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9655094 | 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 | import OAI.Analysis.CircleDomains.Reflection.AnnularReflectionInjection
namespace OAI
/-! Reflection into a moving target circle. The unnormalized bound does
not divide by its radius, so it remains useful along degenerating disks. -/
open Set Function Filter Metric Complex
open scoped Topology ComplexConjugate
noncomputable section
namespace CircleDomainRigidity
def circleReflectionExtension (c:ℂ) (r:ℝ) (f:ℂ → ℂ) (z:ℂ) : ℂ :=
c+(r:ℂ)*annularCircleReflect (fun w => (f w-c)/(r:ℂ)) z
lemma circleReflectionExtension_outer {c z:ℂ} {r:ℝ} {f:ℂ → ℂ}
(hr:0<r) (hz:1≤‖z‖) : circleReflectionExtension c r f z=f z := by
have hrc : (r:ℂ)≠0 := by exact_mod_cast hr.ne'
rw [circleReflectionExtension,annularCircleReflect,ite_eq_left hz]
field_simp
ring
lemma normalized_circle_norm (c:ℂ) {r:ℝ} (hr:0<r) (w:ℂ) :
‖(w-c)/(r:ℂ)‖=‖w-c‖/r := by
simp only [norm_div,Complex.norm_real,Real.norm_eq_abs,abs_of_pos hr]
lemma circleReflectionExtension_differentiableOn {U:Set ℂ} {f:ℂ → ℂ} {c:ℂ} {r:ℝ}
(hr:0<r) (hU:IsOpen U) (hz:∀z∈U,z≠0)
(hs:∀z∈U,unitCircleReflection z∈U)
(hc:ContinuousOn f (U∩{z | 1≤‖z‖}))
(hd:∀z∈U,1<‖z‖ → DifferentiableAt ℂ f z)
(ho:∀z∈U,1≤‖z‖ → r≤‖f z-c‖)
(hb:∀z∈U,‖z‖=1 → ‖f z-c‖=r) :
DifferentiableOn ℂ (circleReflectionExtension c r f) U := by
let g := fun z => (f z-c)/(r:ℂ)
have hg : ContinuousOn g (U∩{z | 1≤‖z‖}) := (hc.sub continuousOn_const).div_const _
have hgd : ∀z∈U,1<‖z‖ → DifferentiableAt ℂ g z :=
fun z hzU hzn => ((hd z hzU hzn).sub_const c).div_const _
have hgn : ∀z∈U,1≤‖z‖ → g z≠0 := by
intro z hzU hzn
apply norm_pos_iff.mp
change 0<‖(f z-c)/(r:ℂ)‖
rw [normalized_circle_norm c hr]
exact div_pos (hr.trans_le (ho z hzU hzn)) hr
have hgb : ∀z∈U,‖z‖=1 → ‖g z‖=1 := by
intro z hzU hzn
change ‖(f z-c)/(r:ℂ)‖=1
rw [normalized_circle_norm c hr,hb z hzU hzn,div_self hr.ne']
exact (differentiableOn_const c).add ((differentiableOn_const (r:ℂ)).mul
(differentiableOn_annularCircleReflect hU hz hs hg hgd hgn hgb))
lemma circleReflectionExtension_norm_le {U:Set ℂ} {f:ℂ → ℂ} {c:ℂ} {r M:ℝ}
(hr:0<r) (hz:∀z∈U,z≠0) (hs:∀z∈U,unitCircleReflection z∈U)
(hM:‖c‖+r≤M)
(hf:∀z∈U,1≤‖z‖ → r≤‖f z-c‖ ∧ ‖f z‖≤M)
{z:ℂ} (hzU:z∈U) : ‖circleReflectionExtension c r f z‖≤M := by
by_cases hzo : 1≤‖z‖
· rw [circleReflectionExtension_outer hr hzo]
exact (hf z hzU hzo).2
· let g := fun w => (f w-c)/(r:ℂ)
have hge : 1≤‖g (unitCircleReflection z)‖ := by
change 1≤‖(f (unitCircleReflection z)-c)/(r:ℂ)‖
rw [normalized_circle_norm c hr]
exact (one_le_div hr).mpr
(hf _ (hs z hzU) (unitCircleReflection_outer (hz z hzU) (lt_of_not_ge hzo).le)).1
have hinner : ‖annularCircleReflect g z‖≤1 := by
rw [annularCircleReflect,ite_eq_right hzo,unitCircleReflection_norm]
exact (inv_le_one₀ (zero_lt_one.trans_le hge)).mpr hge
change ‖c+(r:ℂ)*annularCircleReflect g z‖≤M
calc
_ ≤ ‖c‖+‖(r:ℂ)*annularCircleReflect g z‖ := norm_add_le _ _
_ = ‖c‖+r*‖annularCircleReflect g z‖ := by
rw [norm_mul,Complex.norm_real,Real.norm_eq_abs,abs_of_pos hr]
_ ≤ ‖c‖+r := by nlinarith
_ ≤ M := hM
end CircleDomainRigidity
end
end OAI
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