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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 | import OAI.Geometry.IsometricImmersion.Flows.FlowParameterContinuity
import Mathlib.Analysis.Calculus.Deriv.Slope
namespace OAI
noncomputable section
open Set Metric Filter Function MeasureTheory
open scoped ContDiff Topology Interval
namespace SmoothLocal.Flow
open SmoothLocal.Geometry SmoothLocal.ODE SmoothLocal.Weighted
theorem hasDerivAt_of_exact_secant {f g : ℝ → ℝ} {s : ℝ}
(hg : ContinuousAt g s)
(he : ∀ᶠ u in 𝓝 s, f u - f s = (u - s) * g u) :
HasDerivAt f (g s) s := by
rw [hasDerivAt_iff_tendsto_slope]
have hslope : slope f s =ᶠ[𝓝[≠] s] g := by
filter_upwards [he.filter_mono nhdsWithin_le_nhds, self_mem_nhdsWithin] with u hu hne
have hne' : u ≠ s := by simpa only [mem_compl_iff, mem_singleton_iff] using hne
rw [slope_def_field, hu]
exact mul_div_cancel_left₀ (g u) (sub_ne_zero.mpr hne')
exact (hg.mono_left nhdsWithin_le_nhds).congr' hslope.symm
variable {q : Coord → ℝ} {U : Set Coord} {Y : ℝ → ℝ → ℝ}
theorem flow_difference_formula
(hq : ContDiffOn ℝ ∞ q U) (hU : IsOpen U) (hSU : modelSquare ⊆ U)
(hY : ContinuousOn (uncurry Y) (Icc (-2 : ℝ) 2 ×ˢ Icc (-2 : ℝ) 2))
(hrange : ∀ s ∈ Icc (-2 : ℝ) 2, ∀ t ∈ Icc (-2 : ℝ) 2,
Y s t ∈ Icc (-3 : ℝ) 3)
(hstart : ∀ s ∈ Icc (-2 : ℝ) 2, Y s 0 = s)
(hode : ∀ s ∈ Icc (-2 : ℝ) 2, ∀ t ∈ Icc (-2 : ℝ) 2,
HasDerivWithinAt (Y s) (-q (coordinatePoint t (Y s t))) (Icc (-2 : ℝ) 2) t)
{s0 s1 t : ℝ} (hs0 : s0 ∈ Icc (-2 : ℝ) 2) (hs1 : s1 ∈ Icc (-2 : ℝ) 2)
(ht : t ∈ Ioo (-2 : ℝ) 2) :
Y s1 t - Y s0 t = (s1 - s0) *
Real.exp (-(∫ u in 0..t, flowAverageCoefficient q Y s0 s1 u)) := by
have hcoeff := flowAverageCoefficient_continuous hq hU hSU hY hrange
have ha : ContinuousOn (flowAverageCoefficient q Y s0 s1) (Icc (-2 : ℝ) 2) := by
rw [continuousOn_iff_continuous_domRestrict]
exact hcoeff.comp
((continuous_const (y := (⟨s0, hs0⟩ : CapInterval))).prodMk
((continuous_const (y := (⟨s1, hs1⟩ : CapInterval))).prodMk continuous_id))
have hz (u : ℝ) (hu : u ∈ Ioo (-2 : ℝ) 2) :
HasDerivAt (fun v => Y s1 v - Y s0 v)
(-flowAverageCoefficient q Y s0 s1 u * (Y s1 u - Y s0 u)) u := by
exact (flow_difference_hasDerivWithinAt hq hU hSU ⟨hu.1.le, hu.2.le⟩
(hrange s0 hs0 u ⟨hu.1.le, hu.2.le⟩) (hrange s1 hs1 u ⟨hu.1.le, hu.2.le⟩)
(hode s0 hs0 u ⟨hu.1.le, hu.2.le⟩) (hode s1 hs1 u ⟨hu.1.le, hu.2.le⟩)).hasDerivAt
(Icc_mem_nhds hu.1 hu.2)
have he := scalar_linear_evolution_formula (show (0 : ℝ) < 2 by norm_num)
(ha.mono Ioo_subset_Icc_self) hz ht
simpa only [hstart s0 hs0, hstart s1 hs1] using he
theorem cap_flow_hasDerivAt_initial
(hq : ContDiffOn ℝ ∞ q U) (hU : IsOpen U) (hSU : modelSquare ⊆ U)
(hY : ContinuousOn (uncurry Y) (Icc (-2 : ℝ) 2 ×ˢ Icc (-2 : ℝ) 2))
(hrange : ∀ s ∈ Icc (-2 : ℝ) 2, ∀ t ∈ Icc (-2 : ℝ) 2,
Y s t ∈ Icc (-3 : ℝ) 3)
(hstart : ∀ s ∈ Icc (-2 : ℝ) 2, Y s 0 = s)
(hode : ∀ s ∈ Icc (-2 : ℝ) 2, ∀ t ∈ Icc (-2 : ℝ) 2,
HasDerivWithinAt (Y s) (-q (coordinatePoint t (Y s t))) (Icc (-2 : ℝ) 2) t)
{s t : ℝ} (hs : s ∈ Ioo (-2 : ℝ) 2) (ht : t ∈ Ioo (-2 : ℝ) 2) :
HasDerivAt (fun u => Y u t)
(Real.exp (-(∫ r in 0..t, coordPartial 1 q (coordinatePoint r (Y s r))))) s := by
let G (u : ℝ) := Real.exp (-(∫ r in 0..t, flowAverageCoefficient q Y s u r))
have hG : ContinuousOn G (Icc (-2 : ℝ) 2) := by
rw [continuousOn_iff_continuous_domRestrict]
exact (((flowAverageIntegral_continuous hq hU hSU hY hrange
⟨ht.1.le, ht.2.le⟩).comp
((continuous_const (y := (⟨s, ⟨hs.1.le, hs.2.le⟩⟩ : CapInterval))).prodMk continuous_id)).neg).rexp
have he : ∀ᶠ u in 𝓝 s, Y u t - Y s t = (u - s) * G u := by
filter_upwards [Icc_mem_nhds hs.1 hs.2] with u hu
exact flow_difference_formula hq hU hSU hY hrange hstart hode ⟨hs.1.le, hs.2.le⟩ hu ht
have hd := hasDerivAt_of_exact_secant (hG.continuousAt (Icc_mem_nhds hs.1 hs.2)) he
have hdiag : G s = Real.exp (-(∫ r in 0..t,
coordPartial 1 q (coordinatePoint r (Y s r)))) := by
simp only [G, flowAverageCoefficient, verticalAverageDerivative_diagonal]
rw [hdiag] at hd
exact hd
theorem cap_flow_initial_deriv_pos
(hq : ContDiffOn ℝ ∞ q U) (hU : IsOpen U) (hSU : modelSquare ⊆ U)
(hY : ContinuousOn (uncurry Y) (Icc (-2 : ℝ) 2 ×ˢ Icc (-2 : ℝ) 2))
(hrange : ∀ s ∈ Icc (-2 : ℝ) 2, ∀ t ∈ Icc (-2 : ℝ) 2,
Y s t ∈ Icc (-3 : ℝ) 3)
(hstart : ∀ s ∈ Icc (-2 : ℝ) 2, Y s 0 = s)
(hode : ∀ s ∈ Icc (-2 : ℝ) 2, ∀ t ∈ Icc (-2 : ℝ) 2,
HasDerivWithinAt (Y s) (-q (coordinatePoint t (Y s t))) (Icc (-2 : ℝ) 2) t)
{s t : ℝ} (hs : s ∈ Ioo (-2 : ℝ) 2) (ht : t ∈ Ioo (-2 : ℝ) 2) :
0 < deriv (fun u => Y u t) s := by
rw [(cap_flow_hasDerivAt_initial hq hU hSU hY hrange hstart hode hs ht).deriv]
exact Real.exp_pos _
theorem exists_cap_flow_first_variation
(hq : ContDiffOn ℝ ∞ q U) (hU : IsOpen U) (hSU : modelSquare ⊆ U)
(hsmall : ∀ p ∈ modelSquare, |q p| ≤ (1 : ℝ) / 100) :
∃ Y : ℝ → ℝ → ℝ,
ContinuousOn (uncurry Y) (Icc (-2 : ℝ) 2 ×ˢ Icc (-2 : ℝ) 2) ∧
(∀ s ∈ Icc (-2 : ℝ) 2, ContDiffOn ℝ ∞ (Y s) (Icc (-2 : ℝ) 2)) ∧
(∀ s ∈ Icc (-2 : ℝ) 2, Y s 0 = s) ∧
(∀ s ∈ Icc (-2 : ℝ) 2, ∀ t ∈ Icc (-2 : ℝ) 2,
HasDerivWithinAt (Y s) (-q (coordinatePoint t (Y s t))) (Icc (-2 : ℝ) 2) t) ∧
(∀ s ∈ Icc (-2 : ℝ) 2, ∀ t ∈ Icc (-2 : ℝ) 2,
|Y s t - s| ≤ (1 : ℝ) / 50) ∧
(∀ s ∈ Ioo (-2 : ℝ) 2, ∀ t ∈ Ioo (-2 : ℝ) 2,
HasDerivAt (fun u => Y u t)
(Real.exp (-(∫ r in 0..t, coordPartial 1 q (coordinatePoint r (Y s r))))) s ∧
0 < deriv (fun u => Y u t) s) := by
obtain ⟨Y, hY, htime, hstart, hode, hdisp, hdisp', hrange, hLip⟩ :=
exists_cap_flow hq hU hSU hsmall
have hrange' (s : ℝ) (hs : s ∈ Icc (-2 : ℝ) 2) (t : ℝ)
(ht : t ∈ Icc (-2 : ℝ) 2) : Y s t ∈ Icc (-3 : ℝ) 3 :=
abs_le.mp ((hrange s hs t ht).1.trans (by norm_num))
refine ⟨Y, hY, htime, hstart, hode, hdisp', ?_⟩
intro s hs t ht
exact ⟨cap_flow_hasDerivAt_initial hq hU hSU hY hrange' hstart hode hs ht,
cap_flow_initial_deriv_pos hq hU hSU hY hrange' hstart hode hs ht⟩
end SmoothLocal.Flow
end
end OAI
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