| import OAI.Analysis.RCDHeat.ExistingAnalytic |
| import OAI.Analysis.RCDHeat.WeightedGradientConvex |
|
|
| namespace OAI |
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| /-! # The signed first variation for a compact quadratic contact potential -/ |
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| open Set Filter MeasureTheory Metric |
| open scoped NNReal ENNReal Topology |
| noncomputable section |
| namespace RegularPointCoordinates |
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] |
| [ProperSpace X] (μ : Measure X) [μ.Regular] |
| (hQ : QuadraticCheegerEnergy μ) |
| attribute [local instance] pathBorel pathBorelSpace |
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|
| theorem compact_contact_first_variation (hspace : IsGeodesicSpace X) |
| (p : X) {ψ : X → ℝ} {L : ℝ≥0} (hψ : LipschitzWith L ψ) |
| {R : ℝ} (hzero : ∀ x, R < dist p x → ψ x = 0) |
| (π : Measure C(Time,X)) [IsFiniteMeasure π] |
| (hgeo : ∀ᵐ γ ∂π, IsGeodesic γ) {C : ℝ≥0∞} (hC : C ≠ ∞) |
| (hcomp : ∀ t : Time, π.map (fun γ => γ t) ≤ C • μ) |
| (hD : MemLp (fun γ : C(Time,X) => dist (γ 0) (γ 1)) 2 π) |
| (hc : ∀ᵐ γ ∂π, hopfLax (-ψ) 1 (γ 0)+ψ (γ 1) = dist (γ 0) (γ 1)^2/2) : |
| ∃ u : SobolevSpace μ hQ, |
| (hopfLax (-ψ) 1 =ᵐ[μ] SobolevSpace.toLp μ hQ u) ∧ |
| (∀ᵐ γ ∂π, SobolevSpace.minimalSlope μ hQ u (γ 0) = dist (γ 0) (γ 1)) ∧ |
| ∀ g : SobolevSpace μ hQ, |
| Tendsto (planInitialSecant π (SobolevSpace.toLp μ hQ g)) (𝓝[>] (0:ℝ)) |
| (𝓝 (-weightedGradientPair μ hQ (π.map (fun γ => γ 0)) u g)) := by |
| have hpos : (0 : ℝ) < 1 := by norm_num |
| have hnzero (x : X) (hx : R < dist p x) : (-ψ) x = 0 := by |
| simp only [Pi.neg_apply,hzero x hx,neg_zero] |
| have hχ := hopfLax_lipschitz hψ.neg hspace hpos |
| have hχs := hopfLax_hasCompactSupport hψ.neg p R hnzero hpos |
| let u := SobolevSpace.ofLipschitzCompact μ hQ hχ hχs |
| have hurep : hopfLax (-ψ) 1 =ᵐ[μ] SobolevSpace.toLp μ hQ u := |
| (SobolevSpace.ofLipschitzCompact_coe μ hQ hχ hχs).symm |
| have had (x y : X) : hopfLax (-ψ) 1 x+ψ y ≤ dist x y^2/2 := by |
| have hh := hopfLax_le_cost hψ.neg hpos x y |
| simp only [Pi.neg_apply,mul_one] at hh |
| linarith |
| have hbound : ∀ᵐ x ∂μ, SobolevSpace.minimalSlope μ hQ u x ≤ hopfLeftRadius (-ψ) x := |
| hopfLeftRadius_minimalSlope_bound_proper μ hQ hψ.neg hspace p R hnzero |
| have hstart : ∀ᵐ γ ∂π, |
| SobolevSpace.minimalSlope μ hQ u (γ 0) ≤ hopfLeftRadius (-ψ) (γ 0) := |
| ae_of_ae_map (measurable_path_eval 0).aemeasurable |
| (hbound.filter_mono (Measure.absolutelyContinuous_of_le_smul (hcomp 0)).ae_le) |
| have hb : ∀ᵐ γ ∂π, SobolevSpace.minimalSlope μ hQ u (γ 0) ≤ dist (γ 0) (γ 1) := by |
| filter_upwards [hstart,hc] with γ hγ he |
| apply hγ.trans (hopfLeftRadius_le_minimizer hψ.neg ?_) |
| change hopfLax (-ψ) 1 (γ 0) = -ψ (γ 1)+dist (γ 0) (γ 1)^2/(2*1) |
| linarith |
| refine ⟨u,hurep,?_,?_⟩ |
| · exact (SobolevSpace.minimalSlope_isRelaxed μ hQ u).kantorovich_gradient_eq_displacement |
| μ π hgeo hC hcomp hD hurep had hc hb |
| · intro g |
| exact kantorovich_signed_first_variation μ hQ π hgeo hC hcomp hD u g hurep had hc hb |
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| end RegularPointCoordinates |
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| end |
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| end OAI |
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