.finrank ℝ E-1)*l := by |
| nth_rw 1 [← hn]; rw [pow_succ] |
| have h2pow : (2:ℝ)^Module.finrank ℝ E = 2^(Module.finrank ℝ E-1)*2 := by |
| nth_rw 1 [← hn]; rw [pow_succ] |
| have h4pow : (4:ℝ)^Module.finrank ℝ E = (2^Module.finrank ℝ E)^2 := by |
| rw [← pow_mul, mul_comm _ 2,pow_mul]; norm_num |
| constructor |
| · rw [div_pow,hlpow] at hlor |
| have h4 : (0:ℝ) < 4^Module.finrank ℝ E := by positivity |
| have h2 : (0:ℝ) < 2^(Module.finrank ℝ E-1) := by positivity |
| have hs : (μ (ball (0:E) 1)).toReal * l^(Module.finrank ℝ E-1) * l ≤ |
| (μ.toSphere (ball θ l)).toReal * (2^(Module.finrank ℝ E-1)*l) * 4^Module.finrank ℝ E := by |
| apply (div_le_iff₀ h4).mp |
| simpa only [div_mul_eq_mul_div,mul_comm,mul_left_comm,mul_assoc] using hlor |
| have hh : (μ (ball (0:E) 1)).toReal * l^(Module.finrank ℝ E-1) ≤ |
| (μ.toSphere (ball θ l)).toReal * (4^Module.finrank ℝ E * 2^(Module.finrank ℝ E-1)) := by |
| apply (mul_le_mul_iff_left₀ hl).mp |
| simpa only [mul_comm,mul_left_comm,mul_assoc] using hs |
| calc |
| _ = ((μ (ball (0:E) 1)).toReal * l^(Module.finrank ℝ E-1)) / |
| (4^Module.finrank ℝ E * 2^(Module.finrank ℝ E-1)) := by ring |
| _ ≤ _ := (div_le_iff₀ (mul_pos h4 h2)).mpr hh |
| · apply (mul_le_mul_iff_left₀ (show 0 < (1/2:ℝ)^(Module.finrank ℝ E-1)*(l/2) by positivity)).mp |
| calc |
| _ ≤ (2*l)^Module.finrank ℝ E * (μ (ball (0:E) 1)).toReal := hhir |
| _ = (4^Module.finrank ℝ E * (μ (ball (0:E) 1)).toReal * l^(Module.finrank ℝ E-1)) * |
| ((1/2:ℝ)^(Module.finrank ℝ E-1)*(l/2)) := by |
| rw [mul_pow,hlpow,h4pow,h2pow,div_pow] |
| field_simp |
| ring |
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| omit [BorelSpace E] [Nontrivial E] in |
| lemma cap_constants_pos : |
| 0 < (μ (ball (0:E) 1)).toReal / (4^Module.finrank ℝ E * 2^(Module.finrank ℝ E-1)) ∧ |
| 0 < (4:ℝ)^Module.finrank ℝ E * (μ (ball (0:E) 1)).toReal := by |
| have hp : 0 < (μ (ball (0:E) 1)).toReal := |
| ENNReal.toReal_pos (ne_of_gt (Metric.measure_ball_pos μ 0 (by norm_num))) |
| (isBounded_ball.measure_lt_top (μ := μ)).ne |
| constructor <;> positivity |
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| end HenonLaneEmden.CapGeometry |
| end |
| end CapGeometry |
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| end OAI |
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