Mario Luiz Previatti de Souza Métodos de Maz'ia e Landweber para o Problema de Cauchy elíptico

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▼❛r✐♦ ▲✉✐③ Pr❡✈✐❛tt✐ ❞❡ ❙♦✉③❛

▼ét♦❞♦s ❞❡ ▼❛③✬✐❛ ❡ ▲❛♥❞✇❡❜❡r ♣❛r❛ ♦ Pr♦❜❧❡♠❛

❞❡ ❈❛✉❝❤② ❡❧í♣t✐❝♦

❉✐ss❡rt❛çã♦ s✉❜♠❡t✐❞❛ ❛♦ ❈✉rs♦ ❞❡ Pós✲ ●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ P✉r❛ ❡ ❆♣❧✐✲ ❝❛❞❛✱ ❞♦ ❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❋ìs✐❝❛s ❡ ▼❛t❡✲ ♠át✐❝❛s ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙❛♥t❛ ❈❛t❛r✐♥❛✱ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡s✲ tr❡ ❡♠ ▼❛t❡♠át✐❝❛✱ ❝♦♠ ➪r❡❛ ❞❡ ❈♦♥❝❡♥✲ tr❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❆♣❧✐❝❛❞❛✳

❖r✐❡♥t❛❞♦r✿ Pr♦❢✳➸ ❉r✳ ❆♥t♦♥✐♦ ❈❛r❧♦s ●❛r❞❡❧ ▲❡✐tã♦

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❆❣r❛❞❡❝✐♠❡♥t♦s

❆❣r❛❞❡ç♦ ❛ ❉❡✉s ♣♦r ♠♦str❛r ♦ ❝❛♠✐♥❤♦ ❛ s❡r ♣❡r❝♦rr✐❞♦ ❡ ❢❛③❡r ❡✉ ❡st❛r ♥♦ ❝❛♠✐♥❤♦ ❉❡❧❡ ♠❡s♠♦ s❡♥❞♦ t❡♥t❛❞♦r ❛ t♦♠❛r ❛t❛❧❤♦s ♦✉tr♦r❛ ❞♦ ❧❛❞♦ ♥❡❣r♦ ❞❛ ❢♦rç❛✳

❆❣r❛❞❡ç♦ ❛ ♠✐♥❤❛ ♠ã❡✱ ▼❛r❧② Pr❡✈✐❛tt✐✱ ♣♦r s❡♠♣r❡ ❡st❛r ❛♦ ♠❡✉ ❧❛❞♦ ❡ s❡♠♣r❡ ♠❡ ❧❡♠❜r❛r ❞❡ t♦❞♦s ♦s ♠❡✉s ♦❜❥❡t✐✈♦s ♣❡ss♦❛✐s✱ ❢❛③❡♥❞♦✲ ♠❡ s❡♠♣r❡ ✐r ❡♠ ❜✉s❝❛ ❞❡❧❡s❀ ❛❧é♠ ❞❡ t♦❞❛ ♦r✐❡♥t❛çã♦ ❡ ❛♠♦r✳

❆❣r❛❞❡ç♦ ❛ ♠✐♥❤❛ ❢❛♠í❧✐❛✱ ❡♠ ❡s♣❡❝✐❛❧ ♠❡✉s ♣r✐♠♦s ❖s✈❛✐r Pr❡✈✐❛t❡ ❙③❡♥❝③✉❦ ❡ ❖s✈❛❧❞♦ Pr❡✈✐❛t❡ ❙③❡♥❝③✉❦ ❡ ❛ ♠✐♥❤❛ t✐❛ ▲✉③✐❛ Pr❡✈✐❛tt✐✱ ❞❡✈✐❞♦ à t♦❞❛ ♦r✐❡♥t❛çã♦✱ s✉♣♦rt❡ ❡ r❡❢❡r❡♥❝✐❛❧ q✉❡ sã♦ ❡♠ ♠✐♥❤❛ ❢♦r✲ ♠❛çã♦ ❝♦♠♦ ♣❡ss♦❛❀ ❡ ♣♦r t♦❞❛ ❛ ❝♦♠♣❛♥❤✐❛ ❡ ❛❥✉❞❛ ❢❛♠✐❧✐❛r ❞✉r❛♥t❡ ♦s ♠♦♠❡♥t♦s ❞✐❢í❝❡✐s ❡♥❢r❡♥t❛❞♦s✳

❆❣r❛❞❡ç♦ ❛ ❈áss✐❛ ❆❧✐♥❡ ♣♦r t♦❞❛ ❛ ❝♦♠♣r❡❡♥sã♦✱ ♣❛❝✐ê♥❝✐❛✱ ❞❡❞✐✲ ❝❛çã♦ ❡ t❡♠♣♦ q✉❡ s❡ ❞✐s♣ôs ❛ ♠✐♠❀ ♣♦r t♦❞♦s ♦s ♠♦♠❡♥t♦s ❥✉♥t♦s q✉❡✱ ❞❡ ✉♠❛ ❢♦r♠❛ ♦✉ ❞❡ ♦✉tr❛✱ ✜③❡r❛♠✲♠❡ t❡r♠✐♥❛r ❡ss❛ ❞✐ss❡rt❛çã♦✳

❆❣r❛❞❡ç♦ ❛♦ ♠❡✉ ♦r✐❡♥t❛❞♦r ❆♥t♦♥✐♦ ❈❛r❧♦s ●❛r❞❡❧ ▲❡✐tã♦ ♣♦r t♦❞❛ ♦r✐❡♥t❛çã♦✱ ♣❛❝✐ê♥❝✐❛ ❡ ❝♦♥s❡❧❤♦s q✉❡ só ✉♠ ❛♠✐❣♦ s❛❜❡ ❞❛r✳

❆❣r❛❞❡ç♦ ❛♦s ❞❡♠❛✐s ♣r♦❢❡ss♦r❡s ♣r❡s❡♥t❡s ❡♠ ♠✐♥❤❛ ❢♦r♠❛çã♦ ♠❛✲ t❡♠át✐❝❛ ❡ àq✉❡❧❡s ♣r❡s❡♥t❡s ❡♠ ♠✐♥❤❛ ❢♦r♠❛çã♦ ♣❡ss♦❛❧✳

❆❣r❛❞❡ç♦ ❛♦s ❝♦❧❡❣❛s ❞❡ ♠❡str❛❞♦ ♣♦r ❢❛❝✐❧✐t❛r ♦s ♠♦♠❡♥t♦s ❡str❡s✲ s❛♥t❡s ❞❡ss❡ ♣❡rí♦❞♦✳

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❘❡s✉♠♦

◆❡st❛ ❞✐ss❡rt❛çã♦ ❢♦✐ tr❛❜❛❧❤❛❞♦ ♦ ❝❧áss✐❝♦ ❡①❡♠♣❧♦ ❞❡ ♣r♦❜❧❡♠❛ ♠❛❧ ♣♦st♦✱ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤② ❡❧í♣t✐❝♦ ♣❛r❛ ♦ ♦♣❡r❛❞♦r ❞❡ ▲❛♣❧❛❝❡ s♦❜r❡ ✉♠ ❝♦♥❥✉♥t♦ Ω⊂R2 s✉✜❝✐❡♥t❡♠❡♥t❡ r❡❣✉❧❛r✱ ♦♥❞❡ ♦s ❞❛❞♦s ❞❡ ❈❛✉❝❤② sã♦ ❢♦r♥❡❝✐❞♦s ❛♣❡♥❛s s♦❜r❡ ✉♠❛ ♣❛rt❡ ❞❛ ❢r♦♥t❡✐r❛✱ Γ1 ⊂∂Ω✳

❖ ♦❜❥❡t✐✈♦ é ♦ ❞❡ r❡❝♦♥str✉✐r ♦ tr❛ç♦ ❞❛ H1(Ω)✲s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ ❞❡

▲❛♣❧❛❝❡ s♦❜r❡ ∂Ω\Γ1✳ P❛r❛ t❛❧ ✜♥❛❧✐❞❛❞❡✱ ❢♦✐ ❛♥❛❧✐s❛❞♦ ❞♦✐s ♠ét♦❞♦s

✐t❡r❛t✐✈♦s❀ ♦ ♠ét♦❞♦ ❞❡ ▼❛③✬✐❛ q✉❡ ❝♦♥s✐st❡ ❡♠ r❡s♦❧✈❡r s✉❝❡ss✐✈❛♠❡♥t❡ ♣r♦❜❧❡♠❛s ❞❡ ✈❛❧♦r ❞❡ ❝♦♥t♦r♥♦ ♠✐st♦ ✭q✉❡ sã♦ ❜❡♠ ♣♦st♦s✮ ✉t✐❧✐③❛♥❞♦ ♦s ❞❛❞♦s ❞❡ ❈❛✉❝❤② ❝♦♠♦ ♣❛rt❡ ❞❛s ❝♦♥❞✐çõ❡s ❞❡ ❢r♦♥t❡✐r❛ ❡ ♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r✱ ❜❛s❡❛❞♦ ♥❛ ❡q✉❛çã♦ ♥♦r♠❛❧ ❞❛ ❝♦♥❞✐çã♦ ❞❡ ♦t✐♠❛❧✐❞❛❞❡ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ ♣❛r❛ r❡s♦❧✈❡r ♦ ♣r♦❜❧❡♠❛ ❞❡ ♠í♥✐♠♦s q✉❛❞r❛❞♦s✳ ❆tr❛✈és ❞❡ ✉♠❛ ❛❜♦r❞❛❣❡♠ ✈✐❛ ❛♥á❧✐s❡ ❢✉♥❝✐♦♥❛❧ ❝♦♠ ✉♠❛ t♦♣♦❧♦❣✐❛ ♥ã♦ ✉s✉❛❧ ❢♦✐ ❞❡♠♦♥str❛❞♦ ❛ ❛♥á❧✐s❡ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ♣❛r❛ ♦ ♠ét♦❞♦ ❞❡ ▼❛③✬✐❛ s♦❜ ❞❛❞♦s ❡①❛t♦s❀ ♣♦r ♦✉tr♦ ❧❛❞♦✱ ♣❛r❛ ❞❡♠♦♥str❛r q✉❡ ♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r é ✉♠ ♠ét♦❞♦ ❞❡ r❡❣✉❧❛r✐③❛çã♦ ❡ ♦❜t❡r t❛①❛ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛✱ ❛ t❡♦r✐❛ ❞❡ r❡❣✉❧❛r✐③❛çã♦ ❝❧áss✐❝❛✳ ❆♦ ✜♥❛❧✱ ✉♠❛ r❡❧❛çã♦ ❡♥tr❡ ♦s ♠ét♦❞♦s ❢♦✐ ❡♥❝♦♥tr❛❞❛✱ ❛ ✐❣✉❛❧❞❛❞❡ ❡♥tr❡ ❛s ✐t❡r❛çõ❡s✱ ♣♦s✲ s✐❜✐❧✐t❛♥❞♦✱ ❛ss✐♠✱ ❝♦♥❝❧✉✐r ❛ ❛♥á❧✐s❡ ❞♦ ♠ét♦❞♦ ❞❡ ▼❛③✬✐❛✱ ✐st♦ é✱ s♦❜ ❞❛❞♦s ❝♦♠ r✉í❞♦s✳

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❆❜str❛❝t

❚❤✐s ❞✐ss❡rt❛t✐♦♥ ❞❡❛❧s ✇✐t❤ t❤❡ ❝❧❛ss✐❝❛❧ ✐❧❧✲♣♦s❡❞ ♣r♦❜❧❡♠ ❡①❛♠♣❧❡✱ t❤❡ ❡❧❧✐♣t✐❝ ❈❛✉❝❤② ♣r♦❜❧❡♠ ❢♦r t❤❡ ▲❛♣❧❛❝❡ ♦♣❡r❛t♦r ❛t ❛ s✉✣❝✐❡♥t❧② r❡❣✉❧❛r s❡t Ω⊂R2✱ ✇❤❡r❡ t❤❡ ❈❛✉❝❤② ❞❛t❛ ❛r❡ ❣✐✈❡♥ ♦♥❧② ❛t ♣❛rt ♦❢ t❤❡ ❜♦✉♥❞❛r②✱Γ1⊂∂Ω✳ ❚❤❡ ❣♦❛❧ ✐s t♦ r❡❝♦♥str✉❝t t❤❡ tr❛❝❡ ♦❢H1(Ω)✲

s♦❧✉t✐♦♥ ♦❢ t❤❡ ▲❛♣❧❛❝❡ ❡q✉❛t✐♦♥ ❛t ∂Ω\Γ1✳ ❋♦r s✉❝❤ ♣✉r♣♦s❡✱ t✇♦

✐t❡r❛t✐✈❡ ♠❡t❤♦❞s ❛r❡ ❛♥❛❧②③❡❞❀ t❤❡ ❛❧❣♦r✐t❤♠ ♦❢ ▼❛③✬✐❛ ✐s ❛ ♠❡t❤♦❞ ❜❛s❡❞ ♦♥ s♦❧✈✐♥❣ s✉❝❝❡ss✐✈❡❧② ✇❡❧❧✲♣♦s❡❞ ♠✐①❡❞ ❜♦✉♥❞❛r② ✈❛❧✉❡ ♣r♦✲ ❜❧❡♠s ✉s✐♥❣ t❤❡ ❣✐✈❡♥ ❈❛✉❝❤② ❞❛t❛ ❛s ♣❛rt ♦❢ t❤❡ ❜♦✉♥❞❛r② ❞❛t❛ ❛♥❞ t❤❡ ▲❛♥❞✇❡❜❡r ✐t❡r❛t✐♦♥✱ ✇❤✐❝❤ ✐s ❜❛s❡❞ ♦♥ t❤❡ ♥♦r♠❛❧ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ✜rst ♦r❞❡r ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥ t♦ s♦❧✈❡ t❤❡ ♥♦♥❧✐♥❡❛r ❧❡❛st sq✉❛r❡ ♣r♦✲ ❜❧❡♠✳ ❆♥ ❛♣♣r♦❛❝❤ ✈✐❛ ❢✉♥❝t✐♦♥❛❧ ❛♥❛❧②s✐s ✇✐t❤ ✉♥✉s✉❛❧ t♦♣♦❧♦❣② ✇❛s ✉s❡❞ t♦ ♣r♦♦❢ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ❛♥❛❧②s✐s ✉♥❞❡r ❡①❛❝t ❞❛t❛❀ ♦♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ t♦ s❤♦✇ t❤❛t ▲❛♥❞✇❡❜❡r ✐t❡r❛t✐♦♥ ✐s ❛ r❡❣✉❧❛r✐③❛t✐♦♥ ♠❡t❤♦❞ ❛♥❞ t♦ ♦❜t❛✐♥ ❛ ❝♦♥✈❡r❣❡♥❝❡ r❛t❡✱ t❤❡ ❝❧❛ss✐❝❛❧ r❡❣✉❧❛r✐③❛t✐♦♥ t❤❡♦r② ✇❛s ✇✐❞❡❧② ✉s❡❞✳ ❆t t❤❡ ❡♥❞ ♦❢ t❤✐s ❞✐ss❡rt❛t✐♦♥✱ ❛ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ♠❡t❤♦❞s ✇❛s ❢♦✉♥❞✱ t❤❡ ✐t❡r❛t✐♦♥s ❛r❡ ❡q✉❛❧✱ ❛❧❧♦✇✐♥❣ t♦ ❝♦♠♣❧❡t❡ t❤❡ ▼❛③✬✐❛✬s ♠❡t❤♦❞ ❛♥❛❧②s✐s✱ ✐✳❡✳✱ ✉♥❞❡r ♥♦✐s❡ ❞❛t❛✳

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❙✉♠ár✐♦

■♥tr♦❞✉çã♦ ✶✸

✶ ❖ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤② ❡❧í♣t✐❝♦ ✶✼

✶✳✶ ❙♦❜r❡ ❛ ❜❡♠ ♣♦s✐çã♦ ❞♦ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤② ❡❧í♣t✐❝♦ ✳ ✳ ✶✽

✷ ❉❡s❝r✐çã♦ ❡ ❆♥á❧✐s❡ ❞♦s ▼ét♦❞♦s ❞❡ ▼❛③✬✐❛ ❡ ❞❡ ▲❛♥❞✇❡✲ ❜❡r ♣❛r❛ ♦ Pr♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤② ❊❧í♣t✐❝♦ ✷✸ ✷✳✶ ▼ét♦❞♦ ❞❡ ▼❛③✬✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✶✳✶ ❆♥á❧✐s❡ ❞♦ ♠ét♦❞♦ ❞❡ ▼❛③✬✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✷✳✷ ▼ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✷✳✷✳✶ ❆♥á❧✐s❡ ❞♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺

✸ ❆ ❘❡❧❛çã♦ ❊♥tr❡ ♦s ▼ét♦❞♦s ❞❡ ▼❛③✬✐❛ ❡ ▲❛♥❞✇❡❜❡r ✹✸

❈♦♥s✐❞❡r❛çõ❡s ❋✐♥❛✐s ✹✾

❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✺✶

(12)
(13)

■♥tr♦❞✉çã♦

❆s ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ❛♣❛r❡❝❡♠ ♣❛r❛ ♠♦❞❡❧❛r ❞✐✈❡rs♦s ♣r♦❜❧❡♠❛s ❞❛s ❝✐ê♥❝✐❛s ❛♣❧✐❝❛❞❛s✱ ❡♠ s✐t✉❛çõ❡s ❝♦♠♦ ❝♦♥❞✉t✐✈✐❞❛❞❡ tér♠✐❝❛✱ ❞✐s✲ ♣❡rsã♦ ❞❡ ♣♦❧✉❡♥t❡s✱ ♣r♦❝❡ss❛♠❡♥t♦ ❞❡ ✐♠❛❣❡♥s✱ ✜♥❛♥ç❛s✱ ♦t✐♠✐③❛çã♦✱ ❡t❝✳ ❉❡♥tr❡ ❡ss❛s ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s q✉❡ ♠♦❞❡❧❛♠ ❡ss❡s ♣r♦❜❧❡♠❛s ❤á ❝❛s♦s ❡♠ q✉❡ ♥ã♦ s❡ ❜✉s❝❛ ✉♠❛ s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ ❡✱ s✐♠✱ ❝♦♠♣❧❡✲ t❛r ♦s ❞❛❞♦s ❢❛❧t❛♥t❡s ♣❛r❛ ♠♦❞❡❧❛r ❝♦♠♣❧❡t❛♠❡♥t❡ ❛❧❣✉♠ ❢❡♥ô♠❡♥♦ ♦✉ s✐t✉❛çã♦ ❞❡s❡❥❛❞❛✳

❖ ♣r♦❜❧❡♠❛ ❛ s❡r r❡s♦❧✈✐❞♦ ♣❡❧♦s ♠ét♦❞♦s ❞❡ ▼❛③✬✐❛ ❡ ▲❛♥❞✇❡❜❡r r❡❢❡r❡♥t❡ ❛♦ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤② ❡❧í♣t✐❝♦ ♣♦❞❡ s❡r ❛♣❧✐❝❛❞♦ ❛ ♣r♦❜❧❡♠❛s ❞❡ t♦♠♦❣r❛✜❛ ♦✉ r❡❝♦♥str✉çã♦ ❞❡ ❞❛❞♦s✳ ❈♦♠♦ ♣♦r ❡①❡♠♣❧♦✱ ✈❡r✐✜❝❛r ❛ tr❛♥s❢❡rê♥❝✐❛ ❞❡ ❝❛❧♦r ♥✉♠❛ r❡❣✐ã♦ ❞❡ ✉♠❛ ♣❧❛❝❛ ❡♠ ❡q✉✐❧í❜r✐♦ tér♠✐❝♦ ♦♥❞❡ ♥ã♦ é ♣♦ssí✈❡❧ s❡ ❝❛❧❝✉❧❛r✳

◆❡st❡ tr❛❜❛❧❤♦✱ ❡♥q✉❛❞r❛❞♦ ❞❡♥tr♦ ❞❛ ár❡❛ ❞❡ ♣r♦❜❧❡♠❛s ✐♥✈❡rs♦s✱ ♦ ♠♦❞❡❧♦ ❡st✉❞❛❞♦ é ♦ ❝❧áss✐❝♦ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤② ❡❧í♣t✐❝♦ ❝♦♠ ♦ ✐♥t✉✐t♦ ❞❡ ❛ss♦❝✐á✲❧♦ ❛ ✉♠ ♣r♦❜❧❡♠❛ ✐♥✈❡rs♦✳ ❉❡♥tr❡ ❛❧❣✉♥s ❞♦s ♠ét♦❞♦s ❝❛♣❛③❡s ❞❡ r❡s♦❧✈❡r ♦ ♣r♦❜❧❡♠❛ ✐♥✈❡rs♦ ❡stã♦ ♦ ♠ét♦❞♦ ❞❡ ▼❛③✬✐❛ ❡ ♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r ❧✐♥❡❛r✳

❖ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤② ❡❧í♣t✐❝♦ tr❛❜❛❧❤❛❞♦ é ✉♠ ♣r♦❜❧❡♠❛ ❞❡ ✈❛❧♦r ✐♥✐❝✐❛❧ ✐♥❞❡♣❡♥❞❡♥t❡ ♥♦ t❡♠♣♦✳ ❆q✉✐ ♦ ♦♣❡r❛❞♦r ❞✐❢❡r❡♥❝✐❛❧ é ♦ ♦♣❡r❛❞♦r ❞❡ ▲❛♣❧❛❝❡ s♦❜r❡ ✉♠ s✉❜❝♦♥❥✉♥t♦ Ω⊂R2 ❛❜❡rt♦✱ ❧✐♠✐t❛❞♦ ❡ s✐♠♣❧❡s✲ ♠❡♥t❡ ❝♦♥❡①♦✱ ❝✉❥❛ ❢r♦♥t❡✐r❛ s✉❛✈❡∂Ω♣♦❞❡ s❡r ❞✐✈✐❞✐❞❛ ❡♠ ❞✉❛s ❝♦♠✲

♣♦♥❡♥t❡s ❛❜❡rt❛s✱ ❞✐s❥✉♥t❛s ❡ ❝♦♥❡①❛s Γ1 ❡Γ2 t❛✐s q✉❡Γ1⊔Γ2 =∂Ω✳

❖s ❞❛❞♦s✱ ❝❤❛♠❛❞♦s ❞❡ ❞❛❞♦s ❞❡ ❈❛✉❝❤②✱ ❛s ❝♦♥❞✐çõ❡s ❞❡ ❝♦♥t♦r♥♦ ❞❡ ❉✐r✐❝❤❧❡t ❡ ❞❡ ◆❡✉♠❛♥♥✱ ♥♦ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤② ❡❧í♣t✐❝♦ sã♦ ♣r❡s❝r✐t♦s ❛♣❡♥❛s s♦❜r❡ ✉♠❛ ♣❛rt❡ ❞❛ ❢r♦♥t❡✐r❛✱Γ1✱ ✐st♦ é✱u|Γ1 =f ❡uν|Γ1 =g✱

♦♥❞❡ f ❡g sã♦ ❝♦♥❤❡❝✐❞❛s✳ ❆ss✐♠✱ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤② ❡❧í♣t✐❝♦ ❡♠

q✉❡stã♦ é

(CP) ∆u= 0, Ω; u|Γ1 =f; uν|Γ1 =g.

(14)

✶✹

♥✉❛♠❡♥t❡ ❞♦s ❞❛❞♦s✳ ❙❡ ✉♠❛ ❞❡ss❛s ❝♦♥❞✐çõ❡s é ✈✐♦❧❛❞❛✱ ♦ ♣r♦❜❧❡♠❛ é ❞✐t♦ s❡r ♠❛❧ ♣♦st♦✳ ❖ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤② ❡❧í♣t✐❝♦ é ♠❛❧ ♣♦st♦✱ ♦ ♣ró✲ ♣r✐♦ ❍❛❞❛♠❛r❞ ❡①✐❜✐✉ ✉♠ ❡①❡♠♣❧♦ ❡♠ q✉❡ ♥ã♦ ♦❝♦rr❡ ❛ ❞❡♣❡♥❞ê♥❝✐❛ ❝♦♥tí♥✉❛ ❞♦s ❞❛❞♦s❀ ♥❛ ✈❡r❞❛❞❡✱ ❛ ✉♥✐❝✐❞❛❞❡ ❞❛ s♦❧✉çã♦✱ ❝❛s♦ ❡①✐st❛✱ é ❛ ú♥✐❝❛ ❝♦♥❞✐çã♦ s❛t✐s❢❡✐t❛✳

◆♦ ❡♥t❛♥t♦✱ ♦ ♦❜❥❡t✐✈♦ ❝♦♠ ♦s ♠ét♦❞♦s ✐t❡r❛t✐✈♦s ♥ã♦ é ♦ ❞❡ ❡♥✲ ❝♦♥tr❛r ❛ s♦❧✉çã♦ u❞♦ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤② ❡❧í♣t✐❝♦ (CP)✱ ❛ ♠❡t❛ é

❛ ❞❡ r❡❝♦♥str✉✐r ♦ tr❛ç♦ ❞❛ s♦❧✉çã♦ u ❛♦ ❧♦♥❣♦ ❞❛ ❢r♦♥t❡✐r❛ ♦♥❞❡ ♥❡✲

♥❤✉♠ ❞❛❞♦ é ❢♦r♥❡❝✐❞♦✱ ✐st♦ é✱ ❛✈❛❧✐❛r ♦ tr❛ç♦ ❞❛ s♦❧✉çã♦ u ❞❡ (CP)

s♦❜r❡ Γ2 ⊂∂Ω✱ ❡♠ ♣❛rt✐❝✉❧❛r✱ ♦ tr❛ç♦ ❞❡ ◆❡✉♠❛♥♥✱ ♦✉ s❡❥❛✱ ❝❛❧❝✉❧❛r

uν|Γ2✳ ◆♦t❡ q✉❡ é ❡q✉✐✈❛❧❡♥t❡ r❡s♦❧✈❡r ❡st❡s ❞♦✐s ♣r♦❜❧❡♠❛s✳ ❉❡ ❢❛t♦✱

s❡ ❝❛❧❝✉❧❛rϕ=uν|Γ2✱ ❡♥tã♦ t❡♠✲s❡ ♦ s❡❣✉✐♥t❡ ♣r♦❜❧❡♠❛ ♠✐st♦

∆u= 0, Ω; u|Γ1 =f; uν|Γ2 =ϕ;

q✉❡ é ❜❡♠ ♣♦st♦✱ ❡ ❛ss✐♠✱ ♦❜té♠✲s❡ ❛ s♦❧✉çã♦u❞♦ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤②✳

❊♥❣❧ ❞✐③ q✉❡ ♣♦r ♣r♦❜❧❡♠❛s ✐♥✈❡rs♦s s❡ ❡♥t❡♥❞❡ àq✉❡❧❡s ♣r♦❜❧❡♠❛s q✉❡ ✈✐s❛♠ ❞❡t❡r♠✐♥❛r ❝❛✉s❛s ❛tr❛✈és ❞♦s ❡❢❡✐t♦s ♦❜s❡r✈❛❞♦s✳ ❖✉ s❡❥❛✱ ❞♦✐s ♣r♦❜❧❡♠❛s sã♦ ✐♥✈❡rs♦s ✉♠ ❞♦ ♦✉tr♦ s❡ ♥❛ ❢♦r♠✉❧❛çã♦ ❞❡ ❝❛❞❛ ✉♠ ❡♥✈♦❧✈❡ ❛ s♦❧✉çã♦ ❞♦ ♦✉tr♦✳ ❉❡ss❛ ❢♦r♠❛✱ ♣♦❞❡✲s❡ ❛ss♦❝✐❛r ❛♦ ♣r♦❜❧❡♠❛ ❞✐r❡t♦ ♠❛❧ ♣♦st♦ ✭❡♥❝♦♥tr❛r ❛ s♦❧✉çã♦u✮ ❛ ✉♠ ♣r♦❜❧❡♠❛ ✐♥✈❡rs♦ ✭♠❛❧

♣♦st♦✮ ❞❛ ❢♦r♠❛ F(ϕ) =g✱ ♦♥❞❡ ϕé s♦❧✉çã♦ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ✐♥✈❡rs♦

s❡ϕ =uν|Γ2✱ ✐st♦ é✱ ϕé ♦ tr❛ç♦ ❞❛ s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤②

s♦❜r❡ ❛ ❢r♦♥t❡✐r❛ ♦♥❞❡ ♥ã♦ s❡ t✐♥❤❛ ❞❛❞♦s ♣r❡s❝r✐t♦s ❡F é ✉♠ ♦♣❡r❛❞♦r

❛✜♠ ❡♥tr❡ ❡s♣❛ç♦s ❞❡ ❍✐❧❜❡rt ❛ s❡r ❞❡t❡r♠✐♥❛❞♦ q✉❡ ❞❡♣❡♥❞❡ ❛♣❡♥❛s ❞♦s ❞❛❞♦sf ❡g✳ ◆♦t❡ q✉❡ ♣❛r❛ ❍❛❞❛♠❛r❞✱ ❛ ❡q✉❛çã♦ ❞❡ ♦♣❡r❛❞♦r❡s F(ϕ) =gs❡r✐❛ ❜❡♠ ♣♦st❛ s❡F ❢♦ss❡ ✉♠❛ ❜✐❥❡çã♦ ❝♦♠ ✐♥✈❡rs❛ ❝♦♥tí♥✉❛✳

◆❛ ♣rát✐❝❛✱ ♦s ❞❛❞♦s ♥✉♥❝❛ sã♦ ♣r❡❝✐s♦s✱ ♦ q✉❡ s❡ tê♠✱ ♥❛ ✈❡r❞❛❞❡✱ sã♦ ❞❛❞♦s ♣ró①✐♠♦s✱ ✐st♦ é✱ ✉♠❛ ❛♣r♦①✐♠❛çã♦gδ fδ t❛✐s q✉❡

fδ−f

+

gδ−g

< δ;

♦♥❞❡fδ gδ ♥ã♦ ♣r❡❝✐s❛♠ ❡st❛r ♥❛ ✐♠❛❣❡♠ ❞❡F δ >0é ❝❤❛♠❛❞♦ ♦

♥í✈❡❧ ❞❡ r✉í❞♦✳ ❖ ♥í✈❡❧ ❞❡ r✉í❞♦ ❞❡✈❡✲s❡ ❛ ✐♠♣r❡❝✐sõ❡s ♥♦s ❛♣❛r❡❧❤♦s ❞❡ ♠❡❞✐❞❛s✱ ❡rr♦s ❞❡ ♠♦❞❡❧❛❣❡♥s ❡ ❞❡♠❛✐s ✐♥❝❡rt❡③❛s✳

❙❡ ❛ s♦❧✉çã♦ ❡①✐st❡ ♣❛r❛F(ϕ) =g✱ ❛ ❢❛❧❤❛ ❞❛ t❡r❝❡✐r❛ ❝♦♥❞✐çã♦ ❞❡

(15)

✶✺

❇r❡✈❡♠❡♥t❡✱ ✉♠❛ ❢❛♠í❧✐❛ ❞❡ ♣r♦❜❧❡♠❛s ❜❡♠ ♣♦st♦s q✉❡ s❡ ❛♣r♦①✐♠❛ ❞❡ ✉♠ ♣r♦❜❧❡♠❛ ♠❛❧ ♣♦st♦ é ✉♠ ♠ét♦❞♦ ❞❡ r❡❣✉❧❛r✐③❛çã♦✳ ❆ ❞✐✜❝✉❧❞❛❞❡ ❡♠ ❡♥❝♦♥tr❛r ✉♠❛ s♦❧✉çã♦ϕ∗❞❡F(ϕ) =g✐♥✈❡rt❡♥❞♦ ♦ ♦♣❡r❛❞♦rF❡stá

❡♠ q✉❡ ♣♦❞❡✲s❡ ♦❜t❡r s♦❧✉çõ❡s ✐♥❛❞❡q✉❛❞❛s✱ ❞❡✈✐❞♦ ❛ ✐♥st❛❜✐❧✐❞❛❞❡ ❞♦ ♣r♦❜❧❡♠❛✱ s❡ ❢♦r♥❡❝✐❞♦s s♦♠❡♥t❡ ❞❛❞♦s ♣❡rt✉r❜❛❞♦s✳ ❆ss✐♠✱ ♦ ♦❜❥❡t✐✈♦ é ❡♥❝♦♥tr❛r ✉♠❛ ❛♣r♦①✐♠❛çã♦ϕ∗

kq✉❡ ❞❡♣❡♥❞❡ ❝♦♥t✐♥✉❛♠❡♥t❡ ❞♦s ❞❛❞♦s

♣❡rt✉r❜❛❞♦s ❡ ❝♦♥✈✐r❥❛ ♣❛r❛ ϕ∗✱ s❡ ♦ ♥í✈❡❧ ❞♦ r✉í❞♦ t❡♥❞❡ ❛ ③❡r♦ ❡ ♦

♣❛râ♠❡tr♦ ❞❡ r❡❣✉❧❛r✐③❛çã♦ k é ❡s❝♦❧❤✐❞♦ ❛♣r♦♣r✐❛❞❛♠❡♥t❡✱ ✐st♦ é✱ ϕ∗

k

❞❡✈❡ s❡r ❝❛❧❝✉❧❛❞❛ ❞❡ ♠❛♥❡✐r❛ ❡stá✈❡❧✳ ❆♣ós ❣❛r❛♥t✐r ❡st❛❜✐❧✐❞❛❞❡ ϕδ

k → ϕk✱ s❡ δ → 0✱ ❡ ❛ ❝♦♥✈❡r❣ê♥❝✐❛

ϕδ

k →ϕ∗✱ s❡δ, k→0✱ ❞♦ ♠ét♦❞♦✱ ♦✉ s❡❥❛✱ ❞❡♣♦✐s ❞❡ ❞❡♠♦♥str❛❞♦ q✉❡ é

✉♠ ♠ét♦❞♦ ❞❡ r❡❣✉❧❛r✐③❛çã♦✱ ❤á ❛✐♥❞❛ ❛ ❞✐✜❝✉❧❞❛❞❡ ❞❡ s❡ ❡♥❝♦♥tr❛r t❛①❛ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ❞♦ ♠ét♦❞♦✱ ♦✉ s❡❥❛✱ ❛ t❛①❛ ❝♦♠ q✉❡

ϕδk−ϕ∗ →0✱ s❡δ→0✳

❖s ❞♦✐s ♠ét♦❞♦s✱ ❞❡ ▼❛③✬✐❛ ❡ ❞❡ ▲❛♥❞✇❡❜❡r ❧✐♥❡❛r✱ ❝❛♣❛③❡s ❞❡ r❡✲ s♦❧✈❡r ♦ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤② ❡❧í♣t✐❝♦ ❢♦r❛♠ tr❛❜❛❧❤❛❞♦s ❞❡ ♠❛♥❡✐r❛s ❧✐❣❡✐r❛♠❡♥t❡ ❞✐❢❡r❡♥t❡s✳ ❖ ♣r✐♠❡✐r♦ ❞❡❧❡s✱ ♠ét♦❞♦ ❞❡ ▼❛③✬✐❛✱ ✐♥✐❝✐❛❧✲ ♠❡♥t❡ ❛♣r❡s❡♥t❛❞♦ ❡ ❡st✉❞❛❞♦ ❡♠ ❬✽❪✱ ❢♦✐ ❛❜♦r❞❛❞♦ ♣♦r té❝♥✐❝❛s ❞❡ ❛♥á❧✐s❡ ❢✉♥❝✐♦♥❛❧✱ ❛❞♦t❛♥❞♦ ✉♠❛ ♥♦✈❛ t♦♣♦❧♦❣✐❛ s♦❜r❡ ♦s ❡s♣❛ç♦s ✉t✐✲ ❧✐③❛❞♦s✱ ❝♦♠♦ ❡♠ ❬✹❪✳ ❊st❡ ♠ét♦❞♦ ❢♦✐ ❛♥❛❧✐s❛❞♦✱ ✐♥✐❝✐❛❧♠❡♥t❡✱ ❛♣❡♥❛s s♦❜ ❞❛❞♦s ❡①❛t♦s✱ ✐st♦ é✱ ♣r♦✈❛❞❛ ❛♣❡♥❛s ❛ ❝♦♥✈❡r❣ê♥❝✐❛ ♣❛r❛ ❞❛❞♦s s❡♠ r✉í❞♦✳ ❖ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r ❧✐♥❡❛r ❢♦✐ ✐♥t❡✐r❛♠❡♥t❡ ❛♥❛❧✐s❛❞♦ ♣❡❧❛ t❡♦r✐❛ ❝❧áss✐❝❛ ❞❡ r❡❣✉❧❛r✐③❛çã♦✳ ❋♦✐ ♣r♦✈❛❞♦ q✉❡ é ✉♠ ♠ét♦❞♦ ❞❡ r❡❣✉❧❛r✐③❛çã♦ ✭❛♥á❧✐s❡ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ❡ ❡st❛❜✐❧✐❞❛❞❡ ❞♦ ♠ét♦❞♦✮ ❡ ✉♠❛ t❛①❛ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛✳

P❛r❛ t❡r♠✐♥❛r ❛ ❛♥á❧✐s❡ ❞♦ ♠ét♦❞♦ ❞❡ ▼❛③✬✐❛✱ ✐st♦ é✱ ❞❡♠♦♥str❛r ♦✉ ♦❜s❡r✈❛r ❛ ❡st❛❜✐❧✐❞❛❞❡ ❡ ❛♥á❧✐s❡ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ❞♦ ♠ét♦❞♦✱ ❡st❛❜❡❧❡❝❡r ❛❧❣✉♠ ❝r✐tér✐♦ ❞❡ ♣❛r❛❞❛ ❡ ❝♦♥❝❧✉✐r ❛❧❣✉♠❛ t❛①❛ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛✱ ❝❛s♦ ❢♦r♥❡❝✐❞♦ ❛♣❡♥❛s ❞❛❞♦s ❝♦♠ r✉í❞♦s✱ ❢♦✐ ♥❡❝❡ssár✐❛ ❛ ❞✐s❝✉ssã♦ ❡♥tr❡ ❛ r❡❧❛çã♦ ❞❛s ✐t❡r❛çõ❡s ❞♦s ♠ét♦❞♦s ❞❡ ▼❛③✬✐❛ ❡ ▲❛♥❞✇❡❜❡r ❧✐♥❡❛r✳ ❊st❛ ❞✐s❝✉ssã♦ ❡♥tr❡ ♦s ♠ét♦❞♦s é ❛ ❝♦♥tr✐❜✉✐çã♦ ❞❡st❛ ❞✐ss❡rt❛çã♦✱ ♦♥❞❡ s❡ ❞❡♠♦♥str❛ ❛ ✐❣✉❛❧❞❛❞❡ ❡♥tr❡ ❛s ✐t❡r❛çõ❡s ❞♦s ♠ét♦❞♦s ❞❡ ▼❛③✬✐❛ ❡ ▲❛♥❞✇❡❜❡r ❧✐♥❡❛r✳

(16)

✶✻

(17)

❈❛♣ít✉❧♦ ✶

❖ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤②

❡❧í♣t✐❝♦

❙❡❥❛Ω⊂R2 ✉♠ ❝♦♥❥✉♥t♦ ❛❜❡rt♦✱ ❧✐♠✐t❛❞♦ ❡ s✐♠♣❧❡s♠❡♥t❡ ❝♦♥❡①♦ ❝♦♠ ❢r♦♥t❡✐r❛ s✉❛✈❡ ∂Ω✱ ♦♥❞❡ ∂Ω ♣♦❞❡ s❡r ❞✐✈✐❞✐❞♦ ❡♠ ❞✉❛s ❝♦♠♣♦✲

♥❡♥t❡s ❛❜❡rt❛s ❡ ❝♦♥❡①❛sΓ1✱Γ2t❛✐s q✉❡Γ1∩Γ2=∅❡Γ1∪Γ2=∂Ω✳

❖ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤② ❡❧í♣t✐❝♦ s♦❜r❡ Ω ❝♦♥s✐st❡ ♥♦ ♣r♦❜❧❡♠❛ ❞❡

✈❛❧♦r ✐♥✐❝✐❛❧ ✭✐♥❞❡♣❡♥❞❡♥t❡ ❞♦ t❡♠♣♦✮ ♣❛r❛ ✉♠ ♦♣❡r❛❞♦r ❞✐❢❡r❡♥❝✐❛❧ ❡❧í♣t✐❝♦ s♦❜r❡Ω❡♠ q✉❡ ♦s ❞❛❞♦s ✐♥✐❝✐❛✐s sã♦ ❞❛❞♦s ❛♣❡♥❛s ❡♠Γ1⊂∂Ω✳

❆ss✐♠✱ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤② ❡❧í♣t✐❝♦ ❝♦♥s✐❞❡r❛❞♦ é ♣❛r❛ ♦ ♦♣❡r❛❞♦r ❞❡ ▲❛♣❧❛❝❡✱ ❞❛❞♦ ♣♦r

(CP) ∆u= 0, Ω; u|Γ1 =f; uν|Γ1 =g;

♦♥❞❡(f, g)é ❝❤❛♠❛❞♦ ❞❡ ♣❛r ❞❡ ❞❛❞♦s ❞❡ ❈❛✉❝❤②✳

❈♦♠♦ s♦❧✉çã♦ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤② (CP) ❝♦♥s✐❞❡r❡ ❞✐str✐✲

❜✉✐çõ❡s u ∈ H1(Ω) q✉❡ r❡s♦❧✈❡♠ ♦ ♣r♦❜❧❡♠❛ ♥❛ ❢♦r♠✉❧❛çã♦ ❢r❛❝❛ ❡

t❛♠❜é♠ s❛t✐s❢❛③❡♠ ♦s ❞❛❞♦s ❞❡ ❈❛✉❝❤② (f, g) s♦❜r❡ Γ1 ♥♦ s❡♥t✐❞♦ ❞♦

tr❛ç♦ ❞♦ ♦♣❡r❛❞♦r✳ ❆❧é♠ ❞✐ss♦✱ ♦ ♣❛r ❞❡ ❞❛❞♦s ❞❡ ❈❛✉❝❤② (f, g) ❡stá

♥♦ ❡s♣❛ç♦H1/2

1)×H001/2(Γ1)′ ❬✹❪✳

❖ ♣r♦❜❧❡♠❛ ❛ s❡r ❛♥❛❧✐s❛❞♦ é ♦ ❞❡ r❡❝♦♥str✉✐r ♦ tr❛ç♦ ❞❛ s♦❧✉çã♦ ❞❡ss❡ ♣r♦❜❧❡♠❛ ❞❡ ✈❛❧♦r ✐♥✐❝✐❛❧ s♦❜r❡ ❛ ♣❛rt❡ ❞❛ ❢r♦♥t❡✐r❛ ♦♥❞❡ ♥ã♦ tê♠ ❞❛❞♦s ✐♥✐❝✐❛✐s✱ ✐st♦ é✱ ❛✈❛❧✐❛ruν s♦❜r❡Γ2✳

◆♦t❡ q✉❡ r❡s♦❧✈❡r ❡st❡ ♣r♦❜❧❡♠❛ ✭❛✈❛❧✐❛r ♦ tr❛ç♦ ❞❡ ✉♠❛ s♦❧✉çã♦ s♦❜r❡Γ2✮ é ❡q✉✐✈❛❧❡♥t❡ ❛ ❡♥❝♦♥tr❛r ❡st❛ s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤②✳

❉❡ ❢❛t♦✱ s❡ ϕ=uν|Γ2 é ♦ tr❛ç♦ ❞❛ s♦❧✉çã♦ u❞❡(CP)s♦❜r❡Γ2✱ ❡♥tã♦

(18)

✶✽

❞❡ ✈❛❧♦r ❞❡ ❝♦♥t♦r♥♦ ♠✐st♦ ❜❡♠ ♣♦st♦ ❬❆✳✶❪✿

∆u= 0, Ω; u|Γ1 =f; uν|Γ2 =ϕ.

❆q✉✐✱ϕé ♦ tr❛ç♦ ❞❡ ◆❡✉♠❛♥♥✱ ❛ ❢♦r♠✉❧❛çã♦ ❞♦ ♣r♦❜❧❡♠❛ t❛♠❜é♠ ♣♦❞❡

s❡r ❢❡✐t❛ ❝♦♠ ♦ tr❛ç♦ ❞❡ ❉✐r✐❝❤❧❡t✳ ❊st❡s ♦♣❡r❛❞♦r❡s s❡rã♦ ❞❡✜♥✐❞♦s ❛♦ ❧♦♥❣♦ ❞♦ t❡①t♦✳

◆❛ s❡çã♦ s❡❣✉✐♥t❡✱ ♦s ❛s♣❡❝t♦s s♦❜r❡ ❛ ❜❡♠ ♣♦s✐çã♦ ♣❛r❛ ♦ ♣r♦✲ ❜❧❡♠❛ ❞❡ ❈❛✉❝❤② ❡❧í♣t✐❝♦ s❡❣✉♥❞♦ ❍❛❞❛♠❛r❞ sã♦ ❛♣r❡s❡♥t❛❞♦s✱ ✐st♦ é✱ ❡①✐stê♥❝✐❛✱ ✉♥✐❝✐❞❛❞❡ ❡ ❛ ❞❡♣❡♥❞ê♥❝✐❛ ❝♦♥tí♥✉❛ ❞♦s ❞❛❞♦s✳

✶✳✶ ❙♦❜r❡ ❛ ❜❡♠ ♣♦s✐çã♦ ❞♦ ♣r♦❜❧❡♠❛ ❞❡

❈❛✉❝❤② ❡❧í♣t✐❝♦

❆ s❡çã♦ ❛♥❛❧✐s❛ ❝❛❞❛ ✉♠ ❞♦s três ❝r✐tér✐♦s ❞❡ ❍❛❞❛♠❛r❞ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤②✳ ❆❧é♠ ❞✐ss♦✱ t❛♠❜é♠✱ é ❛♣r❡s❡♥t❛❞♦ ✉♠ r❡s✉❧t❛❞♦ s♦❜r❡ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♣❛r❡s ❞❡ ❞❛❞♦s(f, g)❞❡ ❈❛✉❝❤②✳

❆♣❡s❛r ❞❛ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çã♦ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤② s❡r ❢❡✲ r✐❞❛✱ ♦ ♣r✐♠❡✐r♦ r❡s✉❧t❛❞♦ ❞á ✉♠❛ ❝♦♥❞✐çã♦ ♥❡❝❡ssár✐❛ ♣❛r❛ ❛ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çã♦ ❞❡(CP)❛ r❡s♣❡✐t♦ ❞♦s ❞❛❞♦s ❞❡ ❈❛✉❝❤②✳

Pr♦♣♦s✐çã♦ ✶✳✶ ❈♦♥s✐❞❡r❡(f, g)✉♠ ♣❛r ❞❡ ❞❛❞♦s ❞❡ ❈❛✉❝❤② ❝♦♠f ❡ g❞❡ ❝❧❛ss❡C∞✳ ❙❡f 0❡ ❡①✐st❡ s♦❧✉çã♦ ♣❛r❛(CP)❡♠C2(Ω)✱ ❡♥tã♦

g é ✉♠❛ ❢✉♥çã♦ ❛♥❛❧ít✐❝❛✳

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛u∈C2(Ω)s♦❧✉çã♦ ❞❡(CP)♣❛r❛ ❞❛❞♦s ❞❡ ❈❛✉✲

❝❤②(f, g)✱ ♦♥❞❡f ≡0❡gé ✉♠❛ ❢✉♥çã♦ ❞❡ ❝❧❛ss❡C∞

1)✳ ❊♥tã♦✱ ❝♦♠♦

∆u= 0✱ut❡♠ ❛ s❡❣✉✐♥t❡ r❡♣r❡s❡♥t❛çã♦ ❞❡ ●r❡❡♥ ❬♣✳ ✶✽✱ ✸❪✿

u(y) =

Z

∂Ω

(u(x)Γν(x−y)−Γ(x−y)uν(x))d∂Ω(x), (y∈Ω);

♦♥❞❡Γ(x−y) = 1

2πln|x−y|✱ ♦♥❞❡x∈∂Ω❡y∈Ω✳

❈♦♠♦ ♦ ✐♥t❡❣r❛♥❞♦ é ❛♥❛❧ít✐❝♦ ❝♦♠ r❡s♣❡✐t♦ ❛ y✱ ♣♦✐s ❝♦♠♦ y ∈Ω

❡x∈∂Ω❡Ω é ❛❜❡rt♦✱ t❡♠✲s❡ q✉❡x6=y✱ ♦✉ s❡❥❛✱ ❛ s✐♥❣✉❧❛r✐❞❛❞❡ ❞❡ Γ(x−y)♥ã♦ ♦❝♦rr❡ ❡✱ ❛ss✐♠✱ ❝♦♥❝❧✉✐✲s❡ q✉❡ ué ❛♥❛❧ít✐❝❛ ❡♠Ω✳

P❡❧♦ ♣r✐♥❝í♣✐♦ ❞❡ r❡✢❡①ã♦ ❞❡ ❙❝❤✇❛r③ ❬❆✳✷❪✱ ❛ ❢✉♥çã♦

U(x1, . . . , xn) =

u(x1, . . . , xn), xn≥0;

u(x1, . . . ,−xn), xn<0;

s❛t✐s❢❛③∆U = 0 ❡✱ ❛❧é♠ ❞✐ss♦✱ é ❛♥❛❧ít✐❝❛✳ ❊♥tã♦✱γN,Γ1(U)é ❛♥❛❧ít✐❝❛

(19)

✶✾

❯♠ ♣❛r ❞❡ ❞❛❞♦s ❞❡ ❈❛✉❝❤②(f, g)s❡rá ❞✐t♦ ❝♦♥s✐st❡♥t❡✱ ❝❛s♦ ❡①✐st❛

s♦❧✉çã♦ ♣❛r❛ (CP)✳

❉❛s ❝♦♥❞✐çõ❡s ❞❡ ❍❛❞❛♠❛r❞✱ ❛ ✉♥✐❝✐❞❛❞❡ ❞❛ s♦❧✉çã♦ ❞❡(CP)é ❛

ú♥✐❝❛ ♣♦ssí✈❡❧ ❞❡ s❡r ❞❡♠♦♥str❛❞❛✱ ❝♦♠♦ s❡❣✉❡ ❛❜❛✐①♦✳

Pr♦♣♦s✐çã♦ ✶✳✷ ❙❡❥❛ Ω ⊂R2 ❛❜❡rt♦✱ ❧✐♠✐t❛❞♦✱ s✐♠♣❧❡s♠❡♥t❡ ❝♦♥❡①♦ ❝♦♠ ❢r♦♥t❡✐r❛ ❛♥❛❧ít✐❝❛ ❡ ❝♦♥s✐❞❡r❡ Γ ⊂∂Ω ❛❜❡rt♦ ❡ s✐♠♣❧❡s♠❡♥t❡ ❝♦✲

♥❡①♦✳ ❊♥tã♦✱ ♦ ♣r♦❜❧❡♠❛ (CP) ❝♦♠ ♣❛r ❞❡ ❞❛❞♦s ❞❡ ❈❛✉❝❤② (f, g) ∈ H1/2(Γ)×H1/2

00 (Γ)′ ❝♦♥s✐st❡♥t❡ t❡♠ ♥♦ ♠á①✐♠♦ ✉♠❛ s♦❧✉çã♦✳

❉❡♠♦♥str❛çã♦✿ ❙✉♣♦♥❤❛ q✉❡ ❡①✐st❛♠ ❞✉❛s s♦❧✉çõ❡sw❡v❡♠H1(Ω)

♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤② ❝♦♠ ♣❛r ❞❡ ❞❛❞♦s (f, g) ∈ H1/2(Γ)×

H001/2(Γ)′ ❝♦♥s✐st❡♥t❡✳ ❊♥tã♦✱u:=w−vé s♦❧✉çã♦ ❞❡

∆u= 0, Ω; u|Γ = 0; uν|Γ = 0.

❈♦♠♦∆u= 0✱ t❡♠✲s❡ q✉❡u∈C∞(Ω)✱ ♦✉ s❡❥❛✱ué ✉♠❛ s♦❧✉çã♦ ❝❧áss✐❝❛

♣❛r❛(CP)❝♦♠ ❞❛❞♦s ❤♦♠♦❣ê♥❡♦s ❡♠Γ✳ ❆❧é♠ ❞✐ss♦✱ ❝♦♠♦ ♦ ♦♣❡r❛❞♦r

❧❛♣❧❛❝✐❛♥♦ é ✉♠ ♦♣❡r❛❞♦r ❡❧í♣t✐❝♦ ❡ Γ ❡ ∂Ω\Γ ♥ã♦ sã♦ ✈❛r✐❡❞❛❞❡s

❝❛r❛❝t❡ríst✐❝❛s✱ é ♣♦ssí✈❡❧ ❡♥❝♦♥tr❛r ✉♠❛ ❢❛♠í❧✐❛ ❞❡ ✈❛r✐❡❞❛❞❡s ❛♥❛❧ít✐❝❛s

{Γλ}0≤λ≤1 t❛❧ q✉❡Γ0 = Γ❡Γ1 =∂Ω\Γ ❡Γλ t❡♠ ♦s ♠❡s♠♦s ♣♦♥t♦s

✜♥❛✐s✳ ▲♦❣♦✱ s❡ ✉t✐❧✐③❛r ❡ss❛ ❢❛♠í❧✐❛ {Γλ}0≤λ≤1 ♥❛ ❞❡♠♦♥str❛çã♦ ❞♦

t❡♦r❡♠❛ ❞❡ ❍♦❧♠❣r❡♥ ❬♣✳ ✶✻✻✱ ✼❪✱ ❡♥tã♦u= 0 ❡♠Ω✳ P♦rt❛♥t♦✱w=v✳

❖ ❡①❡♠♣❧♦ ❛ s❡❣✉✐r✱ ❡♥❝♦♥tr❛❞♦ ♣♦r ❍❛❞❛♠❛r❞✱ ♠♦str❛ q✉❡ ✉♠❛ s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤② ♣♦❞❡ ♥ã♦ ❞❡♣❡♥❞❡r ❝♦♥t✐♥✉❛♠❡♥t❡ ❞♦s ❞❛❞♦s ✐♥✐❝✐❛✐s✳

❊①❡♠♣❧♦ ✶✳✶ ❈♦♥s✐❞❡r❡ ❛ s❡❣✉✐♥t❡ ❢❛♠í❧✐❛ ❞❡ ♣r♦❜❧❡♠❛s ❞❡ ❈❛✉❝❤②✿

    

∆uk(x, y) = 0, (x, y)∈Ω := (0,1)×(0,1);

uk(x,0) = 0, x∈(0,1);

∂uk

∂y (x,0) = 1

πksin (πkx), x∈(0,1);

♣❛r❛k∈N✳

Pr✐♠❡✐r❛♠❡♥t❡✱ ♥♦t❡ q✉❡

0≤ 1

πksin (πkx)

≤ 1 πk ||

sin (πkx)|| ≤ π1

1 k .

❈♦♠♦ ♣❛r❛ k → ∞✱ k 1/k k→ 0✱ ❡♥tã♦

(20)

✷✵

k → ∞ t❡♠✲s❡ ✉♠ ♣r♦❜❧❡♠❛ ❞❡ ✈❛❧♦r ✐♥✐❝✐❛❧ ❤♦♠♦❣ê♥❡♦ q✉❡ ❛❞♠✐t❡

❛♣❡♥❛s ❛ s♦❧✉çã♦ tr✐✈✐❛❧✳

❆❣♦r❛✱ ♥♦t❡ q✉❡ ❛s s♦❧✉çõ❡s uk(x, y) =

1

(πk)2sinh (πky) sin (πkx)

❡①✐st❡♠ ♣❛r❛ ❝❛❞❛k∈N❡ sã♦ ú♥✐❝❛s✳ ◆♦ ❡♥t❛♥t♦✱ ♣❛r❛ ❝❛❞❛0< y <1✱ t❡♠✲s❡ q✉❡ exp (πky)

2 (πk)2 sin (πkx)

≤ 1

(πk)2sinh (πky) sin (πkx)

;

♦✉ s❡❥❛✱ uk t♦r♥❛✲s❡ ✐❧✐♠✐t❛❞♦ ❡ ♦s❝✐❧❛ ❝❛❞❛ ✈❡③ ♠❛✐s à ♠❡❞✐❞❛ q✉❡

k→ ∞ ❡✱ ♣♦rt❛♥t♦✱uk 90 ❡♠ ♥❡♥❤✉♠❛ ♥♦r♠❛ r❛③♦á✈❡❧✳

P♦rt❛♥t♦✱ ❛ s♦❧✉çã♦ ♥ã♦ ❞❡♣❡♥❞❡ ❝♦♥t✐♥✉❛♠❡♥t❡ ❞♦s ❞❛❞♦s ✐♥✐❝✐❛✐s✳

❈♦♥❝❧✉✐✲s❡✱ ❡♥tã♦✱ q✉❡ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤②(CP)é ✉♠ ♣r♦❜❧❡♠❛

♠❛❧ ♣♦st♦ s❡❣✉♥❞♦ ❍❛❞❛♠❛r❞✳

❆♣❡s❛r ❞❛ ❞❡♣❡♥❞ê♥❝✐❛ ❝♦♥tí♥✉❛ ❞♦s ❞❛❞♦s ♣❛r❛ (CP)♥ã♦ s❡r ✈❡✲

r✐✜❝❛❞❛ ♣❛r❛ t♦❞♦s ♦s ♣❛r❡s ❞❡ ❞❛❞♦s ❞❡ ❈❛✉❝❤②(f, g)✱ ♦ r❡s✉❧t❛❞♦ ❛

s❡❣✉✐r ❞✐③ q✉❡ s❡ ✜①❛rf✱ ❡♥tã♦ ♦ ❝♦♥❥✉♥t♦ ❞❡ ❞❛❞♦sg♣❛r❛ ♦ q✉❛❧ ❡①✐st❡

s♦❧✉çã♦ ❞❡(CP)é ❞❡♥s♦ ♥♦ ❡s♣❛ç♦ ❞❡ ❙♦❜♦❧❡✈H001/2(Γ1)′❀ ♦ ♠❡s♠♦ ✈❛❧❡

s❡ ✜①❛rg✱ ♦✉ s❡❥❛✱ ♦ ❝♦♥❥✉♥t♦ ❞❡ ❞❛❞♦sf ♣❛r❛ ♦ q✉❛❧ ❡①✐st❡ s♦❧✉çã♦ ❞❡ (CP)é ❞❡♥s♦ ♥♦ ❡s♣❛ç♦ ❞❡ ❙♦❜♦❧❡✈H1/2

1)✳ ❊st❡ r❡s✉❧t❛❞♦ é ♦❜t✐❞♦

❞❡ ❬✶❪✳

Pr♦♣♦s✐çã♦ ✶✳✸ P❛r❛ ♦ ♣❛r ❞❡ ❞❛❞♦s ❞❡ ❈❛✉❝❤②(f, g)∈H1/2 1)×

H001/2(Γ1)′✱ tê♠✲s❡ q✉❡

✶✳ ♣❛r❛ f ∈H1/2

1)✜①♦✱ ♦ ❝♦♥❥✉♥t♦ ❞❡ ❞❛❞♦s g∈H001/2(Γ1)′ ♣❛r❛

♦ q✉❛❧ ❡①✐st❡u∈H1(Ω)s❛t✐s❢❛③❡♥❞♦ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤②(CP)

é ❞❡♥s♦ ❡♠ H001/2(Γ1)′❀

✷✳ ♣❛r❛ g ∈H001/2(Γ1)′ ✜①♦✱ ♦ ❝♦♥❥✉♥t♦ ❞❡ ❞❛❞♦sf ∈H1/2(Γ1)♣❛r❛

♦ q✉❛❧ ❡①✐st❡u∈H1(Ω)s❛t✐s❢❛③❡♥❞♦ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤②(CP)

é ❞❡♥s♦ ❡♠ H1/2 1)✳

❉❡♠♦♥str❛çã♦✿ ◆❡st❛ ❞❡♠♦♥str❛çã♦ ❛♣❡♥❛s ❛ ♣❛rt❡ ✶ é ❛♣r❡s❡♥t❛❞❛ ✉♠❛ ✈❡③ q✉❡ ❛ ♦✉tr❛ ❛✜r♠❛çã♦ s❡❣✉❡ ❛♥❛❧♦❣❛♠❡♥t❡✳ ◆♦t❡ q✉❡ é s✉✜❝✐✲ ❡♥t❡ ❞❡♠♦♥str❛r ♣❛r❛f ≡0✳

❙❡❥❛u∈H1(Ω) s♦❧✉çã♦ ❞❡(CP)✳ ❙✉♣♦♥❤❛✱ ♣♦r ❝♦♥tr❛❞✐çã♦✱ q✉❡ ♦

❝♦♥❥✉♥t♦ ❞❡ ❞❛❞♦s ❞❡g ∈H001/2(Γ1)′ t❛❧ q✉❡ ❡①✐st❡u∈H1(Ω) s❛t✐s❢❛✲

(21)

✷✶

❙❡❥❛G ⊆H001/2(Γ1)′ ♦ ❝♦♥❥✉♥t♦ ❞❡ ❞❛❞♦s ❞❡ g t❛❧ q✉❡ ♦ ♣r♦❜❧❡♠❛

❞❡ ✈❛❧♦r ✐♥✐❝✐❛❧

∆u= 0, Ω; u|Γ1 = 0; uν|Γ1 =g;

t❡♥❤❛ s♦❧✉çã♦✳

◆♦t❡ q✉❡G6=H001/2(Γ1)′✳ ❊♥tã♦✱ ❡①✐st❡ ✉♠❛ ❢♦r♠❛ ❧✐♥❡❛r ❝♦♥tí♥✉❛

♥ã♦ ♥✉❧❛l∈H001/2(Γ1)′ ❡l∈G⊆H1/2(Γ1)t❛❧ q✉❡hl, gi= 0, ∀g∈G✳

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

∆v= 0, Ω; v|Γ1 =l; vν|Γ2 = 0;

♥♦t❡ q✉❡ ❡st❡ ♣r♦❜❧❡♠❛ é ❜❡♠ ♣♦st♦ ❬❆✳✶❪✳ ❆❣♦r❛✱ ♣❡❧❛ s❡❣✉♥❞❛ ❢ór✲ ♠✉❧❛ ❞❡ ●r❡❡♥ R

Ω(v∆u−u∆v)dx =

R

Γ1⊔Γ2(vuν−uvν)d∂Ω ❬♣✳ ✶✼✱

✸❪✱ t❡♠✲s❡ q✉❡ R

Γ1⊔Γ2(vuν−uvν)dx = 0; ❛✐♥❞❛ ♠❛✐s✱ ♣❡❧❛s ❝♦♥❞✐çõ❡s

❞❡ ❢r♦♥t❡✐r❛✱

Z

Γ2

vuνdx= 0.

❈♦♥s✐❞❡r❡ t❛♠❜é♠h∈C∞

2)❡ ♦ s❡❣✉✐♥t❡ ♣r♦❜❧❡♠❛ ❞❡ ✈❛❧♦r ❞❡

❝♦♥t♦r♥♦ ♠✐st♦ ✭❜❡♠ ♣♦st♦ ❬❆✳✶❪✮

∆w= 0, Ω; w|Γ1 = 0; wν|Γ2 =h;

❝✉❥❛ s♦❧✉çã♦ é ú♥✐❝❛ ❡♠H1(Ω)✳ ❆ss✐♠✱ ❛♥❛❧♦❣❛♠❡♥t❡✱ ♣❡❧❛ ❢ór♠✉❧❛ ❞❡

●r❡❡♥ ❛❝✐♠❛ ♣❛r❛ v ❡w❡ ✉t✐❧✐③❛♥❞♦ ❛s ❝♦♥❞✐çõ❡s ❞❡ ❢r♦♥t❡✐r❛✱ t❡♠✲s❡ q✉❡R

Γ2vhdx= 0, ∀h∈C

2)✳

▲♦❣♦✱vs❛t✐s❢❛③ ♦ ♣r♦❜❧❡♠❛ ❤♦♠♦❣ê♥❡♦

∆v= 0, Ω; v|Γ2 = 0; vν|Γ2 = 0;

♦✉ s❡❥❛✱ v ≡0 ♣❡❧♦ t❡♦r❡♠❛ ❞❡ ❍♦❧♠❣r❡♥ ❬♣✳✶✻✻✱ ✼❪ ❡✱ ❛ss✐♠✱ l ≡0✱ ♦

q✉❡ é ✉♠❛ ❝♦♥tr❛❞✐çã♦✳

P♦rt❛♥t♦✱ ♦ ❝♦♥❥✉♥t♦ ❞❡ ❞❛❞♦s g ∈ H001/2(Γ2)′ ♣❛r❛ ♦ q✉❛❧ ❡①✐st❡

u∈H1(Ω) s❛t✐s❢❛③❡♥❞♦(CP)é ❞❡♥s♦ ❡♠ H1/2 00 (Γ1)′✳

❈♦♠ ♦ ♦❜❥❡t✐✈♦ ❞❡ ❡s❝r❡✈❡r ♦ ♣r♦❜❧❡♠❛ ❞❡ ❛✈❛❧✐❛r ♦ tr❛ç♦ ❞❛ s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛ ❞❡ ✈❛❧♦r ✐♥✐❝✐❛❧ s♦❜r❡ ❛ ❢r♦♥t❡✐r❛ Γ2 ✭♦♥❞❡ ♥❡♥❤✉♠ ❞❛❞♦

é ♣r❡s❝r✐t♦✮ ♥❛ ❢♦r♠❛ ❞❡ ✉♠❛ ❡q✉❛çã♦ ❞❡ ♦♣❡r❛❞♦r❡s F(ϕ) =y ❞❡♣❡♥✲

❞❡♥❞♦ ❛♣❡♥❛s ❞♦s ❞❛❞♦s ❞❡ ❈❛✉❝❤② (f, g) ∈ H1/2

1)×H001/2(Γ1)′ ❡

F :H001/2(Γ2)′ −→ H001/2(Γ1)′ é ✉♠ ♦♣❡r❛❞♦r ❡♥tr❡ ❡s♣❛ç♦s ❞❡ ❍✐❧❜❡rt

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✷✷

❼ Uf :H001/2(Γ2)′−→H1(Ω)✱ ♦♥❞❡Uf(ϕ) :=ué s♦❧✉çã♦ ❞❡

∆u= 0, Ω; u|Γ1 =f; uν|Γ2 =ϕ;

♥♦t❡ q✉❡ ❡st❡ ♦♣❡r❛❞♦r ❡stá ❜❡♠ ❞❡✜♥✐❞♦✱ ♣♦✐s ♦ ♣r♦❜❧❡♠❛ ❞❡ ✈❛❧♦r ❞❡ ❝♦♥t♦r♥♦ ♠✐st♦ ❛ s❡r r❡s♦❧✈✐❞♦ é ❜❡♠ ♣♦st♦ ❬❆✳✶❪✳

❼ γN,Γ1 : H

1(Ω) −→ H1/2

00 (Γ1)′✱ ♦♥❞❡ γN,Γ1(u) :=uν|Γ1✱ ❡st❡ é ♦

♦♣❡r❛❞♦r tr❛ç♦ ❞❡ ◆❡✉♠❛♥♥ s♦❜r❡Γ1✳

❆ss✐♠✱ ❞❡✜♥❛F :=γN,Γ1◦Uf :H

1/2

00 (Γ2)′ −→H001/2(Γ1)′ ♣♦r

F(ϕ) :=γN,Γ1(Uf(ϕ)) ;

♥♦t❡ q✉❡

γN,Γ1(Uf(ϕ)) =γN,Γ1(u) =uν|Γ1 .

▲♦❣♦✱ ❞❛❞♦(f, g)∈H1/2

1)×H001/2(Γ1)′ ✉♠ ♣❛r ❞❡ ❞❛❞♦s ❞❡ ❈❛✉✲

❝❤② ❝♦♥s✐st❡♥t❡✱ ♦ ♣r♦❜❧❡♠❛ ❛ s❡r ❛♥❛❧✐s❛❞♦ ✭❡♥❝♦♥tr❛r ♦ tr❛ç♦ ❞❛ s♦✲ ❧✉çã♦ ❞❡(CP)s♦❜r❡Γ2✮ ♣♦❞❡ s❡r ✈✐st♦ ♣❡❧♦ s❡❣✉✐♥t❡ ♣r♦❜❧❡♠❛ ✐♥✈❡rs♦

♠❛❧ ♣♦st♦✱F(ϕ) =g✳

❖❜s❡r✈❡ q✉❡ F é ✉♠ ♦♣❡r❛❞♦r ❛✜♠✱ ✐st♦ é✱ F(ϕ) := Fl(ϕ) +Ff❀

♦♥❞❡ Fl(ϕ) := vν|Γ1 é ❛ ♣❛rt❡ ❧✐♥❡❛r ❡ Ff := wν|Γ1 é ❛ ♣❛rt❡ ❛✜♠✱

❝♦♠v s❡♥❞♦ ❛ s♦❧✉çã♦ ❞❡

∆v= 0, Ω; v|Γ1 = 0; vν|Γ2 =ϕ;

❡w✱ ❛ s♦❧✉çã♦ ❞❡

∆w= 0, Ω; w|Γ1 =f; wν|Γ2 = 0.

❖ ♣ró①✐♠♦ ❝❛♣ít✉❧♦ ❛♣r❡s❡♥t❛ ❞♦✐s ♠ét♦❞♦s ✐t❡r❛t✐✈♦s ❞❡ ♣♦♥t♦ ✜①♦ ♣❛r❛ r❡s♦❧✈❡r ❡ss❡ ♣r♦❜❧❡♠❛ ✐♥✈❡rs♦ ♠❛❧ ♣♦st♦✱ ❡♥❝♦♥tr❛r ♦ tr❛ç♦ ❞❡ ◆❡✉♠❛♥♥ ❞❡u∈H1(Ω)s♦❜r❡Γ

2✱ ♦♥❞❡ué s♦❧✉çã♦ ❞❡

∆u= 0, Ω; u|Γ1 =f; uν|Γ1 =g.

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❈❛♣ít✉❧♦ ✷

❉❡s❝r✐çã♦ ❡ ❆♥á❧✐s❡ ❞♦s

▼ét♦❞♦s ❞❡ ▼❛③✬✐❛ ❡ ❞❡

▲❛♥❞✇❡❜❡r ♣❛r❛ ♦

Pr♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤②

❊❧í♣t✐❝♦

❖s ♠ét♦❞♦s ❛♥❛❧✐s❛❞♦s ♥❡st❡ tr❛❜❛❧❤♦ ♣❛r❛ r❡s♦❧✈❡r ♦ ♣r♦❜❧❡♠❛ ✐♥✲ ✈❡rs♦ ♠❛❧ ♣♦st♦ r❡❢❡r❡♥t❡ ❛♦ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤② ❡❧í♣t✐❝♦ ❞❡s❝r✐t♦s ♥♦ ❝❛♣ít✉❧♦ ❛♥t❡r✐♦r sã♦ ♦s ♠ét♦❞♦s ✐t❡r❛t✐✈♦s ❞❡ ▼❛③✬✐❛ ❡ ❞❡ ▲❛♥❞✇❡❜❡r✳ ❆ ❛♥á❧✐s❡ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ♣❛r❛ ♦ ♠ét♦❞♦ ❞❡ ▼❛③✬✐❛ ❡stá ❢❡✐t❛ ♣♦r ✉♠❛ ❛❜♦r❞❛❣❡♠ ❞❡ ❛♥á❧✐s❡ ❢✉♥❝✐♦♥❛❧ ❡ ✉t✐❧✐③❛♥❞♦ ✉♠❛ t♦♣♦❧♦❣✐❛ ❞✐✲ ❢❡r❡♥t❡ ♣❛r❛ ♦ ❡s♣❛ç♦ ❞❡ ❙♦❜♦❧❡✈ H001/2(Γ2)′✱ t♦r♥❛♥❞♦✲♦s ❡s♣❛ç♦s ❞❡

❍✐❧❜❡rt ❡ ❢❡✐t❛ ❛♣❡♥❛s s♦❜ ❞❛❞♦s ❡①❛t♦s✳ ❈♦♥❝❧✉✐♥❞♦ q✉❡ ♦ ♠ét♦❞♦ ❞❡ ▼❛③✬✐❛ é ✉♠ ♠ét♦❞♦ ❞❡ r❡❣✉❧❛r✐③❛çã♦ ❛♣❡♥❛s ♥♦ t❡r❝❡✐r♦ ❝❛♣ít✉❧♦✳ ❊♥✲ q✉❛♥t♦ q✉❡ ❛ ❛♥á❧✐s❡ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛✱ ❡st❛❜✐❧✐❞❛❞❡ ❞♦ ♠ét♦❞♦ ❡ ❛ t❛①❛ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ✭♠❡s♠♦ q✉❛♥❞♦ ❢♦r♥❡❝✐❞♦s ❞❛❞♦s ♣❡rt✉r❜❛❞♦s✮ ❞♦ ♠é✲ t♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r é ❞❡♠♦♥str❛❞❛ ♣❡❧❛ t❡♦r✐❛ ❝❧áss✐❝❛ ❞❡ r❡❣✉❧❛r✐③❛çã♦✳ ❖ ❝❛♣ít✉❧♦ ❡stá ❛♣r❡s❡♥t❛❞♦ ❡♠ ❞✉❛s s❡çõ❡s✿ ♥❛ ♣r✐♠❡✐r❛ s❡çã♦ ♣❛r❛ ♦ ❛❧❣♦rít♠♦ ❞❡ ▼❛③✬✐❛ ❡ ❛ s❡❣✉♥❞❛✱ ♣❛r❛ ♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r✳ ❆s r❡s♣❡❝t✐✈❛s ❛♥á❧✐s❡s ❡stã♦ ❡♠ s✉❜s❡çõ❡s✳

❉❛❞♦ ✉♠ ♣❛r ❞❡ ❞❛❞♦s ❞❡ ❈❛✉❝❤② (f, g) ∈ H1/2

1)×H001/2(Γ1)′

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✷✹

◆❡✉♠❛♥♥ ❞❡u∈H1(Ω)♥❛ ❢r♦♥t❡✐r❛Γ

2✭♦♥❞❡ ♥❡♥❤✉♠ ❞❛❞♦ é ❢♦r♥❡❝✐❞♦

♣r❡✈✐❛♠❡♥t❡✮✱ ♦♥❞❡ué ❛ s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤②(CP)✱ ✐st♦ é✱

❡♥❝♦♥tr❛ruν|Γ2✳

❊♠ r❡❧❛çã♦ ❛ ❞✐♠❡♥sã♦✱ ❛♣❡s❛r ❞❡ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤② ❡❧í♣t✐❝♦ ❡ ♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r ♣♦❞❡r❡♠ s❡r ❛♥❛❧✐s❛❞♦s ❡♠ três ❞✐♠❡♥sõ❡s✱ ❛ ❢♦r♠✉❧❛çã♦ ❞♦ ♠ét♦❞♦ ❞❡ ▼❛③✬✐❛ ♠✉❞❛ ❞❡✈✐❞♦ ❛♦ ❢❛t♦ ❞❡ ♦s t❡♦r❡♠❛s ❞❡ tr❛ç♦ s❡r❡♠ ❞❡♣❡♥❞❡♥t❡s ❞❛ ❞✐♠❡♥sã♦ ❞♦ ❡s♣❛ç♦✳ P♦rt❛♥t♦✱ ♥❡ss❛ ❞✐ss❡rt❛çã♦ é ✈✐st♦ ❛♣❡♥❛s ❡♠ ❞✉❛s ❞✐♠❡♥sõ❡s✳

✷✳✶ ▼ét♦❞♦ ❞❡ ▼❛③✬✐❛

❖ ❛❧❣♦rít♠♦ ❞❡ ▼❛③✬✐❛ ❝♦♥s✐st❡ ❡♠ ❞❛❞♦ ✉♠❛ ❛♣r♦①✐♠❛çã♦ ✐♥✐❝✐❛❧

ϕ0∈H001/2(Γ2)′ ♣❛r❛ uν|Γ2✱ ♦♥❞❡u∈H

1(Ω)é s♦❧✉çã♦ ❞❡(CP)✱ ❣❡r❛r

❛ s❡❣✉✐♥t❡ ✐t❡r❛çã♦{ϕk}k∈N✿

      

w∈H1(Ω)resolve: ∆w= 0, Ω; w|

Γ1 =f; wν|Γ2 =ϕk;

ψk:=w|Γ2 ;

v∈H1(Ω)resolve: ∆v= 0, Ω; v

ν|Γ1 =g; v|Γ2 =ψk;

ϕk+1:=vν|Γ2;

♣❛r❛ ✉♠ ♣❛r ❞❡ ❞❛❞♦s ❞❡ ❈❛✉❝❤②(f, g)∈H1/2

1)×H001/2(Γ1)′ ❝♦♥s✐s✲

t❡♥t❡❀ ❛ ✐❞❡✐❛ é ❛ ❞❡ r❡s♦❧✈❡r ♣r♦❜❧❡♠❛s ❞❡ ✈❛❧♦r ❞❡ ❝♦♥t♦r♥♦ ♠✐st♦s ❜❡♠ ♣♦st♦s ❬❆✳✶❪ s✉❝❡ss✐✈❛♠❡♥t❡ ✉s❛♥❞♦ ♦s ❞❛❞♦s ❞❡ ❈❛✉❝❤② ❝♦♠♦ ♣❛rt❡ ❞❛s ❝♦♥❞✐çõ❡s ❞❡ ❝♦♥t♦r♥♦✳

◆♦t❡ q✉❡ ♥❛ ✐t❡r❛çã♦ ❛❝✐♠❛ ❞♦✐s ♣r♦❜❧❡♠❛s ❞❡ ✈❛❧♦r ❞❡ ❝♦♥t♦r♥♦ sã♦ r❡s♦❧✈✐❞♦s ❡ ❞♦✐s tr❛ç♦s ❞❡ s♦❧✉çõ❡s sã♦ ❝❛❧❝✉❧❛❞♦s✳ P♦rt❛♥t♦✱ ❞✉❛s s❡q✉ê♥❝✐❛s sã♦ ❣❡r❛❞❛s✳ ❆ ♣r✐♠❡✐r❛✱ ❞❡ tr❛ç♦s ❞❡ ❉✐r✐❝❤❧❡t ❡ ❛ s❡❣✉♥❞❛✱ ❞❡ tr❛ç♦s ❞❡ ◆❡✉♠❛♥♥✱ ❛♠❜❛s ❞❡✜♥✐❞❛s s♦❜r❡ Γ2❀ ❞❡✜♥✐❞❛s ❛♦ ❧♦♥❣♦

❞♦ t❡①t♦✳ ❆❧é♠ ❞✐ss♦✱ ❝♦♠♦ w, v ∈ H1(Ω)✱ tê♠✲s❡ q✉❡ {ϕ

k}k∈N ⊆

H001/2(Γ2)′ ❬❆✳✸❪ ❡{ψk}k∈N⊆H1/2(Γ2)❬✹❪✳

❈♦♠ ♦ ♦❜❥❡t✐✈♦ ❞❡ ❢❛③❡r ❛ ❛♥á❧✐s❡ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ❞❡st❡ ♠ét♦❞♦✱ ♦❜s❡r✈❡ q✉❡ ❛ ✐t❡r❛çã♦ ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞❛ ♣♦r ♣♦tê♥❝✐❛s ❞❡ ✉♠ ♦♣❡✲ r❛❞♦r T : H001/2(Γ2)′ −→ H001/2(Γ2)′✱ ❝♦♥str✉✐♥❞♦ ❡ss❡ ♦♣❡r❛❞♦r T ❞❛

s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿

Pr✐♠❡✐r♦✱ ❞❡✜♥❛ ♦s s❡❣✉✐♥t❡s ♦♣❡r❛❞♦r❡s✿

❼ LN :H001/2(Γ2)′ −→H1(Ω), LN(ϕ) :=w, ♦♥❞❡wé s♦❧✉çã♦ ❞❡

∆w= 0, Ω; w|Γ1 =f; wν|Γ2 =ϕ;

❼ LD:H1/2(Γ2)−→H1(Ω), LD(ψ) :=v,♦♥❞❡v é s♦❧✉çã♦ ❞❡

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✷✺

♥♦t❡ q✉❡ ❡st❡s ♦♣❡r❛❞♦r❡s ❡stã♦ ❜❡♠ ❞❡✜♥✐❞♦s ♣♦✐s ♦s ♣r♦❜❧❡♠❛s ❞❡ ✈❛❧♦r ❞❡ ❝♦♥t♦r♥♦ ♠✐st♦s sã♦ ❜❡♠ ♣♦st♦s ❬❆✳✶❪✳

❆ss✐♠✱ ❛ ✐t❡r❛çã♦ ❞♦ ♠ét♦❞♦ ❞❡ ▼❛③✬✐❛ ♣♦❞❡ s❡r r❡❡s❝r✐t❛ ♣♦r

w=LN(ϕk); ψk=γD,Γ2(w);

v=LD(ψk); ϕk+1=γN,Γ2(v);

♦♥❞❡

γN,Γ2 :H

1(Ω)

−→H001/2(Γ2)′, γN,Γ2(u) :=uν|Γ2 ;

é ♦ tr❛ç♦ ❞❡ ◆❡✉♠❛♥♥ s♦❜r❡Γ2❡

γD,Γ2 :H

1(Ω)

−→H1/2(Γ2), γD,Γ2(u) :=u|Γ2 ;

é ♦ tr❛ç♦ ❞❡ ❉✐r✐❝❤❧❡t s♦❜r❡Γ2✳

▲♦❣♦✱ ♣❛r❛ r❡♣r❡s❡♥t❛r ❛❧❣♦rít♠♦ ♣♦r ♣♦tê♥❝✐❛s ❞❡ ✉♠ ♦♣❡r❛❞♦r

T :H001/2(Γ2)′−→H001/2(Γ2)′✱ ❜❛st❛ ❞❡✜♥í✲❧♦ ♣♦r

T :=γN,Γ2◦LD◦γD,Γ2◦LN.

◆♦t❡ q✉❡T s❛t✐s❢❛③✱ ♣♦r ❞❡✜♥✐çã♦✱

ϕk+1=T(ϕk) =Tk+1(ϕ0), k∈N;

♦✉ s❡❥❛✱ ♦ ♠ét♦❞♦ ❞❡ ▼❛③✬✐❛ é ✉♠❛ ✐t❡r❛çã♦ ❞❡ ♣♦♥t♦ ✜①♦ q✉❡ ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞❛ ♣♦r ♣♦tê♥❝✐❛s ❞❡ ✉♠ ♦♣❡r❛❞♦r ❛✜♠ ✭♣♦✐s LD ❡ LN

❝❧❛r❛♠❡♥t❡ sã♦ ♦♣❡r❛❞♦r❡s ❛✜♥s✮✳

❈♦♠♦T é ✉♠ ♦♣❡r❛❞♦r ❛✜♠✱ ❡♥tã♦ϕk+1=T(ϕk) :=Tl(ϕk) +T0✱

k ∈ N✱ ♦♥❞❡ Tlk) =γN,Γ2 LlDγD,Γ2LlNk) é ❛ ♣❛rt❡ ❧✐♥❡❛r✱ ❝♦♠Ll

D❡LlN s❡♥❞♦ ❛s ♣❛rt❡s ❧✐♥❡❛r❡s ❞❡LD ❡LN✱ r❡s♣❡❝t✐✈❛♠❡♥t❡❀ ❡

T0=γN,Γ2◦L

l

D◦γD,Γ2(wf) +γN,Γ2(vg)s❡♥❞♦ ❛ ♣❛rt❡ ❛✜♠✱ ❝♦♠ wf

❡ vg ❡♠ H1(Ω) s❡♥❞♦ ❛s ♣❛rt❡s ❛✜♥s ❞❡ LN ❡ LD✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❡

❝♦♠wf ❞❡♣❡♥❞❡♥❞♦ ❛♣❡♥❛s ❞❡f ❡vg ❞❡♣❡♥❞❡♥❞♦ ❛♣❡♥❛s ❞❡g✳

❆ss✐♠✱ r❡❝✉rs✐✈❛♠❡♥t❡✱ t❡♠✲s❡ q✉❡

ϕk+1=Tlk+1(ϕ0) +

k

X

j=0

Tlj(T0), k∈N.

❖❜s❡r✈❛çã♦ ✷✳✶ ◆♦t❡ q✉❡ s❡ϕ:=γN,Γ2(u)✱ ♦♥❞❡ué s♦❧✉çã♦ ❞❡(CP)

❝♦♠ ❞❛❞♦s ❞❡ ❈❛✉❝❤②(f, g)∈H1/2

1)×H001/2(Γ1)′ ❝♦♥s✐st❡♥t❡✱ ❡♥tã♦

T ϕ=ϕ✱ ♣♦r ❞❡✜♥✐çã♦ ❞❡T✳ ❆❣♦r❛✱ s❡ϕé ✉♠ ♣♦♥t♦ ✜①♦ ❞♦ ♦♣❡r❛❞♦rT✱

❛s ❢✉♥çõ❡sw❡v tê♠ ♦s ♠❡s♠♦s tr❛ç♦s s♦❜r❡ Γ2✳ ▲♦❣♦✱ ♣❡❧❛ ✉♥✐❝✐❞❛❞❡

❞❛ s♦❧✉çã♦✱ w=v é s♦❧✉çã♦ ❞❡

∆u= 0, Ω; u|Γ1 =f; uν|Γ1 =g.

◆❛ s✉❜s❡çã♦ ❡stá ❛♣r❡s❡♥t❛❞♦ ❛s ♣r♦♣r✐❡❞❛❞❡s s♦❜r❡ ♦ ♦♣❡r❛❞♦rTl

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✷✻

✷✳✶✳✶ ❆♥á❧✐s❡ ❞♦ ♠ét♦❞♦ ❞❡ ▼❛③✬✐❛

◆❡st❛ s✉❜s❡çã♦✱ ❛❧é♠ ❞❛ ❛♥á❧✐s❡ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ❞♦ ♠ét♦❞♦ ❞❡ ▼❛③✬✐❛✱ ♦ r❡s✉❧t❛❞♦ r❡❝í♣r♦❝♦ t❛♠❜é♠ é ♣r♦✈❛❞♦✱ ✐st♦ é✱ s❡ ❛ s❡q✉ê♥❝✐❛

{ϕk}k∈N={Tk(ϕ0)}k∈N ❝♦♥✈❡r❣❡✱ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤② ❛ss♦❝✐❛❞♦ é

❝♦♥s✐st❡♥t❡ ❡ ϕ := limk→∞ϕk é ♦ tr❛ç♦ ❞❡ ◆❡✉♠❛♥♥ ❞❛ s♦❧✉çã♦ ❞❡

(CP)✳

❖s ❞♦✐s ♣r✐♠❡✐r♦s r❡s✉❧t❛❞♦s ❞❡st❛ s❡çã♦ ❢♦r♥❡❝❡♠ ❛s ❢❡rr❛♠❡♥t❛s ♣❛r❛ ❛ ❞❡♠♦♥str❛çã♦ ❞❛ ❝♦♥✈❡r❣ê♥❝✐❛ ❞♦ ♠ét♦❞♦✱ ♦✉ s❡❥❛✱ ♣r♦♣r✐❡❞❛❞❡s s♦❜r❡ ♦ ♦♣❡r❛❞♦rTl✳

❆ ♣r✐♠❡✐r❛ ♣r♦♣♦s✐çã♦ ❞á ✐♥❢♦r♠❛çõ❡s ❛ r❡s♣❡✐t♦ ❞♦ ♦♣❡r❛❞♦rTl✱ ❛

♣♦s✐t✐✈✐❞❛❞❡ ❡ ❛ ✐♥❥❡t✐✈✐❞❛❞❡ ❞♦ ♦♣❡r❛❞♦rTl✱ 1♥ã♦ é ❛✉t♦✈❛❧♦r ❞❡Tl❡

♦ ♦♣❡r❛❞♦rTlé ❛✉t♦❛❞❥✉♥t♦✳

P❛r❛ ❛ ❞❡♠♦♥str❛çã♦ ❞❛ ♣r♦♣♦s✐çã♦ ❛ s❡❣✉✐r ❞❡✈❡✲s❡ ❡st❛❜❡❧❡❝❡r ♣r✐✲ ♠❡✐r♦ ❛❧❣✉♠❛s ✐❞❡♥t✐❞❛❞❡s✳

❉❡✜♥❛ ♦ ♦♣❡r❛❞♦rW :H001/2(Γ2)′ −→H1(Ω) ♣♦r

W(ϕ) :=LlD◦γD,Γ2◦L

l N(ϕ);

❡♠ q✉❡Ll

D✱LlN ❡γD,Γ2sã♦ ♦s ♦♣❡r❛❞♦r❡s ❞❡✜♥✐❞♦s ❛♥t❡r✐♦r♠❡♥t❡ ♥❡st❛

s❡çã♦✳

❆ ❢❡rr❛♠❡♥t❛ ♣r✐♥❝✐♣❛❧ ❞❡st❛ ❞❡♠♦♥str❛çã♦ é ❛ ❢ór♠✉❧❛ ❞❡ ●r❡❡♥ ❬❆✳✹❪

Z

∆uvdx=

Z

Γ1⊔Γ2

uνvd∂Ω−

Z

(∇v)t(∇u)dx.

❆ss✐♠✱ ❝♦♠ ❡ss❛ ❢ór♠✉❧❛ ❡ ❝♦♥s✐❞❡r❛♥❞♦ ϕ, ψ ∈H001/2(Γ2)′✱ t❡♠✲s❡

q✉❡

Z

Ω ∇

LlNTl(ϕ)− ∇W(ϕ) t

∇LlN(ψ)dx=

Z

Γ1⊔Γ2

LlNTl(ϕ)−W(ϕ)νLlN(ψ)d∂Ω

Z

∆ LlNTl(ϕ)−W(ϕ)LlN(ψ)dx

❆❣♦r❛✱ s❡ ✉s❛r ❛ ♥♦t❛çã♦ w:= Ll

N(ϕ)✱ w:=LlN(ψ)❡ v :=W(ϕ)✱

❝♦♥❝❧✉✐✲s❡ q✉❡

Z

Ω ∇

LlNTl(ϕ)− ∇W(ϕ) t

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✷✼

♣♦✐s ♣❡❧❛ ❞❡✜♥✐çã♦ ❞❡v✱ ❞❡w❡ ❞❡w✱ tê♠✲s❡ q✉❡w|Γ1 = 0❡ q✉❡w|Γ2 =

v|Γ2 ❥á q✉❡ L

l

NTl(ϕ) = LlN(vν|Γ2) ❡ t❛♥t♦ w q✉❛♥t♦ v s❛t✐s❢❛③❡♠ ❛

❡q✉❛çã♦ ❞❡ ▲❛♣❧❛❝❡ ❡♠Ω✳

▲♦❣♦✱ ✈❛❧❡ ❛ ✐❣✉❛❧❞❛❞❡ Z

Ω ∇

LlNTl(ϕ)

t

∇LlN(ψ)dx=

Z

(∇W(ϕ))t∇LlN(ψ)dx;

❡✱ ♣♦rt❛♥t♦✱

hTl(ϕ), ψi∗=h∇LlNTl(ϕ),∇LlN(ψ)i=h∇W(ϕ),∇LlN(ψ)i;

♦♥❞❡h., .ié ♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ✐♥t❡r♥♦ s♦❜r❡L2(Ω)

h., .i∗:=

Z

Ω ∇

LlN(ϕ)

t

∇LlN(ϕ)dx;

é ♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦ s♦❜r❡H001/2(Γ2)′✳

❆♥❛❧♦❣❛♠❡♥t❡✱ ❝♦♠ ❛ ♠❡s♠❛ ♥♦t❛çã♦✱ t❡♠✲s❡ q✉❡

h∇LlN(ϕ),∇W(ψ)i=h∇W(ϕ),∇W(ψ)i;

✐st♦ é✱

Z

Ω ∇

W(ϕ)− ∇LlN(ϕ)

t

∇W(ψ)dx= 0;

♣♦✐s✱ ♣❡❧❛ ❞❡✜♥✐çã♦ ❞❡v❡ ❞❡w✱ tê♠✲s❡ q✉❡vν|Γ1 =w|Γ1 = 0✱v|Γ2 =

w|Γ2 ❡v ❡ws❛t✐s❢❛③❡♠ ❛ ❡q✉❛çã♦ ❞❡ ▲❛♣❧❛❝❡✳

❆❧é♠ ❞✐ss♦✱ ✈❛❧❡ ❛s s❡❣✉✐♥t❡s ✐❣✉❛❧❞❛❞❡s✿

h∇LlN(ϕ),∇LlNTl(ψ)i=h∇LlN(ϕ),∇W(ψ)i;

h∇W(ϕ),∇W(ψ)i=h∇W(ϕ),∇LlN(ψ)i;

♦❜t✐❞❛s✱ t❛♠❜é♠✱ ❛tr❛✈és ❞❛ ❢ór♠✉❧❛ ❞❡ ●r❡❡♥ ❬❆✳✹❪ ❡ ♣❡❧❛ ❞❡✜♥✐çã♦ ❞❡

v✱ ❞❡w❡ ❞❡w✳

❯♠❛ ♥♦t❛ ✐♠♣♦rt❛♥t❡ ❛ s❡r ❢❡✐t❛ ❛♥t❡s ❞❡ ❡♥✉♥❝✐❛r ❛ ♣r♦♣♦s✐çã♦ é q✉❡ ❛ ♥♦r♠❛ ||.||∗ ♣r♦✈❡♥✐❡♥t❡ ❞♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦ h., .i∗ é ❡q✉✐✈❛❧❡♥t❡

❛ ♥♦r♠❛ ✉s✉❛❧ ❞❡ H001/2(Γ2)′✳ ❆❧é♠ ❞✐ss♦✱ H001/2(Γ2)′ é ✉♠ ❡s♣❛ç♦ ❞❡

❍✐❧❜❡rt ❝♦♠ ❡st❡ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❬✹❪✳ ❆ ❡q✉✐✈❛❧ê♥❝✐❛ ❡♥tr❡ ❡ss❛ ♥♦r♠❛ ❡ ❛ ♥♦r♠❛ ✉s✉❛❧ ❞♦ ❡s♣❛ç♦ ❡♥❝♦♥tr❛✲s❡ ❡♠ ❬✹❪✳

❆ss✐♠✱ ♣❛r❛ ♦s ♣ró①✐♠♦s r❡s✉❧t❛❞♦s✱ ❛ ❡str✉t✉r❛ ❛ s❡r ❝♦♥s✐❞❡r❛❞❛ s♦❜r❡ ♦ ❡s♣❛ç♦ ❞❡ ❙♦❜♦❧❡✈ H001/2(Γ2)′ é ♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦ h., .i∗✱ ❛✜♠

❞❡ t♦r♥á✲❧♦ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt✳ ❊ ❛ ♥♦r♠❛ ♣r♦✈❡♥✐❡♥t❡ ❞❡ss❡ ♣r♦❞✉t♦ ✐♥t❡r♥♦ é ❞❡♥♦t❛❞❛ ♣♦r ||.||∗✳

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✷✽

Pr♦♣♦s✐çã♦ ✷✳✶ ❙❡❥❛Tl∈L(H001/2(Γ2)′)♦ ♦♣❡r❛❞♦r ❞❡✜♥✐❞♦ ❝♦♠♦ ♥❛

❞❡s❝r✐çã♦ ❞♦ ♠ét♦❞♦ ❞❡ ▼❛③✬✐❛✳ ❊♥tã♦✱ tê♠✲s❡ q✉❡Tl é ♣♦s✐t✐✈♦✱ ❛✉t♦✲

❛❞❥✉♥t♦✱ 1♥ã♦ é ❛✉t♦✈❛❧♦r ❞❡Tl ❡ ✐♥❥❡t✐✈♦✳

❉❡♠♦♥str❛çã♦✿ P❛rt❡ ✶ ✲ Tl é ♣♦s✐t✐✈♦✿ ❈♦♠ ❛s ✐❞❡♥t✐❞❛❞❡s ❡st❛✲

❜❡❧❡❝✐❞❛s ❛♥t❡r✐♦r♠❡♥t❡ à ❞❡♠♦♥str❛çã♦✱ ❝♦♥❝❧✉✐✲s❡ ❢❛❝✐❧♠❡♥t❡ q✉❡

hTl(ϕ), ϕi∗ = h∇LNl Tl(ϕ),∇LlN(ϕ)i

= h∇W(ϕ),∇LlN(ϕ)i

= h∇W(ϕ),∇W(ϕ)i ≥ c||W(ϕ)||2H1(Ω);

∀ϕ∈H001/2(Γ2)′ ❡ ❛❧❣✉♠❛ ❝♦♥st❛♥t❡c∈R✳

P♦rt❛♥t♦✱Tlé ♣♦s✐t✐✈♦✳

P❛rt❡ ✷ ✲1♥ã♦ é ❛✉t♦✈❛❧♦r ❞❡Tl✿ P♦r ❝♦♥tr❛❞✐çã♦✱ s✉♣♦♥❤❛ q✉❡

1é ❛✉t♦✈❛❧♦r ❞❡Tl✱ ✐st♦ é✱∃ϕ∈H001/2(Γ2)′ ♥ã♦ ♥✉❧♦ t❛❧ q✉❡Tl(ϕ) =ϕ✳

❈♦♠ ❛ ♠❡s♠❛ ♥♦t❛çã♦ ♣r❡❝❡❞❡♥t❡ à ♣r♦♣♦s✐çã♦✱ ❧❡♠❜r❡ q✉❡ws❛t✐s❢❛③

∆w= 0, Ω; w|Γ1 = 0; wν|Γ2 =ϕ;

❡ q✉❡v s❛t✐s❢❛③

∆v= 0, Ω; vν|Γ1 = 0; v|Γ2 =w|Γ2 .

❆❧é♠ ❞✐ss♦✱ ♦❜s❡r✈❡ q✉❡ ❛ ❞✐❢❡r❡♥ç❛ (v −w) é s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛

❤♦♠♦❣ê♥❡♦

∆(v−w) = 0, Ω; (v−w)|Γ2 = 0; (v−w)ν|Γ2 = 0;

♣♦✐s v|Γ2 = w|Γ2❀ ♦✉ s❡❥❛✱ ♣❡❧❛ ✉♥✐❝✐❞❛❞❡ ❞❛ s♦❧✉çã♦✱ ❞❡✈❡♠♦s t❡r

v =w❀ ♣❡❧♦ ❢❛t♦ ❞❡ q✉❡ ❛ ú♥✐❝❛ s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛ ❤♦♠♦❣ê♥❡♦ é ❛

s♦❧✉çã♦ tr✐✈✐❛❧✳ P❡❧❛ ❞❡✜♥✐çã♦ ❞❡ v ❡ ❞❡ w✱ tê♠✲s❡ w|Γ1 = 0 ❡ 0 =

vν|Γ1 = wν|Γ1❀ ♣♦✐s w = v✳ P♦rt❛♥t♦✱ ❝♦♠♦ w t❛♠❜é♠ s❛t✐s❢❛③ ♦

♣r♦❜❧❡♠❛ ❤♦♠♦❣ê♥❡♦✱ w ❞❡✈❡ s❡r ✐❞❡♥t✐❝❛♠❡♥t❡ ♥✉❧♦✳ ▲♦❣♦✱ ϕ ≡ 0✱

❝♦♥tr❛❞✐çã♦✳

▲♦❣♦✱1 ♥ã♦ é ❛✉t♦✈❛❧♦r ❞❡Tl✳

P❛rt❡ ✸ ✲ Tl é ❛✉t♦❛❞❥✉♥t♦✿ P❛r❛ϕ, ψ ∈H001/2(Γ2)′✱ ♣❡❧❛s ✐❞❡♥✲

t✐❞❛❞❡s ❡st❛❜❡❧❡❝✐❞❛s ♣r❡✈✐❛♠❡♥t❡✱ tê♠✲s❡ q✉❡

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✷✾

❡ q✉❡

h∇W(ϕ),∇W(ψ)i=h∇W(ϕ),∇LlN(ψ)i;

♦✉ s❡❥❛✱ ❡st❛s ✐❣✉❛❧❞❛❞❡s ✐♠♣❧✐❝❛♠ q✉❡hTl(ϕ), ψi∗=hϕ, Tl(ψ)i∗✳

P♦rt❛♥t♦✱Tl é ❛✉t♦❛❞❥✉♥t♦✳

P❛rt❡ ✹ ✲ Tl é ✐♥❥❡t✐✈❛✿ ❙❡❥❛♠ ϕ1, ϕ2 ∈ H001/2(Γ2)′ t❛✐s q✉❡

Tl(ϕ1) = Tl(ϕ2) ❡ ❞❡✜♥❛ w := LlN(ϕ1 −ϕ2) ❡ v := W(ϕ1−ϕ2)✳

◆♦t❡ q✉❡ ❛ ❤✐♣ót❡s❡ Tl(ϕ1 −ϕ2) = 0 ✐♠♣❧✐❝❛ ❡♠ vν|Γ2 = 0✱ ♣♦✐s

Tl(ϕ1−ϕ2) =γN,Γ2(v) =vν|Γ2 ❡ ♥♦t❡ q✉❡vs❛t✐s❢❛③

∆v= 0, Ω; vν|Γ1 = 0; v|Γ2 =w|Γ2 .

❈♦♠♦v é ❝♦♥st❛♥t❡✱ ♣♦✐s s❛t✐s❢❛③

∆v= 0, Ω; vν= 0, ∂Ω;

❞❡✈❡✲s❡ t❡r q✉❡ v ≡ 0✱ ♣♦✐s ❛ ú♥✐❝❛ ❢✉♥çã♦ ❝♦♥st❛♥t❡ ❡♠ H001/2(Γ2)′

é ❛ ❢✉♥çã♦ ♥✉❧❛ ❬✹❪✳ ▲♦❣♦✱ w ≡ 0 ❡♠ Ω ❡ s❡❣✉❡ q✉❡ ϕ1 = ϕ2✱ ♣♦✐s

0 =wν|Γ2 =ϕ1−ϕ2✳

P♦rt❛♥t♦✱Tl é ✐♥❥❡t✐✈❛✳

◆❛ ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r ❢♦✐ ♣r♦✈❛❞♦ q✉❡ 1 ♥ã♦ é ❛✉t♦✈❛❧♦r ❞❡ Tl✳

❆❣♦r❛✱ ❛ ♣r♦♣♦s✐çã♦ ❛ s❡❣✉✐r ❞✐③ q✉❡✱ ❞❡ ❢❛t♦✱ ♦ ♦♣❡r❛❞♦r é ♥ã♦ ❡①♣❛♥✲ s✐✈♦ ❡✱ ❛❧é♠ ❞✐ss♦✱ ❛ ♣r♦✈❛ ❞❡ q✉❡Tlé r❡❣✉❧❛r ❛ss✐♥tót✐❝♦ ❡♠H001/2(Γ2)′

é ❛♣r❡s❡♥t❛❞❛✳

Pr♦♣♦s✐çã♦ ✷✳✷ ❙❡❥❛Tl∈L(H001/2(Γ2)′)♦ ♦♣❡r❛❞♦r ❞❡✜♥✐❞♦ ❝♦♠♦ ♥❛

❞❡s❝r✐çã♦ ❞♦ ♠ét♦❞♦ ❞❡ ▼❛③✬✐❛✳ ❊♥tã♦✱ tê♠✲s❡ q✉❡ Tl é r❡❣✉❧❛r ❛ss✐♥✲

tót✐❝♦ ❡ ♥ã♦ ❡①♣❛♥s✐✈♦ ❡♠H001/2(Γ2)′✳

❉❡♠♦♥str❛çã♦✿ P❛rt❡ ✶ ✲ Tl é r❡❣✉❧❛r ❛ss✐♥tót✐❝♦✿ P❛r❛ ♣r♦✈❛r

q✉❡Tlé r❡❣✉❧❛r ❛ss✐♥tót✐❝♦✱ ❞❡✈❡✲s❡ ♠♦str❛r q✉❡

lim

k→∞

Tlk+1(ϕ)−Tlk(ϕ)

∗= 0;

∀ϕ∈H001/2(Γ2)′✳

❆ss✐♠✱ ❜❛st❛ ✈❡r✐✜❝❛r q✉❡Tk

l (ϕ0)→0, ∀ϕ0∈R(Tl−I)✱ ♣♦✐s ✈❛❧❡

❛ ✐❞❡♥t✐❞❛❞❡

Tlk+1(ϕ)−Tlk(ϕ)

=Tlk(Tl−I) (ϕ).

❉❡ ❢❛t♦✱ s❡❥❛ ψ ∈ H001/2(Γ2)′ t❛❧ q✉❡ (Tl−I) (ψ) = ϕ✳ P❛r❛ ❡ss❛

❞❡♠♦♥str❛çã♦✱ ♦❜s❡r✈❡ q✉❡ ❛ ✐t❡r❛çã♦Tk

l(ψ)♣♦❞❡ s❡r r❡❡s❝r✐t❛ ♣♦r

wk(ψ) =LlN(γN,Γ2(vk−1(ψ))), k≥1;

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✸✵

❝♦♠w0(ψ) =LlN(ψ)✳

❆ss✐♠✱ ♣♦r ❡ss❛ ❞❡s❝r✐çã♦✱ s❡❣✉❡ q✉❡ ❝♦♠♦wk s❛t✐s❢❛③

∆wk = 0, Ω; wk|Γ1 = 0; (wk)ν|Γ2 = (vk−1)ν|Γ2;

❡vk é s♦❧✉çã♦ ❞❡

∆vk= 0, Ω; (vk)ν|Γ1 = 0; vk|Γ2 =wk|Γ2 ;

tê♠✲s❡ q✉❡wk|Γ1 = (vk)ν|Γ1 = 0❡ ❡♠Γ2✱vk =wk ❡(wk)ν= (vk−1)ν✳

❈♦♠ ✐ss♦ ❡ ✉t✐❧✐③❛♥❞♦ ❛ ❢ór♠✉❧❛ ❞❡ ●r❡❡♥ ❬❆✳✹❪✱ ❝♦♥❝❧✉✐✲s❡ ❛s s❡✲ ❣✉✐♥t❡s ✐❣✉❛❧❞❛❞❡s✿

Z

(∇vk)t(∇vk)dx =

Z

Γ2

vk(vk)νd∂Ω;

Z

(∇wk)t(∇wk)dx =

Z

Γ2

wk(wk)νd∂Ω;

Z

(∇vk)t(∇wk)dx =

Z

Γ2

wk(vk)νd∂Ω;

Z

(∇vk−1)t(∇wk)dx =

Z

Γ2

wk(vk−1)νd∂Ω;

❝♦♠ ❛ ♠❡s♠❛ ❛r❣✉♠❡♥t❛çã♦ ❞❛s ✐❞❡♥t✐❞❛❞❡s ♦❜t✐❞❛s ♣❛r❛ ❛ ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r✳

❊ss❛s q✉❛tr♦ ✐❞❡♥t✐❞❛❞❡s✱ ❛♥❛❧♦❣❛♠❡♥t❡✱ ❢♦r♥❡❝❡♠ q✉❡

Z

Ω∇

(wk−vk−1)t∇(wk−vk−1)dx=

Z

(∇vk−1)t(∇vk−1)−(∇wk)t(∇wk)

dx;

Z

Ω∇

(vk−wk)t∇(vk−wk)dx=

Z

(∇wk)t(∇wk)−(∇vk)t(∇vk)

dx.

P❡❧❛ ❞❡✜♥✐çã♦ ❞❡ϕ❡ψ✱ t❡♠✲s❡ q✉❡wk(ϕ) =wk+1(ψ)−wk(ψ)✱ ♣❛r❛

✈❡r ✐ss♦ ❜❛st❛ ♦❜s❡r✈❛r q✉❡

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✸✶

◆♦t❡✱ ♣❡❧♦ q✉❡ ❢♦✐ ❢❡✐t♦ ❛❝✐♠❛✱ q✉❡ Z

(∇wk(ϕ))t∇wk(ϕ)dx

≤2

Z

Ω∇

(wk+1(ψ)−wk(ψ))t∇(wk+1(ψ)−wk(ψ))dx

= 2

Z

(∇wk(ψ))t(∇wk(ψ))−(∇wk+1(ψ))t(∇wk+1(ψ))

dx;

♦✉ s❡❥❛✱ ✈❛❧❡ q✉❡

Tlk(ϕ)

2

∗≤2

Tlk(ψ)

2 ∗−

Tlk+1(ψ) 2 ∗

, ∀k∈N;

♣❡❧❛ ❞❡✜♥✐çã♦ wk(.) :=LlN(.)✳

❆❧é♠ ❞✐ss♦✱ ♦❜s❡r✈❡ q✉❡ Tlk(ψ)

2 ∗−

Tlk+1(ψ)

2

∗≥0✱∀k∈N✱ ✐st♦

é✱ ❛ s❡q✉ê♥❝✐❛ Tlk(ψ)

∗ k∈N ♥ã♦ é ❝r❡s❝❡♥t❡✳

▲♦❣♦✱limk→∞

Tlk(ϕ)

∗= 0.

P♦rt❛♥t♦✱Tl é r❡❣✉❧❛r ❛ss✐♥tót✐❝♦✳

P❛rt❡ ✷ ✲ Tl é ♥ã♦ ❡①♣❛♥s✐✈♦✿ P❛r❛ ❞❡♠♦♥str❛r q✉❡ Tl é ♥ã♦

❡①♣❛♥s✐✈♦✱ ✐st♦ é✱ ||Tl||L(H1/2 00 (Γ′2)) ≤

1✱ ❞❡✜♥❛ ♣❛r❛ ϕ ∈ H001/2(Γ2)′✱

w:=Ll

N(ϕ)❡v:=LlD◦γD,Γ2◦L

l N(ϕ)✳

❆❣♦r❛✱ ❝♦♠♦Tl(ϕ) =γN,Γ2(v)❡ ♣❡❧❛ ❢ór♠✉❧❛ ❞❡ ●r❡❡♥ ❬❆✳✹❪✱ t❡♠✲s❡

q✉❡

hTl(ϕ), Tl(ϕ)i∗=

Z

Ω ∇

LlN(γD,Γ2(v))

t

∇LlN(γD,Γ2(v))dx

=

Z

Γ1⊔Γ2

LlN(γD,Γ2(v))

νL l

N(γD,Γ2(v))d∂Ω

+

Z

∆ Ll

N(γD,Γ2(v))

Ll

N(γD,Γ2(v))dx

=

Z

Γ1⊔Γ2

vνLlN(γD,Γ2(v))d∂Ω

+

Z

∆(v)LlN(γD,Γ2(v))dx

=

Z

(∇v)t∇LlN(γD,Γ2(v))dx

Z

(∇v)t(∇v)dx

1/2

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✸✷

♦✉ s❡❥❛✱ ✈❛❧❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡

hTl(ϕ), Tl(ϕ)i∗≤

Z

(∇v)t(∇v)dx.

❚❛♠❜é♠✱ ♣❡❧❛ ❞❡✜♥✐çã♦ ❞❡ v ❡ w ❡ ♣❡❧❛ ❢ór♠✉❧❛ ❞❡ ●r❡❡♥ ❬❆✳✹❪✱

s❡❣✉❡ q✉❡

Z

(∇v)t(∇v)dx=

Z

Γ2

vνvd∂Ω

=

Z

Γ1⊔Γ2

vνwd∂Ω

=

Z

(∇v)t(∇w)dx

Z

(∇v)t(∇v)dx

1/2Z

(∇w)t(∇w)dx

1/2

;

✐st♦ é✱

Z

(∇v)t(∇v)dx≤

Z

(∇w)t(∇w)dx.

▲♦❣♦✱ ♣❛r❛ ✈❡r q✉❡Tlé ♥ã♦ ❡①♣❛♥s✐✈♦✱ ♦❜s❡r✈❡ q✉❡

||Tl(ϕ)||∗≤

Z

(∇w)t(∇w)dx

1/2

=hϕ, ϕi1∗/2=||ϕ||∗;

♣♦✐sw:=Ll N(ϕ)✳

P♦rt❛♥t♦✱Tlé ♥ã♦ ❡①♣❛♥s✐✈♦✳

❆♣ós ❡ss❛ ❛♥á❧✐s❡ s♦❜r❡ ♦ ♦♣❡r❛❞♦r Tl✱ ♦ tr❛❜❛❧❤♦ ❞❡ ❞❡♠♦♥str❛r

❛ ❛♥á❧✐s❡ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ♣❛r❛ ♦ ♠ét♦❞♦ ❞❡ ▼❛③✬✐❛ é ❢❛❝✐❧✐t❛❞♦❀ ❝♦♠♦ ❛❜❛✐①♦ ♦ t❡♦r❡♠❛ ❞❡st❛ s❡çã♦✳

❚❡♦r❡♠❛ ✷✳✶ ❙❡❥❛♠T ❡Tl♦s ♦♣❡r❛❞♦r❡s ❞❡✜♥✐❞♦s ❝♦♠♦ ♥❛ ❞❡s❝r✐çã♦

❞♦ ♠ét♦❞♦ ❞❡ ▼❛③✬✐❛✳ ❙❡(f, g)∈H1/2

1)×H001/2(Γ1)′ é ✉♠ ♣❛r ❞❡ ❞❛✲

❞♦s ❞❡ ❈❛✉❝❤② ❝♦♥s✐st❡♥t❡✱ ❡♥tã♦ ❛ s❡q✉ê♥❝✐❛{ϕk}k∈N={Tk(ϕ0)}kN

❝♦♥✈❡r❣❡ ♣❛r❛ ♦ tr❛ç♦ ❞❡ ◆❡✉♠❛♥♥ ❞❛ s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤②✱ ✐st♦ é✱ ♣❛r❛uν|Γ2✱ ♣❛r❛ t♦❞♦ϕ0∈H

Figure

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