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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 | import OAI.Geometry.IsometricImmersion.Caps.VaryingCapInduction
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 lowerCapMetricRequests {bStar : ℝ} : ℕ → LowerCapRectangle bStar →
List (ℕ × LowerCapRectangle bStar)
| 0, _ => []
| k+1, r => (k+8,r.energyOuter)::(k+7,r)::lowerCapMetricRequests k r.energyOuter.expand
def FiniteCapMetricJets {bStar : ℝ} (g : MetricField) (Y : ℝ → ℝ → ℝ)
(J : CapMetricJetBudget bStar) (L : List (ℕ × LowerCapRectangle bStar)) : Prop :=
∀ b ∈ L, ∀ (i j : Fin 2) (n : ℕ), n ≤ b.1+2 →
∀ p ∈ b.2.image Y, ‖iteratedFDeriv ℝ n (fun q => g q i j) p‖ ≤ J b.1 b.2
theorem lowerCapMetricRequests_length {bStar : ℝ} (k : ℕ) (r : LowerCapRectangle bStar) :
(lowerCapMetricRequests k r).length = 2*k := by
induction k generalizing r with
| zero => rfl
| succ k ih => simp only [lowerCapMetricRequests,List.length_cons,ih]; omega
theorem lowerCapMetricRequests_order_le {bStar : ℝ} (k : ℕ)
(r : LowerCapRectangle bStar) (b : ℕ × LowerCapRectangle bStar)
(hb : b ∈ lowerCapMetricRequests k r) : b.1 ≤ k+7 := by
induction k generalizing r with
| zero => simp [lowerCapMetricRequests] at hb
| succ k ih =>
simp only [lowerCapMetricRequests,List.mem_cons] at hb
rcases hb with rfl | rfl | hb
· omega
· omega
· have h := ih r.energyOuter.expand hb
omega
theorem lowerCapMetricRequests_full_order_le {bStar : ℝ} (k : ℕ)
(r : LowerCapRectangle bStar) (b : ℕ × LowerCapRectangle bStar)
(hb : b ∈ lowerCapMetricRequests k r) {n : ℕ} (hn : n ≤ b.1+2) : n ≤ k+9 := by
have h := lowerCapMetricRequests_order_le k r b hb
omega
theorem uniform_finite_varying_lowerCap_L2
(G Z d c e0 kappa : ℝ) (hG : 0 ≤ G) (hZ : 0 ≤ Z) (hd : 0 < d) (hc : 0 < c)
{bStar : ℝ} (J : CapMetricJetBudget bStar) :
∀ k : ℕ, ∀ r : LowerCapRectangle bStar,
(∀ b ∈ lowerCapMetricRequests k r, 0 ≤ J b.1 b.2) → ∃ H : ℝ≥0,
∀ (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 (lowerCapMetricRequests k r) → CoordinateL2Bound z (r.image Y) (k+8) H := by
intro k
induction k with
| zero =>
intro r hJ
refine ⟨originalC8L2Budget Z, ?_⟩
intro g U z Y W hg hU hSU hgB hdet hh hf hlocal
exact hh.original_L2 hU hSU hZ (hf.image_properties r).2.1 (hf.image_subset_square r)
| succ k ih =>
intro r hJ
let T := r.energyOuter
have hsourceRequest : (k+8,T) ∈ lowerCapMetricRequests (k+1) r := by
simp [lowerCapMetricRequests,T]
have hmixedRequest : (k+7,r) ∈ lowerCapMetricRequests (k+1) r := by
simp [lowerCapMetricRequests]
have htail : ∀ b, b ∈ lowerCapMetricRequests k T.expand →
b ∈ lowerCapMetricRequests (k+1) r := by
intro b hb
change b ∈ (k+8,T)::(k+7,r)::lowerCapMetricRequests k T.expand
exact List.mem_cons_of_mem _ (List.mem_cons_of_mem _ hb)
obtain ⟨H, hH⟩ := ih T.expand (fun b hb => hJ b (htail b hb))
obtain ⟨CR, hCR0, hCR⟩ := exists_localized_actualHighRemainder_bound
G Z (J (k+8) T) hG hZ (hJ (k+8,T) hsourceRequest) hd hc (k+5) (by omega)
obtain ⟨CP, hCP0, hCP⟩ := exists_same_region_solution_P_jet_bound
(J (k+7) r) Z (hJ (k+7,r) hmixedRequest) hd hc (k+7)
let M := heightQuotientJetBound G Z d c
let B := max 1 ((squareSobolevWeight (flowInteriorRadius M T.margin T.margin)+1)*(H : ℝ))
have hB : 1 ≤ B := le_max_left _ _
let Rraw := actualHighRemainderL2Budget CR (heightPFirstBound G Z d c) B H modelSquare (k+5)
have hRfinite : Rraw < (⊤ : ℝ≥0∞) := actualHighRemainderL2Budget_lt_top CR
(heightPFirstBound G Z d c) B H modelSquare_isCompact.measure_lt_top (k+5)
let RB : ℝ≥0 := Rraw.toNNReal
have hRB : (RB : ℝ≥0∞) = Rraw := ENNReal.coe_toNNReal hRfinite.ne
let Ccap := lowerCapConstant G Z d c e0 kappa (k+8) r
have hCcap : 0 < Ccap := lowerCapConstant_pos G Z d c e0 kappa (k+8) r
let Htop : ℝ≥0 := ⟨Real.sqrt (Ccap*((RB : ℝ)^2+(H : ℝ)^2)), Real.sqrt_nonneg _⟩
have hTopSq : (Htop : ℝ)^2 = Ccap*((RB : ℝ)^2+(H : ℝ)^2) :=
Real.sq_sqrt (mul_nonneg hCcap.le (add_nonneg (sq_nonneg _) (sq_nonneg _)))
refine ⟨uniformMixedOrderBudget CP B H Htop (k+9), ?_⟩
intro g U z Y W hg hU hSU hgB hdet hh hf hlocal
have hlocalTail : FiniteCapMetricJets g Y J (lowerCapMetricRequests k T.expand) :=
fun b hb => hlocal b (htail b hb)
have hLarge : CoordinateL2Bound z (T.expand.image Y) (k+8) H := hH g U z Y W hg hU hSU hgB hdet hh hf hlocalTail
have hT : CoordinateL2Bound z (T.image Y) (k+8) H :=
hLarge.restrict (T.image_subset_expand Y)
have hR : CoordinateL2Bound z (r.image Y) (k+8) H :=
hT.restrict (r.image_subset_energyOuter Y)
have hlowT0 : CoordinateBound z (T.image Y) (k+6)
((squareSobolevWeight (flowInteriorRadius M T.margin T.margin)+1)*(H : ℝ)) := by
apply CapInductionFlow.pointwise_from_larger_L2 hh hf hG hZ hd hc T
convert hLarge using 1
have hlowT : CoordinateBound z (T.image Y) (k+6) B :=
hlowT0.mono le_rfl (le_max_right _ _)
obtain ⟨hTcompact, hTmeas, hTfinite, hTO⟩ := hf.image_properties T
obtain ⟨hRcompact, hRmeas, hRfiniteImage, hRO⟩ := hf.image_properties r
have hTS : T.image Y ⊆ modelSquare := hTO.trans modelOpenSquare_subset
have hRS : r.image Y ⊆ modelSquare := hRO.trans modelOpenSquare_subset
have hTU : T.image Y ⊆ U := hTS.trans hSU
have hRU : r.image Y ⊆ U := hRS.trans hSU
have hlowFactors : ∀ n j, n + heightStateBaseOrder j ≤ (k+5)+1 →
∀ p ∈ T.image Y, |heightStateFactor z n j p| ≤ B := by
intro n j hn
exact coordinateBound_heightFactors hlowT n j (by omega)
have hL2Factors : ∀ n j, n + heightStateBaseOrder j ≤ (k+5)+3 →
eLpNorm (heightStateFactor z n j) 2 (volume.restrict (T.image Y)) ≤ (H : ℝ≥0∞) := by
intro n j hn
exact hT.heightFactors_of_aestronglyMeasurable n j (by omega)
(((heightStateFactor_contDiffOn hU hh.smooth n j).continuousOn.mono
hTU).aestronglyMeasurable hTmeas)
have hgHighT : ∀ i j n, n ≤ (k+5)+5 → ∀ p ∈ T.image Y,
‖iteratedFDeriv ℝ n (fun q => g q i j) p‖ ≤ J (k+8) T := by
intro i j n hn p hp
exact hlocal (k+8,T) hsourceRequest i j n (by omega) p hp
obtain ⟨hsource, _, hsourceMem⟩ := hCR g z U (T.image Y) hg hU hSU hgB hdet
hh.smooth hh.lowJet hh.denominator hTmeas hTS hTfinite hgHighT B hB H hlowFactors hL2Factors
have hsourceRB : eLpNorm (actualHighRemainder g z (k+5)) 2
(volume.restrict (T.image Y)) ≤ (RB : ℝ≥0∞) := by
rw [hRB]
exact hsource.trans (actualHighRemainderL2Budget_mono_area CR
(heightPFirstBound G Z d c) B H (measure_mono hTS) (k+5))
have hsourceIntegral : (∫ p in T.image Y, (actualHighRemainder g z (k+5) p)^2) ≤ (RB : ℝ)^2 :=
integral_sq_le_of_eLpNorm_two_le hsourceMem hsourceRB
have huIntegral : (∫ p in T.image Y, (verticalJet z (k+8) p)^2) ≤ (H : ℝ)^2 := by
have hu := hT.integral_sq hh.smooth hU hTmeas hTU (List.replicate (k+8) 1) (by simp)
simpa only [orderedPartial_replicate_y] using hu
have hcap : (∫ p in r.image Y,
(coordPartial 0 (verticalJet z (k+8)) p)^2+(verticalJet z (k+9) p)^2) ≤
Ccap*((∫ p in T.image Y, (actualHighRemainder g z (k+5) p)^2)+
(∫ p in T.image Y, (verticalJet z (k+8) p)^2)) := by
simpa only [Nat.add_assoc, show 5+3=8 by omega, show 5+4=9 by omega] using hf.physicalEstimate (k+5) r
have hgradient : (∫ p in r.image Y,
(coordPartial 0 (verticalJet z (k+8)) p)^2+(verticalJet z (k+9) p)^2) ≤ (Htop : ℝ)^2 := by
rw [hTopSq]
exact hcap.trans (mul_le_mul_of_nonneg_left (add_le_add hsourceIntegral huIntegral) hCcap.le)
have hxcont : ContinuousOn (coordPartial 0 (verticalJet z (k+8))) (r.image Y) :=
((partial_contDiffOn (verticalJet_contDiffOn hU hh.smooth (k+8)) hU 0).continuousOn).mono hRU
have hycont : ContinuousOn (verticalJet z (k+9)) (r.image Y) :=
(verticalJet_contDiffOn hU hh.smooth (k+9)).continuousOn.mono hRU
have hparts := integral_each_sq_le_sum hRcompact hxcont hycont
have hxTop : eLpNorm (coordPartial 0 (verticalJet z (k+8))) 2
(volume.restrict (r.image Y)) ≤ (Htop : ℝ≥0∞) :=
eLpNorm_two_le_of_integral_sq_le (compact_continuous_memLp_two hRcompact hxcont)
(hparts.1.trans hgradient)
have hyTop : eLpNorm (verticalJet z (k+9)) 2
(volume.restrict (r.image Y)) ≤ (Htop : ℝ≥0∞) :=
eLpNorm_two_le_of_integral_sq_le (compact_continuous_memLp_two hRcompact hycont)
(hparts.2.trans hgradient)
have hz2 : CoordinateBound z modelSquare 2 Z :=
(coordinateBound_five_of_frechet_eight hh.smooth hU hSU hh.lowJet).mono (by norm_num) le_rfl
have hP : ∀ n ≤ (k+9)-2, ∀ p ∈ r.image Y,
‖iteratedFDeriv ℝ n (sixVariableP g) (solutionJet z p)‖ ≤ CP := by
intro n hn p hp
exact hCP g z U (r.image Y) hg hU hRU hRS
(fun i j n hn p hp => hlocal (k+7,r) hmixedRequest i j n hn p hp)
(hz2.restrict_domain hRS) (fun p hp => hdet p (hRS hp))
(fun p hp => hh.denominator p (hRS hp)) n (by omega) p hp
have hlowR : CoordinateBound z (r.image Y) ((k+9)-3) B := by
intro word hword p hp
exact hlowT word (by omega) p (r.image_subset_energyOuter Y hp)
have hPrev : CoordinateL2Bound z (r.image Y) ((k+9)-1) H := by
convert hR using 1
omega
have hOU : modelOpenSquare ⊆ U := modelOpenSquare_subset.trans hSU
have hgO : SmoothPositiveOn g modelOpenSquare :=
⟨fun i j => (hg.1 i j).mono hOU, fun p hp => hg.2 p (hOU hp)⟩
have hDo : ∀ p ∈ modelOpenSquare,
(covHessian g z p).det = gaussianCurvature g p*heightEnergy g z p :=
fun p hp => hh.darboux p (modelOpenSquare_subset hp)
have hyyO : ∀ p ∈ modelOpenSquare, covHessian g z p 1 1 ≠ 0 :=
fun p hp => LowQuotient.denominator_ne_zero hc hh.denominator p (modelOpenSquare_subset hp)
have hfinal := coordinateL2Bound_order_step hgO modelOpenSquare_isOpen (hh.smooth.mono hOU)
hDo hyyO hRmeas hRO hRS (by omega : 7 ≤ k+9) hCP0 hB hP hlowR hPrev hyTop
(by simpa only [show (k+9)-1=k+8 by omega] using hxTop)
convert hfinal using 1
end SmoothLocal.Flow
end
end OAI
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