| import Mathlib |
| import OAI.Analysis.BiholderTransport.Regularity.ParametricLowerSequence |
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| namespace OAI |
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| noncomputable section |
| open Set Filter Manifold Bundle |
| open scoped Topology ContDiff NNReal |
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| 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] |
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| 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 |
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| end WeakMTWTransport |
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| end |
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| end OAI |
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