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2fca478 | 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 66 67 68 69 70 71 | import OAI.Geometry.Perimeter.Regularity.EllipticScaleStep
namespace OAI
noncomputable section
open Set Filter MeasureTheory
open scoped Topology ENNReal ContDiff InnerProductSpace
namespace RPISingularHarnack
variable {n : ℕ}
def LinearContractionProperty (n : ℕ) (θ δ K : ℝ) : Prop :=
∀ (A : Euc n → Euc n →L[ℝ] Euc n), Continuous A →
∀ (f : Euc n → ℝ), ContDiff ℝ 1 f → ∀ F : Euc n → Euc n, Continuous F →
∀ (p : Euc n) (r : ℝ), 0 < r → WeakLinearEquation A f F (Metric.ball p r) →
∀ (A₀ : Euc n →L[ℝ] Euc n) (F₀ q : Euc n) (α β : ℝ), 0 ≤ α → 0 ≤ β →
(∀ x∈Metric.ball p r, ‖ContinuousLinearMap.id ℝ (Euc n)-A x‖ ≤ δ ∧
‖A₀-A x‖ ≤ α ∧ ‖F x-F₀‖ ≤ β) →
∃ q' : Euc n, ‖q'-q‖^2 ≤ K*scaledGradientEnergy f p r q ∧
scaledGradientEnergy f p (θ*r) q' ≤ scaledGradientEnergy f p r q/16+K*(α*‖q‖+β)^2
theorem exists_linear_contraction (hn : 0 < n) {ρ : ℝ} (hρ : 0 < ρ) :
∃ θ δ K : ℝ, 0 < θ ∧ θ < min ρ (1/2) ∧ 0 < δ ∧ 1 ≤ K ∧ LinearContractionProperty n θ δ K := by
obtain ⟨C,hC,hB⟩ := linear_gradient_scaled_step hn
let V := volume.real (Metric.ball (0:Euc n) 1)
have hV : 0 ≤ V := measureReal_nonneg
have htend : Tendsto (fun t : ℝ ↦ 2*C^2*t^2*V) (𝓝 0) (𝓝 0) := by
have hh : ContinuousAt (fun t : ℝ ↦ 2*C^2*t^2*V) 0 := by fun_prop
simpa only [zero_pow (by norm_num : 2≠0),mul_zero,zero_mul] using hh.tendsto
obtain ⟨s,hs,hsb⟩ := Metric.tendsto_nhds_nhds.mp htend (1/32) (by norm_num)
let θ := min (s/2) (min (ρ/2) (1/4))
have hθ : 0 < θ := lt_min (by positivity) (lt_min (by positivity) (by norm_num))
have hθs : θ < s := (min_le_left _ _).trans_lt (by linarith)
have hθρ : θ < ρ := ((min_le_right _ _).trans (min_le_left _ _)).trans_lt (by linarith)
have hθhalf : θ < 1/2 := ((min_le_right _ _).trans (min_le_right _ _)).trans_lt (by norm_num)
have hsmall : 2*C^2*θ^2*V ≤ 1/32 := by
have hh := hsb (x := θ) (by simpa only [dist_zero_right,Real.norm_eq_abs,abs_of_pos hθ] using hθs)
rw [dist_zero_right,Real.norm_eq_abs,abs_of_nonneg (by positivity)] at hh
exact hh.le
let δ := Real.sqrt (θ^n/128)
have hδ : 0 < δ := Real.sqrt_pos.mpr (by positivity)
have hδsq : δ^2=θ^n/128 := Real.sq_sqrt (by positivity)
have hdpart : 4*δ^2*(θ^n)⁻¹=1/32 := by rw [hδsq]; field_simp; norm_num
let K := max 1 (max (C^2) (4*(θ^n)⁻¹*V))
have hK : 1 ≤ K := le_max_left _ _
have hKC : C^2 ≤ K := (le_max_left _ _).trans (le_max_right _ _)
have hKV : 4*(θ^n)⁻¹*V ≤ K := (le_max_right _ _).trans (le_max_right _ _)
refine ⟨θ,δ,K,hθ,lt_min hθρ hθhalf,hδ,hK,?_⟩
intro A hA f hf F hF p r hr h A₀ F₀ q α β hα hβ hb
obtain ⟨q',hqp,hqE⟩ := hB A hA f hf F hF p r hr h A₀ F₀ q δ α β hδ.le hα hβ hb
have hE : 0 ≤ scaledGradientEnergy f p r q := scaledGradientEnergy_nonneg f p hr.le q
refine ⟨q',hqp.trans (mul_le_mul_of_nonneg_right hKC hE),?_⟩
have hqE' := hqE θ hθ hθhalf.le
rw [hdpart] at hqE'
have h1 := mul_le_mul_of_nonneg_right hsmall hE
have h2 := mul_le_mul_of_nonneg_right hKV (sq_nonneg (α*‖q‖+β))
change _ ≤ _ at hqE'
dsimp only [V] at h1 h2
nlinarith
theorem WeakLinearEquation.mono {A : Euc n → Euc n →L[ℝ] Euc n}
{f : Euc n → ℝ} {F : Euc n → Euc n} {U W : Set (Euc n)}
(h : WeakLinearEquation A f F U) (hWU : W⊆U) : WeakLinearEquation A f F W := by
intro φ
exact h ⟨φ.val,φ.property.1,φ.property.2.1,φ.property.2.2.trans hWU⟩
end RPISingularHarnack
end
end OAI
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