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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 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 | import OAI.Geometry.Perimeter.Regularity.EllipticContinuousComparison
namespace OAI
noncomputable section
open Set Filter MeasureTheory
open scoped Topology ENNReal ContDiff InnerProductSpace
namespace RPISingularHarnack
variable {n : ℕ}
def WeakLinearEquation (A : Euc n → Euc n →L[ℝ] Euc n) (f : Euc n → ℝ)
(F : Euc n → Euc n) (U : Set (Euc n)) : Prop :=
∀ φ : SmoothLocalTest U,
(∫ x, inner ℝ (A x (gradient f x)) (gradient φ.val x)) =
∫ x, inner ℝ (F x) (gradient φ.val x)
def linearResidual (A : Euc n → Euc n →L[ℝ] Euc n) (f : Euc n → ℝ)
(F : Euc n → Euc n) (A₀ : Euc n →L[ℝ] Euc n) (F₀ q : Euc n) (x : Euc n) :=
(gradient f x-q) - A x (gradient f x) + F x + (A₀ q-F₀)
theorem linearResidual_continuous {A : Euc n → Euc n →L[ℝ] Euc n}
{f : Euc n → ℝ} {F : Euc n → Euc n} (hA : Continuous A) (hf : ContDiff ℝ 1 f)
(hF : Continuous F) (A₀ : Euc n →L[ℝ] Euc n) (F₀ q : Euc n) :
Continuous (linearResidual A f F A₀ F₀ q) :=
((((gradient_continuous_C1 hf).sub continuous_const).sub
(hA.clm_apply (gradient_continuous_C1 hf))).add hF).add continuous_const
theorem linearResidual_pairing {A : Euc n → Euc n →L[ℝ] Euc n}
{f : Euc n → ℝ} {F : Euc n → Euc n} {U : Set (Euc n)}
(hA : Continuous A) (hf : ContDiff ℝ 1 f) (hF : Continuous F)
(h : WeakLinearEquation A f F U) (A₀ : Euc n →L[ℝ] Euc n) (F₀ q : Euc n)
(φ : SmoothLocalTest U) :
(∫ x, inner ℝ (gradient (fun y ↦ f y-inner ℝ q y) x) (gradient φ.val x)) =
∫ x, inner ℝ (linearResidual A f F A₀ F₀ q x) (gradient φ.val x) := by
have hg := inner_gradient_pairing_integrable (gradient_continuous_C1 hf) φ
have hq := inner_gradient_pairing_integrable (continuous_const : Continuous (fun _ : Euc n ↦ q)) φ
have hAf := inner_gradient_pairing_integrable (hA.clm_apply (gradient_continuous_C1 hf)) φ
have hFi := inner_gradient_pairing_integrable hF φ
have hc := inner_gradient_pairing_integrable
(continuous_const : Continuous (fun _ : Euc n ↦ A₀ q-F₀)) φ
simp_rw [gradient_sub_linear (hf.differentiable (by simp)),linearResidual,inner_add_left,inner_sub_left]
simp_rw [← inner_sub_left (A₀ q) F₀]
erw [integral_add ((hg.sub hq).sub hAf |>.add hFi) hc,
integral_add ((hg.sub hq).sub hAf) hFi,integral_sub (hg.sub hq) hAf,
constant_gradient_pairing_zero φ (A₀ q-F₀),h φ]
ring_nf
rfl
theorem linearResidual_bound {A : Euc n → Euc n →L[ℝ] Euc n}
{f : Euc n → ℝ} {F : Euc n → Euc n} (A₀ : Euc n →L[ℝ] Euc n) (F₀ q x : Euc n)
{δ α β : ℝ} (hδ : ‖ContinuousLinearMap.id ℝ (Euc n)-A x‖≤δ)
(hα : ‖A₀-A x‖≤α) (hβ : ‖F x-F₀‖≤β) :
‖linearResidual A f F A₀ F₀ q x‖≤δ*‖gradient f x-q‖+α*‖q‖+β := by
have he : linearResidual A f F A₀ F₀ q x =
(ContinuousLinearMap.id ℝ (Euc n)-A x) (gradient f x-q)+(A₀-A x) q+(F x-F₀) := by
simp only [linearResidual,sub_apply,ContinuousLinearMap.id_apply,map_sub]
abel
rw [he]
have h1 := (ContinuousLinearMap.id ℝ (Euc n)-A x).le_opNorm (gradient f x-q)
have h2 := (A₀-A x).le_opNorm q
have hh := norm_add_le ((ContinuousLinearMap.id ℝ (Euc n)-A x) (gradient f x-q)+(A₀-A x) q) (F x-F₀)
have hh' := norm_add_le ((ContinuousLinearMap.id ℝ (Euc n)-A x) (gradient f x-q)) ((A₀-A x) q)
have h1' := mul_le_mul_of_nonneg_right hδ (norm_nonneg (gradient f x-q))
have h2' := mul_le_mul_of_nonneg_right hα (norm_nonneg q)
linarith
theorem continuous_residual_energy_bound {e W : Euc n → Euc n} (he : Continuous e)
(hW : Continuous W) {δ b : ℝ} (hδ : 0≤δ) (hb : 0≤b)
(h : ∀ x∈Metric.ball (0:Euc n) 1, ‖e x‖≤δ*‖W x‖+b) :
(∫ x in Metric.ball (0:Euc n) 1, ‖e x‖^2) ≤
2*δ^2*(∫ x in Metric.ball (0:Euc n) 1, ‖W x‖^2)+2*b^2*volume.real (Metric.ball (0:Euc n) 1) := by
have hI (G : Euc n → Euc n) (hG : Continuous G) :
IntegrableOn (fun x ↦ ‖G x‖^2) (Metric.ball (0:Euc n) 1) volume :=
((hG.norm.pow 2).continuousOn.integrableOn_compact (isCompact_closedBall 0 1)).mono_set Metric.ball_subset_closedBall
have : IsFiniteMeasure (volume.restrict (Metric.ball (0:Euc n) 1)) := ⟨by
rw [Measure.restrict_apply_univ]; exact measure_ball_lt_top⟩
calc
_ ≤ ∫ x in Metric.ball (0:Euc n) 1, 2*δ^2*‖W x‖^2+2*b^2 := by
apply integral_mono_ae (hI e he) (((hI W hW).const_mul _).add (integrable_const _))
filter_upwards [ae_restrict_mem measurableSet_ball] with x hx
have hsq := (sq_le_sq₀ (norm_nonneg (e x))
(add_nonneg (mul_nonneg hδ (norm_nonneg (W x))) hb)).2 (h x hx)
change ‖e x‖^2≤2*δ^2*‖W x‖^2+2*b^2
nlinarith [sq_nonneg (δ*‖W x‖-b)]
_ = _ := by rw [integral_add ((hI W hW).const_mul _) (integrable_const _),integral_const_mul,integral_const]; simp [smul_eq_mul,mul_comm]
end RPISingularHarnack
end
end OAI
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