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a97875e | 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 | import OAI.Analysis.CircleDomains.Uniformization.ComplexRectangleLimit
namespace OAI
/-! A local rectangular Morera passage, allowing the smooth approximants'
poles outside the fixed open set and allowing a finite initial segment. -/
open Set Function Filter MeasureTheory Complex
open scoped Topology Interval
noncomputable section
namespace CircleDomainRigidity
open CannonBeltrami
lemma locally_uniform_curve_integral_on {U : Set ℂ} (hU : IsOpen U)
{f : ℕ → ℂ → ℂ} {g : ℂ → ℂ} (hf : ∀ᶠn in atTop,ContinuousOn (f n) U)
(hl : TendstoLocallyUniformlyOn f g atTop U) (γ : ℝ → ℂ) (hγ : Continuous γ)
(a b : ℝ) (hsub : MapsTo γ (uIcc a b) U) :
Tendsto (fun n => ∫t in a..b,f n (γ t)) atTop (𝓝 (∫t in a..b,g (γ t))) := by
have ht := (tendstoLocallyUniformlyOn_iff_forall_isCompact hU).mp hl
(γ '' uIcc a b) (image_subset_iff.mpr hsub) (isCompact_uIcc.image hγ)
have hc : TendstoUniformlyOn (fun n t => f n (γ t)) (fun t => g (γ t)) atTop (uIcc a b) :=
(ht.comp γ).mono (fun t ht => mem_image_of_mem γ ht)
exact hc.tendsto_intervalIntegral_of_continuousOn
(hf.mono fun n hn => hn.comp hγ.continuousOn hsub)
lemma complexRectangleFlux_tendsto_on {U : Set ℂ} (hU : IsOpen U)
{f : ℕ → ℂ → ℂ} {g : ℂ → ℂ} (hf : ∀ᶠn in atTop,ContinuousOn (f n) U)
(hl : TendstoLocallyUniformlyOn f g atTop U) (z w : ℂ) (hzw : Rectangle z w⊆U) :
Tendsto (fun n => complexRectangleFlux (f n) z w) atTop (𝓝 (complexRectangleFlux g z w)) := by
have h₁ := locally_uniform_curve_integral_on hU hf hl (fun x => x+z.im*I)
(by fun_prop) z.re w.re (fun x hx => hzw (by change _ ∈ _ ∧ _ ∈ _; simpa using And.intro hx (left_mem_uIcc : z.im∈uIcc z.im w.im)))
have h₂ := locally_uniform_curve_integral_on hU hf hl (fun x => x+w.im*I)
(by fun_prop) z.re w.re (fun x hx => hzw (by change _ ∈ _ ∧ _ ∈ _; simpa using And.intro hx (right_mem_uIcc : w.im∈uIcc z.im w.im)))
have h₃ := locally_uniform_curve_integral_on hU hf hl (fun y => w.re+y*I)
(by fun_prop) z.im w.im (fun y hy => hzw (by change _ ∈ _ ∧ _ ∈ _; simpa using And.intro (right_mem_uIcc : w.re∈uIcc z.re w.re) hy))
have h₄ := locally_uniform_curve_integral_on hU hf hl (fun y => z.re+y*I)
(by fun_prop) z.im w.im (fun y hy => hzw (by change _ ∈ _ ∧ _ ∈ _; simpa using And.intro (left_mem_uIcc : z.re∈uIcc z.re w.re) hy))
exact ((h₁.sub h₂).add (h₃.const_mul I)).sub (h₄.const_mul I)
lemma complexRectangleFlux_eq_bar_integral_on {U : Set ℂ} (hU : IsOpen U)
{f : ℂ → ℂ} (hf : ContDiffOn ℝ 1 f U) (z w : ℂ) (hzw : Rectangle z w⊆U) :
complexRectangleFlux f z w=(2*I)*(∫x : ℝ in z.re..w.re,
∫y : ℝ in z.im..w.im,partialBar f (x+y*I)) := by
have hd := hf.continuousOn_fderiv_of_isOpen hU (by norm_num)
have hi : IntegrableOn (fun p => I • fderiv ℝ f p 1-fderiv ℝ f p I) (Rectangle z w) :=
((hd.clm_apply continuousOn_const).const_smul I |>.sub
(hd.clm_apply continuousOn_const)).mono hzw |>.integrableOn_compact
(isCompact_uIcc.reProdIm isCompact_uIcc)
have he := integral_boundary_rect_of_differentiableOn_real f z w
((hf.differentiableOn (by norm_num)).mono hzw) hi
have hid (p : ℂ) : I • fderiv ℝ f p 1-fderiv ℝ f p I=(2*I)*partialBar f p := by
simp only [smul_eq_mul,partialBar]
ring_nf
simp [I_sq]
ring
change complexRectangleFlux f z w=_ at he
rw [he]
simp_rw [hid,intervalIntegral.integral_const_mul]
lemma differentiableOn_of_local_rectangle_errors {U : Set ℂ} (hU : IsOpen U)
{f : ℕ → ℂ → ℂ} {g : ℂ → ℂ}
(hf : ∀ᶠn in atTop,ContDiffOn ℝ 1 (f n) U)
(hl : TendstoLocallyUniformlyOn f g atTop U)
(he : ∀z w,Rectangle z w⊆U →
Tendsto (fun n => ∫x : ℝ in z.re..w.re,
∫y : ℝ in z.im..w.im,partialBar (f n) (x+y*I)) atTop (𝓝 0)) :
DifferentiableOn ℂ g U := by
apply (isConservativeOn_and_continuousOn_iff_isDifferentiableOn hU).mp
refine ⟨?_,hl.continuousOn (hf.frequently.mono fun n hn => hn.continuousOn)⟩
intro z w hzw
have hz : complexRectangleFlux g z w=0 := by
apply tendsto_nhds_unique (complexRectangleFlux_tendsto_on hU
(hf.mono fun n hn => hn.continuousOn) hl z w hzw)
have ht := (he z w hzw).const_mul (2*I)
simp only [mul_zero] at ht
apply ht.congr'
filter_upwards [hf] with n hn
exact (complexRectangleFlux_eq_bar_integral_on hU hn z w hzw).symm
rw [←add_eq_zero_iff_eq_neg,wedgeIntegral_add_wedgeIntegral_eq]
exact hz
end CircleDomainRigidity
end
end OAI
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