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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 | import OAI.Geometry.Perimeter.Regularity.EllipticIteration
import OAI.Geometry.Perimeter.Regularity.EllipticGradientLimit
namespace OAI
noncomputable section
open Set Filter MeasureTheory
open scoped Topology ENNReal ContDiff InnerProductSpace
namespace RPISingularHarnack
variable {n : ℕ}
theorem linear_pointwise_iteration {θ δ K a B H R : ℝ}
(hθ : 0<θ) (hθ1 : θ<1) (hK : 1≤K)
(hc : LinearContractionProperty n θ δ K)
(ha : 0≤a) (hB : 0≤B) (hH : 0≤H) (hR : 0<R)
(hsmall : 32*K^2*a≤1) (hforce : 16*K*B≤H)
{A : Euc n → Euc n →L[ℝ] Euc n} (hA : Continuous A)
{f : Euc n → ℝ} (hf : ContDiff ℝ 1 f) {F : Euc n → Euc n} (hF : Continuous F)
(p : Euc n) (h : WeakLinearEquation A f F (Metric.ball p R))
(hnear : ∀ x∈Metric.ball p R, ‖ContinuousLinearMap.id ℝ (Euc n)-A x‖≤δ)
(hosc : ∀ j x, x∈Metric.ball p (θ^j*R) →
‖A p-A x‖≤a*(1/2)^j ∧ ‖F x-F p‖≤B*(1/2)^j)
(hinit : scaledGradientEnergy f p R 0≤H^2) :
∃ q : ℕ → Euc n, (∀ j, ‖q j-gradient f p‖≤2*K*H*(1/2)^j) ∧
(∀ j, scaledGradientEnergy f p (θ^j*R) (q j)≤H^2*((1/2)^j)^2) ∧
‖gradient f p‖≤2*K*H := by
obtain ⟨q,hq0,hqb,hqE,hqinc⟩ := linear_energy_iteration hθ hθ1 hK hc ha hB hH hR
hsmall hforce hA hf hF p h hnear hosc hinit
have hdist (j : ℕ) : dist (q j) (q (j+1))≤(K*H)*(1/2:ℝ)^j := by
rw [dist_eq_norm,norm_sub_rev]; exact hqinc j
obtain ⟨w,hw⟩ := cauchySeq_tendsto_of_complete
(cauchySeq_of_le_geometric (1/2) (K*H) (by norm_num) hdist)
have hwf : gradient f p=w := gradient_eq_of_scaled_energy hf p
(fun j ↦ by positivity)
(by simpa only [zero_mul] using (tendsto_pow_atTop_nhds_zero_of_lt_one hθ.le hθ1).mul_const R)
(by simpa only [zero_pow (by norm_num : (2:ℕ)≠0),mul_zero] using
((tendsto_pow_atTop_nhds_zero_of_lt_one (by norm_num : (0:ℝ)≤1/2) (by norm_num)).pow 2).const_mul (H^2))
hw hqE
have herror (j : ℕ) : ‖q j-gradient f p‖≤2*K*H*(1/2)^j := by
have hh := dist_le_of_le_geometric_of_tendsto (1/2) (K*H) (by norm_num) hdist hw j
rw [dist_eq_norm,←hwf] at hh
convert hh using 1
ring
refine ⟨q,herror,hqE,?_⟩
have hh := herror 0
simpa only [hq0,zero_sub,norm_neg,pow_zero,mul_one] using hh
end RPISingularHarnack
end
end OAI
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