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110f4f3 | 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 | import Mathlib
import OAI.Analysis.BiholderTransport.Coordinates.LocalMetricCharts
import OAI.Analysis.BiholderTransport.Coordinates.PoleCoordinates
namespace OAI
noncomputable section
open Set Filter Metric Manifold Bundle
open scoped Topology ContDiff NNReal
namespace WeakMTWTransport
variable {n : ℕ} {M : Type*} [MetricSpace M] [CompactSpace M] [Nonempty M]
[ChartedSpace (Model n) M] [IsManifold 𝓘(ℝ,Model n) ∞ M]
[RiemannianBundle (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)]
[IsContMDiffRiemannianBundle 𝓘(ℝ,Model n) ∞ (Model n)
(fun x : M => TangentSpace 𝓘(ℝ,Model n) x)]
[IsRiemannianManifold 𝓘(ℝ,Model n) M]
lemma exists_chart_hopf_gradient_bound (c:M) :
∃K≥0,∀ᶠ z in 𝓝 (extChartAt 𝓘(ℝ,Model n) c c),∀t:ℝ,1/2 ≤ t →
∀u:M → ℝ,Continuous u →
‖fderiv ℝ (fun w=>hopfLax t u ((extChartAt 𝓘(ℝ,Model n) c).symm w)) z‖ ≤ K := by
let : IsContinuousRiemannianBundle (Model n)
(fun x : M => TangentSpace 𝓘(ℝ,Model n) x) :=
continuousRiemannianBundle_of_smooth (IB := 𝓘(ℝ,Model n))
obtain ⟨L,r,hr,hT,hL⟩ := exists_lipschitzOn_inverse_chart (E := Model n) c
let D:ℝ≥0:=Real.toNNReal (Metric.diam (univ:Set M))
have hD:∀x y:M,dist x y ≤ D := by
intro x y
rw [Real.coe_toNNReal _ Metric.diam_nonneg]
exact Metric.dist_le_diam_of_mem isCompact_univ.isBounded (mem_univ x) (mem_univ y)
refine ⟨2*(D:ℝ)*L,by positivity,?_⟩
filter_upwards [ball_mem_nhds (extChartAt 𝓘(ℝ,Model n) c c) hr] with z hz
intro t ht u hu
have ht0:0 < t := by linarith
have hn:=norm_fderiv_le_of_lipschitzOn ℝ (isOpen_ball.mem_nhds hz)
((hopfLax_lipschitz hu hD ht0).comp_lipschitzOnWith hL)
have hdiv:(D:ℝ)/t ≤ 2*D := (div_le_iff₀ ht0).mpr (by nlinarith [D.coe_nonneg])
calc
_ ≤ ↑((D/Real.toNNReal t)*L:ℝ≥0) := hn
_ = (D:ℝ)/t*L := by rw [NNReal.coe_mul,NNReal.coe_div,Real.coe_toNNReal _ ht0.le]
_ ≤ _ := mul_le_mul_of_nonneg_right hdiv L.coe_nonneg
end WeakMTWTransport
end
end OAI
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