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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 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 | import OAI.Geometry.IsometricImmersion.Energy.MovingIntervalSourceL2
import OAI.Geometry.IsometricImmersion.Calculus.CoordinateJetBounds
import Mathlib.Analysis.Calculus.FDeriv.Measurable
namespace OAI
noncomputable section
open Set Filter MeasureTheory
open scoped ContDiff Topology Interval ENNReal NNReal
namespace SmoothLocal.Hyperbolic
open SmoothLocal.Geometry SmoothLocal.Weighted SmoothLocal.ODE
def spatialFirstJet (z : Coord → ℝ) (i : Fin 2) : Coord → ℝ := coordPartial i z
theorem spatialJet_firstJet_eq_word (z : Coord → ℝ) (n : ℕ) (i : Fin 2) :
spatialJet (spatialFirstJet z i) n =
iteratedCoordPartial (List.replicate n 0 ++ [i]) z := by
induction n with
| zero => rfl
| succ n hn =>
simp only [spatialJet, List.replicate_succ, List.cons_append, iteratedCoordPartial, hn]
theorem coordinateBound_spatialFirstJet
{z : Coord → ℝ} {V : Set Coord} {N n : ℕ} {Z : ℝ}
(hz : CoordinateBound z V (N + 1) Z) (hn : n ≤ N) (i : Fin 2) {p : Coord} (hp : p ∈ V) :
|spatialJet (spatialFirstJet z i) n p| ≤ Z := by
rw [spatialJet_firstJet_eq_word]
apply hz _ _ p hp
simp only [List.length_append, List.length_replicate, List.length_singleton]
omega
def firstJetSliceBudget (Z widthUpper : ℝ) : ℝ≥0 :=
(‖Z‖ₑ * (ENNReal.ofReal widthUpper) ^ (1 / (2 : ℝ))).toNNReal
theorem firstJetSliceBudget_coe (Z widthUpper : ℝ) :
(firstJetSliceBudget Z widthUpper : ℝ≥0∞) =
‖Z‖ₑ * (ENNReal.ofReal widthUpper) ^ (1 / (2 : ℝ)) := by
exact ENNReal.coe_toNNReal (by finiteness)
theorem coordinateBound_to_spatialFirstJet_L2
{z : Coord → ℝ} {V : Set Coord} {theta left right Z widthUpper : ℝ} {N : ℕ}
(hz : CoordinateBound z V (N + 1) Z) (hZ : 0 ≤ Z)
(hseg : ∀ x ∈ Icc left right, coordinatePoint x theta ∈ V)
(hwidth : right - left ≤ widthUpper) :
SpatialSliceL2Bound (spatialFirstJet z) theta left right N
(firstJetSliceBudget Z widthUpper) := by
intro i n hn
have hmeas : Measurable (spatialJet (spatialFirstJet z i) n) := by
cases n with
| zero => exact measurable_fderiv_apply_const ℝ z (Pi.single i 1)
| succ order =>
exact measurable_fderiv_apply_const ℝ
(spatialJet (spatialFirstJet z i) order) (Pi.single 0 1)
have hpoint : Continuous (fun x : ℝ => coordinatePoint x theta) := by
unfold coordinatePoint
fun_prop
have hdom : ∀ᵐ x ∂volume.restrict (Icc left right),
‖spatialJet (spatialFirstJet z i) n (coordinatePoint x theta)‖ ≤ ‖Z‖ := by
filter_upwards [ae_restrict_mem measurableSet_Icc] with x hx
rw [Real.norm_eq_abs, Real.norm_of_nonneg hZ]
exact coordinateBound_spatialFirstJet hz hn i (hseg x hx)
have hvol : volume (Icc left right) ≤ ENNReal.ofReal widthUpper := by
rw [Real.volume_Icc]
exact ENNReal.ofReal_le_ofReal hwidth
calc
_ ≤ eLpNorm (fun _ : ℝ => Z) 2 (volume.restrict (Icc left right)) :=
eLpNorm_mono_ae (hmeas.comp hpoint.measurable).aestronglyMeasurable hdom
_ = ‖Z‖ₑ * (volume (Icc left right)) ^ (1 / (2 : ℝ)) := by
rw [eLpNorm_const' Z (by norm_num) (by norm_num)]
simp only [Measure.restrict_apply_univ, ENNReal.toReal_ofNat]
_ ≤ ‖Z‖ₑ * (ENNReal.ofReal widthUpper) ^ (1 / (2 : ℝ)) :=
mul_le_mul' le_rfl (ENNReal.rpow_le_rpow hvol (by norm_num))
_ = _ := (firstJetSliceBudget_coe Z widthUpper).symm
theorem coordinateC8_to_spatialFirstJet_L2_seven
{z : Coord → ℝ} {V : Set Coord} {theta left right Z widthUpper : ℝ}
(hz : CoordinateBound z V 8 Z) (hZ : 0 ≤ Z)
(hseg : ∀ x ∈ Icc left right, coordinatePoint x theta ∈ V)
(hwidth : right - left ≤ widthUpper) :
SpatialSliceL2Bound (spatialFirstJet z) theta left right 7
(firstJetSliceBudget Z widthUpper) :=
coordinateBound_to_spatialFirstJet_L2 hz hZ hseg hwidth
theorem spatialFirstJet_slice_memLp_two
{z : Coord → ℝ} {U : Set Coord} {theta left right : ℝ}
(hU : IsOpen U) (hz : ContDiffOn ℝ ∞ z U)
(hseg : ∀ x ∈ Icc left right, coordinatePoint x theta ∈ U) (i : Fin 2) (n : ℕ) :
MemLp (fun x => spatialJet (spatialFirstJet z i) n (coordinatePoint x theta)) 2
(volume.restrict (Icc left right)) :=
smooth_slice_memLp_two
(spatialJet_contDiffOn hU (partial_contDiffOn hz hU i) n).continuousOn hseg
end SmoothLocal.Hyperbolic
end
end OAI
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