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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 50 51 52 53 54 55 | import Mathlib
import OAI.Analysis.BiholderTransport.Regularity.ParametricLowerSequence
namespace OAI
noncomputable section
open Set Filter 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]
variable {P : Type*} [NormedAddCommGroup P] [NormedSpace ℝ P] [CompleteSpace P]
lemma parametric_short_gradient_tendsto {a : M} {ψ : P×Model n → ℝ}
{p : P} {x : Model n} (L : ℝ≥0)
(hx : x∈(extChartAt 𝓘(ℝ,Model n) a).target) (hψ : ContDiffAt ℝ ∞ ψ (p,x))
{pj : ℕ → P} {xj : ℕ → Model n} {tj : ℕ → ℝ} {f : ℕ → M → ℝ}
(hp : Tendsto pj atTop (𝓝 p)) (hxx : Tendsto xj atTop (𝓝 x))
(ht : Tendsto tj atTop (𝓝 0)) (htpos : ∀ᶠ i in atTop,0<tj i)
(hlip : ∀ᶠ i in atTop,LipschitzWith L (f i))
(hbelow : ∀ᶠ i in atTop,∀ w,ψ (pj i,w)≤f i ((extChartAt 𝓘(ℝ,Model n) a).symm w))
(htouch : ∀ᶠ i in atTop,f i ((extChartAt 𝓘(ℝ,Model n) a).symm (xj i))=ψ (pj i,xj i))
(hd : ∀ᶠ i in atTop,DifferentiableAt ℝ
(fun w=>hopfLax (tj i) (f i) ((extChartAt 𝓘(ℝ,Model n) a).symm w))
(parametricShortForward a ψ ((pj i,tj i),xj i))) :
Tendsto (fun i=>fderiv ℝ
(fun w=>hopfLax (tj i) (f i) ((extChartAt 𝓘(ℝ,Model n) a).symm w))
(parametricShortForward a ψ ((pj i,tj i),xj i))) atTop
(𝓝 (fderiv ℝ (fun w=>ψ (p,w)) x)) := by
obtain ⟨U,hU,hD,_⟩ := exists_parametric_short_lower_sequence L hx hψ hp hxx ht htpos hlip hbelow htouch
apply hD.congr'
filter_upwards [hU,hd] with i hi hdi
let z := parametricShortForward a ψ ((pj i,tj i),xj i)
let F := fun w=>hopfLax (tj i) (f i) ((extChartAt 𝓘(ℝ,Model n) a).symm w)
have hm : IsLocalMin (fun w=>F w-U i w) z := by
filter_upwards [hi.2.2] with w hw
change F z-U i z≤F w-U i w
rw [hi.2.1]
linarith only [hw]
have he := hm.fderiv_eq_zero
rw [fderiv_fun_sub hdi (hi.1.differentiableAt (by norm_num))] at he
exact (sub_eq_zero.mp he).symm
end WeakMTWTransport
end
end OAI
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