File size: 2,830 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 | import OAI.Geometry.IsometricImmersion.Comparison.ActualComparisonEnergy
namespace OAI
noncomputable section
open Set MeasureTheory
open scoped ContDiff Topology Interval
namespace SmoothLocal.Hyperbolic
open SmoothLocal.Geometry SmoothLocal.Weighted SmoothLocal.ODE
theorem movingSourceNorm_le_of_pointwise_bound
{f : Coord → ℝ} {xl xr a b speed t F : ℝ}
(hspeed : 0 ≤ speed) (ht : t ∈ Icc a b)
(horder : inwardLeft xl a speed t ≤ inwardRight xr a speed t)
(hF : 0 ≤ F)
(hbound : ∀ p ∈ shrinkingSlab xl xr a b speed, |f p| ≤ F) :
movingSourceNorm xl xr a speed f t ≤ F*Real.sqrt (xr-xl) := by
let l := inwardLeft xl a speed t
let r := inwardRight xr a speed t
have hlen : r-l ≤ xr-xl := by
have hi : 0 ≤ speed*(t-a) := mul_nonneg hspeed (sub_nonneg.mpr ht.1)
dsimp [l,r,inwardLeft,inwardRight]
linarith
have hi := intervalIntegral.norm_integral_le_of_norm_le_const
(a := l) (b := r) (C := F^2) (f := fun x => f (coordinatePoint x t)^2) (by
intro x hx
have hx0 : x ∈ Ioc l r := by
rw [uIoc_of_le (show l ≤ r from horder)] at hx
exact hx
have hb := hbound _ (point_mem_shrinkingSlab ht ⟨hx0.1.le,hx0.2⟩)
rw [Real.norm_eq_abs,abs_of_nonneg (sq_nonneg _)]
simpa only [sq_abs] using (sq_le_sq₀ (abs_nonneg _) hF).mpr hb)
have hsquare : (∫ x in l..r, f (coordinatePoint x t)^2) ≤ F^2*(xr-xl) := by
apply (le_abs_self _).trans
apply hi.trans
rw [abs_of_nonneg (sub_nonneg.mpr horder)]
exact mul_le_mul_of_nonneg_left hlen (sq_nonneg F)
have hsqrt := Real.sqrt_le_sqrt hsquare
simpa only [movingSourceNorm,movingIntervalIntegral,l,r,
Real.sqrt_mul (sq_nonneg F),Real.sqrt_sq hF] using hsqrt
theorem integrated_movingSourceNorm_le_of_pointwise_bound
{f : Coord → ℝ} {xl xr a b speed F : ℝ}
(hab : a ≤ b) (hspeed : 0 ≤ speed) (hF : 0 ≤ F)
(horder : ∀ t ∈ Icc a b, inwardLeft xl a speed t ≤ inwardRight xr a speed t)
(hbound : ∀ p ∈ shrinkingSlab xl xr a b speed, |f p| ≤ F) :
(∫ t in a..b, movingSourceNorm xl xr a speed f t) ≤
F*Real.sqrt (xr-xl)*(b-a) := by
have hi := intervalIntegral.norm_integral_le_of_norm_le_const
(a := a) (b := b) (C := F*Real.sqrt (xr-xl))
(f := movingSourceNorm xl xr a speed f) (by
intro t ht
have ht0 : t ∈ Ioc a b := by simpa only [uIoc_of_le hab] using ht
have ht1 : t ∈ Icc a b := ⟨ht0.1.le,ht0.2⟩
rw [Real.norm_eq_abs,abs_of_nonneg (show 0 ≤ movingSourceNorm xl xr a speed f t from Real.sqrt_nonneg _)]
exact movingSourceNorm_le_of_pointwise_bound hspeed ht1 (horder t ht1) hF hbound)
exact (le_abs_self _).trans (by simpa only [Real.norm_eq_abs,abs_of_nonneg (sub_nonneg.mpr hab)] using hi)
end SmoothLocal.Hyperbolic
end
end OAI
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