File size: 2,490 Bytes
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 | import OAI.Geometry.Perimeter.Regularity.EllipticLinearStep
namespace OAI
noncomputable section
open Set Filter MeasureTheory
open scoped Topology ENNReal ContDiff InnerProductSpace
namespace RPISingularHarnack
variable {n : ℕ}
theorem linear_gradient_excess_step (hn : 0<n) :
∃ C : ℝ, 0≤C ∧ ∀ (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 →
WeakLinearEquation A f F (Metric.ball 0 1) →
∀ (A₀ : Euc n →L[ℝ] Euc n) (F₀ q : Euc n) (δ α β : ℝ),
0≤δ → 0≤α → 0≤β →
(∀ x∈Metric.ball (0:Euc n) 1, ‖ContinuousLinearMap.id ℝ (Euc n)-A x‖≤δ ∧
‖A₀-A x‖≤α ∧ ‖F x-F₀‖≤β) →
∃ q' : Euc n, ‖q'-q‖^2≤C^2*gradientEnergy f 0 1 q ∧
∀ θ : ℝ, θ≤1/2 → gradientEnergy f 0 θ q' ≤
(4*δ^2+2*C^2*θ^2*volume.real (Metric.ball (0:Euc n) θ))*gradientEnergy f 0 1 q+
4*(α*‖q‖+β)^2*volume.real (Metric.ball (0:Euc n) 1) := by
obtain ⟨C,hC,hB⟩ := continuous_gradient_one_scale hn
refine ⟨C,hC,fun A hA f hf F hF h A₀ F₀ q δ α β hδ hα hβ hb ↦ ?_⟩
let g : Euc n → ℝ := fun x ↦ f x-inner ℝ q x
have hg : ContDiff ℝ 1 g := hf.sub (innerSL ℝ q).contDiff
let e := linearResidual A f F A₀ F₀ q
have he : Continuous e := linearResidual_continuous hA hf hF _ _ _
obtain ⟨z,hz,hzE⟩ := hB g hg e he (linearResidual_pairing hA hf hF h A₀ F₀ q)
have hgrad (x : Euc n) : gradient g x=gradient f x-q := gradient_sub_linear (hf.differentiable (by simp)) q x
have hE (r : ℝ) (w : Euc n) : gradientEnergy g 0 r w=gradientEnergy f 0 r (q+w) := by
unfold gradientEnergy
simp_rw [hgrad,sub_sub]
have hBnd := continuous_residual_energy_bound he ((gradient_continuous_C1 hf).sub continuous_const)
hδ (add_nonneg (mul_nonneg hα (norm_nonneg _)) hβ) (fun x hx ↦ by
have hx' := hb x hx
simpa only [add_assoc,Pi.sub_apply,e] using linearResidual_bound (f := f) A₀ F₀ q x hx'.1 hx'.2.1 hx'.2.2)
change (∫ x in Metric.ball (0:Euc n) 1, ‖e x‖^2)≤
2*δ^2*gradientEnergy f 0 1 q+2*(α*‖q‖+β)^2*volume.real (Metric.ball (0:Euc n) 1) at hBnd
refine ⟨q+z,?_,fun θ hθ ↦ ?_⟩
· simpa only [add_sub_cancel_left,hE,add_zero] using hz
· have hh := hzE θ hθ
rw [hE,hE,add_zero] at hh
nlinarith
end RPISingularHarnack
end
end OAI
|