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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 | import OAI.Geometry.IsometricImmersion.Caps.NestedCapRectangles
import OAI.Geometry.IsometricImmersion.Caps.AllOrderPhysicalCap
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
variable {bStar : ℝ}
structure CapInductionHeight (g : MetricField) (U : Set Coord) (Z c e0 : ℝ)
(z : Coord → ℝ) : Prop where
smooth : ContDiffOn ℝ ∞ z U
lowJet : ∀ k ≤ 8, ∀ p ∈ modelSquare, ‖iteratedFDeriv ℝ k z p‖ ≤ Z
denominator : ∀ p ∈ modelSquare, c ≤ |covHessian g z p 1 1|
energy : ∀ p ∈ modelSquare, e0 ≤ heightEnergy g z p
quotient : ∀ p ∈ modelSquare, |hessianQuotient g z p| ≤ (1 : ℝ)/100
darboux : ∀ p ∈ modelSquare,
(covHessian g z p).det = gaussianCurvature g p*heightEnergy g z p
def lowerCapConstant (G Z d c e0 kappa : ℝ) (ell : ℕ) (r : LowerCapRectangle bStar) : ℝ :=
heightPhysicalCapConstant G Z d c e0 kappa ell r.left r.right r.bottom r.top ((r.top+bStar)/2)
theorem lowerCapConstant_pos (G Z d c e0 kappa : ℝ) (ell : ℕ) (r : LowerCapRectangle bStar) :
0 < lowerCapConstant G Z d c e0 kappa ell r :=
heightPhysicalCapConstant_pos G Z d c e0 kappa ell r.left r.right r.bottom r.top ((r.top+bStar)/2)
structure CapInductionFlow (g : MetricField) (U : Set Coord) (G Z d c e0 kappa : ℝ)
(z : Coord → ℝ) (Y : ℝ → ℝ → ℝ) (W : Set Coord) : Prop where
domainOpen : IsOpen W
squareSubset : modelSquare ⊆ W
domainSubset : W ⊆ U
quotientSmooth : ContDiffOn ℝ ∞ (hessianQuotient g z) W
quotientBound : CoordinateBound (hessianQuotient g z) modelSquare 3 (heightQuotientJetBound G Z d c)
continuous : ContinuousOn (uncurry Y) (Icc (-2 : ℝ) 2 ×ˢ Icc (-2 : ℝ) 2)
jointSmooth : ContDiffOn ℝ ∞ (fun p : ℝ × ℝ => Y p.2 p.1) (pairRectangle 2 (-2) 2)
start : ∀ s ∈ Icc (-2 : ℝ) 2, Y s 0 = s
ode : ∀ s ∈ Icc (-2 : ℝ) 2, ∀ t ∈ Icc (-2 : ℝ) 2,
HasDerivWithinAt (Y s) (-hessianQuotient g z (coordinatePoint t (Y s t))) (Icc (-2 : ℝ) 2) t
range : ∀ s ∈ Icc (-2 : ℝ) 2, ∀ t ∈ Icc (-2 : ℝ) 2, Y s t ∈ Icc (-3 : ℝ) 3
displacement : ∀ p ∈ capChartDomain, |capFlowHeight Y p-p 1| ≤ (1 : ℝ)/50
physicalEstimate : ∀ m : ℕ, ∀ {bStar : ℝ}, ∀ r : LowerCapRectangle bStar,
(∫ p in r.image Y, (coordPartial 0 (verticalJet z (m+3)) p)^2+(verticalJet z (m+4) p)^2) ≤
lowerCapConstant G Z d c e0 kappa (m+3) r *
((∫ p in r.energyOuter.image Y, (actualHighRemainder g z m p)^2)+
(∫ p in r.energyOuter.image Y, (verticalJet z (m+3) p)^2))
theorem exists_capInductionFlow
{g0 eta : MetricField} {z : Coord → ℝ} {U : Set Coord} {G Z d c e0 kappa : ℝ}
(hg : SmoothPositiveOn (g0+eta) U) (hU : IsOpen U) (hSU : modelSquare ⊆ U)
(hG : 0 ≤ G) (hZ : 0 ≤ Z) (hd : 0 < d) (hc : 0 < c) (he0 : 0 < e0)
(hgB : ∀ i j : Fin 2, ∀ k ≤ 8, ∀ p ∈ modelSquare,
‖iteratedFDeriv ℝ k (fun q => (g0+eta) q i j) p‖ ≤ G)
(hdet : ∀ p ∈ modelSquare, d ≤ |((g0+eta) p).det|)
(hh : CapInductionHeight (g0+eta) U Z c e0 z)
(hk : 0 < kappa) (hbackground : ∀ p ∈ U, gaussianCurvature g0 p = modelCurvature kappa p)
(hsupport : tsupport eta ⊆ patchBox)
(hcentral : ∀ p ∈ centralBox, gaussianCurvature (g0+eta) p < -kappa/2) :
∃ (W : Set Coord) (Y : ℝ → ℝ → ℝ), CapInductionFlow (g0+eta) U G Z d c e0 kappa z Y W := by
obtain ⟨V,Y,hV,hSV,hVU,hq,hqB,hchart,hY,hYs,hvar,hstart,hode,hrange,hdisp50,hdisp,himages,hcap⟩ :=
exists_C8_one_flow_physical_cap_estimates hg hU hSU hh.smooth hG hZ hd hc he0 hgB hh.lowJet
hdet hh.denominator hh.energy hh.quotient hh.darboux hk hbackground hsupport hcentral
refine ⟨V, Y, hV, hSV, hVU, hq, hqB, hY, hYs, hstart, hode, hrange, hdisp, ?_⟩
intro m bStar r
exact ((hcap m).2 r.left r.right r.bottom r.top ((r.top+bStar)/2) r.left_gt r.right_lt r.bottom_gt
r.horizontal r.vertical (by linarith [r.top_lt]) (by linarith [r.top_lt,r.ceiling_le])).2
theorem CapInductionFlow.image_properties
{g : MetricField} {U W : Set Coord} {G Z d c e0 kappa : ℝ} {z : Coord → ℝ} {Y : ℝ → ℝ → ℝ}
(hf : CapInductionFlow g U G Z d c e0 kappa z Y W) (r : LowerCapRectangle bStar) :
IsCompact (r.image Y) ∧ MeasurableSet (r.image Y) ∧ volume (r.image Y) < ⊤ ∧
r.image Y ⊆ modelOpenSquare :=
cap_closed_image_properties hf.jointSmooth hf.displacement r.region_subset_domain
theorem CapInductionFlow.image_subset_square
{g : MetricField} {U W : Set Coord} {G Z d c e0 kappa : ℝ} {z : Coord → ℝ} {Y : ℝ → ℝ → ℝ}
(hf : CapInductionFlow g U G Z d c e0 kappa z Y W) (r : LowerCapRectangle bStar) :
r.image Y ⊆ modelSquare := (hf.image_properties r).2.2.2.trans modelOpenSquare_subset
theorem CapInductionFlow.pointwise_from_larger_L2
{g : MetricField} {U W : Set Coord} {G Z d c e0 kappa : ℝ} {z : Coord → ℝ} {Y : ℝ → ℝ → ℝ}
(hh : CapInductionHeight g U Z c e0 z) (hf : CapInductionFlow g U G Z d c e0 kappa z Y W)
(hG : 0 ≤ G) (hZ : 0 ≤ Z) (hd : 0 < d) (hc : 0 < c)
(r : LowerCapRectangle bStar) {n : ℕ} {H : ℝ≥0}
(hL : CoordinateL2Bound z (r.expand.image Y) (n+2) H) :
CoordinateBound z (r.image Y) n
((squareSobolevWeight (flowInteriorRadius (heightQuotientJetBound G Z d c) r.margin r.margin)+1)*(H : ℝ)) := by
apply coordinateL2Bound_to_expanded_cap_coordinateBound hf.quotientSmooth hf.domainOpen hf.squareSubset
hf.continuous hf.range hf.start hf.ode (heightQuotientJetBound_nonneg hG hZ hd hc)
hh.quotient (fun p hp => hf.quotientBound [1] (by norm_num) p hp)
(hh.smooth.mono hf.domainSubset) r hL
def originalC8L2Budget (Z : ℝ) : ℝ≥0 :=
(‖Z‖ₑ * (volume modelSquare) ^ (1/(2 : ℝ))).toNNReal
theorem originalC8L2Budget_coe (Z : ℝ) :
(originalC8L2Budget Z : ℝ≥0∞) = ‖Z‖ₑ * (volume modelSquare) ^ (1/(2 : ℝ)) := by
apply ENNReal.coe_toNNReal
exact (ENNReal.mul_lt_top (by finiteness)
(ENNReal.rpow_lt_top_of_nonneg (by norm_num) modelSquare_isCompact.measure_lt_top.ne)).ne
theorem CapInductionHeight.original_L2
{g : MetricField} {U : Set Coord} {Z c e0 : ℝ} {z : Coord → ℝ}
(hh : CapInductionHeight g U Z c e0 z) (hU : IsOpen U) (hSU : modelSquare ⊆ U)
(hZ : 0 ≤ Z) {V : Set Coord} (hV : MeasurableSet V) (hVS : V ⊆ modelSquare) :
CoordinateL2Bound z V 8 (originalC8L2Budget Z) := by
have hC := coordinateBound_of_frechet_bounds hh.smooth hU hSU hh.lowJet
intro ds hds
rw [originalC8L2Budget_coe]
exact (bounded_real_eLpNorm_two hV hZ (fun p hp => by
rw [Real.norm_eq_abs]
exact hC ds hds p (hVS hp))).trans
(mul_le_mul' le_rfl (ENNReal.rpow_le_rpow (z := 1/(2 : ℝ)) (measure_mono hVS) (by norm_num)))
theorem compact_continuous_memLp_two {f : Coord → ℝ} {V : Set Coord}
(hV : IsCompact V) (hf : ContinuousOn f V) : MemLp f 2 (volume.restrict V) :=
(memLp_two_iff_integrable_sq (hf.aestronglyMeasurable hV.isClosed.measurableSet)).mpr
((hf.pow 2).integrableOn_compact hV)
theorem integral_each_sq_le_sum {f h : Coord → ℝ} {V : Set Coord}
(hV : IsCompact V) (hf : ContinuousOn f V) (hh : ContinuousOn h V) :
(∫ p in V, (f p)^2) ≤ ∫ p in V, (f p)^2+(h p)^2 ∧
(∫ p in V, (h p)^2) ≤ ∫ p in V, (f p)^2+(h p)^2 := by
have hfI : IntegrableOn (fun p => (f p)^2) V volume := (hf.pow 2).integrableOn_compact hV
have hhI : IntegrableOn (fun p => (h p)^2) V volume := (hh.pow 2).integrableOn_compact hV
exact ⟨setIntegral_mono_on hfI (hfI.add hhI) hV.isClosed.measurableSet
(fun p hp => le_add_of_nonneg_right (sq_nonneg _)),
setIntegral_mono_on hhI (hfI.add hhI) hV.isClosed.measurableSet
(fun p hp => le_add_of_nonneg_left (sq_nonneg _))⟩
end SmoothLocal.Flow
end
end OAI
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