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8e4d533 | 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 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 | import Mathlib
import OAI.Geometry.CAT0Fillings.Bubble.Moments
import OAI.Geometry.CAT0Fillings.Transport.Integrals
import OAI.Geometry.CAT0Fillings.Transport.Cauchy
namespace OAI
section
open Set Filter MeasureTheory
open scoped Topology NNReal
namespace CAT0Fillings.RadialSobolev
lemma finite_transport_estimate {n : ℕ} {ω p q R D : ℝ} {f g : ℝ → ℝ} {K : ℝ≥0}
(hR : 0 ≤ R) (hD : 0 < D) (hn : 2 ≤ n) (hω : 0 < ω)
(hf : LipschitzWith K f) (hg : Continuous g) (hp : 0 < p) (hq : 1 ≤ q)
(hfp : ∀ r ∈ Ioo 0 R, 0 < f r) (hgp : ∀ r, 0 < g r) (hfR : f R = 0)
(hmass : radialCDF n ω f p R = radialCDF n ω g p D)
(hexp : q+p/(n:ℝ)=p) (hexp2 : 2*(q-1)=p) :
(n:ℝ)^2*(∫ s in (0:ℝ)..D,s^(n-1)*(g s)^q)^2 ≤
q^2*(∫ r in (0:ℝ)..R,r^(n-1)*(deriv f r)^2)*
(∫ s in (0:ℝ)..D,s^(n-1)*(g s)^p*s^2) := by
have hn0 : 0 < n := by omega
let t := radialTransport (f := f) hD.le hn0 hω hg hp (fun r _ => hgp r)
have ht : Continuous t := radialTransport_continuous _ _ _ hf.continuous hg hp _
have hz : Continuous (fun r => (f r)^(q-1)*t r) :=
(hf.continuous.rpow_const (fun _ => Or.inr (sub_nonneg.mpr hq))).mul ht
have hcs := weighted_interval_cs hR (continuous_id.pow (n-1)) hz hf
(fun r hr => pow_nonneg hr.1 (n-1))
have hI : (∫ r in (0:ℝ)..R,r^(n-1)*deriv f r*((f r)^(q-1)*t r)) =
∫ r in (0:ℝ)..R,r^(n-1)*(f r)^(q-1)*deriv f r*t r := by
apply intervalIntegral.integral_congr
intro r hr
ring
have hM : (∫ r in (0:ℝ)..R,r^(n-1)*((f r)^(q-1)*t r)^2) =
∫ s in (0:ℝ)..D,s^(n-1)*(g s)^p*s^2 := by
rw [radialTransport_integral_moment_unscaled hR hD hn0 hω hf.continuous hg hp hfp
(fun r _ => hgp r) hmass (fun s => s^2)]
apply intervalIntegral.integral_congr_Ioo_of_le hR
intro r hr
dsimp only
rw [mul_pow,←Real.rpow_mul_natCast (hfp r hr).le]
have he : (q-1)*(2:ℝ)=p := by linarith
rw [Nat.cast_ofNat,he]
ring
dsimp only [Pi.pow_apply, id_eq] at hcs
rw [hI,hM] at hcs
have hibp := finite_transport_ibp hR hD hn hω hf hg hp hq hfp hgp hfR hmass hexp
have hJ0 : 0 ≤ (n:ℝ)*(∫ s in (0:ℝ)..D,s^(n-1)*(g s)^q) := by
apply mul_nonneg (Nat.cast_nonneg n)
apply intervalIntegral.integral_nonneg hD.le
intro s hs
exact mul_nonneg (pow_nonneg hs.1 _) (Real.rpow_nonneg (hgp s).le _)
have hsq := (sq_le_sq₀ hJ0 (hJ0.trans hibp)).mpr hibp
have hsq' := mul_le_mul_of_nonneg_left hcs (sq_nonneg q)
calc
_ ≤ (-q*(∫ r in (0:ℝ)..R,r^(n-1)*(f r)^(q-1)*deriv f r*t r))^2 := by
simpa only [mul_pow] using hsq
_ = q^2*(∫ r in (0:ℝ)..R,r^(n-1)*(f r)^(q-1)*deriv f r*t r)^2 := by ring
_ ≤ _ := by simpa only [mul_assoc] using hsq'
end CAT0Fillings.RadialSobolev
end
section
open Real Set Filter MeasureTheory
open scoped Topology NNReal
namespace CAT0Fillings.RadialSobolev
lemma radial_integral_smul (n : ℕ) (p A D : ℝ) (g : ℝ → ℝ)
(hA : 0 ≤ A) (hg : ∀ r, 0 ≤ g r) :
(∫ r in (0:ℝ)..D,r^(n-1)*(A*g r)^p) =
A^p*∫ r in (0:ℝ)..D,r^(n-1)*(g r)^p := by
rw [←intervalIntegral.integral_const_mul]
apply intervalIntegral.integral_congr
intro r hr
dsimp only
rw [Real.mul_rpow hA (hg r)]
ring
lemma radial_second_moment_smul (n : ℕ) (p A D : ℝ) (g : ℝ → ℝ)
(hA : 0 ≤ A) (hg : ∀ r, 0 ≤ g r) :
(∫ r in (0:ℝ)..D,r^(n-1)*(A*g r)^p*r^2) =
A^p*∫ r in (0:ℝ)..D,r^(n-1)*(g r)^p*r^2 := by
rw [←intervalIntegral.integral_const_mul]
apply intervalIntegral.integral_congr
intro r hr
dsimp only
rw [Real.mul_rpow hA (hg r)]
ring
lemma radial_integral_pos {n : ℕ} {p R : ℝ} {f : ℝ → ℝ}
(hn : 0 < n) (hp : 0 < p) (hR : 0 < R) (hf : Continuous f)
(hfp : ∀ r ∈ Ioo 0 R, 0 < f r) :
0 < ∫ r in (0:ℝ)..R,r^(n-1)*(f r)^p := by
have h := radialCDF_strictMonoOn (ω := 1) hn zero_lt_one hf hp hfp
(show (0:ℝ) ∈ Icc 0 R from ⟨le_rfl,hR.le⟩) (show R ∈ Icc 0 R from ⟨hR.le,le_rfl⟩) hR
rw [radialCDF_zero,radialCDF, mul_one] at h
exact (mul_pos_iff.mp h).resolve_right (by rintro ⟨hneg,_⟩; exact (not_lt_of_ge (Nat.cast_nonneg _)) hneg) |>.2
lemma normalization_power {p F I : ℝ} (hp : 0 < p) (hF : 0 < F) (hI : 0 < I) :
((F/I)^(1/p))^p = F/I := by
rw [←Real.rpow_mul (div_nonneg hF.le hI.le),one_div_mul_cancel hp.ne',Real.rpow_one]
lemma normalize_transport_estimate {n : ℕ} {R D : ℝ} {f : ℝ → ℝ} {K : ℝ≥0}
(hn : 2 < n) (hR : 0 < R) (hD : 0 < D) (hf : LipschitzWith K f)
(hfp : ∀ r ∈ Ioo 0 R, 0 < f r) (hfR : f R = 0) :
let F := ∫ r in (0:ℝ)..R,r^(n-1)*(f r)^(sobolevP n)
let I := ∫ r in (0:ℝ)..D,r^(n-1)*(bubble n r)^(sobolevP n)
let J := ∫ r in (0:ℝ)..D,r^(n-1)*(bubble n r)^(sobolevQ n)
let M := ∫ r in (0:ℝ)..D,r^(n-1)*(bubble n r)^(sobolevP n)*r^2
(n:ℝ)^2*J^2*(F/I)^(2/sobolevP n) ≤
(sobolevQ n)^2*(∫ r in (0:ℝ)..R,r^(n-1)*(deriv f r)^2)*M := by
dsimp only
let F := ∫ r in (0:ℝ)..R,r^(n-1)*(f r)^(sobolevP n)
let I := ∫ r in (0:ℝ)..D,r^(n-1)*(bubble n r)^(sobolevP n)
let A := (F/I)^(1/sobolevP n)
have hs := sobolev_exponents hn
have hF : 0 < F := radial_integral_pos (by omega) hs.1 hR hf.continuous hfp
have hI : 0 < I := radial_integral_pos (by omega) hs.1 hD (bubble_continuous n) (fun r _ => bubble_pos n r)
have hA : 0 < A := Real.rpow_pos_of_pos (div_pos hF hI) _
have hAp : A^(sobolevP n) = F/I := normalization_power hs.1 hF hI
have hm : radialCDF n 1 f (sobolevP n) R =
radialCDF n 1 (fun r => A*bubble n r) (sobolevP n) D := by
rw [radialCDF,radialCDF,radial_integral_smul n _ A D _ hA.le (fun r => (bubble_pos n r).le),hAp]
change (n:ℝ)*1*F = (n:ℝ)*1*(F/I*I)
rw [div_mul_cancel₀ _ hI.ne']
have ht := finite_transport_estimate hR.le hD (by omega) zero_lt_one hf
((bubble_continuous n).const_mul A) hs.1 hs.2.1 hfp
(fun r => mul_pos hA (bubble_pos n r)) hfR hm hs.2.2.1 hs.2.2.2.1
rw [radial_integral_smul _ _ _ _ _ hA.le (fun r => (bubble_pos n r).le),
radial_second_moment_smul _ _ _ _ _ hA.le (fun r => (bubble_pos n r).le),mul_pow] at ht
have he : (A^(sobolevQ n))^2 = A^(sobolevP n)*A^2 := by
rw [←Real.rpow_mul_natCast hA.le, ←Real.rpow_natCast, ←Real.rpow_add hA]
congr 1
norm_num only [Nat.cast_ofNat]
linarith [hs.2.2.2.2]
have heA : A^2 = (F/I)^(2/sobolevP n) := by
rw [←Real.rpow_mul_natCast (div_nonneg hF.le hI.le)]
change (F/I)^((1/sobolevP n)*(2:ℝ)) = _
congr 1
ring
rw [he] at ht
have hout := (mul_le_mul_iff_right₀ (Real.rpow_pos_of_pos hA (sobolevP n))).mp
(show A^(sobolevP n)*((n:ℝ)^2*
(∫ r in (0:ℝ)..D,r^(n-1)*(bubble n r)^(sobolevQ n))^2*A^2) ≤
A^(sobolevP n)*((sobolevQ n)^2*(∫ r in (0:ℝ)..R,r^(n-1)*(deriv f r)^2)*
(∫ r in (0:ℝ)..D,r^(n-1)*(bubble n r)^(sobolevP n)*r^2)) by convert ht using 1 <;> first | rfl | ring)
rwa [heA] at hout
end CAT0Fillings.RadialSobolev
end
end OAI
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