File size: 5,996 Bytes
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 | import Mathlib
import OAI.Geometry.CAT0Fillings.Minimizers.Minimum
import OAI.Geometry.CAT0Fillings.Radial.ClosedInequality
import OAI.Geometry.CAT0Fillings.Powers.PolynomialBound
import OAI.Geometry.CAT0Fillings.Euler.Contradiction
namespace OAI
section
open Set Filter MeasureTheory Metric TopologicalSpace
open scoped Topology NNReal ENNReal
namespace CAT0Fillings
open ChartGeometry AnalyticMinimizer MassMeasure Rearrangement RadialSobolev
variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X]
[CompactSpace X] [Nonempty X]
theorem extremal_euler_powers {k : ℕ} (hk : 0 < k) (hX : IsCAT0 X)
{T : Functional X (k+2)} (hT : IsIntegral (k+2) T) (hz : boundarySucc T = 0)
(hm : 0 < mass T) (q : ChartGeometry hT.1)
{d r : ℝ} (hd : 0 < d) (hr : 1 < r)
(hfill : ∀ P : Functional X (k+1), IsIntegral (k+1) P → boundarySucc P = 0 →
∃ R : Functional X (k+2), IsIntegral (k+2) R ∧ boundarySucc R = P ∧
mass R ≤ fillingCoefficient (k+1)*(mass P)^(fillingPower (k+1)))
(hext : ∀ B : Functional X (k+2), IsIntegral (k+2) B → boundarySucc B = 0 →
d*((mass T)^r-(mass B)^r) ≤ fillingVolume (T-B))
(v : q.Sobolev)
(hv : ∀ᵐ x ∂currentMassMeasure hT.1, 0 ≤ q.inclusion v x)
(heuler : ∀ ψ : q.Sobolev,
(4/((k+2:ℝ)-2))*inner ℝ (q.closedGradient v) (q.closedGradient ψ) +
(k+2:ℝ)*inner ℝ (q.inclusion v) (q.inclusion ψ) =
(k+2:ℝ)*(∫ x, ((q.inclusion v) x)^(sobolevP (k+2)-1)*(q.inclusion ψ) x ∂currentMassMeasure hT.1)) :
(∀ b : ℝ, 0 < b → MemLp (q.inclusion v) (ENNReal.ofReal b) (currentMassMeasure hT.1)) ∧
(∀ γ : ℝ, 1 ≤ γ → ∃ R : q.Sobolev,
(q.inclusion R : X → ℝ) =ᵐ[currentMassMeasure hT.1] (fun x => (q.inclusion v x)^γ) ∧
(q.closedGradient R : _ → _) =ᵐ[q.atlasMeasure]
(fun w => (γ*(q.inclusion v (q.atlasParam w))^(γ-1)) • q.closedGradient v w)) := by
have hn : (2:ℝ) < k+2 := by exact_mod_cast (show 2 < k+2 by omega)
have hk0 : 0 < (k+2:ℝ)-2 := by linarith
have hp : 2 < sobolevP (k+2) := by
dsimp [sobolevP]
push_cast
exact (lt_div_iff₀ hk0).mpr (by linarith)
have hB : 0 < (k+2:ℝ) := by positivity
have hsphere : 0 < sphereArea (k+2) := by
unfold sphereArea
exact mul_pos (by positivity) (omega_pos (by omega))
let S := (k+2:ℝ)*sphereArea (k+2)^(2/(k+2:ℝ))
have hS : 0 < S := mul_pos hB (Real.rpow_pos_of_pos hsphere _)
obtain ⟨C,hC,hb⟩ := extremal_closed_sobolev hk hX hT hz hm q hd hr hfill hext
(show 0 < S/2 by positivity) (show S/2 < S by linarith)
exact q.closed_all_powers hz v hp (div_pos (by norm_num) hk0) hB
(show 0 < S-S/2 by linarith) hC hv hb heuler
end CAT0Fillings
end
section
open Set Filter MeasureTheory Metric
open scoped Topology NNReal ENNReal
namespace CAT0Fillings
open ChartGeometry AnalyticMinimizer MassMeasure Rearrangement RadialSobolev
variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X]
[CompactSpace X] [Nonempty X]
theorem normalized_extremal_contradiction {k : ℕ} (hk : 0 < k) (hX : IsCAT0 X)
{T : Functional X (k+2)} (hT : IsIntegral (k+2) T) (hz : boundarySucc T = 0)
(hm : 0 < mass T) (hms : mass T < sphereArea (k+2))
{d r : ℝ} (hd : 0 < d) (hr : 1 < r)
(hnorm : d*r*(mass T)^(r-1) = 1/(k+2:ℝ))
(hfill : ∀ P : Functional X (k+1), IsIntegral (k+1) P → boundarySucc P = 0 →
∃ R : Functional X (k+2), IsIntegral (k+2) R ∧ boundarySucc R = P ∧
mass R ≤ fillingCoefficient (k+1)*(mass P)^(fillingPower (k+1)))
(hext : ∀ B : Functional X (k+2), IsIntegral (k+2) B → boundarySucc B = 0 →
d*((mass T)^r-(mass B)^r) ≤ fillingVolume (T-B)) : False := by
obtain ⟨q⟩ := exists_chartGeometry hX hT.1 hT.2.1
obtain ⟨v,hv,hW,hWs,hE,heuler,hmin⟩ := extremal_normalized_minimum hk hX hT hz hm hms q hd hr hfill hext
obtain ⟨hmom,hpow⟩ := extremal_euler_powers hk hX hT hz hm q hd hr hfill hext v hv heuler
let β : ℝ := 1/((k+2:ℝ)-2)
have hk0 : 0 < (k:ℝ) := Nat.cast_pos.mpr hk
have hb : 0 < β := by dsimp [β]; exact div_pos (by norm_num) (by linarith)
have hnb : (k+2:ℝ)*β = 1+2*β := by dsimp [β]; simp only [add_sub_cancel_right]; field_simp [hk0.ne']
have hp : sobolevP (k+2) = 2+4*β := by
dsimp [sobolevP,β]
push_cast
simp only [add_sub_cancel_right]
field_simp [hk0.ne']
ring
have ha : 4/((k+2:ℝ)-2) = 4*β := by dsimp [β]; ring
have hψ : ∀ ψ : q.Sobolev,
4*β*inner ℝ (q.closedGradient v) (q.closedGradient ψ) +
(k+2:ℝ)*inner ℝ (q.inclusion v) (q.inclusion ψ) =
(k+2:ℝ)*(∫ x, ((q.inclusion v) x)^(1+4*β)*(q.inclusion ψ) x ∂currentMassMeasure hT.1) := by
intro ψ
simpa only [hp,ha,show 2+4*β-1 = 1+4*β by ring] using heuler ψ
have hrad (o : X) (g : X → ℝ) (K : ℝ≥0) (hg : LipschitzWith K g)
(hg0 : ∀ x, 0 ≤ g x) (hg1 : ∀ x, g x ≤ 1) :
q.radialVariation o g ≤ (k+2:ℝ)*q.sweptMass o g := by
have h := radial_inequality_of_extremal hX hT hz q hd.le (by linarith : 0 ≤ r) hm hext o hg hg0 hg1
rw [hnorm] at h
have hh := mul_le_mul_of_nonneg_left h (show 0 ≤ (k+2:ℝ) by positivity)
simpa only [←mul_assoc,mul_one_div_cancel (show (k+2:ℝ) ≠ 0 by positivity),one_mul] using hh
apply q.euler_chord_contradiction hX (by omega) hz v hb ?_ hv hmom hpow ?_ ?_ ?_ ?_ ?_ ?_
· simpa only [Nat.cast_add,Nat.cast_one,add_assoc,show (1:ℝ)+1=2 by norm_num] using hnb
· simpa only [Nat.cast_add,Nat.cast_one,add_assoc,show (1:ℝ)+1=2 by norm_num] using hψ
· simpa only [Nat.cast_add,Nat.cast_one,add_assoc,show (1:ℝ)+1=2 by norm_num] using hrad
· rwa [hp] at hW
· rwa [hp] at hWs
· simpa only [hp,ha,Nat.cast_add,Nat.cast_one,add_assoc,show (1:ℝ)+1=2 by norm_num] using hE
· simpa only [hp,ha,Nat.cast_add,Nat.cast_one,add_assoc,show (1:ℝ)+1=2 by norm_num] using hmin
end CAT0Fillings
end
end OAI
|