File size: 4,218 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 | import OAI.Geometry.IsometricImmersion.Estimates.MovingSourceBounds
import OAI.Geometry.IsometricImmersion.Energy.MovingEnergyGradient
namespace OAI
noncomputable section
open Set Filter MeasureTheory
open scoped ContDiff Topology Interval
namespace SmoothLocal.Hyperbolic
open SmoothLocal.Geometry SmoothLocal.Weighted SmoothLocal.ODE
theorem fixed_slice_gradient_le_moving_energy
{S v : Coord → ℝ} {V : Set Coord} {xl xr a speed t pulseRadius s0 : ℝ}
(hV : IsOpen V) (hS : ContinuousOn S V) (hv : ContDiffOn ℝ ∞ v V)
(hs0 : 0 < s0) (hpulse : 0 ≤ pulseRadius)
(hleft : inwardLeft xl a speed t ≤ -pulseRadius)
(hright : pulseRadius ≤ inwardRight xr a speed t)
(hmem : ∀ x ∈ Icc (inwardLeft xl a speed t) (inwardRight xr a speed t),
coordinatePoint x t ∈ V)
(hfloor : ∀ x ∈ Icc (inwardLeft xl a speed t) (inwardRight xr a speed t),
s0 ≤ S (coordinatePoint x t)) (i : Fin 2) :
Real.sqrt (∫ x in Icc (-pulseRadius) pulseRadius, (coordPartial i v (boxPoint x t))^2) ≤
Real.sqrt (2*(1+1/s0))*Real.sqrt (movingEnergy xl xr a speed S v t) := by
let l := inwardLeft xl a speed t
let r := inwardRight xr a speed t
let K := 2*(1+1/s0)
have hlr : l ≤ r := hleft.trans ((by linarith : -pulseRadius ≤ pulseRadius).trans hright)
have hK : 0 ≤ K := by dsimp [K]; positivity
have hpointContinuous : Continuous (fun x : ℝ => coordinatePoint x t) := by
unfold coordinatePoint
fun_prop
have hvi : ContinuousOn (fun x => coordPartial i v (coordinatePoint x t)) (Icc l r) :=
(partial_contDiffOn hv hV i).continuousOn.comp hpointContinuous.continuousOn hmem
have hvt : ContinuousOn (fun x => coordPartial 1 v (coordinatePoint x t)) (Icc l r) :=
(partial_contDiffOn hv hV 1).continuousOn.comp hpointContinuous.continuousOn hmem
have hvx : ContinuousOn (fun x => coordPartial 0 v (coordinatePoint x t)) (Icc l r) :=
(partial_contDiffOn hv hV 0).continuousOn.comp hpointContinuous.continuousOn hmem
have hSc : ContinuousOn (fun x => S (coordinatePoint x t)) (Icc l r) :=
hS.comp hpointContinuous.continuousOn hmem
have he : ContinuousOn (fun x => energyDensity S v (coordinatePoint x t)) (Icc l r) :=
((hvt.pow 2).add (hSc.mul (hvx.pow 2))).div_const 2
have hsmall : (∫ x in -pulseRadius..pulseRadius, (coordPartial i v (coordinatePoint x t))^2) ≤
∫ x in l..r, (coordPartial i v (coordinatePoint x t))^2 :=
intervalIntegral.integral_mono_interval hleft (by linarith) hright
(Filter.Eventually.of_forall (fun _ => sq_nonneg _))
((hvi.pow 2).intervalIntegrable_of_Icc hlr)
have henergy : (∫ x in l..r, (coordPartial i v (coordinatePoint x t))^2) ≤
K*movingEnergy xl xr a speed S v t := by
have hmono := intervalIntegral.integral_mono_on (μ := volume) hlr
((hvi.pow 2).intervalIntegrable_of_Icc hlr)
((continuousOn_const.mul he).intervalIntegrable_of_Icc hlr) (fun x hx => by
have hscalar := scalar_gradient_le_energy hs0 (hfloor x hx)
(a := coordPartial 1 v (coordinatePoint x t)) (b := coordPartial 0 v (coordinatePoint x t))
have hi : (coordPartial i v (coordinatePoint x t))^2 ≤
(coordPartial 1 v (coordinatePoint x t))^2+(coordPartial 0 v (coordinatePoint x t))^2 := by
fin_cases i <;> simp only [Fin.zero_eta, Fin.mk_one] <;> nlinarith [sq_nonneg (coordPartial 0 v (coordinatePoint x t)),
sq_nonneg (coordPartial 1 v (coordinatePoint x t))]
exact hi.trans hscalar)
dsimp only [Pi.pow_apply,Pi.mul_apply] at hmono
simpa only [intervalIntegral.integral_const_mul,movingEnergy,movingIntervalIntegral,l,r,K] using hmono
have hroot := Real.sqrt_le_sqrt (hsmall.trans henergy)
rw [Real.sqrt_mul hK] at hroot
have hpoint : (fun x => coordPartial i v (coordinatePoint x t)^2) =
(fun x => coordPartial i v (boxPoint x t)^2) := by
funext x
have he : coordinatePoint x t=boxPoint x t := by ext j; fin_cases j <;> simp [coordinatePoint,boxPoint]
rw [he]
simpa only [intervalIntegral.integral_of_le (by linarith : -pulseRadius ≤ pulseRadius),
← integral_Icc_eq_integral_Ioc,hpoint,K] using hroot
end SmoothLocal.Hyperbolic
end
end OAI
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