File size: 3,344 Bytes
59d1c80 | 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 | import OAI.Geometry.IsometricImmersion.Caps.FiniteCapInduction
namespace OAI
noncomputable section
open Set Filter Function MeasureTheory
open scoped ContDiff Topology BigOperators ENNReal NNReal
namespace SmoothLocal.Flow
open SmoothLocal.Geometry SmoothLocal.ODE SmoothLocal.Weighted SmoothLocal.Model
open SmoothLocal.HighEquation SmoothLocal.Analytic SmoothLocal.Sobolev
def lowerCapPointwiseRequests {bStar : ℝ} (n : ℕ) (r : LowerCapRectangle bStar) :
List (ℕ × LowerCapRectangle bStar) := lowerCapMetricRequests (n-6) r.expand
theorem lowerCapPointwiseRequests_length {bStar : ℝ} (n : ℕ)
(r : LowerCapRectangle bStar) :
(lowerCapPointwiseRequests n r).length = 2*(n-6) :=
lowerCapMetricRequests_length (n-6) r.expand
theorem lowerCapPointwiseRequests_full_order_le {bStar : ℝ} (n : ℕ)
(r : LowerCapRectangle bStar) (b : ℕ × LowerCapRectangle bStar)
(hb : b ∈ lowerCapPointwiseRequests n r) {k : ℕ} (hk : k ≤ b.1+2) :
k ≤ max 8 (n+3) := by
by_cases hn : n ≤ 6
· have hn0 : n-6=0 := Nat.sub_eq_zero_of_le hn
simp [lowerCapPointwiseRequests,hn0,lowerCapMetricRequests] at hb
· have h := lowerCapMetricRequests_full_order_le (n-6) r.expand b hb hk
omega
theorem uniform_finite_varying_lowerCap_pointwise
(G Z d c e0 kappa : ℝ) (hG : 0 ≤ G) (hZ : 0 ≤ Z) (hd : 0 < d) (hc : 0 < c)
{bStar : ℝ} (J : CapMetricJetBudget bStar) (n : ℕ) (r : LowerCapRectangle bStar)
(hJ : ∀ b ∈ lowerCapPointwiseRequests n r, 0 ≤ J b.1 b.2) :
∃ B : ℝ, 0 ≤ B ∧
∀ (g : MetricField) (U : Set Coord) (z : Coord → ℝ) (Y : ℝ → ℝ → ℝ) (W : Set Coord),
SmoothPositiveOn g U → IsOpen U → modelSquare ⊆ U →
(∀ i j : Fin 2, ∀ k ≤ 4, ∀ p ∈ modelSquare,
‖iteratedFDeriv ℝ k (fun q => g q i j) p‖ ≤ G) →
(∀ p ∈ modelSquare, d ≤ |(g p).det|) →
CapInductionHeight g U Z c e0 z → CapInductionFlow g U G Z d c e0 kappa z Y W →
FiniteCapMetricJets g Y J (lowerCapPointwiseRequests n r) →
CoordinateBound z (r.image Y) n B := by
obtain ⟨H,hH⟩ := uniform_finite_varying_lowerCap_L2 G Z d c e0 kappa hG hZ hd hc
J (n-6) r.expand hJ
let M := heightQuotientJetBound G Z d c
let B := (squareSobolevWeight (flowInteriorRadius M r.margin r.margin)+1)*(H : ℝ)
have hr := flowInteriorRadius_pos M r.margin_pos r.margin_pos
have hB : 0 ≤ B := by
dsimp only [B,squareSobolevWeight]
positivity
refine ⟨B,hB,?_⟩
intro g U z Y W hg hU hSU hgB hdet hh hf hlocal
exact CapInductionFlow.pointwise_from_larger_L2 hh hf hG hZ hd hc r
((hH g U z Y W hg hU hSU hgB hdet hh hf hlocal).mono
(by omega : n+2 ≤ (n-6)+8) le_rfl)
theorem coordinateL2Bound_of_pointwise_in_modelSquare
{z : Coord → ℝ} {V : Set Coord} {n : ℕ} {B : ℝ}
(hV : MeasurableSet V) (hVS : V ⊆ modelSquare) (hB : 0 ≤ B)
(hbound : CoordinateBound z V n B) :
CoordinateL2Bound z V n (originalC8L2Budget B) := by
intro ds hds
rw [originalC8L2Budget_coe]
exact (bounded_real_eLpNorm_two hV hB (fun p hp => by
rw [Real.norm_eq_abs]
exact hbound ds hds p hp)).trans
(mul_le_mul' le_rfl (ENNReal.rpow_le_rpow (z := 1/(2 : ℝ))
(measure_mono hVS) (by norm_num)))
end SmoothLocal.Flow
end
end OAI
|