| import Mathlib |
| import OAI.Geometry.CAT0Fillings.Bubble.Euclidean |
| import OAI.Geometry.CAT0Fillings.Transport.Normalization |
|
|
| namespace OAI |
|
|
| section |
|
|
| open Set Filter MeasureTheory |
| open scoped Topology NNReal |
|
|
| namespace CAT0Fillings.RadialSobolev |
|
|
| lemma bubble_intervals_tendsto {n : ℕ} (hn : 2 < n) : |
| Tendsto (fun D : ℝ => ∫ r in (0:ℝ)..D, r^(n-1)*(bubble n r)^(sobolevP n)) atTop (𝓝 (bubbleMass n)) ∧ |
| Tendsto (fun D : ℝ => ∫ r in (0:ℝ)..D, r^(n-1)*(bubble n r)^(sobolevQ n)) atTop (𝓝 (sobolevQ n*bubbleMass n)) ∧ |
| Tendsto (fun D : ℝ => ∫ r in (0:ℝ)..D, r^(n-1)*(bubble n r)^(sobolevP n)*r^2) atTop |
| (𝓝 ((n:ℝ)/(n-2)*bubbleMass n)) := by |
| have hi := bubble_moments_integrable hn |
| have he := bubble_moments hn |
| constructor |
| · simpa only [id_eq,he.1] using intervalIntegral_tendsto_integral_Ioi 0 hi.1 tendsto_id |
| constructor |
| · simpa only [id_eq,he.2.1] using intervalIntegral_tendsto_integral_Ioi 0 hi.2.1 tendsto_id |
| · simpa only [id_eq,he.2.2] using intervalIntegral_tendsto_integral_Ioi 0 hi.2.2 tendsto_id |
|
|
| lemma sobolev_power_complement {n : ℕ} (hn : 2 < n) : 1-2/sobolevP n = 2/(n:ℝ) := by |
| have h : (0:ℝ) < n := by exact_mod_cast (show 0 < n by omega) |
| have hd : (n:ℝ)-2 ≠ 0 := ne_of_gt (sub_pos.mpr (by exact_mod_cast hn)) |
| dsimp [sobolevP] |
| field_simp |
| ring |
|
|
| lemma bubble_normalization_energy {n : ℕ} {F : ℝ} (hn : 2 < n) (hF : 0 < F) : |
| bubbleMass n*(F/bubbleMass n)^(2/sobolevP n) = |
| (bubbleMass n)^(2/(n:ℝ))*F^(2/sobolevP n) := by |
| have hI := bubble_mass_pos hn |
| rw [Real.div_rpow hF.le hI.le] |
| calc |
| _ = ((bubbleMass n)^(1:ℝ)/(bubbleMass n)^(2/sobolevP n))*F^(2/sobolevP n) := by rw [Real.rpow_one]; ring |
| _ = _ := by rw [←Real.rpow_sub hI,sobolev_power_complement hn] |
|
|
| theorem radial_sharp {n : ℕ} {R : ℝ} {f : ℝ → ℝ} {K : ℝ≥0} |
| (hn : 2 < n) (hR : 0 < R) (hf : LipschitzWith K f) |
| (hfp : ∀ r ∈ Ioo 0 R, 0 < f r) (hfR : f R = 0) : |
| (n:ℝ)*(n-2)*(bubbleMass n)^(2/(n:ℝ))* |
| (∫ r in (0:ℝ)..R,r^(n-1)*(f r)^(sobolevP n))^(2/sobolevP n) ≤ |
| ∫ r in (0:ℝ)..R,r^(n-1)*(deriv f r)^2 := by |
| let F := ∫ r in (0:ℝ)..R,r^(n-1)*(f r)^(sobolevP n) |
| let E := ∫ r in (0:ℝ)..R,r^(n-1)*(deriv f r)^2 |
| have hs := sobolev_exponents hn |
| have hF : 0 < F := radial_integral_pos (by omega) hs.1 hR hf.continuous hfp |
| have hI := bubble_mass_pos hn |
| have hi := bubble_intervals_tendsto hn |
| have hdiv : Tendsto (fun D : ℝ => F/(∫ r in (0:ℝ)..D,r^(n-1)*(bubble n r)^(sobolevP n))) |
| atTop (𝓝 (F/bubbleMass n)) := tendsto_const_nhds.div hi.1 hI.ne' |
| have hp : ContinuousAt (fun x : ℝ => x^(2/sobolevP n)) (F/bubbleMass n) := |
| (Real.continuousAt_rpow_const _ _ (Or.inl (div_pos hF hI).ne')) |
| have hl := ((hi.2.1.pow 2).const_mul ((n:ℝ)^2)).mul (hp.tendsto.comp hdiv) |
| have hr := hi.2.2.const_mul ((sobolevQ n)^2*E) |
| have ht := le_of_tendsto_of_tendsto hl hr (by |
| filter_upwards [eventually_gt_atTop (0:ℝ)] with D hD |
| exact normalize_transport_estimate hn hR hD hf hfp hfR) |
| have hnR : (2:ℝ) < n := by exact_mod_cast hn |
| have hq : 0 < sobolevQ n := lt_of_lt_of_le zero_lt_one hs.2.1 |
| have he : (n:ℝ)*(n-2)*bubbleMass n*(F/bubbleMass n)^(2/sobolevP n) ≤ E := by |
| have hfactor : 0 < (sobolevQ n)^2*(n:ℝ)/(n-2)*bubbleMass n := by positivity |
| apply (mul_le_mul_iff_right₀ hfactor).mp |
| convert ht using 1 <;> dsimp [F,E] <;> field_simp [ne_of_gt (sub_pos.mpr hnR)] |
| rw [mul_assoc ((n:ℝ)*(n-2)),bubble_normalization_energy hn hF] at he |
| simpa only [mul_assoc] using he |
|
|
| end CAT0Fillings.RadialSobolev |
| end |
|
|
| section |
|
|
| open Set Filter MeasureTheory |
| open scoped Topology NNReal |
|
|
| namespace CAT0Fillings.RadialSobolev |
|
|
| lemma omega_pos {n : ℕ} (hn : 0 < n) : 0 < omega n := by |
| rw [omega_gamma hn] |
| have h : (0:ℝ) < n := by exact_mod_cast hn |
| exact div_pos (pow_pos (Real.sqrt_pos.mpr Real.pi_pos) _) (Real.Gamma_pos_of_pos (by positivity)) |
|
|
| lemma sphere_rpow_bubble {n : ℕ} (hn : 2 < n) : |
| (sphereArea n)^(2/(n:ℝ)) = |
| 4*((n:ℝ)*omega n*bubbleMass n)^(2/(n:ℝ)) := by |
| have hn0 : (0:ℝ) < n := by exact_mod_cast (show 0 < n by omega) |
| have hω := omega_pos (show 0 < n by omega) |
| rw [←bubble_mass_sphere hn, Real.mul_rpow (by positivity) |
| (mul_nonneg (mul_nonneg hn0.le hω.le) (bubble_mass_pos hn).le)] |
| have he : ((2:ℝ)^n)^(2/(n:ℝ)) = 4 := by |
| rw [←Real.rpow_natCast, ←Real.rpow_mul (by norm_num : (0:ℝ) ≤ 2)] |
| have hx : (n:ℝ)*(2/(n:ℝ)) = 2 := by field_simp |
| rw [hx,Real.rpow_two] |
| norm_num |
| rw [he] |
|
|
| lemma polar_sharp_coefficient {n : ℕ} {F E : ℝ} (hn : 2 < n) (hF : 0 ≤ F) |
| (hE : (n:ℝ)*(n-2)*(bubbleMass n)^(2/(n:ℝ))*F^(2/sobolevP n) ≤ E) : |
| (n:ℝ)*(sphereArea n)^(2/(n:ℝ))*((n:ℝ)*omega n*F)^(2/sobolevP n) ≤ |
| 4/(n-2)*((n:ℝ)*omega n*E) := by |
| have hn0 : (0:ℝ) < n := by exact_mod_cast (show 0 < n by omega) |
| have hn2 : (0:ℝ) < (n:ℝ)-2 := sub_pos.mpr (by exact_mod_cast hn) |
| have hω := omega_pos (show 0 < n by omega) |
| have hc : 0 < (n:ℝ)*omega n := mul_pos hn0 hω |
| have he : 2/(n:ℝ)+2/sobolevP n = 1 := by linarith [sobolev_power_complement hn] |
| have hp : ((n:ℝ)*omega n)^(2/(n:ℝ))*((n:ℝ)*omega n)^(2/sobolevP n) = (n:ℝ)*omega n := by |
| rw [←Real.rpow_add hc,he,Real.rpow_one] |
| have hh := mul_le_mul_of_nonneg_left hE (show 0 ≤ 4*((n:ℝ)*omega n)/(n-2) by positivity) |
| rw [sphere_rpow_bubble hn,Real.mul_rpow hc.le (bubble_mass_pos hn).le, |
| Real.mul_rpow hc.le hF] |
| have hh' : 4*(n:ℝ)*((n:ℝ)*omega n)*(bubbleMass n)^(2/(n:ℝ))*F^(2/sobolevP n) ≤ |
| 4/(n-2)*((n:ℝ)*omega n*E) := by |
| convert hh using 1 <;> first | rfl | field_simp [hn2.ne'] |
| calc |
| _ = 4*(n:ℝ)* |
| (((n:ℝ)*omega n)^(2/(n:ℝ))*((n:ℝ)*omega n)^(2/sobolevP n))* |
| (bubbleMass n)^(2/(n:ℝ))*F^(2/sobolevP n) := by ring |
| _ ≤ _ := by rw [hp]; exact hh' |
| |
| theorem radial_sharp_polar {n : ℕ} {R : ℝ} {f : ℝ → ℝ} {K : ℝ≥0} |
| (hn : 2 < n) (hR : 0 < R) (hf : LipschitzWith K f) |
| (hfp : ∀ r ∈ Ioo 0 R, 0 < f r) (hfR : f R = 0) : |
| (n:ℝ)*(sphereArea n)^(2/(n:ℝ))* |
| ((n:ℝ)*omega n*(∫ r in (0:ℝ)..R,r^(n-1)*(f r)^(sobolevP n)))^(2/sobolevP n) ≤ |
| 4/(n-2)*((n:ℝ)*omega n*(∫ r in (0:ℝ)..R,r^(n-1)*(deriv f r)^2)) := by |
| exact polar_sharp_coefficient hn |
| (radial_integral_pos (by omega) (sobolev_exponents hn).1 hR hf.continuous hfp).le |
| (radial_sharp hn hR hf hfp hfR) |
| |
| end CAT0Fillings.RadialSobolev |
| end |
| |
| end OAI |
| |
