| import OAI.Geometry.RuledCovering.CapAsymptotics |
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| namespace OAI |
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| noncomputable section |
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| open Set MeasureTheory Filter Metric |
| open scoped Topology ContDiff Pointwise |
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| namespace RuledApproximation |
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| def singularEnergy (p : ℝ) (z : Plane) : ℝ := |
| singularField p z 0^2+2*singularField p z 1^2 |
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| lemma singularEnergy_nonneg (p : ℝ) (z : Plane) : 0 ≤ singularEnergy p z := by |
| unfold singularEnergy |
| positivity |
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| lemma singularEnergy_measurable (p : ℝ) : Measurable (singularEnergy p) := by |
| unfold singularEnergy singularField |
| simp only [vec2_apply_zero,vec2_apply_one] |
| fun_prop |
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| lemma singularField_zero {p : ℝ} (hp : 1 < p) : singularField p 0 = 0 := by |
| ext i |
| fin_cases i <;> simp [singularField,vec2,Real.zero_rpow (show p-1 ≠ 0 by linarith)] |
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| lemma singularEnergy_zero {p : ℝ} (hp : 1 < p) : singularEnergy p 0 = 0 := by |
| simp [singularEnergy,singularField_zero hp] |
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| lemma singularEnergy_bounded {p : ℝ} (hp : 1 < p) : |
| ∃ C : ℝ, 0 ≤ C ∧ ∀ z ∈ coneTriangle 2, singularEnergy p z ≤ C := by |
| obtain ⟨C,hC,hbound⟩ := singularField_homogeneous_bounds p |
| let B := C*(2:ℝ)^(p-1) |
| have hB : 0 ≤ B := mul_nonneg (by linarith) (by positivity) |
| refine ⟨3*B^2,by positivity,?_⟩ |
| intro z hz |
| have hv : ‖singularField p z‖ ≤ B := by |
| by_cases hzr : z 1 = 0 |
| · have he : z = 0 := by |
| have hb : z 0 = 0 := abs_nonpos_iff.mp (hz.2.2.trans_eq hzr) |
| ext i |
| fin_cases i <;> assumption |
| rw [he,singularField_zero hp,norm_zero] |
| exact hB |
| · have hz1 : 0 < z 1 := lt_of_le_of_ne hz.1 (Ne.symm hzr) |
| exact (hbound z hz1 (hz.2.2.trans (by linarith))).1.trans |
| (mul_le_mul_of_nonneg_left (Real.rpow_le_rpow hz.1 hz.2.1 (by linarith)) (by linarith)) |
| have hx := (plane_coordinate_le (singularField p z) 0).trans hv |
| have hy := (plane_coordinate_le (singularField p z) 1).trans hv |
| have hx' := sq_le_sq₀ (abs_nonneg _) hB |>.mpr hx |
| have hy' := sq_le_sq₀ (abs_nonneg _) hB |>.mpr hy |
| simp only [sq_abs] at hx' hy' |
| unfold singularEnergy |
| nlinarith only [hx',hy'] |
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| lemma singularEnergy_integrable {p : ℝ} (hp : 1 < p) : |
| IntegrableOn (singularEnergy p) (coneTriangle 2) := by |
| obtain ⟨C,_,hC⟩ := singularEnergy_bounded hp |
| apply IntegrableOn.of_bound (coneTriangle_isCompact 2).measure_lt_top |
| (singularEnergy_measurable p).aestronglyMeasurable C |
| filter_upwards [ae_restrict_mem (coneTriangle_isCompact 2).measurableSet] with z hz |
| simpa only [Real.norm_eq_abs,abs_of_nonneg (singularEnergy_nonneg p z)] using hC z hz |
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| def energyCorrection (p c : ℝ) (z : Plane) : ℝ := |
| -singularEnergy p z / |
| (Real.sqrt (1+c^2*singularEnergy p z)*(Real.sqrt (1+c^2*singularEnergy p z)+1)) |
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| lemma energyCorrection_measurable (p c : ℝ) : Measurable (energyCorrection p c) := by |
| unfold energyCorrection |
| have hm := singularEnergy_measurable p |
| fun_prop |
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| lemma energy_sqrt_ge_one (p c : ℝ) (z : Plane) : |
| 1 ≤ Real.sqrt (1+c^2*singularEnergy p z) := by |
| have he := Real.sqrt_le_sqrt (show (1:ℝ) ≤ 1+c^2*singularEnergy p z by |
| nlinarith only [mul_nonneg (sq_nonneg c) (singularEnergy_nonneg p z)]) |
| simpa only [Real.sqrt_one] using he |
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| lemma energyCorrection_norm (p c : ℝ) (z : Plane) : |
| ‖energyCorrection p c z‖ ≤ singularEnergy p z/2 := by |
| have hH := singularEnergy_nonneg p z |
| have hs := energy_sqrt_ge_one p c z |
| unfold energyCorrection |
| rw [Real.norm_eq_abs,abs_div,abs_neg,abs_of_nonneg hH, |
| abs_of_pos (by nlinarith : 0 < Real.sqrt (1+c^2*singularEnergy p z)*(Real.sqrt (1+c^2*singularEnergy p z)+1))] |
| apply div_le_div_of_nonneg_left hH (by norm_num) |
| nlinarith |
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| lemma physicalWeight_energy (p c : ℝ) (z : Plane) : |
| physicalWeight (Real.sqrt 2) (c • singularField p z) = |
| 1/Real.sqrt (1+c^2*singularEnergy p z) := by |
| unfold physicalWeight singularEnergy |
| simp only [PiLp.smul_apply,smul_eq_mul,Real.sq_sqrt (show (0:ℝ) ≤ 2 by norm_num)] |
| congr 2 |
| ring |
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| lemma physicalWeight_energy_expansion (p c : ℝ) (z : Plane) : |
| physicalWeight (Real.sqrt 2) (c • singularField p z) = 1+c^2*energyCorrection p c z := by |
| have hH := singularEnergy_nonneg p z |
| have hs := energy_sqrt_ge_one p c z |
| have hs0 : Real.sqrt (1+c^2*singularEnergy p z) ≠ 0 := by linarith |
| have hs1 : Real.sqrt (1+c^2*singularEnergy p z)+1 ≠ 0 := by linarith |
| have he := Real.sq_sqrt (show 0 ≤ 1+c^2*singularEnergy p z by positivity) |
| rw [physicalWeight_energy] |
| unfold energyCorrection |
| field_simp |
| nlinarith only [he] |
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| lemma energyCorrection_tendsto (p : ℝ) (z : Plane) : |
| Tendsto (fun c : ℝ => energyCorrection p c z) (𝓝[>] 0) (𝓝 (-singularEnergy p z/2)) := by |
| have h0 : Tendsto (fun c : ℝ => c) (𝓝[>] 0) (𝓝 0) := tendsto_id.mono_left nhdsWithin_le_nhds |
| have hs : Tendsto (fun c : ℝ => Real.sqrt (1+c^2*singularEnergy p z)) (𝓝[>] 0) (𝓝 1) := by |
| have h : Tendsto (fun c : ℝ => 1+c^2*singularEnergy p z) (𝓝[>] 0) (𝓝 1) := by |
| simpa using ((h0.pow 2).mul_const (singularEnergy p z)).const_add 1 |
| simpa only [Real.sqrt_one] using h.sqrt |
| have hd : Tendsto (fun c : ℝ => Real.sqrt (1+c^2*singularEnergy p z)*(Real.sqrt (1+c^2*singularEnergy p z)+1)) (𝓝[>] 0) (𝓝 2) := by |
| convert hs.mul (hs.add (tendsto_const_nhds (x := (1:ℝ)))) using 1 |
| norm_num |
| exact tendsto_const_nhds.div hd (by norm_num) |
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| lemma singularDomain_subset_cone (p gamma c : ℝ) : singularDomain p gamma c ⊆ coneTriangle 2 := by |
| intro z hz |
| exact ⟨(abs_nonneg _).trans hz.2.2.1,hz.2.1,hz.2.2.1⟩ |
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| lemma singularDomain_eventually_mem {p gamma : ℝ} (hp : 0 ≤ p) (hp2 : p ≤ 2) (hg : 0 < gamma) |
| {z : Plane} (hz : z ∈ coneTriangle 2) (hz1 : 0 < z 1) : |
| ∀ᶠ c : ℝ in 𝓝[>] 0, z ∈ singularDomain p gamma c ↔ z ∈ coneTriangle (1/2) := by |
| by_cases hr : z 1 ≤ 1/2 |
| · filter_upwards [self_mem_nhdsWithin, |
| (right_rpow_tendsto_zero hg).eventually (gt_mem_nhds hz1)] with c hc hd |
| have hm : z ∈ singularDomain p gamma c := by |
| refine ⟨hd.le,hz.2.1,hz.2.2,hr.trans ?_⟩ |
| have hn := ceiling_nonneg hp hp2 (z 0) |
| have hn' : 0 ≤ c^2*ceilingPolynomial p (z 0)+c^(5/2:ℝ) := |
| add_nonneg (mul_nonneg (sq_nonneg c) hn) (Real.rpow_nonneg hc.le _) |
| linarith |
| exact ⟨fun _ => ⟨hz.1,hr,hz.2.2⟩,fun _ => hm⟩ |
| · have ht : Tendsto (fun c : ℝ => (1/2:ℝ)+c^2*ceilingPolynomial p (z 0)+c^(5/2:ℝ)) |
| (𝓝[>] 0) (𝓝 (1/2:ℝ)) := by |
| have h0 : Tendsto (fun c : ℝ => c) (𝓝[>] 0) (𝓝 0) := tendsto_id.mono_left nhdsWithin_le_nhds |
| simpa using (tendsto_const_nhds.add ((h0.pow 2).mul_const (ceilingPolynomial p (z 0)))).add |
| (right_rpow_tendsto_zero (q := (5/2:ℝ)) (by norm_num)) |
| filter_upwards [ht.eventually (gt_mem_nhds (lt_of_not_ge hr))] with c hc |
| exact ⟨fun h => False.elim (not_le_of_gt hc h.2.2.2),fun h => False.elim (hr h.2.1)⟩ |
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| lemma energyCorrection_integral_tendsto {p gamma : ℝ} (hp : 1 < p) (hp2 : p ≤ 2) (hg : 0 < gamma) : |
| Tendsto (fun c : ℝ => ∫ z in singularDomain p gamma c, energyCorrection p c z) (𝓝[>] 0) |
| (𝓝 (-(∫ z in coneTriangle (1/2), singularEnergy p z)/2)) := by |
| have ht : Tendsto (fun c : ℝ => ∫ z in coneTriangle 2, |
| (singularDomain p gamma c).indicator (energyCorrection p c) z) (𝓝[>] 0) |
| (𝓝 (∫ z in coneTriangle 2, (coneTriangle (1/2)).indicator (fun z => -singularEnergy p z/2) z)) := by |
| apply tendsto_integral_filter_of_dominated_convergence (fun z => singularEnergy p z/2) |
| · exact Filter.Eventually.of_forall (fun c => |
| ((energyCorrection_measurable p c).indicator (singularDomain_isCompact p gamma c).measurableSet).aestronglyMeasurable) |
| · exact Filter.Eventually.of_forall (fun c => Filter.Eventually.of_forall (fun z => by |
| by_cases hz : z ∈ singularDomain p gamma c |
| · rw [Set.indicator_of_mem hz] |
| exact energyCorrection_norm p c z |
| · rw [Set.indicator_of_notMem hz,norm_zero] |
| exact div_nonneg (singularEnergy_nonneg p z) (by norm_num))) |
| · exact (singularEnergy_integrable hp).div_const 2 |
| · filter_upwards [ae_restrict_mem (coneTriangle_isCompact 2).measurableSet] with z hz |
| by_cases hzr : z 1 = 0 |
| · have he : z = 0 := by |
| have hb : z 0 = 0 := abs_nonpos_iff.mp (hz.2.2.trans_eq hzr) |
| ext i |
| fin_cases i <;> assumption |
| subst z |
| have he : ∀ c : ℝ, energyCorrection p c 0 = 0 := by simp [energyCorrection,singularEnergy_zero hp] |
| simp [Set.indicator,he,singularEnergy_zero hp] |
| · have he := singularDomain_eventually_mem (le_trans (by norm_num) hp.le) hp2 hg hz |
| (lt_of_le_of_ne hz.1 (Ne.symm hzr)) |
| by_cases hza : z ∈ coneTriangle (1/2) |
| · rw [Set.indicator_of_mem hza] |
| apply (energyCorrection_tendsto p z).congr' |
| filter_upwards [he] with c hc |
| exact (Set.indicator_of_mem (hc.mpr hza) _).symm |
| · rw [Set.indicator_of_notMem hza] |
| apply tendsto_const_nhds.congr' |
| filter_upwards [he] with c hc |
| exact (Set.indicator_of_notMem (fun h => hza (hc.mp h)) _).symm |
| have he₁ : ∀ c : ℝ, (∫ z in coneTriangle 2, (singularDomain p gamma c).indicator (energyCorrection p c) z) = |
| ∫ z in singularDomain p gamma c, energyCorrection p c z := by |
| intro c |
| rw [integral_indicator (singularDomain_isCompact p gamma c).measurableSet, |
| Measure.restrict_restrict (singularDomain_isCompact p gamma c).measurableSet, |
| Set.inter_eq_left.mpr (singularDomain_subset_cone p gamma c)] |
| have he₂ : (∫ z in coneTriangle 2, (coneTriangle (1/2)).indicator (fun z => -singularEnergy p z/2) z) = |
| -(∫ z in coneTriangle (1/2), singularEnergy p z)/2 := by |
| rw [integral_indicator (coneTriangle_isCompact (1/2)).measurableSet, |
| Measure.restrict_restrict (coneTriangle_isCompact (1/2)).measurableSet, |
| Set.inter_eq_left.mpr (show coneTriangle (1/2) ⊆ coneTriangle 2 from fun z hz => |
| ⟨hz.1,hz.2.1.trans (by norm_num),hz.2.2⟩),integral_div,integral_neg] |
| simpa only [he₁,he₂] using ht |
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| end RuledApproximation |
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| end |
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| end OAI |
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