| 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 |
|
|
