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■♥tr♦❞✉❝t✐♦♥✱ ♠♦t✐✈❛t✐♦♥ ❚✇♦ ❡①❛♠♣❧❡s ❢r♦♠ t❤❡ ❧✐tt❡r❛t✉r❡

• r✉s❤✐♥✲t②♣❡ ♦♣❡r❛t♦rs

❑♦❧♠♦❣♦r♦✈✲t②♣❡ ♦♣❡r❛t♦rs

❈♦♥tr♦❧❧❛❜✐❧✐t② r❡s✉❧ts ❢♦r ❞❡❣❡♥❡r❛t❡ ♣❛r❛❜♦❧✐❝ ♦♣❡r❛t♦rs

❑❛r✐♥❡ ❇❡❛✉❝❤❛r❞∗

∗❈◆❘❙✱ ❈▼▲❙✱ ❊❝♦❧❡ P♦❧②t❡❝❤♥✐q✉❡✳

❍❨P ✷✵✶✷✱ P❛❞♦✈❛✱ ❏✉♥❡ ✷✻

❑✳ ❇❡❛✉❝❤❛r❞

■♥tr♦❞✉❝t✐♦♥✱ ♠♦t✐✈❛t✐♦♥ ❚✇♦ ❡①❛♠♣❧❡s ❢r♦♠ t❤❡ ❧✐tt❡r❛t✉r❡

• r✉s❤✐♥✲t②♣❡ ♦♣❡r❛t♦rs

❑♦❧♠♦❣♦r♦✈✲t②♣❡ ♦♣❡r❛t♦rs

P❧❛♥

✶

■♥tr♦❞✉❝t✐♦♥✱ ♠♦t✐✈❛t✐♦♥

✷

❚✇♦ ❡①❛♠♣❧❡s ❢r♦♠ t❤❡ ❧✐tt❡r❛t✉r❡

✸

• r✉s❤✐♥✲t②♣❡ ♦♣❡r❛t♦rs

✹

❑♦❧♠♦❣♦r♦✈✲t②♣❡ ♦♣❡r❛t♦rs

❑✳ ❇❡❛✉❝❤❛r❞

■♥tr♦❞✉❝t✐♦♥✱ ♠♦t✐✈❛t✐♦♥ ❚✇♦ ❡①❛♠♣❧❡s ❢r♦♠ t❤❡ ❧✐tt❡r❛t✉r❡

• r✉s❤✐♥✲t②♣❡ ♦♣❡r❛t♦rs

❑♦❧♠♦❣♦r♦✈✲t②♣❡ ♦♣❡r❛t♦rs

❉❡❣❡♥❡r❛t❡ ♣❛r❛❜♦❧✐❝ ♦♣❡r❛t♦r

Ω ⊂ R♥ ❜♦✉♥❞❡❞✱ ❆(①) ✵ ✐♥ Ω    ✉t − ❞✐✈[❆(①)∇✉] + ❜(t, ①).∇✉ = ❢ (t, ①) ✐♥ Ω × (✵, ❚) ✉(✵, ①) = ✉✵(①) ① ∈ Ω ❇.❈. ♦♥ (✵, ❚) × Γ P♦ss✐❜❧❡ ❞❡❣❡♥❡r❛❝② ✿ ❛t t❤❡ ❜♦✉♥❞❛r② ✿ ❆(.)∇❞Γ(.) = ✵ ♦♥ Γ✵ ⊂ Γ✱ ✐♥ t❤❡ ✐♥t❡r✐♦r ✿ ❆(①) ✐s ♥♦t > ✵✱ ∀① ∈ ❉ ⊂ Ω ❖❢ ♣❛rt✐❝✉❧❛r ✐♥t❡r❡st ✿ ❤②♣♦❡❧❧✐♣t✐❝ ♦♣❡r❛t♦rs ❆♣♣❧✐❝❛t✐♦♥s ✿ ❝❧✐♠❛t♦❧♦❣② ✭❇✉❞②❦♦✲❙❡❧❧❡rs✮ ♣❛rt✐❝❧❡s s②st❡♠s ✭❑♦❧♠♦❣♦r♦✈✮ ❇✉t t❤❡ ♠♦t✐✈❛t✐♦♥s ♦❢ t❤✐s t❛❧❦ ❛r❡ r❛t❤❡r t❤❡♦r❡t✐❝❛❧✳✳✳

❑✳ ❇❡❛✉❝❤❛r❞

■♥tr♦❞✉❝t✐♦♥✱ ♠♦t✐✈❛t✐♦♥ ❚✇♦ ❡①❛♠♣❧❡s ❢r♦♠ t❤❡ ❧✐tt❡r❛t✉r❡

• r✉s❤✐♥✲t②♣❡ ♦♣❡r❛t♦rs

❑♦❧♠♦❣♦r♦✈✲t②♣❡ ♦♣❡r❛t♦rs

❚❤❡ ♥✉❧❧ ❝♦♥tr♦❧❧❛❜✐❧✐t② ♣r♦❜❧❡♠

▲❡t ✉s t❛❦❡ ❛ ❧♦❝❛❧❧② ❞✐str✐❜✉t❡❞ ❝♦♥tr♦❧ ❛s ❛ s♦✉r❝❡ t❡r♠ ✿    ✉t − ❞✐✈[❆(①)∇✉] + ❜(t, ①).∇✉ = ✶ω(①)❢ (t, ①) ✐♥ (✵, ❚) × Ω ✉(✵, ①) = ✉✵(①) ① ∈ ω + ❇✳❈✳ ♦♥ ∂Ω

ω Ω Γ

◗✉❡st✐♦♥ ✿ ●✐✈❡♥ ✉✵ ∈ ▲✷(Ω) ❛♥❞ ❚ > ✵ ❞♦❡s t❤❡r❡ ❡①✐st ❢ ∈ ▲✷((✵, ❚) × Ω) s✉❝❤ t❤❛t ✉(❚, .) = ✵ ❄

❑✳ ❇❡❛✉❝❤❛r❞

■♥tr♦❞✉❝t✐♦♥✱ ♠♦t✐✈❛t✐♦♥ ❚✇♦ ❡①❛♠♣❧❡s ❢r♦♠ t❤❡ ❧✐tt❡r❛t✉r❡

• r✉s❤✐♥✲t②♣❡ ♦♣❡r❛t♦rs

❑♦❧♠♦❣♦r♦✈✲t②♣❡ ♦♣❡r❛t♦rs

❈❧❛ss✐❝❛❧ r❡s✉❧ts ❛❜♦✉t ❡①❛❝t ❝♦♥tr♦❧❧❛❜✐❧✐t②

P❛r❛❜♦❧✐❝ ❝❛s❡ ✿ ♥✉❧❧ ❝♦♥tr♦❧❧❛❜✐❧✐t② ♦❢ t❤❡ ❤❡❛t ❡q✉❛t✐♦♥ ✉t − ∆✉ = ✶ω❢ (t, ①), ① ∈ Ω ❤♦❧❞s ∀ω ⊂ Ω ❛♥❞ ∀❚ > ✵✳ ✭✐♥✜♥✐t❡ ♣r♦♣❛❣❛t✐♦♥ s♣❡❡❞✮ ❬▲❡❜❡❛✉✲❘♦❜❜✐❛♥♦✱ ❋✉rs✐❦♦✈✲■♠♠❛♥✉✈✐❧♦✈✭✶✾✾✺✮❪ ❍②♣❡r❜♦❧✐❝ ❝❛s❡ ✿ ❝♦♥tr♦❧❧❛❜✐❧✐t② ♦❢ t❤❡ ✇❛✈❡ ❡q r❡q✉✐r❡s ✉tt − ∆✉ = ✶ω❢ (t, ①), ① ∈ Ω ❣❡♦♠❡tr✐❝ ❝♦♥❞✐t✐♦♥ ♦♥ (Ω, ω) ✿ ω ♠❡❡ts ❡✈❡r② r❛② ♦❢ ❣❡♦♠❡tr✐❝ ♦♣t✐❝s ❚ > ❚♠✐♥❂ ♠✐♥✐♠❛❧ t✐♠❡ ❢♦r ❛♥ ♦♣t✐❝ r❛② t♦ ♠❡❡t ω✳ ✭✜♥✐t❡ ♣r♦♣❛❣❛t✐♦♥ s♣❡❡❞✮ ❬❇❛r❞♦s✲▲❡❜❡❛✉✲❘❛✉❝❤✭✶✾✾✷✮❪ ◗✉ ✿ ❲❤❛t ❛❜♦✉t ❞❡❣❡♥❡r❛t❡ ♣❛r❛❜♦❧✐❝ s②st❡♠s ❄

❑✳ ❇❡❛✉❝❤❛r❞

■♥tr♦❞✉❝t✐♦♥✱ ♠♦t✐✈❛t✐♦♥ ❚✇♦ ❡①❛♠♣❧❡s ❢r♦♠ t❤❡ ❧✐tt❡r❛t✉r❡

• r✉s❤✐♥✲t②♣❡ ♦♣❡r❛t♦rs

❑♦❧♠♦❣♦r♦✈✲t②♣❡ ♦♣❡r❛t♦rs

❚❤❡ ❞✉❛❧✐t② ♠❡t❤♦❞ ✐♥ ❛♥ ❛❜str❛❝t ❢r❛♠❡ ✭❍❯▼✮

❝♦♥tr♦❧ s②st❡♠ ❛❞❥♦✐♥t s②st❡♠ ❞②

❞t = ❆② + ❇✉

②(✵) = ②✵

• − ❞φ

❞t = ❆∗φ

φ(❚) = φ❚ ■❢ ②✵ = ✵ t❤❡♥ ②(❚), φ❚ = ❚

✵

✉(t), ❇∗φ(t)❞t ❊♥❞✲♣♦✐♥t ♠❛♣ ✿ F❚ : ▲✷(✵, ❚) → ❍ ✉ → ②(❚) F∗

❚ :

❍ → ▲✷(✵, ❚) φ❚ → ❇∗φ(.) ❆♣♣r♦①✐♠❛t❡ ❝♦♥tr♦❧ ⇔ ❘❛♥❣❡(F❚) ❞❡♥s❡ ⇔ F∗

❚ ✐♥❥❡❝t✐✈❡

❊①❛❝t ❝♦♥tr♦❧ ⇔ F❚ s✉r❥❡❝t✐✈❡ ⇔ F∗

❚(φ❚)✷ ❝φ❚✷

◆✉❧❧ ❝♦♥tr♦❧❧❛❜✐❧✐t② ⇔ F∗

❚(φ❚)✷ ❝φ(✵)✷

❑✳ ❇❡❛✉❝❤❛r❞

■♥tr♦❞✉❝t✐♦♥✱ ♠♦t✐✈❛t✐♦♥ ❚✇♦ ❡①❛♠♣❧❡s ❢r♦♠ t❤❡ ❧✐tt❡r❛t✉r❡

• r✉s❤✐♥✲t②♣❡ ♦♣❡r❛t♦rs

❑♦❧♠♦❣♦r♦✈✲t②♣❡ ♦♣❡r❛t♦rs

❚♦ s✉♠♠❛r✐③❡ ✿ ❛♣♣r♦①✐♠❛t❡ ❝♦♥tr♦❧❧❛❜✐❧✐t②

   ✉t − ❞✐✈[❆(①)∇✉] = ✶ω(①)❢ (t, ①) ✐♥ (✵, ❚) × Ω ✉(✵, ①) = ✉✵(①) ① ∈ ω ✉(t, .) = ✵ ♦♥ Γ ❛♣♣r♦①✐♠❛t❡ ❝♦♥tr♦❧❧❛❜✐❧✐t② ✐♥ t✐♠❡ ❚ > ✵ ∀✉✵, ✉✶ ∈ ▲✷(Ω) ∀ǫ > ✵ ∃❢ ∈ ▲✷(◗❚) : ✉(❚, .) − ✉✶▲✷(Ω) < ǫ

• ◗❚ |❢ |✷ ❈❚
• Ω |✉✵ − ✉✶|✷

❜② ❞✉❛❧✐t② ❡q✉✐✈❛❧❡♥t t♦ ✉♥✐q✉❡ ❝♦♥t✐♥✉❛t✐♦♥ ❢r♦♠ (✵, ❚) × ω ✈t + ❞✐✈[❆(①)∇✈] = ✵ ✐♥ (✵, ❚) × Ω ✈(t, .) = ✵ ♦♥ Γ s❛t✐s✜❡s ✈ ≡ ✵ ♦♥ (✵, ❚) × ω ⇒ ✈ ≡ ✵ ✐♥ ◗❚

❑✳ ❇❡❛✉❝❤❛r❞

■♥tr♦❞✉❝t✐♦♥✱ ♠♦t✐✈❛t✐♦♥ ❚✇♦ ❡①❛♠♣❧❡s ❢r♦♠ t❤❡ ❧✐tt❡r❛t✉r❡

• r✉s❤✐♥✲t②♣❡ ♦♣❡r❛t♦rs

❑♦❧♠♦❣♦r♦✈✲t②♣❡ ♦♣❡r❛t♦rs

❚♦ s✉♠♠❛r✐③❡ ✿ ♥✉❧❧ ❝♦♥tr♦❧❧❛❜✐❧✐t②

   ✉t − ❞✐✈[❆(①)∇✉] = ✶ω❢ (t, ①) ✐♥ (✵, ❚) × Ω ✉(✵, ①) = ✉✵(①) ① ∈ ω ✉(t, .) = ✵ ♦♥ Γ ♥✉❧❧ ❝♦♥tr♦❧❧❛❜✐❧✐t② ✐♥ t✐♠❡ ❚ > ✵ ✿ ∀✉✵ ∈ ▲✷(Ω) ∃❢ ∈ ▲✷(◗❚) : ✉(❚, .) = ✵

• ◗❚ |❢ |✷ ❈❚
• Ω |✉✵|✷

❜② ❞✉❛❧✐t② ❡q✉✐✈❛❧❡♥t t♦ ♦❜s❡r✈❛❜✐❧✐t② ✐♥❡q✉❛❧✐t② ♦♥ (✵, ❚) × ω ✈t + ❞✐✈[❆(①)∇✈] = ✵ ✐♥ (✵, ❚) × Ω ✈(t, .) = ✵ ♦♥ Γ s❛t✐s✜❡s

• Ω

✈(✵, ①)✷❞① ❈❚ ❚

✵

• ω

✈(t, ①)✷❞①❞t ❚❤✐s ✐s ❛ ✬q✉❛♥t✐✜❡❞ ✈❡rs✐♦♥✬ ♦❢ t❤❡ ✉♥✐q✉❡ ❝♦♥t✐♥✉❛t✐♦♥✳

❑✳ ❇❡❛✉❝❤❛r❞

■♥tr♦❞✉❝t✐♦♥✱ ♠♦t✐✈❛t✐♦♥ ❚✇♦ ❡①❛♠♣❧❡s ❢r♦♠ t❤❡ ❧✐tt❡r❛t✉r❡

• r✉s❤✐♥✲t②♣❡ ♦♣❡r❛t♦rs

❑♦❧♠♦❣♦r♦✈✲t②♣❡ ♦♣❡r❛t♦rs

❘♦❛❞♠❛♣ t♦ ♦❜s❡r✈❛❜✐❧✐t② ❢♦r ❤❡❛t ❡q✉❛t✐♦♥s

❋❛tt♦r✐♥✐✲❘✉ss❡❧✭✶✾✼✶✮✱ ❘✉ss❡❧✭✶✾✼✽✮ ✿ ❘✐❡s③ ❜❛s✐s ❛♣♣r♦❛❝❤ ▲❡❜❡❛✉✲❘♦❜❜✐❛♥♦✭✶✾✾✺✮ ✿ ❘✐❡s③ ❜❛s✐s ✰ ❧♦❝❛❧ ❈❛r❧❡♠❛♥ ❡st✐♠❛t❡s ❋✉rs✐❦♦✈ ❛♥❞ ■♠❛♥✉✈✐❧♦✈✭✶✾✾✻✮ ✿ ❣❧♦❜❛❧ ❈❛r❧❡♠❛♥ ❡st✐♠❛t❡ ✈t + ❞✐✈[❆(①)∇✈] = ✵ ✐♥ (✵, ❚) × Ω ✈(t, .) = ✵ ♦♥ Γ ❚

✵

• Ω

τ ✸ [t(❚ − t)]✸ ✈ ✷❡✷τφ(t,①)❞①❞t ❈ ❚

✵

• ω

✈ ✷❞①❞t ✇❤❡r❡ τ >> ✶✱ φ(t, ①) = ❡ψ(①)−❡✷ψ∞

t(❚−t)

✱ ∇ψ = ✵ ✐♥ Ω − ω✳ ❲❤❛t ❝❤❛♥❣❡s ✐♥ t❤❡ ❞❡❣❡♥❡r❛t❡ ❝❛s❡ ❄ ♦❜s❡r✈❛❜✐❧✐t②✴♥✉❧❧ ❝♦♥tr♦❧❧❛❜✐❧✐t② ♠❛② ❢❛✐❧ ✭❢♦r ✈✐♦❧❡♥t ❞❡❣❡♥❡r❛❝②✮ φ ♠✉st ❜❡ ❛❞❛♣t❡❞ t♦ ❞❡❣❡♥❡r❛❝② ■s t❤❡ ❈❛r❧❡♠❛♥ ❡st✐♠❛t❡ tr✉❡ ❄ ✇❤✐❝❤ φ ❄✳✳✳

❑✳ ❇❡❛✉❝❤❛r❞

■♥tr♦❞✉❝t✐♦♥✱ ♠♦t✐✈❛t✐♦♥ ❚✇♦ ❡①❛♠♣❧❡s ❢r♦♠ t❤❡ ❧✐tt❡r❛t✉r❡

• r✉s❤✐♥✲t②♣❡ ♦♣❡r❛t♦rs

❑♦❧♠♦❣♦r♦✈✲t②♣❡ ♦♣❡r❛t♦rs

✷♥❞ ♣❛rt

❚✇♦ ❡①❛♠♣❧❡s ❢r♦♠ t❤❡ ❧✐tt❡r❛t✉r❡

❑✳ ❇❡❛✉❝❤❛r❞

■♥tr♦❞✉❝t✐♦♥✱ ♠♦t✐✈❛t✐♦♥ ❚✇♦ ❡①❛♠♣❧❡s ❢r♦♠ t❤❡ ❧✐tt❡r❛t✉r❡

• r✉s❤✐♥✲t②♣❡ ♦♣❡r❛t♦rs

❑♦❧♠♦❣♦r♦✈✲t②♣❡ ♦♣❡r❛t♦rs

❊① ✶ ✿ ❇♦✉♥❞❛r② ❞❡❣❡♥❡r❛❝②✱ s✐♠♣❧❡st ❝❛s❡

   ✉t + (①α✉①)① = ✶ω(①)❢ (t, ①) (t, ①) ∈ (✵, ❚) × (✵, ✶) ✉(t, ✶) = ✵, ✉(t, ✵) = ✵ ✐❢ ✵ α < ✶; (①α✉①)(t, ✵) = ✵ ✐❢ ✶ α ❚❤❡♦r❡♠ ❬❈❛♥♥❛rs❛✲▼❛rt✐♥❡③✲❱❛♥❝♦st❡♥♦❜❧❡✭✷✵✵✽✮❪ ◆✉❧❧ ❝♦♥tr♦❧❧❛❜✐❧✐t② ✐s

• ❢❛❧s❡

α ✷ (→ r❡❣✐♦♥❛❧ ♥✉❧❧ ❝♦♥tr♦❧❧❛❜✐❧✐t②) tr✉❡ ✵ α < ✷

T 1 ω

regional

x t

❚❡❝❤♥✐❝s ✿ ❈❱❆❘✴❈❛r❧❡♠❛♥ ❡st✐♠❛t❡s ❛♥❞ ❍❛r❞②✬s ✐♥❡q✉❛❧✐t②

❑✳ ❇❡❛✉❝❤❛r❞

■♥tr♦❞✉❝t✐♦♥✱ ♠♦t✐✈❛t✐♦♥ ❚✇♦ ❡①❛♠♣❧❡s ❢r♦♠ t❤❡ ❧✐tt❡r❛t✉r❡

• r✉s❤✐♥✲t②♣❡ ♦♣❡r❛t♦rs

❑♦❧♠♦❣♦r♦✈✲t②♣❡ ♦♣❡r❛t♦rs

❊①t❡♥s✐♦♥s

▼♦r❡ ❣❡♥❡r❛❧ ✶❉ ♣r♦❜❧❡♠s ❞✐✈❡r❣❡♥❝❡ ❢♦r♠

▼❛rt✐♥❡③✲❱❛♥❝♦st❡♥♦❜❧❡✭✷✵✵✻✮ ✉t − (❛(①)✉①)① = ✶ω❢ ❆❧❛❜❛✉✲❈❛♥♥❛rs❛✲❋r❛❣♥❡❧❧✐✭✷✵✵✻✮ ✉t − (❛(①)✉①)① + ❣(✉) = ✶ω❢ ❋❧♦r❡s✲❞❡ ❚❡r❡s❛✭✷✵✶✵✮ ✉t − (①θ✉①)① + ①σ❜(①, t)✉① = ✶ω❢

♥♦♥ ❞✐✈❡r❣❡♥❝❡ ❢♦r♠ ❈❛♥♥❛rs❛✲❋r❛❣♥❡❧❧✐✲❘♦❝❝❤❡tt✐✭✷✵✵✼✱✷✵✵✽✮ ✉t − ❛(①)✉①① − ❜(①)✉① = ✶ω❢ ❞❡❣❡♥❡r❛t❡✴s✐♥❣✉❧❛r ♣r♦❜❧❡♠s ❱❛♥❝♦st❡♥♦❜❧❡✲❩✉❛③✉❛✭✷✵✵✽✱ ✷✵✵✾✮ ✉t − (①θ✉①)① − λ

①σ ✉ = ✶ω❢

s②st❡♠s

❈❛♥♥❛rs❛✲❞❡ ❚❡r❡s❛✭✷✵✵✾✮ ❝❛s❝❛❞❡ ✷ × ✷ ▼❛♥✐❛r ❡t ❛❧✳✭✷✵✵✶✮ ❣❡♥❡r❛❧ ✷ × ✷

❊①t❡♥s✐♦♥ ✐♥ ✷❉ ✿ ❈❛♥♥❛rs❛✲▼❛rt✐♥❡③✲❱❛♥❝♦st❡♥♦❜❧❡✭✷✵✵✾✮

❑✳ ❇❡❛✉❝❤❛r❞

■♥tr♦❞✉❝t✐♦♥✱ ♠♦t✐✈❛t✐♦♥ ❚✇♦ ❡①❛♠♣❧❡s ❢r♦♠ t❤❡ ❧✐tt❡r❛t✉r❡

• r✉s❤✐♥✲t②♣❡ ♦♣❡r❛t♦rs

❑♦❧♠♦❣♦r♦✈✲t②♣❡ ♦♣❡r❛t♦rs

❊① ✷ ✿ ◆✉❧❧ ❝♦♥tr♦❧❧❛❜✐❧✐t② ♦❢ ❈r♦❝❝♦✲t②♣❡ ❡q✉❛t✐♦♥s

✉t + ✉① − ✉②② = ✶ω(①, ②)❢ (t, ①, ②) (t, ①, ✈) ∈ (✵, ❚) × (✵, ▲) × (✵, ✶), ❉✐r✐❝❤❧❡t ✐♥ ②, ♣❡r✐♦❞✐❝ ✐♥ ① ❉❡❝♦✉♣❧✐♥❣ ❜❡t✇❡❡♥ tr❛♥s♣♦rt ✐♥ ① ❛♥❞ ❞✐✛✉s✐♦♥ ✐♥ ② ⇒ ❘❡❣✐♦♥❛❧ ♥✉❧❧ ❝♦♥tr♦❧❧❛❜✐❧✐t② ❬▼❛rt✐♥❡③✲❘❛②♠♦♥❞✲❱❛♥❝♦st❡♥♦❜❧❡ ✷✵✵✸❪ ω = (❛, ❜)① × ω② ⇒ Ω❈ = (❛, ❜ + ❚)① × (✵, ✶)②

y x t T ωy a b b+T 1 L

❚❡❝❤♥✐❝s ✿ ✶❉ ❈❛r❧❡♠❛♥ ❡st✐♠❛t❡s ❛❧♦♥❣ ❝❤❛r❛❝t❡r✐st✐❝s

❑✳ ❇❡❛✉❝❤❛r❞

■♥tr♦❞✉❝t✐♦♥✱ ♠♦t✐✈❛t✐♦♥ ❚✇♦ ❡①❛♠♣❧❡s ❢r♦♠ t❤❡ ❧✐tt❡r❛t✉r❡

• r✉s❤✐♥✲t②♣❡ ♦♣❡r❛t♦rs

❑♦❧♠♦❣♦r♦✈✲t②♣❡ ♦♣❡r❛t♦rs

✸r❞ ♣❛rt

❈♦♥tr♦❧❧❛❜✐❧✐t② ♦❢

• r✉s❤✐♥ t②♣❡ ♦♣❡r❛t♦rs

❑✳ ❇❡❛✉❝❤❛r❞

■♥tr♦❞✉❝t✐♦♥✱ ♠♦t✐✈❛t✐♦♥ ❚✇♦ ❡①❛♠♣❧❡s ❢r♦♠ t❤❡ ❧✐tt❡r❛t✉r❡

• r✉s❤✐♥✲t②♣❡ ♦♣❡r❛t♦rs

❑♦❧♠♦❣♦r♦✈✲t②♣❡ ♦♣❡r❛t♦rs

• r✉s❤✐♥✲t②♣❡ ♦♣❡r❛t♦rs

γ > ✵    ✉t − ✉①① − |①|✷γ✉②② = ✶ω(①, ②)❢ (t, ①, ②) ✐♥ Ω ✉(t, ., .) = ✵ ♦♥ ∂Ω ✉(✵, ①, ②) = ✉✵(①, ②) ω Ω

• 1

1 1 x y ❘❦ ✿ ♥♦ ❞✐s♣❡rs✐♦♥ ✇❤❡♥ γ = ✶ ❢♦r t❤❡ ❛ss♦❝✐❛t❡❞ ❙❝❤rö❞✐♥❣❡r ❡q ♦♥ R✷ ❬●ér❛r❞✲●ré❧❧✐❡r ✷✵✶✵❪

❑✳ ❇❡❛✉❝❤❛r❞

■♥tr♦❞✉❝t✐♦♥✱ ♠♦t✐✈❛t✐♦♥ ❚✇♦ ❡①❛♠♣❧❡s ❢r♦♠ t❤❡ ❧✐tt❡r❛t✉r❡

• r✉s❤✐♥✲t②♣❡ ♦♣❡r❛t♦rs

❑♦❧♠♦❣♦r♦✈✲t②♣❡ ♦♣❡r❛t♦rs

❆♣♣r♦①✐♠❛t❡ ❝♦♥tr♦❧✴✉♥✐q✉❡ ❝♦♥t✐♥✉❛t✐♦♥ ❤♦❧❞s ∀γ✱ ∀ω

❊❧❧✐♣t✐❝ ❝❛s❡ ✿ ❬●❛r♦❢❛❧♦ ✶✾✾✸❪ ✈t − ✈①① − |①|✷γ✈②② = ✵ ✐♥ Ω ✈(t, ., .) = ✵ ♦♥ ∂Ω

ω Ω

• 1

1 1 x y

✈ ≡ ✵ ♦♥ (✵, ❚) × ω ⇒ ✈ ≡ ✵ ♦♥ (✵, ❚) × (✵, ✶) × (✵, ✶) ✭❯❈ ❢♦r ♣❛r❛❜♦❧✐❝ ♦♣❡r❛t♦rs✮ ⇒ ✈ ♥ ≡ ✵ ♦♥ (✵, ❚) × (✵, ✶)✱ ∀♥ ⇒ ✈ ♥ ≡ ✵ ♦♥ (✵, ❚) × (−✶, ✶)✱ ∀♥ ✐✳❡✳ ✈ ≡ ✵ ♦♥ (✵, ❚) × Ω ✈ ♥(t, ①) := ✶

✵

✈(t, ①, ②) s✐♥(♥π②)❞② ✈ ♥

t − ✈ ♥ ①① + (♥π)✷|①|✷γ✈ ♥ = ✵

❑✳ ❇❡❛✉❝❤❛r❞

■♥tr♦❞✉❝t✐♦♥✱ ♠♦t✐✈❛t✐♦♥ ❚✇♦ ❡①❛♠♣❧❡s ❢r♦♠ t❤❡ ❧✐tt❡r❛t✉r❡

• r✉s❤✐♥✲t②♣❡ ♦♣❡r❛t♦rs

❑♦❧♠♦❣♦r♦✈✲t②♣❡ ♦♣❡r❛t♦rs

❚❤❡♦r❡♠ ❬❑❇✲❈❛♥♥❛rs❛✲●✉❣❧✐❡❧♠✐✱ ✷✵✶✷❪

(●)    ✉t − ✉①① − |①|✷γ✉②② = ✶ω(①, ②)❢ (t, ①, ②) ✐♥ Ω ✉(t, ., .) = ✵ ♦♥ ∂Ω ✉(✵, ①, ②) = ✉✵(①, ②)

ω Ω

• 1

1 1 x y

✵ < γ < ✶ ✿ ✭●✮ ♥✉❧❧ ❝♦♥tr♦❧❧❛❜❧❡ ∀❚ > ✵✱ ∀ω γ > ✶ ✿ ✭●✮ ♥♦t ♥✉❧❧ ❝♦♥tr♦❧❧❛❜❧❡ γ = ✶ ❛♥❞ ω = (❛, ❜) × (✵, ✶) ✿ ∃❚ ∗ ❛✷/✷ s✉❝❤ t❤❛t ✭●✮ ✐s ♥✉❧❧ ❝♦♥tr♦❧❧❛❜❧❡ ∀❚ > ❚ ∗ ♥♦t ♥✉❧❧ ❝♦♥tr♦❧❧❛❜❧❡ ∀❚ < ❚ ∗

❑✳ ❇❡❛✉❝❤❛r❞

■♥tr♦❞✉❝t✐♦♥✱ ♠♦t✐✈❛t✐♦♥ ❚✇♦ ❡①❛♠♣❧❡s ❢r♦♠ t❤❡ ❧✐tt❡r❛t✉r❡

• r✉s❤✐♥✲t②♣❡ ♦♣❡r❛t♦rs

❑♦❧♠♦❣♦r♦✈✲t②♣❡ ♦♣❡r❛t♦rs

◆✉❧❧ ❝♦♥tr♦❧❧❛❜✐❧✐t② ⇔ ♦❜s❡r✈❛❜✐❧✐t② ✐♥❡q✉❛❧✐t②

(● ∗)    ✈t − ✈①① − |①|✷γ✈②② = ✵ ✈(t, ., .) = ✵ ♦♥ ∂Ω ✈(✵, ①, ②) = ✈✵(①, ②) ♦❜s❡r✈❛❜❧❡ ✐♥ [✵, ❚] × ω : ∃❈❚ > ✵ s✉❝❤ t❤❛t ∀✈✵ ∈ ▲✷(Ω)

• Ω

✈(❚, ①, ②)✷❞①❞② ❈❚ ❚

✵

• ω

✈(t, ①, ②)✷❞①❞② ❚❤❡♦r❡♠ ❬❑❇✲❈❛♥♥❛rs❛✲●✉❣❧✐❡❧♠✐✱ ✷✵✶✷❪ ✵ < γ < ✶ ✿ (● ∗) ♦❜s❡r✈❛❜❧❡ ∀❚ > ✵✱ ∀ω γ > ✶ ✿ (● ∗) ♥♦t ♦❜s❡r✈❛❜❧❡ γ = ✶ ❛♥❞ ω = (❛, ❜) × (✵, ✶) ✿ ∃❚ ∗ ❛✷/✷ s✉❝❤ t❤❛t (● ∗) ✐s ♦❜s❡r✈❛❜❧❡ ∀❚ > ❚ ∗ ♥♦t ♦❜s❡r✈❛❜❧❡ ∀❚ < ❚ ∗

❑✳ ❇❡❛✉❝❤❛r❞

■♥tr♦❞✉❝t✐♦♥✱ ♠♦t✐✈❛t✐♦♥ ❚✇♦ ❡①❛♠♣❧❡s ❢r♦♠ t❤❡ ❧✐tt❡r❛t✉r❡

• r✉s❤✐♥✲t②♣❡ ♦♣❡r❛t♦rs

❑♦❧♠♦❣♦r♦✈✲t②♣❡ ♦♣❡r❛t♦rs

❲❤❡r❡ t❤❡ ❝❧❛ss✐❝❛❧ ❛♣♣r♦❛❝❤ ❝❛✉s❡s ♣r♦❜❧❡♠s

❋♦r t❤❡ ❤❡❛t ❡q✉❛t✐♦♥✱ t❤❡ ❈❛r❧❡♠❛♥ ❡st✐♠❛t❡ ✐s ♣r♦✈❡❞ ✇✐t❤ ❛ ✇❡✐❣❤t ♦❢ t❤❡ ❢♦r♠ ϕ(①) = ❡λψ(①) − ❡λ❑✱ ✇❤❡r❡ ❑ > ψ∞✱ λ >> ✶ s✉✣❝✐❡♥t❧② ❧❛r❣❡✱ ψ > ✵ ♦♥ Ω ❛♥❞ ψ|∂Ω = ✵ ❛♥❞ ∇ψ(①) = ✵, ∀① ∈ Ω − ω. ❲✐t❤ t❤❡ s❛♠❡ str❛t❡❣②✱ ❢♦r ●r✉s❤✐♥✱ ♦♥❡ ✇♦✉❧❞ ♥❡❡❞

• ψ①

|①|γψ②

• = ✵, ♦♥ Ω − ω ✳✳✳

ω Ω

• 1

1 1 x y

❑✳ ❇❡❛✉❝❤❛r❞

■♥tr♦❞✉❝t✐♦♥✱ ♠♦t✐✈❛t✐♦♥ ❚✇♦ ❡①❛♠♣❧❡s ❢r♦♠ t❤❡ ❧✐tt❡r❛t✉r❡

• r✉s❤✐♥✲t②♣❡ ♦♣❡r❛t♦rs

❑♦❧♠♦❣♦r♦✈✲t②♣❡ ♦♣❡r❛t♦rs

Pr♦♦❢ ✇❤❡♥ ω ✐s ❛ str✐♣ ✿ ❋♦✉r✐❡r ❞❡❝♦♠♣♦s✐t✐♦♥

ω Ω

• 1

1 1 x y a b

✈(t, ①, ②) =

∞

• ♥=✶

✈♥(t, ①) s✐♥(♥π②) ✈t − ✈①① − |①|✷γ✈②② = ✵ ✈(t, ., .) = ✵ ♦♥ ∂Ω (● ∗

♥ )

∂t✈♥ − ∂✷

①✈♥ + (♥π)✷|①|✷γ✈♥ = ✵

✈♥(t, ±✶) = ✵

• ♦❛❧ ✿
• Ω ✈(❚, ①, ②)✷❞①❞② ❈

❚

✵

• ω ✈(t, ①, ②)✷❞①❞②❞t

⇔ ∞

♥=✶

✶

−✶ ✈♥(❚, ①)✷❞① ❈ ∞ ♥=✶

❚

✵

❜

❛ ✈♥(t, ①)✷❞①❞t

⇔ ❯♥✐❢♦r♠ ♦❜s❡r✈❛❜✐❧✐t② ♦❢ (● ∗

♥ ) ✇rt ♥ ✿

✶

−✶

✈♥(❚, ①)✷❞① ❈ ❚

✵

❜

❛

✈♥(t, ①)✷❞①❞t

❑✳ ❇❡❛✉❝❤❛r❞

■♥tr♦❞✉❝t✐♦♥✱ ♠♦t✐✈❛t✐♦♥ ❚✇♦ ❡①❛♠♣❧❡s ❢r♦♠ t❤❡ ❧✐tt❡r❛t✉r❡

• r✉s❤✐♥✲t②♣❡ ♦♣❡r❛t♦rs

❑♦❧♠♦❣♦r♦✈✲t②♣❡ ♦♣❡r❛t♦rs

Pr♦♦❢ ♦❢ ✭❯❖✮ ✇❤❡♥ ✵ < γ < ✶ ♦r ✭γ = ✶✱❚ ❧❛r❣❡✮

(● ∗

♥ )

• ∂t✈♥ − ∂✷

①✈♥ + (♥π)✷|①|✷γ✈♥ = ✵

✈♥(t, ±✶) = ✵

✶

❊①♣❧✐❝✐t ❞❡❝❛② r❛t❡ ❢♦r t❤❡ ❋♦✉r✐❡r ❝♦♠♣♦♥❡♥ts ✶

−✶

✈♥(❚, ①)✷❞① ❡−❝♥

✷ ✶+γ (❚−t)

✶

−✶

✈♥(t, ①)✷❞①, ∀t ∈ [✵, ❚]

✷

❈❛r❧❡♠❛♥ ❡st✐♠❛t❡ ✇✐t❤ ✇❡✐❣❤t ❡

♥❚✷ψ(①) t(❚−t) ✭❛❞❛♣t❡❞ ✵ < γ < ✶/✷✮

✷❚/✸

❚/✸

✶

−✶

✈♥(t, ①)✷❞①❞t ❡❈♥ ❚

✵

❜

❛

✈♥(t, ①)✷❞①❞t ∀♥ ♥❚ ❚ ✸ ✶

−✶

✈♥(❚, ①)✷❞① ❡❈♥−❝♥

✷ ✶+γ ❚ ✸

❚

✵

❜

❛

✈♥(t, ①)✷❞①❞t ∀♥ ♥❚ ❈♥ − ❝♥

✷ ✶+γ ❚

✸ → −∞ ✇❤❡♥ (✵ < γ < ✶, ∀❚ > ✵) ♦r (γ = ✶, ❚ > ✸/❝)

❑✳ ❇❡❛✉❝❤❛r❞

■♥tr♦❞✉❝t✐♦♥✱ ♠♦t✐✈❛t✐♦♥ ❚✇♦ ❡①❛♠♣❧❡s ❢r♦♠ t❤❡ ❧✐tt❡r❛t✉r❡

• r✉s❤✐♥✲t②♣❡ ♦♣❡r❛t♦rs

❑♦❧♠♦❣♦r♦✈✲t②♣❡ ♦♣❡r❛t♦rs

❲❤❡♥ ω ✐s ♥♦t ❛ str✐♣ ✿ ❝♦♥str✉❝t✐♦♥ ♦❢ t❤❡ ❝♦♥tr♦❧

▲❡❜❡❛✉✲❘♦❜❜✐❛♥♦✬s str❛t❡❣② ✿ ✵ = ❚✵ < ❚✶ < ... < ❚❥ → ❚✱ ❚❥+✶ = ❚❥ + ✷❛❥ ♦♥ [❚❥, ❚❥ + ❛❥]✱ ♦♥❡ ❛♣♣❧✐❡s ❛ ❝♦♥tr♦❧ t❤❛t st❡❡rs t❤❡ ✷❥✲✜rt ❝♦♠♣♦♥❡♥ts t♦ ③❡r♦ ✿ ❝♦st ❡❈✷❥ ♦♥ [❚❥ + ❛❥, ❚❥+✶]✱ ♥♦ ❝♦♥tr♦❧ → ❞✐ss✐♣❛t✐♦♥ ❡−❝(✷❥ )

✷ ✶+γ ❚❥

❑❡② ♣♦✐♥t ✿ ∃❈ > ✵✱ st ∀◆✱ ◆

❦=✶ |❜❦|✷ ❡❈◆ ❞ ❝

• ◆

❦=✶ ❜❦ s✐♥(❦②)

• ✷

❞②

• Ω ✈(❚, ①, ②)✷❞①❞②

=

✷❥

• ♥=✶

✶

−✶ ✈♥(❚, ①)✷❞①

❈

✷❥

• ♥=✶

❚

✵

❜

❛ ✈♥(t, ①)✷❞①❞t

❡❈✷❥ ❚

✵

❜

❛

❞

❝

• ✷❥
• ♥=✶

✈♥(t, ①) s✐♥(❦π②)

• ✷

❞②❞①❞t = ❡❈✷❥ ❚

✵

• ω ✈(t, ①, ②)✷❞①❞②❞t

❑✳ ❇❡❛✉❝❤❛r❞

■♥tr♦❞✉❝t✐♦♥✱ ♠♦t✐✈❛t✐♦♥ ❚✇♦ ❡①❛♠♣❧❡s ❢r♦♠ t❤❡ ❧✐tt❡r❛t✉r❡

• r✉s❤✐♥✲t②♣❡ ♦♣❡r❛t♦rs

❑♦❧♠♦❣♦r♦✈✲t②♣❡ ♦♣❡r❛t♦rs

❋❛✐❧✉r❡ ♦❢ ❯❖ ✇❤❡♥ γ > ✶ ♦r (γ = ✶, ❚ s♠❛❧❧) ✿ str❛t❡❣②

❚❛❦❡ t❤❡ ✜rst ❡✐❣❡♥❢✉♥❝t✐♦♥s ✇♥ ✇✐t❤ λ♥

• −✇ ′′

♥ (①) + [(♥π)✷|①|✷γ − λ♥]✇♥(①) = ✵ ,

① ∈ (−✶, ✶) , ♥ ∈ N∗ , ✇♥(±✶) = ✵ , ✇♥ ≥ ✵ , ✇♥▲✷(−✶,✶) = ✶ ✈♥(t, ①) := ✇♥(①)❡−λ♥t ✐s ❛ ♣❛rt✐❝✉❧❛r s♦❧✉t✐♦♥ ♦❢ ∂t✈♥ − ∂✷

①✈♥ + (♥π)✷|①|✷γ✈♥ = ✵

(t, ①) ∈ (✵, ❚) × (−✶, ✶) , ✈♥(t, ±✶) = ✵ t ∈ (✵, ❚) , ✭❯❖✮ ❢❛✐❧s ✐❢ ✇❡ ❝❛♥ ♣r♦✈✐❞❡ ✉♣♣❡r ❜♦✉♥❞ s✉❝❤ t❤❛t ❚

✵

❜

❛ ✈♥(t, ①)✷❞①❞t

✶

−✶ ✈♥(❚, ①)✷❞①

= ❡✷λ♥❚ − ✶ ✷λ♥ ❜

❛

✇♥(①)✷❞① − →

♥→+∞ ✵

❚❤✐s ✐s t❡❝❤♥✐❝❛❧ ❜❡❝❛✉s❡ [(♥π)✷|①|✷γ − λ♥] ❝❤❛♥❣❡s s✐❣♥ ✐♥ (−✶, ✶)✳

❑✳ ❇❡❛✉❝❤❛r❞

■♥tr♦❞✉❝t✐♦♥✱ ♠♦t✐✈❛t✐♦♥ ❚✇♦ ❡①❛♠♣❧❡s ❢r♦♠ t❤❡ ❧✐tt❡r❛t✉r❡

• r✉s❤✐♥✲t②♣❡ ♦♣❡r❛t♦rs

❑♦❧♠♦❣♦r♦✈✲t②♣❡ ♦♣❡r❛t♦rs

❋❛✐❧✉r❡ ♦❢ ❯❖ ✇❤❡♥ γ = ✶ ❛♥❞ ❚ ❛✷/✷

❲❡ ❝❛♥ ♣❡r❢♦r♠ ❡①♣❧✐❝✐t ❝♦♠♣✉t❛t✐♦♥s ❜❡❝❛✉s❡ −✇ ′′

♥ + (♥π)✷|①|✷✇♥ = λ♥✇♥,

① ∈ (−✶, ✶) ✇♥(±✶) = ✵ ✐s ✬❝❧♦s❡✬ t♦ t❤❡ ❤❛r♠♦♥✐❝ ♦s❝✐❧❧❛t♦r ✿ ✇♥(①) ∼

✹

√♥❡− ♥π①✷

✷ ,

λ♥ ∼ ♥π. ❚❤✉s ❚

✵

❜

❛ ✈♥(t, ①)✷❞①❞t

✶

−✶ ✈♥(❚, ①)✷❞①

∼ ❡♥π(✷❚−❛✷) ✷❛π✷♥✸/✷ t❡♥❞s t♦ ③❡r♦ ✇❤❡♥ ❚ ❛✷/✷✳

❑✳ ❇❡❛✉❝❤❛r❞

■♥tr♦❞✉❝t✐♦♥✱ ♠♦t✐✈❛t✐♦♥ ❚✇♦ ❡①❛♠♣❧❡s ❢r♦♠ t❤❡ ❧✐tt❡r❛t✉r❡

• r✉s❤✐♥✲t②♣❡ ♦♣❡r❛t♦rs

❑♦❧♠♦❣♦r♦✈✲t②♣❡ ♦♣❡r❛t♦rs

❋❛✐❧✉r❡ ♦❢ ❯❖ ✇❤❡♥ γ > ✶ ✿ ❝♦♠♣❛r✐s♦♥ ❛r❣✉♠❡♥t

−✇ ′′

♥ (①) + [(♥π)✷|①|✷γ − λ♥]✇♥(①) = ✵, ① ∈ (−✶, ✶)

✇♥(±✶) = ✵ r❡str✐❝t t♦ [①♥, ✶] ✇❤❡r❡ ①♥ = [λ♥/(♥π)✷]✶/✷γ → ✵ ❡q✉❛t✐♦♥ ②✐❡❧❞s ✉♣♣❡r ❜♦✉♥❞ |✇ ′

♥(①♥)| √①♥λ♥

❜② ❝♦♠♣❛r✐s♦♥ ❛r❣✉♠❡♥t    −❲ ′′

♥ (①) + [(♥π)✷|①|✷γ − λ♥]❲♥(①) ✵

❲♥(✶) ✵ ❲ ′

♥(①♥) < −√①♥λ♥

⇒ ✇♥ ❲♥ ♦♥ [①♥, ✶] ❝♦♥str✉❝t ❈♥ > ✵ s✉❝❤ t❤❛t ❲♥(①) := ❈♥❡−❈γ♥①γ+✶ s❛t✐s✜❡s ❡✷λ♥❚ − ✶ ✷λ♥ ❜

❛

✇♥(①)✷❞① ❡✷λ♥❚ − ✶ ✷λ♥ ❜

❛

❲♥(①)✷❞① ❡✷♥(

λ♥ ♥ ❚−❈γ)❘(♥)

❝♦♥❝❧✉❞❡ ✇✐t❤ ❞✐ss✐♣❛t✐♦♥ s♣❡❡❞ λ♥ ❝♥

✷ ✶+γ

❑✳ ❇❡❛✉❝❤❛r❞

■♥tr♦❞✉❝t✐♦♥✱ ♠♦t✐✈❛t✐♦♥ ❚✇♦ ❡①❛♠♣❧❡s ❢r♦♠ t❤❡ ❧✐tt❡r❛t✉r❡

• r✉s❤✐♥✲t②♣❡ ♦♣❡r❛t♦rs

❑♦❧♠♦❣♦r♦✈✲t②♣❡ ♦♣❡r❛t♦rs

❈♦♥❝❧✉❞✐♥❣ r❡♠❛r❦s ❛♥❞ ♦♣❡♥ ♣r♦❜❧❡♠s ❛❜♦✉t ●r✉s❤✐♥

✉t − ✉①① − |①|✷γ✉②② = ✶ω(①, ②)❢ (t, ①, ②)

ω Ω

• 1

1 1 x y

❲❡ ❤❛✈❡ ♣r♦✈❡❞ t❤❛t ♥✉❧❧ ❝♦♥tr♦❧❧❛❜✐❧✐t② ❤♦❧❞s ∀❚ > ✵✱ ∀ω ✇❤❡♥ γ ∈ (✵, ✶) ❤♦❧❞s ♦♥❧② ❢♦r ❚ >> ✶ ✇❤❡♥ γ = ✶ ❛♥❞ ω = (❛, ❜) × (✵, ✶) ❞♦❡s ♥♦t ❤♦❧❞ ✇❤❡♥ ❞❡❣❡♥❡r❛❝② ✐s t♦♦ str♦♥❣✱ ✐✳❡✳ γ > ✶✳ ❖♣❡♥ ♣r♦❜❧❡♠s ✿ ✇❤❡♥ γ = ✶ ✿ ■s ❚♠✐♥ = ❛✷/✷ ❄ ❲❤❡♥ ❚ > ❛✷/✷✱ ∀ω ❄ ♠♦r❡ ❣❡♥❡r❛❧ ❛♥❞✴♦r ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ Ω ❄ ❜♦✉♥❞❛r② ❝♦♥tr♦❧ ❄

❑✳ ❇❡❛✉❝❤❛r❞

■♥tr♦❞✉❝t✐♦♥✱ ♠♦t✐✈❛t✐♦♥ ❚✇♦ ❡①❛♠♣❧❡s ❢r♦♠ t❤❡ ❧✐tt❡r❛t✉r❡

• r✉s❤✐♥✲t②♣❡ ♦♣❡r❛t♦rs

❑♦❧♠♦❣♦r♦✈✲t②♣❡ ♦♣❡r❛t♦rs

✹t❤ ♣❛rt

❈♦♥tr♦❧❧❛❜✐❧✐t② ♦❢ ❑♦❧♠♦❣♦r♦✈✲t②♣❡ ❡q✉❛t✐♦♥s

❑✳ ❇❡❛✉❝❤❛r❞

■♥tr♦❞✉❝t✐♦♥✱ ♠♦t✐✈❛t✐♦♥ ❚✇♦ ❡①❛♠♣❧❡s ❢r♦♠ t❤❡ ❧✐tt❡r❛t✉r❡

• r✉s❤✐♥✲t②♣❡ ♦♣❡r❛t♦rs

❑♦❧♠♦❣♦r♦✈✲t②♣❡ ♦♣❡r❛t♦rs

❑♦❧♠♦❣♦r♦✈✲t②♣❡ ❡q✉❛t✐♦♥s

γ > ✵ ✉t + ✈ γ✉① − ✉✈✈ = ✶ω(①, ✈)❢ (t, ①, ✈) ① ∈ T ✈ ∈ (−✶, ✶) ❇✳❈✳ ❛t ✈ = ±✶ ω Ω

• 1

1 x v 2π ❘❦ ✿ ■❢ γ = ✶✱ ✉(t, ①, ✈) = ❤(t, ① − ✈t, ✈) ✇❤❡r❡ ∂t❤ − (∂✈ − t∂①)✷❤ = ✵

❑✳ ❇❡❛✉❝❤❛r❞

■♥tr♦❞✉❝t✐♦♥✱ ♠♦t✐✈❛t✐♦♥ ❚✇♦ ❡①❛♠♣❧❡s ❢r♦♠ t❤❡ ❧✐tt❡r❛t✉r❡

• r✉s❤✐♥✲t②♣❡ ♦♣❡r❛t♦rs

❑♦❧♠♦❣♦r♦✈✲t②♣❡ ♦♣❡r❛t♦rs

❯♥✐q✉❡ ❝♦♥t✐♥✉❛t✐♦♥ ❢♦r ❑♦❧♠♦❣♦r♦✈✲t②♣❡ ❡q✉❛t✐♦♥s

❆❧✐♥❤❛❝✲❩✉✐❧②✭✶✾✽✵✮ ✿ ■♥ t❤❡ ❡❧❧✐♣t✐❝ ❝❛s❡ ✭∂✷

✈ + ✈∂①✮✱ ✉♥✐q✉❡ ❝♦♥t✐♥✉❛t✐♦♥ ❤♦❧❞s✳

■♥ t❤❡ ♣❛r❛❜♦❧✐❝ ❝❛s❡✱ t❤❡r❡ ❡①✐sts ❛ ③❡r♦✲♦r❞❡r ❈ ∞✲♣❡rt✉r❜❛t✐♦♥ ✇✐t❤♦✉t ✉♥✐q✉❡ ❝♦♥t✐♥✉❛t✐♦♥ ✿ ∀γ ∈ N∗✱ ∃❛(t, ①, ✈)✱ ✉(t, ①, ✈) ❈ ∞✲❢✉♥❝t✐♦♥s ♦♥ ❛ ♥❡✐❣❤❜♦r❤♦♦❞ ❱ ♦❢ ✵ ✐♥ R✸ t❤❛t ✈❛♥✐s❤ ❢♦r ✈ < ✵ s✉❝❤ t❤❛t ✉t − ✈ γ✉① − ✉✈✈ − ❛✉ = ✵ ✐♥ ❱ ✵ ∈ ❙✉♣♣(✉)✳

❑✳ ❇❡❛✉❝❤❛r❞

■♥tr♦❞✉❝t✐♦♥✱ ♠♦t✐✈❛t✐♦♥ ❚✇♦ ❡①❛♠♣❧❡s ❢r♦♠ t❤❡ ❧✐tt❡r❛t✉r❡

• r✉s❤✐♥✲t②♣❡ ♦♣❡r❛t♦rs

❑♦❧♠♦❣♦r♦✈✲t②♣❡ ♦♣❡r❛t♦rs

❚❤❡♦r❡♠ ❬❑❇✱ ✷✵✶✷❪

(❑) : ✉t + ✈ γ✉① − ✉✈✈ = ✶ω(①, ✈)❢ (t, ①, ✈) ① ∈ T ✈ ∈ (−✶, ✶)

ω Ω

• 1

1 x v 2π

P❡r✐♦❞✐❝✲t②♣❡ ❇❈ ❛♥❞ γ = ✶ ✿ ✭❑✮ ✐s ♥✉❧❧ ❝♦♥tr♦❧❧❛❜❧❡ ∀❚ > ✵✱ ∀ω ❉✐r✐❝❤❧❡t ❇❈ ❛♥❞ ω = T × (❛, ❜) γ = ✶ ✿ ✭❑✮ ✐s ♥✉❧❧ ❝♦♥tr♦❧❧❛❜❧❡ ∀❚ > ✵ γ = ✷ ✿ t❤❡r❡ ❡①✐sts ❚ ∗ ❛✷/✷ s✳t✳

✭❑✮ ✐s ♥✉❧❧ ❝♦♥tr♦❧❧❛❜❧❡ ∀❚ > ❚ ∗ ✭❑✮ ✐s ♥♦t ♥✉❧❧ ❝♦♥tr♦❧❧❛❜❧❡ ∀❚ < ❚ ∗

❑✳ ❇❡❛✉❝❤❛r❞

■♥tr♦❞✉❝t✐♦♥✱ ♠♦t✐✈❛t✐♦♥ ❚✇♦ ❡①❛♠♣❧❡s ❢r♦♠ t❤❡ ❧✐tt❡r❛t✉r❡

• r✉s❤✐♥✲t②♣❡ ♦♣❡r❛t♦rs

❑♦❧♠♦❣♦r♦✈✲t②♣❡ ♦♣❡r❛t♦rs

Pr♦♦❢

✉t + ✈ γ✉① − ✉✈✈ = ✵ ① ∈ T ✈ ∈ (−✶, ✶) + ❇❈ ❋♦✉r✐❡r ❞❡❝♦♠♣♦s✐t✐♦♥ ✿ ✉(t, ①, ✈) = ✉♥(t, ✈)❡✐♥① ✇❤❡r❡ ∂t✉♥ + ✐♥✈ γ✉♥ − ∂✷

✈✉♥ = ✵

✈ ∈ (−✶, ✶) + ❇❈ ❈❛r❧❡♠❛♥ ❡st✐♠❛t❡ ✇✐t❤ ✇❡✐❣❤t ❡

√♥❚✷ψ(✈) t(❚−t)

✿ → ❝♦st ❡❈√♥ ❉✐ss✐♣❛t✐♦♥ ❡st✐♠❛t❡ ✿ P❡r✐♦❞✐❝✱ γ = ✶ ✿ ❡−♥✷t✸/✶✷ ✭❡①♣❧✐❝✐t s♦❧✉t✐♦♥✮ ❉✐r✐❝❤❧❡t✱ γ = ✶ ✿ ❡−♥✷/✸t ✭▲②❛♣✉♥♦✈ ❢♥✮ ❉✐r✐❝❤❧❡t✱ γ = ✷ ✿ ❡−δ√♥t ✭▲②❛♣✉♥♦✈ ❢♥✮ ❋❛✐❧✉r❡ ♦❢ ❯❖ ✭γ = ✷✱ ❚ ❛✷/✷✮ ✿ ●♥(t, ✈) := ❡−

√ ✐♥(t+✈ ✷/✷)

❚

✵

❜

❛ |●♥(t, ✈)|✷❞✈❞t

✶

−✶ |●♥(❚, ✈)|✷❞✈

∼

♥→+∞

❡

√ ✷♥(❚−❛✷/✷)

♥✸/✹

❑✳ ❇❡❛✉❝❤❛r❞

■♥tr♦❞✉❝t✐♦♥✱ ♠♦t✐✈❛t✐♦♥ ❚✇♦ ❡①❛♠♣❧❡s ❢r♦♠ t❤❡ ❧✐tt❡r❛t✉r❡

• r✉s❤✐♥✲t②♣❡ ♦♣❡r❛t♦rs

❑♦❧♠♦❣♦r♦✈✲t②♣❡ ♦♣❡r❛t♦rs

❈♦♥❝❧✉❞✐♥❣ r❡♠❛r❦s ❛♥❞ ♦♣❡♥ ♣r♦❜❧❡♠s ❛❜♦✉t ❑♦❧♠♦❣♦r♦✈

✉t − ✈ γ✉① − ✉✈✈ = ✶ω(①, ②)❢ (t, ①, ②)

ω Ω

• 1

1 x v 2π

❲❡ ❤❛✈❡ ♣r♦✈❡❞ t❤❛t ♥✉❧❧ ❝♦♥tr♦❧❧❛❜✐❧✐t② ❤♦❧❞s ∀❚ > ✵✱ ∀ω ✇✐t❤ γ = ✶ ❛♥❞ ♣❡r✐♦❞✐❝ ❇❈ ❤♦❧❞s ∀❚ > ✵ ✇✐t❤ ω = T × (❛, ❜)✱ γ = ✶ ❛♥❞ ❉✐r✐❝❤❧❡t ❇❈ ❤♦❧❞s ❢♦r ❚ > ❚ ∗ > ✵ ✇✐t❤ ω = T × (❛, ❜)✱ γ = ✷ ❛♥❞ ❉✐r✐❝❤❧❡t ❇❈ ❖♣❡♥ ♣r♦❜❧❡♠s ✿ ✇✐t❤ ❉✐r✐❝❤❧❡t ❇❈ ✿ ∀ω ❄ ✇❤❡♥ γ = ✷✱ ❚ ∗ = ❛✷/✷ ❄ ✇❤❡♥ γ ✸ ❄ ♠♦r❡ ❣❡♥❡r❛❧ ❛♥❞✴♦r ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ Ω ❄

❑✳ ❇❡❛✉❝❤❛r❞

■♥tr♦❞✉❝t✐♦♥✱ ♠♦t✐✈❛t✐♦♥ ❚✇♦ ❡①❛♠♣❧❡s ❢r♦♠ t❤❡ ❧✐tt❡r❛t✉r❡

• r✉s❤✐♥✲t②♣❡ ♦♣❡r❛t♦rs

❑♦❧♠♦❣♦r♦✈✲t②♣❡ ♦♣❡r❛t♦rs

❈♦♥❝❧✉❞✐♥❣ r❦ ✿ ❍ör♠❛♥❞❡r✬s ❝♦♥❞✐t✐♦♥ ❢♦r ❤②♣♦❡❧❧✐♣t✐❝✐t②

❚❤❡♦r❡♠ ❬❍ör♠❛♥❞❡r✱ ✶✾✻✼❪ ▲❡t P := r

❥=✶ ❳ ✷ ❥ + ❳✵ + ❝✱ ✇❤❡r❡

❳✵, ..., ❳r ❛r❡ ✶st ♦r❞❡r ❤♦♠♦❣❡♥❡♦✉s ❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦rs ✇✐t❤ ❈ ∞ ❝♦❡✣❝✐❡♥ts ✐♥ ❛♥ ♦♣❡♥ s❡t Ω ⊂ R♥ ❛♥❞ ❝ ∈ ❈ ∞(Ω)✳ ■❢ t❤❡r❡ ❡①✐sts ♥ ♦♣❡r❛t♦rs ❛♠♦♥❣ ❳❥✶✱ [❳❥✶, ❳❥✷]✱✳✳✳✱[❳❥✶, [❳❥✷, [❳❥✸, [..., ❳❥❦]...]]]✱✳✳✳ ✇❤❡r❡ ❥✐ ∈ {✵, ✶, ..., r}✱ ✇❤✐❝❤ ❛r❡ ❧✐♥❡❛r❧② ✐♥❞❡♣❡♥❞❡♥t ❛t ❛♥② ♣♦✐♥t ✐♥ Ω✱ t❤❡♥✱ P ✐s ❤②♣♦❡❧❧✐♣t✐❝ ✿ ❛♥② ❞✐str✐❜✉t✐♦♥ ✉ ✐♥ Ω✱ ✐s ❛ ❈ ∞ ❢✉♥❝t✐♦♥ ✇❤❡r❡ s♦ ✐s P✉✳

• r✉s❤✐♥ ❛♥❞ ❑♦❧♠♦❣♦r♦✈ ❛r❡ ♣r♦t♦t②♣❡s ♦❢ ❤②♣♦❡❧❧✐♣t✐❝ ♦♣❡r❛t♦rs ✿

❳✶(①, ②) :=

• ✶

✵

• ,

❳✷(①, ②) :=

• ✵

①γ

• .

[❳✶, ❳✷](①, ②) =

• ✵

γ①γ−✶

• , [❳✶, [❳✶, ❳✷]](①, ②) =
• ✵

γ(γ − ✶)①γ−✷

• , . . .
• r✉s❤✐♥ ✿ ◆❈ ❤♦❧❞s ♦♥❧② ✇❤❡♥ t❤❡ ✜rst ❜r❛❝❦❡t ✐s s✉✣❝✐❡♥t

❑♦❧♠♦❣♦r♦✈ ✿ ◆❈ ❤♦❧❞s ✇❤❡♥ t❤❡ ✷ ✜rst ❜r❛❝❦❡ts ❛r❡ s✉✣❝✐❡♥t ▲✐♥❦ ◆❈✴♥❜ ♦❢ ✐t❡r❛t❡❞ ▲✐❡ ❜r❛❝❦❡ts ✐♥ ❍ör♠❛♥❞❡r✬s ❝❞t ❄

❑✳ ❇❡❛✉❝❤❛r❞