| import OAI.Geometry.IsometricImmersion.Caps.VaryingCapInduction |
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| namespace OAI |
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| noncomputable section |
| open Set Filter Function MeasureTheory |
| open scoped ContDiff Topology BigOperators ENNReal NNReal |
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| namespace SmoothLocal.Flow |
| open SmoothLocal.Geometry SmoothLocal.ODE SmoothLocal.Weighted SmoothLocal.Model |
| open SmoothLocal.HighEquation SmoothLocal.Analytic SmoothLocal.Sobolev |
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| 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 |
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| 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 |
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| 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 |
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| 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 |
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| 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 |
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| 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 |
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| end SmoothLocal.Flow |
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| end |
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| end OAI |
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