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❙♠♦♦t❤❡rs ❢♦r ❡✣❝✐❡♥t ♠✉❧t✐❣r✐❞ ♠❡t❤♦❞s ✐♥ ■●❆ ❈❧❡♠❡♥s ❍♦❢r❡✐t❤❡r✱ ❙t❡❢❛♥ ❚❛❦❛❝s✱ ❲❛❧t❡r ❩✉❧❡❤♥❡r ❉❉✷✸✱ ❏✉❧② ✷✵✶✺ s✉♣♣♦rt❡❞ ❜② ❚❤❡ ✇♦r❦ ✇❛s ❢✉♥❞❡❞ ❜② t❤❡ ❆✉str✐❛♥ ❙❝✐❡♥❝❡ ❋✉♥❞ ✭❋❲❋✮✿ ◆❋◆ ❙✶✶✼ ✭✜rst ❛♥❞ t❤✐r❞ ❛✉t❤♦r✮ ❛♥❞ ❏✸✸✻✷✲◆✷✺ ✭s❡❝♦♥❞ ❛✉t❤♦r✮✳

❖✉t❧✐♥❡ Pr❡❧✐♠✐♥❛r✐❡s ▼♦❞❡❧ ♣r♦❜❧❡♠ ■●❆ ❞✐s❝r❡t✐③❛t✐♦♥ ▼✉❧t✐❣r✐❞ ❛❧❣♦r✐t❤♠ ▼✉❧t✐❣r✐❞ ❢r❛♠❡✇♦r❦ ❇❛s✐s✲✐♥❞❡♣❡♥❞❡♥t s♠♦♦t❤❡r ❙♠♦♦t❤❡r ❢♦r ♦♥❡ ❞✐♠❡♥s✐♦♥ ❙♠♦♦t❤❡r ❢♦r t✇♦ ❞✐♠❡♥s✐♦♥s ◆✉♠❡r✐❝❛❧ r❡s✉❧ts ❈♦♥❝❧✉s✐♦♥s ❛♥❞ ❖✉t❧♦♦❦

❖✉t❧✐♥❡ Pr❡❧✐♠✐♥❛r✐❡s ▼♦❞❡❧ ♣r♦❜❧❡♠ ■●❆ ❞✐s❝r❡t✐③❛t✐♦♥ ▼✉❧t✐❣r✐❞ ❛❧❣♦r✐t❤♠ ▼✉❧t✐❣r✐❞ ❢r❛♠❡✇♦r❦ ❇❛s✐s✲✐♥❞❡♣❡♥❞❡♥t s♠♦♦t❤❡r ❙♠♦♦t❤❡r ❢♦r ♦♥❡ ❞✐♠❡♥s✐♦♥ ❙♠♦♦t❤❡r ❢♦r t✇♦ ❞✐♠❡♥s✐♦♥s ◆✉♠❡r✐❝❛❧ r❡s✉❧ts ❈♦♥❝❧✉s✐♦♥s ❛♥❞ ❖✉t❧♦♦❦

P♦✐ss♦♥ ♠♦❞❡❧ ♣r♦❜❧❡♠ ❉♦♠❛✐♥ Ω ∈ R ❞ ●✐✈❡♥ ❢✉♥❝t✐♦♥ ❢ ∈ ▲ ✷ (Ω) ❋✐♥❞ ✉ ∈ ❍ ✶ (Ω) s✉❝❤ t❤❛t − ∆ ✉ = ❢ ✐♥ Ω ∂ ✉ ∂ ♥ = ✵ ♦♥ ∂ Ω

❱❛r✐❛t✐♦♥❛❧ ❢♦r♠✉❧❛t✐♦♥ ❉♦♠❛✐♥ Ω ∈ R ❞ ●✐✈❡♥ ❢✉♥❝t✐♦♥ ❢ ∈ ▲ ✷ (Ω) ❋✐♥❞ ✉ ∈ ❍ ✶ (Ω) s✉❝❤ t❤❛t ( ✉ , ˜ ✉ ) ❍ ✶ (Ω) = ( ❢ , ˜ ✉ ) ▲ ✷ (Ω) ✉ ∈ ❍ ✶ (Ω) ✳ ❢♦r ❛❧❧ ˜

❖✉t❧✐♥❡ Pr❡❧✐♠✐♥❛r✐❡s ▼♦❞❡❧ ♣r♦❜❧❡♠ ■●❆ ❞✐s❝r❡t✐③❛t✐♦♥ ▼✉❧t✐❣r✐❞ ❛❧❣♦r✐t❤♠ ▼✉❧t✐❣r✐❞ ❢r❛♠❡✇♦r❦ ❇❛s✐s✲✐♥❞❡♣❡♥❞❡♥t s♠♦♦t❤❡r ❙♠♦♦t❤❡r ❢♦r ♦♥❡ ❞✐♠❡♥s✐♦♥ ❙♠♦♦t❤❡r ❢♦r t✇♦ ❞✐♠❡♥s✐♦♥s ◆✉♠❡r✐❝❛❧ r❡s✉❧ts ❈♦♥❝❧✉s✐♦♥s ❛♥❞ ❖✉t❧♦♦❦

✐s t❤❡ ✉♥✐❢♦r♠ s✉❜❞✐✈✐s✐♦♥ ♦❢ ✵ ✶ ✐♥ ♥ ♥ ✵ ✷ s✉❜✐♥t❡r✈❛❧s ❚ ✐ ♦❢ ❧❡♥❣t❤ ❤ ❤ ✵ ✷ ❢♦r ✵ ✶ ✷ ❙♣❧✐♥❡ s♣❛❝❡ ❙ ♣ ❦ ✐s t❤❡ s♣❛❝❡ ♦❢ ❛❧❧ s♣❧✐♥❡ ❢✉♥❝t✐♦♥s ✐♥ ❈ ❦ ✱ ✇❤✐❝❤ ❛r❡ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧s ♦❢ ❞❡❣r❡❡ ♣ ♦♥ ❡❛❝❤ s✉❜✐♥t❡r✈❛❧ ✐♥ ✳ ▼❛①✐♠✉♠ s♠♦♦t❤♥❡ss✿ ❙ ♣ ❙ ♣ ♣ ✶ ✶ ♠ ❙t❛♥❞❛r❞ ❇✲s♣❧✐♥❡ ❜❛s✐s✿ ① ① ♣ ♣ ❙♣❧✐♥❡ s♣❛❝❡s ✐♥ ♦♥❡ ❞✐♠❡♥s✐♦♥ ◮ ▲❡t Ω = ( ✵ , ✶ )

❙♣❧✐♥❡ s♣❛❝❡ ❙ ♣ ❦ ✐s t❤❡ s♣❛❝❡ ♦❢ ❛❧❧ s♣❧✐♥❡ ❢✉♥❝t✐♦♥s ✐♥ ❈ ❦ ✱ ✇❤✐❝❤ ❛r❡ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧s ♦❢ ❞❡❣r❡❡ ♣ ♦♥ ❡❛❝❤ s✉❜✐♥t❡r✈❛❧ ✐♥ ✳ ▼❛①✐♠✉♠ s♠♦♦t❤♥❡ss✿ ❙ ♣ ❙ ♣ ♣ ✶ ✶ ♠ ❙t❛♥❞❛r❞ ❇✲s♣❧✐♥❡ ❜❛s✐s✿ ① ① ♣ ♣ ❙♣❧✐♥❡ s♣❛❝❡s ✐♥ ♦♥❡ ❞✐♠❡♥s✐♦♥ ◮ ▲❡t Ω = ( ✵ , ✶ ) ◮ M ℓ ✐s t❤❡ ✉♥✐❢♦r♠ s✉❜❞✐✈✐s✐♦♥ ♦❢ Ω = ( ✵ , ✶ ) ✐♥ ♥ ℓ = ♥ ✵ ✷ ℓ s✉❜✐♥t❡r✈❛❧s ❚ ✐ ♦❢ ❧❡♥❣t❤ ❤ ℓ := ❤ ✵ ✷ − ℓ ❢♦r ℓ = ✵ , ✶ , ✷ , . . .

▼❛①✐♠✉♠ s♠♦♦t❤♥❡ss✿ ❙ ♣ ❙ ♣ ♣ ✶ ✶ ♠ ❙t❛♥❞❛r❞ ❇✲s♣❧✐♥❡ ❜❛s✐s✿ ① ① ♣ ♣ ❙♣❧✐♥❡ s♣❛❝❡s ✐♥ ♦♥❡ ❞✐♠❡♥s✐♦♥ ◮ ▲❡t Ω = ( ✵ , ✶ ) ◮ M ℓ ✐s t❤❡ ✉♥✐❢♦r♠ s✉❜❞✐✈✐s✐♦♥ ♦❢ Ω = ( ✵ , ✶ ) ✐♥ ♥ ℓ = ♥ ✵ ✷ ℓ s✉❜✐♥t❡r✈❛❧s ❚ ✐ ♦❢ ❧❡♥❣t❤ ❤ ℓ := ❤ ✵ ✷ − ℓ ❢♦r ℓ = ✵ , ✶ , ✷ , . . . ◮ ❙♣❧✐♥❡ s♣❛❝❡ ❙ ♣ , ❦ ,ℓ (Ω) ✐s t❤❡ s♣❛❝❡ ♦❢ ❛❧❧ s♣❧✐♥❡ ❢✉♥❝t✐♦♥s ✐♥ ❈ ❦ (Ω) ✱ ✇❤✐❝❤ ❛r❡ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧s ♦❢ ❞❡❣r❡❡ ♣ ♦♥ ❡❛❝❤ s✉❜✐♥t❡r✈❛❧ ✐♥ M ℓ ✳

✶ ♠ ❙t❛♥❞❛r❞ ❇✲s♣❧✐♥❡ ❜❛s✐s✿ ① ① ♣ ♣ ❙♣❧✐♥❡ s♣❛❝❡s ✐♥ ♦♥❡ ❞✐♠❡♥s✐♦♥ ◮ ▲❡t Ω = ( ✵ , ✶ ) ◮ M ℓ ✐s t❤❡ ✉♥✐❢♦r♠ s✉❜❞✐✈✐s✐♦♥ ♦❢ Ω = ( ✵ , ✶ ) ✐♥ ♥ ℓ = ♥ ✵ ✷ ℓ s✉❜✐♥t❡r✈❛❧s ❚ ✐ ♦❢ ❧❡♥❣t❤ ❤ ℓ := ❤ ✵ ✷ − ℓ ❢♦r ℓ = ✵ , ✶ , ✷ , . . . ◮ ❙♣❧✐♥❡ s♣❛❝❡ ❙ ♣ , ❦ ,ℓ (Ω) ✐s t❤❡ s♣❛❝❡ ♦❢ ❛❧❧ s♣❧✐♥❡ ❢✉♥❝t✐♦♥s ✐♥ ❈ ❦ (Ω) ✱ ✇❤✐❝❤ ❛r❡ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧s ♦❢ ❞❡❣r❡❡ ♣ ♦♥ ❡❛❝❤ s✉❜✐♥t❡r✈❛❧ ✐♥ M ℓ ✳ ◮ ▼❛①✐♠✉♠ s♠♦♦t❤♥❡ss✿ ❙ ♣ ,ℓ (Ω) := ❙ ♣ , ♣ − ✶ ,ℓ (Ω)

❙♣❧✐♥❡ s♣❛❝❡s ✐♥ ♦♥❡ ❞✐♠❡♥s✐♦♥ ◮ ▲❡t Ω = ( ✵ , ✶ ) ◮ M ℓ ✐s t❤❡ ✉♥✐❢♦r♠ s✉❜❞✐✈✐s✐♦♥ ♦❢ Ω = ( ✵ , ✶ ) ✐♥ ♥ ℓ = ♥ ✵ ✷ ℓ s✉❜✐♥t❡r✈❛❧s ❚ ✐ ♦❢ ❧❡♥❣t❤ ❤ ℓ := ❤ ✵ ✷ − ℓ ❢♦r ℓ = ✵ , ✶ , ✷ , . . . ◮ ❙♣❧✐♥❡ s♣❛❝❡ ❙ ♣ , ❦ ,ℓ (Ω) ✐s t❤❡ s♣❛❝❡ ♦❢ ❛❧❧ s♣❧✐♥❡ ❢✉♥❝t✐♦♥s ✐♥ ❈ ❦ (Ω) ✱ ✇❤✐❝❤ ❛r❡ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧s ♦❢ ❞❡❣r❡❡ ♣ ♦♥ ❡❛❝❤ s✉❜✐♥t❡r✈❛❧ ✐♥ M ℓ ✳ ◮ ▼❛①✐♠✉♠ s♠♦♦t❤♥❡ss✿ ❙ ♣ ,ℓ (Ω) := ❙ ♣ , ♣ − ✶ ,ℓ (Ω) ◮ ❙t❛♥❞❛r❞ ❇✲s♣❧✐♥❡ ❜❛s✐s✿ φ ( ✶ ) ♣ ,ℓ ( ① ) , . . . , φ ( ♠ ℓ ) ♣ ,ℓ ( ① )

✐ ♠ ❥ ✐ ❥ ❚❡♥s♦r ♣r♦❞✉❝t ❇✲s♣❧✐♥❡s✿ ① ② ① ② ♣ ♣ ♣ ✵ ✶ ❞ ✇✐t❤ ❞ ❋♦r ✶✿ ❙ ♣ ❞❡♥♦t❡s t❡♥s♦r ♣r♦❞✉❝t s♣❧✐♥❡ s♣❛❝❡ ▼♦r❡ ❣❡♥❡r❛❧ ❞♦♠❛✐♥s✿ ❣❡♦♠❡tr② ♠❛♣♣✐♥❣ ❋♦r r❡❣✉❧❛r ❣❡♦♠❡tr② ♠❛♣♣✐♥❣s✿ ♠✉❧t✐❣r✐❞ ❢♦r ♣❛r❛♠❡t❡r ❞♦♠❛✐♥ ❝❛♥ ❜❡ ✉s❡❞ ❛s ♣r❡❝♦♥❞✐t✐♦♥❡r ❙♣❧✐♥❡ s♣❛❝❡s ✐♥ t✇♦ ❛♥❞ ♠♦r❡ ❞✐♠❡♥s✐♦♥s ◮ ▲❡t Ω = ( ✵ , ✶ ) ❞

✵ ✶ ❞ ✇✐t❤ ❞ ❋♦r ✶✿ ❙ ♣ ❞❡♥♦t❡s t❡♥s♦r ♣r♦❞✉❝t s♣❧✐♥❡ s♣❛❝❡ ▼♦r❡ ❣❡♥❡r❛❧ ❞♦♠❛✐♥s✿ ❣❡♦♠❡tr② ♠❛♣♣✐♥❣ ❋♦r r❡❣✉❧❛r ❣❡♦♠❡tr② ♠❛♣♣✐♥❣s✿ ♠✉❧t✐❣r✐❞ ❢♦r ♣❛r❛♠❡t❡r ❞♦♠❛✐♥ ❝❛♥ ❜❡ ✉s❡❞ ❛s ♣r❡❝♦♥❞✐t✐♦♥❡r ❙♣❧✐♥❡ s♣❛❝❡s ✐♥ t✇♦ ❛♥❞ ♠♦r❡ ❞✐♠❡♥s✐♦♥s ◮ ▲❡t Ω = ( ✵ , ✶ ) ❞ ◮ ❚❡♥s♦r ♣r♦❞✉❝t ❇✲s♣❧✐♥❡s✿ ϕ ( ✐ + ♠ ℓ ❥ ) ( ① , ② ) = φ ( ✐ ) ♣ ,ℓ ( ① ) φ ( ❥ ) ♣ ,ℓ ( ② ) ♣ ,ℓ

▼♦r❡ ❣❡♥❡r❛❧ ❞♦♠❛✐♥s✿ ❣❡♦♠❡tr② ♠❛♣♣✐♥❣ ❋♦r r❡❣✉❧❛r ❣❡♦♠❡tr② ♠❛♣♣✐♥❣s✿ ♠✉❧t✐❣r✐❞ ❢♦r ♣❛r❛♠❡t❡r ❞♦♠❛✐♥ ❝❛♥ ❜❡ ✉s❡❞ ❛s ♣r❡❝♦♥❞✐t✐♦♥❡r ❙♣❧✐♥❡ s♣❛❝❡s ✐♥ t✇♦ ❛♥❞ ♠♦r❡ ❞✐♠❡♥s✐♦♥s ◮ ▲❡t Ω = ( ✵ , ✶ ) ❞ ◮ ❚❡♥s♦r ♣r♦❞✉❝t ❇✲s♣❧✐♥❡s✿ ϕ ( ✐ + ♠ ℓ ❥ ) ( ① , ② ) = φ ( ✐ ) ♣ ,ℓ ( ① ) φ ( ❥ ) ♣ ,ℓ ( ② ) ♣ ,ℓ ◮ ❋♦r Ω = ( ✵ , ✶ ) ❞ ✇✐t❤ ❞ > ✶✿ ❙ ♣ ,ℓ (Ω) ❞❡♥♦t❡s t❡♥s♦r ♣r♦❞✉❝t s♣❧✐♥❡ s♣❛❝❡