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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 | import OAI.Geometry.IsometricImmersion.Caps.ConstructedCapArea
import OAI.Geometry.IsometricImmersion.Caps.CapGradientTransport
namespace OAI
noncomputable section
open Set Filter Function MeasureTheory
open scoped ContDiff Topology Interval
namespace SmoothLocal.Flow
open SmoothLocal.Geometry SmoothLocal.ODE SmoothLocal.Weighted
theorem capChart_mapsTo_modelSquare_of_range {Y : ℝ → ℝ → ℝ}
(hrange : ∀ s ∈ Icc (-2 : ℝ) 2, ∀ t ∈ Icc (-2 : ℝ) 2, Y s t ∈ Icc (-3 : ℝ) 3) :
MapsTo (capChart Y) capChartDomain modelSquare := by
intro p hp
apply coordinatePoint_mem_modelSquare
· exact ⟨by linarith [hp.1.1],by linarith [hp.1.2]⟩
· exact hrange (p 1) ⟨hp.2.1.le,hp.2.2.le⟩ (p 0) ⟨hp.1.1.le,hp.1.2.le⟩
theorem coordinate_partial_sq_le_gradient (f : Coord → ℝ) (p : Coord) (i : Fin 2) :
(coordPartial i f p)^2 ≤ coordinateGradientSquare f p := by
fin_cases i <;> simp only [Fin.mk_zero,Fin.mk_one,coordinateGradientSquare]
· exact le_add_of_nonneg_right (sq_nonneg _)
· exact le_add_of_nonneg_left (sq_nonneg _)
theorem cap_image_partial_square_le_gradient
{Y : ℝ → ℝ → ℝ} {f : Coord → ℝ} {U : Set Coord}
(hYs : ContDiffOn ℝ ∞ (fun p : ℝ × ℝ => Y p.2 p.1) (pairRectangle 2 (-2) 2))
(hf : ContDiffOn ℝ ∞ f U) (hU : IsOpen U)
(hmap : MapsTo (capChart Y) capChartDomain U)
{tl tr sb st : ℝ} (hbox : closedRectangle tl tr sb st ⊆ capChartDomain)
(i : Fin 2) :
(∫ p in capChart Y '' closedRectangle tl tr sb st, (coordPartial i f p)^2) ≤
∫ p in capChart Y '' closedRectangle tl tr sb st, coordinateGradientSquare f p := by
have hc : IsCompact (capChart Y '' closedRectangle tl tr sb st) :=
(capRectangle_isCompact _ _ _ _).image_of_continuousOn
((capChart_contDiffOn hYs).continuousOn.mono hbox)
have hsub : capChart Y '' closedRectangle tl tr sb st ⊆ U := by
rintro _ ⟨p,hp,rfl⟩
exact hmap (hbox hp)
have hi : ContinuousOn (fun p => (coordPartial i f p)^2)
(capChart Y '' closedRectangle tl tr sb st) :=
((partial_contDiffOn hf hU i).continuousOn.mono hsub).pow 2
have hg : ContinuousOn (coordinateGradientSquare f)
(capChart Y '' closedRectangle tl tr sb st) :=
(coordinateGradientSquare_contDiffOn hf hU).continuousOn.mono hsub
exact setIntegral_mono_on (hi.integrableOn_compact hc) (hg.integrableOn_compact hc)
hc.isClosed.measurableSet (fun p _hp => coordinate_partial_sq_le_gradient f p i)
theorem constructed_cap_gradient_integral_bounds
{q f : Coord → ℝ} {U : Set Coord} {Y : ℝ → ℝ → ℝ} {M : ℝ}
(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)
(hM : 0 ≤ M) (hMq : ∀ p ∈ modelSquare, |coordPartial 1 q p| ≤ M)
(hq0 : ∀ p ∈ modelSquare, |q p| ≤ 1)
(hf : ContDiffOn ℝ ∞ f U)
{tl tr sb st : ℝ} (ht : tl ≤ tr) (hs : sb ≤ st)
(hbox : closedRectangle tl tr sb st ⊆ capChartDomain) :
rectangleIntegral tl tr sb st (coordinateGradientSquare (capPullback Y f)) ≤
((2+(Real.exp (2*M))^2)*Real.exp (2*M))*
(∫ p in capChart Y '' closedRectangle tl tr sb st, coordinateGradientSquare f p) ∧
(∫ p in capChart Y '' closedRectangle tl tr sb st, coordinateGradientSquare f p) ≤
(Real.exp (2*M)*(2+3*(Real.exp (2*M))^2))*
rectangleIntegral tl tr sb st (coordinateGradientSquare (capPullback Y f)) ∧
∀ i : Fin 2,
(∫ p in capChart Y '' closedRectangle tl tr sb st, (coordPartial i f p)^2) ≤
(Real.exp (2*M)*(2+3*(Real.exp (2*M))^2))*
rectangleIntegral tl tr sb st (coordinateGradientSquare (capPullback Y f)) := by
obtain ⟨e,hsource,htarget,heq,hforward,hinverse⟩ :=
exists_global_triangularFlow_chart hq hU hSU hY hrange hstart hode
have hYs := cap_flow_joint_contDiffOn hq hU hSU hY hrange hstart hode
have hvar : ∀ s ∈ Ioo (-2 : ℝ) 2, ∀ t ∈ Ioo (-2 : ℝ) 2, 0 < deriv (fun r => Y r t) s :=
fun s hs t ht => cap_flow_initial_deriv_pos hq hU hSU hY hrange hstart hode hs ht
have hmapS := capChart_mapsTo_modelSquare_of_range hrange
have hmap : MapsTo (capChart Y) capChartDomain U := fun _ hp => hSU (hmapS hp)
have hJ := fun p (hp : p ∈ closedRectangle tl tr sb st) =>
capChart_jacobian_bounds hq hU hSU hY hrange hstart hode hM hMq (hbox hp)
obtain ⟨hfwd,hinv⟩ := cap_gradient_integral_bounds hYs hvar hode hsource heq hf hU hmap ht hs hbox
(fun p hp => hq0 _ (hmapS (hbox hp))) hJ
refine ⟨hfwd,hinv,?_⟩
intro i
exact (cap_image_partial_square_le_gradient hYs hf hU hmap hbox i).trans hinv
end SmoothLocal.Flow
end
end OAI
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