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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 | import OAI.Analysis.CircleDomains.Reflection.RoundReflectionCoefficient
namespace OAI
/-! The reflection-invariant extension remains compactly supported: every
non-base tile lies inside one of the original finitely many disks. -/
open Set Function MeasureTheory Metric Bornology
open scoped Topology OnePoint
noncomputable section
namespace CircleDomainRigidity
attribute [local instance] sphereMetric sphereBorel
local instance : BorelSpace Sphere := ⟨rfl⟩
lemma exists_compact_reflection_coefficient {I : Type*} [Finite I]
(c : I → ℂ) (r : I → ℝ) (hr : ∀i,0<r i)
(hd : Pairwise (fun i j => Disjoint
(((↑) : ℂ → Sphere) '' closedBall (c i) (r i))
(((↑) : ℂ → Sphere) '' closedBall (c j) (r j))))
(μ : ℂ → ℂ) (hm : Measurable μ) {k R : ℝ}
(hk : 0≤k) (hb : ∀z,‖μ z‖≤k) (hz : ∀z,R≤‖z‖ → μ z=0) :
∃ (E : Sphere → ℂ) (S : ℝ),Measurable E ∧ 0<S ∧ (∀p,‖E p‖≤k) ∧
(∀z : ℂ,S≤‖z‖ → E (z : Sphere)=0) ∧
EqOn E (sphereCoefficient 0 μ)
(⋃i,((↑) : ℂ → Sphere) '' closedBall (c i) (r i))ᶜ ∧
∀i p,E p=roundReflectionCoefficient (c i) p
(E (roundDiskReflection (c i) (r i) (hr i) p)) := by
have hμb (p : Sphere) : ‖sphereCoefficient 0 μ p‖≤k := by
induction p using OnePoint.rec with
| infty => simpa only [sphereCoefficient_infty,norm_zero] using hk
| coe z => exact hb z
obtain ⟨E,hEm,hEb,hbase,hEinv⟩ := exists_round_reflection_coefficient c r hr hd
(sphereCoefficient 0 μ) (sphereCoefficient_measurable 0 hm) hk hμb
obtain ⟨B,hB,hbound⟩ := (isCompact_iUnion (fun i => isCompact_closedBall (c i) (r i))).isBounded.exists_pos_norm_lt
let S := max B R
have hS : 0<S := hB.trans_le (le_max_left B R)
refine ⟨E,S,hEm,hS,hEb,?_,hbase,hEinv⟩
intro z hsz
have hout : (z : Sphere)∉⋃i,((↑) : ℂ → Sphere) '' closedBall (c i) (r i) := by
intro hz'
obtain ⟨i,w,hw,he⟩ := mem_iUnion.mp hz'
have hwz : w=z := OnePoint.coe_injective he
subst w
have hlt := hbound z (mem_iUnion.mpr ⟨i,hw⟩)
exact (not_lt_of_ge ((le_max_left B R).trans hsz)) hlt
rw [hbase hout,sphereCoefficient_coe,hz z ((le_max_right B R).trans hsz)]
end CircleDomainRigidity
end
end OAI
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