| 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 |
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|
| end |
|
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| end OAI |
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|
