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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 | import OAI.Geometry.IsometricImmersion.Caps.CapAreaFormula
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 exists_actual_cap_area_chart
{q : 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) :
∃ e : OpenPartialHomeomorph (ℝ × ℝ) (ℝ × ℝ),
e.source = pairRectangle 2 (-2) 2 ∧
e.target = triangularFlow Y '' pairRectangle 2 (-2) 2 ∧
(e : (ℝ × ℝ) → (ℝ × ℝ)) = triangularFlow Y ∧
ContDiffOn ℝ ∞ e e.source ∧ ContDiffOn ℝ ∞ e.symm e.target ∧
(∀ p ∈ capChartDomain,
(fderiv ℝ (capChart Y) p).det = coordPartial 1 (capFlowHeight Y) p ∧
Real.exp (-2*M) ≤ coordPartial 1 (capFlowHeight Y) p ∧
coordPartial 1 (capFlowHeight Y) p ≤ Real.exp (2*M)) ∧
∀ tl tr sb st : ℝ, tl ≤ tr → sb ≤ st →
closedRectangle tl tr sb st ⊆ capChartDomain → ∀ f : Coord → ℝ,
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) ∧
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) ∧
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
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 hJ (p : Coord) (hp : p ∈ capChartDomain) :=
capChart_jacobian_bounds hq hU hSU hY hrange hstart hode hM hMq hp
refine ⟨e,hsource,htarget,heq,hforward,hinverse,?_,?_⟩
· intro p hp
exact ⟨capChart_det_fderiv hYs hp,hJ p hp⟩
· intro tl tr sb st ht hs hbox f hf
have hJrect := fun p (hp : p ∈ closedRectangle tl tr sb st) => hJ p (hbox hp)
obtain ⟨hlower,hupper⟩ := cap_rectangle_square_integral_bounds hYs hvar hsource heq ht hs hbox hJrect hf
exact ⟨cap_image_integral_formula hYs hvar hsource heq ht hs hbox hf,
hlower,hupper,cap_pullback_square_integral_le_image hYs hvar hsource heq ht hs hbox hJrect hf⟩
theorem constructed_cap_pullback_square_le_region
{q : Coord → ℝ} {U V : 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)
{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))
(himage : capChart Y '' closedRectangle tl tr sb st ⊆ V)
(hLp : MemLp f 2 (volume.restrict V)) :
rectangleIntegral tl tr sb st (fun p => (capPullback Y f p)^2) ≤
Real.exp (2*M)*(∫ p in V, (f p)^2) := by
obtain ⟨e,hsource,htarget,heq,hforward,hinverse,hJ,hbridge⟩ :=
exists_actual_cap_area_chart hq hU hSU hY hrange hstart hode hM hMq
have hcomp := (hbridge tl tr sb st ht hs hbox f hf).2.2.2
have hregion : (∫ p in capChart Y '' closedRectangle tl tr sb st, (f p)^2) ≤
∫ p in V, (f p)^2 :=
setIntegral_mono_set hLp.integrable_sq (Filter.Eventually.of_forall (fun p => sq_nonneg (f p)))
himage.eventuallyLE
exact hcomp.trans (mul_le_mul_of_nonneg_left hregion (Real.exp_pos _).le)
end SmoothLocal.Flow
end
end OAI
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