| 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 |
|
|
