File size: 3,129 Bytes
e4d9d2b | 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.Analysis.RCDHeat.ExistingAnalytic
import OAI.Analysis.RCDHeat.WeightedGradientConvex
namespace OAI
/-! # The signed first variation for a compact quadratic contact potential -/
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
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
end RegularPointCoordinates
end
end OAI
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