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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 | import OAI.Geometry.IsometricImmersion.Darboux.QSameStripMargins
import OAI.Geometry.IsometricImmersion.Pulses.ForcingCoefficientSign
namespace OAI
noncomputable section
open Set Filter
open scoped ContDiff Topology Matrix Matrix.Norms.Elementwise
namespace SmoothLocal.HighEquation
open SmoothLocal.Geometry SmoothLocal.Weighted SmoothLocal.Pulse
theorem actual_det_upper_of_entry_bound (g : MetricField) (p : Coord) {G : ℝ}
(hG : 0 ≤ G) (hg : ∀ i j, |g p i j| ≤ G) : (g p).det ≤ 2*G^2 := by
rw [Matrix.det_fin_two]
calc
g p 0 0*g p 1 1-g p 0 1*g p 1 0 ≤
|g p 0 0*g p 1 1|+|g p 0 1*g p 1 0| := by
linarith [le_abs_self (g p 0 0*g p 1 1),
neg_le_abs (g p 0 1*g p 1 0)]
_ = |g p 0 0| *|g p 1 1|+|g p 0 1| *|g p 1 0| := by rw [abs_mul,abs_mul]
_ ≤ G*G+G*G := add_le_add
(mul_le_mul (hg 0 0) (hg 1 1) (abs_nonneg _) hG)
(mul_le_mul (hg 0 1) (hg 1 0) (abs_nonneg _) hG)
_ = _ := by ring
theorem exists_uniform_reference_forcing_margins (G R M M0 : ℝ)
{d nu c eta kappa : ℝ} (hG : 0 ≤ G) (hR : 0 ≤ R)
(hM : 0 ≤ M) (hM0 : 0 ≤ M0)
(hd : 0 < d) (hnu : 0 < nu) (hc : 0 < c) (heta : 0 < eta) (hkappa : 0 < kappa) :
∃ epsilon A : ℝ, 0 < epsilon ∧ 1 ≤ A ∧
∀ (g g0 : MetricField) (z z0 : Coord → ℝ) (U : Set Coord),
SmoothPositiveOn g U → SmoothPositiveOn g0 U → IsOpen U →
ContDiffOn ℝ ∞ z U → ContDiffOn ℝ ∞ z0 U →
∀ left right a b : ℝ, a ≤ b →
closedRectangle left right a b ⊆ U →
(∀ p ∈ closedRectangle left right a b, ‖p‖ ≤ R) →
(∀ i j k, k ≤ 2 → ∀ p ∈ closedRectangle left right a b,
‖iteratedFDeriv ℝ k (fun x => g x i j) p‖ ≤ G) →
(∀ i j k, k ≤ 2 → ∀ p ∈ closedRectangle left right a b,
‖iteratedFDeriv ℝ k (fun x => g0 x i j) p‖ ≤ G) →
CoordinateBound z (closedRectangle left right a b) 3 M →
CoordinateBound z0 (closedRectangle left right a b) 3 M0 →
(∀ p ∈ closedRectangle left right a b, d ≤ |(g p).det|) →
(∀ p ∈ closedRectangle left right a b, d ≤ |(g0 p).det|) →
(∀ i j k, k ≤ 2 → ∀ p ∈ closedRectangle left right a b,
‖iteratedFDeriv ℝ k (fun x => g0 x i j - g x i j) p‖ ≤ epsilon) →
(M + M0) * (b - a) ≤ epsilon →
(∀ x ∈ Ioo left right, z0 (boxPoint x a) = z (boxPoint x a)) →
(∀ x ∈ Ioo left right,
coordPartial 1 z0 (boxPoint x a) = coordPartial 1 z (boxPoint x a)) →
(∀ p ∈ closedRectangle left right a b, nu ≤ |covHessian g z p 0 0|) →
(∀ p ∈ closedRectangle left right a b, c ≤ |covHessian g z p 1 1|) →
(∀ p ∈ closedRectangle left right a b, eta ≤ heightEnergy g z p) →
(∀ p ∈ closedRectangle left right a b, gaussianCurvature g p ≤ -kappa) →
(∀ p ∈ closedRectangle left right a b,
(covHessian g z p).det = gaussianCurvature g p * heightEnergy g z p) →
(∀ p ∈ closedRectangle left right a b,
(hessianQuotient g z p)^2 + gaussianCurvature g p * darbouxG g z p < 0) →
∀ l r : ℝ, left < l → r < right →
∀ p ∈ closedRectangle l r a b,
eta/2 ≤ heightEnergy g0 z p ∧
nu/2 ≤ |covHessian g0 z p 0 0| ∧
|covHessian g0 z p 0 0| ≤ A ∧
(g0 p).det ≤ 2*G^2 ∧
forcingFloor (eta/2) (2*G^2) A ≤ |forcingCoefficient g0 z p| := by
obtain ⟨epsilon,A,hepsilon,hA,hmargin⟩ :=
exists_uniform_Q_same_strip_margins G R M M0 hG hR hM hM0 hd hnu hc heta hkappa
refine ⟨epsilon,A,hepsilon,hA,?_⟩
intro g g0 z z0 U hg hg0 hU hz hz0 left right a b hab hbox hpoint
hgB hg0B hzB hz0B hdet hdet0 hmetric hwidth hvalue hvelocity
hxx hyy henergy hK hD htime l r hleft hright p hp
have hp0 : p 0 ∈ Ioo left right :=
⟨hleft.trans_le hp.1.1, hp.1.2.trans_lt hright⟩
have hpointEq : boxPoint (p 0) (p 1) = p := by
ext i
fin_cases i <;> simp [boxPoint]
have hpbox : p ∈ closedRectangle left right a b :=
⟨⟨hp0.1.le,hp0.2.le⟩,hp.2⟩
have hm := ((hmargin g g0 z z0 U hg hg0 hU hz hz0 left right a b hab hbox hpoint
hgB hg0B hzB hz0B hdet hdet0 hmetric hwidth hvalue hvelocity
hxx hyy henergy hK hD htime (p 0) hp0 (p 1) hp.2).1 1 (by norm_num))
simp only [qHeightJetSegment,stateSegment_one,hpointEq,
stateQDenominator_qSolutionJet] at hm
have hE : eta/2 ≤ heightEnergy g0 z p := by
have hs := hm.2.2.1
unfold stateEnergy at hs
rw [statePoint_qSolutionJet,stateGradient_qSolutionJet,
jetEnergy_at_height hg0 (hbox hpbox) z] at hs
exact hs
have hentry (i j : Fin 2) : |g0 p i j| ≤ G := by
have hh := hg0B i j 0 (by norm_num) p hpbox
simpa only [norm_iteratedFDeriv_zero,Real.norm_eq_abs] using hh
have hdetup := actual_det_upper_of_entry_bound g0 p hG hentry
exact ⟨hE,hm.1,hm.2.1,hdetup,
forcingCoefficient_floor_of_reference_margins hg0 (hbox hpbox)
(half_pos heta) (half_pos hnu) hE hdetup hm.1 hm.2.1⟩
end SmoothLocal.HighEquation
end
end OAI
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