File size: 4,017 Bytes
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 | import Mathlib.Analysis.Complex.HasPrimitives
import OAI.Geometry.Cannon.Beltrami.CauchySymbol
namespace OAI
/-! Rectangular Morera passage for smooth maps with vanishing antiholomorphic
error. Uniform convergence is needed only on the four compact edges. -/
open Set Function Filter MeasureTheory Complex
open scoped Topology Interval
noncomputable section
namespace CircleDomainRigidity
open CannonBeltrami
def complexRectangleFlux (f : ℂ → ℂ) (z w : ℂ) : ℂ :=
(∫x : ℝ in z.re..w.re,f (x+z.im*I))-(∫x : ℝ in z.re..w.re,f (x+w.im*I))+
I*(∫y : ℝ in z.im..w.im,f (w.re+y*I))-I*(∫y : ℝ in z.im..w.im,f (z.re+y*I))
lemma locally_uniform_curve_integral {f : ℕ → ℂ → ℂ} {g : ℂ → ℂ}
(hf : ∀n,Continuous (f n)) (hl : TendstoLocallyUniformly f g atTop)
(γ : ℝ → ℂ) (hγ : Continuous γ) (a b : ℝ) :
Tendsto (fun n => ∫t in a..b,f n (γ t)) atTop (𝓝 (∫t in a..b,g (γ t))) := by
have ht := tendstoLocallyUniformly_iff_forall_isCompact.mp hl
(γ '' uIcc a b) (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 (.of_forall fun n => (hf n |>.comp hγ).continuousOn)
lemma complexRectangleFlux_tendsto {f : ℕ → ℂ → ℂ} {g : ℂ → ℂ}
(hf : ∀n,Continuous (f n)) (hl : TendstoLocallyUniformly f g atTop) (z w : ℂ) :
Tendsto (fun n => complexRectangleFlux (f n) z w) atTop (𝓝 (complexRectangleFlux g z w)) := by
have h₁ := locally_uniform_curve_integral hf hl (fun x => x+z.im*I) (by fun_prop) z.re w.re
have h₂ := locally_uniform_curve_integral hf hl (fun x => x+w.im*I) (by fun_prop) z.re w.re
have h₃ := locally_uniform_curve_integral hf hl (fun y => w.re+y*I) (by fun_prop) z.im w.im
have h₄ := locally_uniform_curve_integral hf hl (fun y => z.re+y*I) (by fun_prop) z.im w.im
exact ((h₁.sub h₂).add (h₃.const_mul I)).sub (h₄.const_mul I)
lemma complexRectangleFlux_eq_bar_integral {f : ℂ → ℂ}
(hf : ContDiff ℝ 1 f) (z w : ℂ) :
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 hi : IntegrableOn (fun p => I • fderiv ℝ f p 1-fderiv ℝ f p I)
(uIcc z.re w.re ×ℂ uIcc z.im w.im) :=
(((hf.continuous_fderiv (by norm_num)).clm_apply continuous_const).const_smul I |>.sub
((hf.continuous_fderiv (by norm_num)).clm_apply continuous_const)).continuousOn.integrableOn_compact
(isCompact_uIcc.reProdIm isCompact_uIcc)
have he := integral_boundary_rect_of_differentiableOn_real f z w
(hf.differentiable (by norm_num)).differentiableOn 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_vanishing_rectangle_errors {U : Set ℂ} (hU : IsOpen U)
{f : ℕ → ℂ → ℂ} {g : ℂ → ℂ} (hf : ∀n,ContDiff ℝ 1 (f n))
(hl : TendstoLocallyUniformly f g atTop)
(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.continuous (.of_forall fun n => (hf n).continuous)).continuousOn⟩
intro z w hzw
have hz : complexRectangleFlux g z w=0 := by
apply tendsto_nhds_unique (complexRectangleFlux_tendsto (fun n => (hf n).continuous) hl z w)
have ht := (he z w hzw).const_mul (2*I)
simpa only [complexRectangleFlux_eq_bar_integral (hf _),mul_zero] using ht
rw [←add_eq_zero_iff_eq_neg,wedgeIntegral_add_wedgeIntegral_eq]
exact hz
end CircleDomainRigidity
end
end OAI
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