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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 | import OAI.Geometry.Perimeter.Regularity.EllipticCurl
namespace OAI
noncomputable section
open Set Filter MeasureTheory
open scoped Topology ENNReal ContDiff InnerProductSpace
namespace RPISingularHarnack
variable {n : ℕ}
def derivativeSecant (G : Euc n → Euc n) (a b : Euc n) : Euc n →L[ℝ] Euc n :=
∫ t in (0:ℝ)..1, fderiv ℝ G (a+t•(b-a))
theorem derivativeSecant_continuous {G : Euc n → Euc n} (hG : ContDiff ℝ 1 G) :
Continuous (fun z : Euc n × Euc n ↦ derivativeSecant G z.1 z.2) := by
apply intervalIntegral.continuous_parametric_intervalIntegral_of_continuous'
exact (hG.continuous_fderiv (by simp)).comp
(continuous_fst.fst.add (continuous_snd.smul (continuous_fst.snd.sub continuous_fst.fst)))
theorem derivativeSecant_apply {G : Euc n → Euc n} (hG : ContDiff ℝ 1 G) (a b : Euc n) :
derivativeSecant G a b (b-a)=G b-G a := by
have hc : Continuous (fun t : ℝ ↦ fderiv ℝ G (a+t•(b-a))) :=
(hG.continuous_fderiv (by simp)).comp (continuous_const.add (continuous_id.smul continuous_const))
rw [derivativeSecant,ContinuousLinearMap.intervalIntegral_apply (hc.intervalIntegrable 0 1)]
have hd (t : ℝ) : HasDerivAt (fun s : ℝ ↦ G (a+s•(b-a)))
(fderiv ℝ G (a+t•(b-a)) (b-a)) t := by
apply (hG.differentiable (by simp) _).hasFDerivAt.comp_hasDerivAt t
simpa using ((hasDerivAt_id t).smul_const (b-a) |>.const_add a)
simpa only [zero_smul,add_zero,one_smul,add_sub_cancel] using
intervalIntegral.integral_eq_sub_of_hasDerivAt (fun t _ ↦ hd t)
((hc.clm_apply continuous_const).intervalIntegrable 0 1)
theorem derivativeSecant_near {G : Euc n → Euc n} (hG : ContDiff ℝ 1 G)
(a b : Euc n) (A : Euc n →L[ℝ] Euc n) (δ : ℝ)
(hb : ∀ t∈Icc (0:ℝ) 1, ‖A-fderiv ℝ G (a+t•(b-a))‖≤δ) :
‖A-derivativeSecant G a b‖≤δ := by
have hc : Continuous (fun t : ℝ ↦ fderiv ℝ G (a+t•(b-a))) :=
(hG.continuous_fderiv (by simp)).comp (continuous_const.add (continuous_id.smul continuous_const))
have he : A-derivativeSecant G a b=∫ t in (0:ℝ)..1, A-fderiv ℝ G (a+t•(b-a)) := by
let : CompleteSpace (Euc n →L[ℝ] Euc n) := ContinuousLinearMap.instCompleteSpace
rw [intervalIntegral.integral_sub intervalIntegrable_const (hc.intervalIntegrable 0 1)]
simp only [derivativeSecant, intervalIntegral.integral_const, sub_zero, one_smul]
rw [he]
have hh := intervalIntegral.norm_integral_le_of_norm_le_const (fun t ht ↦
hb t (by
rw [Set.uIoc_of_le (by norm_num : (0:ℝ)≤1)] at ht
exact Set.Ioc_subset_Icc_self ht))
simpa using hh
theorem derivativeSecant_difference {G : Euc n → Euc n} (hG : ContDiff ℝ 1 G)
(a b a' b' : Euc n) (C : ℝ)
(hb : ∀ t∈Icc (0:ℝ) 1,
‖fderiv ℝ G (a+t•(b-a))-fderiv ℝ G (a'+t•(b'-a'))‖≤C) :
‖derivativeSecant G a b-derivativeSecant G a' b'‖≤C := by
have hc (a b : Euc n) : Continuous (fun t : ℝ ↦ fderiv ℝ G (a+t•(b-a))) :=
(hG.continuous_fderiv (by simp)).comp (continuous_const.add (continuous_id.smul continuous_const))
rw [derivativeSecant,derivativeSecant,←intervalIntegral.integral_sub
((hc a b).intervalIntegrable 0 1) ((hc a' b').intervalIntegrable 0 1)]
have hh := intervalIntegral.norm_integral_le_of_norm_le_const (fun t ht ↦
hb t (by
rw [Set.uIoc_of_le (by norm_num : (0:ℝ)≤1)] at ht
exact Set.Ioc_subset_Icc_self ht))
simpa using hh
end RPISingularHarnack
end
end OAI
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