| 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 |
|
|
