File size: 7,575 Bytes
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 | import OAI.Geometry.IsometricImmersion.Caps.CapJacobian
import OAI.Geometry.IsometricImmersion.Energy.RegionSobolev
import OAI.Geometry.IsometricImmersion.Caps.EllipticGradient
namespace OAI
noncomputable section
open Set Filter Function MeasureTheory
open scoped ContDiff Topology Interval
namespace SmoothLocal.Flow
open SmoothLocal.Geometry SmoothLocal.ODE SmoothLocal.Weighted SmoothLocal.Sobolev
theorem capRectangle_isCompact (tl tr sb st : ℝ) : IsCompact (closedRectangle tl tr sb st) := by
have he : closedRectangle tl tr sb st =
(fun q : ℝ × ℝ => boxPoint q.1 q.2) '' (Icc tl tr ×ˢ Icc sb st) := by
ext p
constructor
· intro hp
exact ⟨(p 0,p 1),⟨hp.1,hp.2⟩,boxPoint_eta p⟩
· rintro ⟨q,hq,rfl⟩
exact boxPoint_mem hq.1 hq.2
rw [he]
exact (isCompact_Icc.prod isCompact_Icc).image (by unfold boxPoint; fun_prop)
theorem rectangleIntegral_eq_coordinate_closed_area
{tl tr sb st : ℝ} {F : Coord → ℝ} (ht : tl ≤ tr) (hs : sb ≤ st)
(hF : ContinuousOn F (closedRectangle tl tr sb st)) :
rectangleIntegral tl tr sb st F = ∫ p in closedRectangle tl tr sb st, F p := by
rw [rectangleIntegral_eq_closed_area ht hs hF]
have hh := (volume_preserving_finTwoArrow ℝ).setIntegral_preimage_emb
(MeasurableEquiv.finTwoArrow (α := ℝ)).measurableEmbedding
(fun q : ℝ × ℝ => F (boxPoint q.1 q.2)) (Icc tl tr ×ˢ Icc sb st)
change (∫ p in closedRectangle tl tr sb st, F (boxPoint (p 0) (p 1))) =
∫ q in Icc tl tr ×ˢ Icc sb st, F (boxPoint q.1 q.2) at hh
simpa only [boxPoint_eta] using hh.symm
theorem cap_image_integral_formula
{Y : ℝ → ℝ → ℝ} {e : OpenPartialHomeomorph (ℝ × ℝ) (ℝ × ℝ)}
(hYs : ContDiffOn ℝ ∞ (fun p : ℝ × ℝ => Y p.2 p.1) (pairRectangle 2 (-2) 2))
(hvar : ∀ s ∈ Ioo (-2 : ℝ) 2, ∀ t ∈ Ioo (-2 : ℝ) 2, 0 < deriv (fun r => Y r t) s)
(hsource : e.source = pairRectangle 2 (-2) 2)
(heq : (e : (ℝ × ℝ) → (ℝ × ℝ)) = triangularFlow Y)
{tl tr sb st : ℝ} (ht : tl ≤ tr) (hs : sb ≤ st)
(hbox : closedRectangle tl tr sb st ⊆ capChartDomain)
{f : Coord → ℝ} (hf : ContinuousOn f (capChart Y '' closedRectangle tl tr sb st)) :
(∫ p in capChart Y '' closedRectangle tl tr sb st, f p) =
rectangleIntegral tl tr sb st (fun p => coordPartial 1 (capFlowHeight Y) p*capPullback Y f p) := by
have hrect : MeasurableSet (closedRectangle tl tr sb st) := (capRectangle_isCompact _ _ _ _).isClosed.measurableSet
have hchart := capChart_contDiffOn hYs
have hd (p : Coord) (hp : p ∈ closedRectangle tl tr sb st) :
HasFDerivWithinAt (capChart Y) (fderiv ℝ (capChart Y) p) (closedRectangle tl tr sb st) p :=
((hchart.contDiffAt (capChartDomain_isOpen.mem_nhds (hbox hp))).differentiableAt (by simp)).hasFDerivAt.hasFDerivWithinAt
have hinj := (capChart_injOn_of_pairChart hsource heq).mono hbox
have hcomp : ContinuousOn (capPullback Y f) (closedRectangle tl tr sb st) :=
hf.comp (hchart.continuousOn.mono hbox) (fun p hp => ⟨p,hp,rfl⟩)
have hJ : ContinuousOn (coordPartial 1 (capFlowHeight Y)) (closedRectangle tl tr sb st) :=
(partial_contDiffOn (capFlowHeight_contDiffOn hYs) capChartDomain_isOpen 1).continuousOn.mono hbox
calc
_ = ∫ p in closedRectangle tl tr sb st, |(fderiv ℝ (capChart Y) p).det| • f (capChart Y p) :=
integral_image_eq_integral_abs_det_fderiv_smul volume hrect hd hinj f
_ = ∫ p in closedRectangle tl tr sb st, coordPartial 1 (capFlowHeight Y) p*capPullback Y f p := by
apply setIntegral_congr_fun hrect
intro p hp
change |(fderiv ℝ (capChart Y) p).det| * f (capChart Y p) = _
rw [capChart_abs_det_fderiv hYs hvar (hbox hp)]
rfl
_ = _ := (rectangleIntegral_eq_coordinate_closed_area ht hs (hJ.mul hcomp)).symm
theorem cap_rectangle_square_integral_bounds
{Y : ℝ → ℝ → ℝ} {e : OpenPartialHomeomorph (ℝ × ℝ) (ℝ × ℝ)}
(hYs : ContDiffOn ℝ ∞ (fun p : ℝ × ℝ => Y p.2 p.1) (pairRectangle 2 (-2) 2))
(hvar : ∀ s ∈ Ioo (-2 : ℝ) 2, ∀ t ∈ Ioo (-2 : ℝ) 2, 0 < deriv (fun r => Y r t) s)
(hsource : e.source = pairRectangle 2 (-2) 2)
(heq : (e : (ℝ × ℝ) → (ℝ × ℝ)) = triangularFlow Y)
{tl tr sb st M : ℝ} (ht : tl ≤ tr) (hs : sb ≤ st)
(hbox : closedRectangle tl tr sb st ⊆ capChartDomain)
(hJ : ∀ p ∈ closedRectangle tl tr sb st,
Real.exp (-2*M) ≤ coordPartial 1 (capFlowHeight Y) p ∧
coordPartial 1 (capFlowHeight Y) p ≤ Real.exp (2*M))
{f : Coord → ℝ} (hf : ContinuousOn f (capChart Y '' closedRectangle tl tr sb st)) :
Real.exp (-2*M)*rectangleIntegral tl tr sb st (fun p => (capPullback Y f p)^2) ≤
∫ p in capChart Y '' closedRectangle tl tr sb st, (f p)^2 ∧
(∫ p in capChart Y '' closedRectangle tl tr sb st, (f p)^2) ≤
Real.exp (2*M)*rectangleIntegral tl tr sb st (fun p => (capPullback Y f p)^2) := by
have hchart := capChart_contDiffOn hYs
have hfc : ContinuousOn (fun p => (capPullback Y f p)^2) (closedRectangle tl tr sb st) :=
(hf.comp (hchart.continuousOn.mono hbox) (fun p hp => ⟨p,hp,rfl⟩)).pow 2
have hJc : ContinuousOn (coordPartial 1 (capFlowHeight Y)) (closedRectangle tl tr sb st) :=
(partial_contDiffOn (capFlowHeight_contDiffOn hYs) capChartDomain_isOpen 1).continuousOn.mono hbox
have hformula := cap_image_integral_formula hYs hvar hsource heq ht hs hbox (hf.pow 2)
change (∫ p in capChart Y '' closedRectangle tl tr sb st, (f p)^2) =
rectangleIntegral tl tr sb st (fun p => coordPartial 1 (capFlowHeight Y) p*(capPullback Y f p)^2) at hformula
rw [hformula]
constructor
· have hh := rectangleIntegral_mono ht hs (continuousOn_const.mul hfc) (hJc.mul hfc)
(fun p hp => mul_le_mul_of_nonneg_right (hJ p hp).1 (sq_nonneg _))
simpa only [Pi.mul_def, rectangleIntegral_const_mul] using hh
· have hh := rectangleIntegral_mono ht hs (hJc.mul hfc) (continuousOn_const.mul hfc)
(fun p hp => mul_le_mul_of_nonneg_right (hJ p hp).2 (sq_nonneg _))
simpa only [Pi.mul_def, rectangleIntegral_const_mul] using hh
theorem cap_pullback_square_integral_le_image
{Y : ℝ → ℝ → ℝ} {e : OpenPartialHomeomorph (ℝ × ℝ) (ℝ × ℝ)}
(hYs : ContDiffOn ℝ ∞ (fun p : ℝ × ℝ => Y p.2 p.1) (pairRectangle 2 (-2) 2))
(hvar : ∀ s ∈ Ioo (-2 : ℝ) 2, ∀ t ∈ Ioo (-2 : ℝ) 2, 0 < deriv (fun r => Y r t) s)
(hsource : e.source = pairRectangle 2 (-2) 2)
(heq : (e : (ℝ × ℝ) → (ℝ × ℝ)) = triangularFlow Y)
{tl tr sb st M : ℝ} (ht : tl ≤ tr) (hs : sb ≤ st)
(hbox : closedRectangle tl tr sb st ⊆ capChartDomain)
(hJ : ∀ p ∈ closedRectangle tl tr sb st,
Real.exp (-2*M) ≤ coordPartial 1 (capFlowHeight Y) p ∧
coordPartial 1 (capFlowHeight Y) p ≤ Real.exp (2*M))
{f : Coord → ℝ} (hf : ContinuousOn f (capChart Y '' closedRectangle tl tr sb st)) :
rectangleIntegral tl tr sb st (fun p => (capPullback Y f p)^2) ≤
Real.exp (2*M)*(∫ p in capChart Y '' closedRectangle tl tr sb st, (f p)^2) := by
have hh := (cap_rectangle_square_integral_bounds hYs hvar hsource heq ht hs hbox hJ hf).1
have hscale : Real.exp (2*M)*Real.exp (-2*M) = 1 := by rw [←Real.exp_add]; simp
calc
_ = Real.exp (2*M)*(Real.exp (-2*M)*rectangleIntegral tl tr sb st (fun p => (capPullback Y f p)^2)) := by
rw [←mul_assoc,hscale,one_mul]
_ ≤ _ := mul_le_mul_of_nonneg_left hh (Real.exp_pos _).le
end SmoothLocal.Flow
end
end OAI
|