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♦

  ❋❧♦r✐❛♥ó♣♦❧✐s ✷✵✶✺

  ❆❣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♦❞♦✳ ●r❛t♦ ♣❡❧♦ s✉♣♦rt❡ ✜♥❛♥❝❡✐r♦ ❢♦r♥❡❝✐❞♦ ♣❡❧❛ ❈❆P❊❙ ♣♦r ❢❛❝✐❧✐t❛r

  ❡ss❛ ❥♦r♥❛❞❛✳

  ❘❡s✉♠♦

  ◆❡st❛ ❞✐ss❡rt❛çã♦ ❢♦✐ tr❛❜❛❧❤❛❞♦ ♦ ❝❧áss✐❝♦ ❡①❡♠♣❧♦ ❞❡ ♣r♦❜❧❡♠❛ ♠❛❧ ♣♦st♦✱ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤② ❡❧í♣t✐❝♦ ♣❛r❛ ♦ ♦♣❡r❛❞♦r ❞❡ ▲❛♣❧❛❝❡

  2

  s♦❜r❡ ✉♠ ❝♦♥❥✉♥t♦ Ω ⊂ R s✉✜❝✐❡♥t❡♠❡♥t❡ r❡❣✉❧❛r✱ ♦♥❞❡ ♦s ❞❛❞♦s ❞❡

  1

  ❈❛✉❝❤② sã♦ ❢♦r♥❡❝✐❞♦s ❛♣❡♥❛s s♦❜r❡ ✉♠❛ ♣❛rt❡ ❞❛ ❢r♦♥t❡✐r❛✱ Γ ⊂ ∂Ω✳

  1

  (Ω) ❖ ♦❜❥❡t✐✈♦ é ♦ ❞❡ r❡❝♦♥str✉✐r ♦ tr❛ç♦ ❞❛ H ✲s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ ❞❡ ▲❛♣❧❛❝❡ s♦❜r❡ ∂Ω\Γ

  1 ✳ 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✳

  P❛❧❛✈r❛s✲❝❤❛✈❡✿ Pr♦❜❧❡♠❛s ✐♥✈❡rs♦s✱ ♣r♦❜❧❡♠❛s ♠❛❧ ♣♦st♦s✱ ♣r♦✲ ❜❧❡♠❛ ❞❡ ❈❛✉❝❤② ❡❧í♣t✐❝♦✱ ♠ét♦❞♦s ✐t❡r❛t✐✈♦s ❞❡ r❡❣✉❧❛r✐③❛çã♦✱ ♠ét♦❞♦ ❞❡ ▼❛③✬✐❛✱ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r✳

  ❆❜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

  2

  r❡❣✉❧❛r s❡t Ω ⊂ R ✱ ✇❤❡r❡ t❤❡ ❈❛✉❝❤② ❞❛t❛ ❛r❡ ❣✐✈❡♥ ♦♥❧② ❛t ♣❛rt ♦❢

  1

  1

  (Ω) t❤❡ ❜♦✉♥❞❛r②✱ Γ ⊂ ∂Ω✳ ❚❤❡ ❣♦❛❧ ✐s t♦ r❡❝♦♥str✉❝t t❤❡ tr❛❝❡ ♦❢ H ✲

  1

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

  ❑❡②✇♦r❞s✿ ■♥✈❡rs❡ ♣r♦❜❧❡♠s✱ ✐❧❧✲♣♦s❡❞ ♣r♦❜❧❡♠s✱ ❡❧❧✐♣t✐❝ ❈❛✉❝❤② ♣r♦❜❧❡♠✱ ✐t❡r❛t✐✈❡ r❡❣✉❧❛r✐③❛t✐♦♥ ♠❡t❤♦❞s✱ ▼❛③✬✐❛ ❛❧❣♦r✐t❤♠✱ ▲❛♥❞✇❡✲ ❜❡r ✐t❡r❛t✐♦♥✳

  ❙✉♠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 ✺✶ ❆ ❘❡s✉❧t❛❞♦s ❆✉①✐❧✐❛r❡s ✺✸

  ■♥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

  2

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

  1

  2

  1 2 = ∂Ω

  ♣♦♥❡♥t❡s ❛❜❡rt❛s✱ ❞✐s❥✉♥t❛s ❡ ❝♦♥❡①❛s Γ ❡ Γ t❛✐s q✉❡ Γ ⊔ Γ ✳ ❖s ❞❛❞♦s✱ ❝❤❛♠❛❞♦s ❞❡ ❞❛❞♦s ❞❡ ❈❛✉❝❤②✱ ❛s ❝♦♥❞✐çõ❡s ❞❡ ❝♦♥t♦r♥♦ ❞❡ ❉✐r✐❝❤❧❡t ❡ ❞❡ ◆❡✉♠❛♥♥✱ ♥♦ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤② ❡❧í♣t✐❝♦ sã♦ ♣r❡s❝r✐t♦s 1 = f 1 = g ❛♣❡♥❛s s♦❜r❡ ✉♠❛ ♣❛rt❡ ❞❛ ❢r♦♥t❡✐r❛✱ Γ

  1 ✱ ✐st♦ é✱ u | Γ ❡ u ν | Γ ✱

  ♦♥❞❡ f ❡ g sã♦ ❝♦♥❤❡❝✐❞❛s✳ ❆ss✐♠✱ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤② ❡❧í♣t✐❝♦ ❡♠ q✉❡stã♦ é 1 1 Γ = f ; u ν Γ = g.

  (CP ) ∆u = 0, Ω; u | | ❙❡❣✉♥❞♦ ❍❛❞❛♠❛r❞✱ ✉♠ ♣r♦❜❧❡♠❛ é ❜❡♠ ♣♦st♦ s❡ ♣❛r❛ t♦❞♦s ♦s

  ❞❛❞♦s ❛❞♠✐ssí✈❡✐s✱ ❡①✐st❡ ✉♠❛ s♦❧✉çã♦✱ ❛ q✉❛❧ é ú♥✐❝❛ ❡ ❞❡♣❡♥❞❡ ❝♦♥t✐✲

  ✶✹ ❞✐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❡ Γ

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

  u 2

  

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

2

  s❡ ❝❛❧❝✉❧❛r ϕ = u ν | Γ ✱ ❡♥tã♦ t❡♠✲s❡ ♦ s❡❣✉✐♥t❡ ♣r♦❜❧❡♠❛ ♠✐st♦ 1 2

  Γ = f ; u ν Γ = ϕ;

  ∆u = 0, Ω; u | | 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✱ ♦♥❞❡ ϕ é s♦❧✉çã♦ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ✐♥✈❡rs♦ 2 s❡ ϕ = u ν | Γ ✱ ✐st♦ é✱ ϕ é ♦ 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 ❞❛❞♦s f ❡ g✳ ◆♦t❡ q✉❡ ♣❛r❛ ❍❛❞❛♠❛r❞✱ ❛ ❡q✉❛çã♦ ❞❡ ♦♣❡r❛❞♦r❡s F (ϕ) = g s❡r✐❛ ❜❡♠ ♣♦st❛ s❡ F ❢♦ss❡ ✉♠❛ ❜✐❥❡çã♦ ❝♦♠ ✐♥✈❡rs❛ ❝♦♥tí♥✉❛✳

  ◆❛ ♣rát✐❝❛✱ ♦s ❞❛❞♦s ♥✉♥❝❛ sã♦ ♣r❡❝✐s♦s✱ ♦ q✉❡ s❡ tê♠✱ ♥❛ ✈❡r❞❛❞❡✱

  δ δ

  sã♦ ❞❛❞♦s ♣ró①✐♠♦s✱ ✐st♦ é✱ ✉♠❛ ❛♣r♦①✐♠❛çã♦ g ❡ f t❛✐s q✉❡

  δ δ

  • g f < δ; − f − g

  δ δ

  ♦♥❞❡ f ❡ g ♥ã♦ ♣r❡❝✐s❛♠ ❡st❛r ♥❛ ✐♠❛❣❡♠ ❞❡ F ❡ δ > 0 é ❝❤❛♠❛❞♦ ♦ ♥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♦ ❞❡ ❍❛❞❛♠❛r❞ ♣❛r❛ ✉♠ ♣r♦❜❧❡♠❛ s❡r ❜❡♠ ♣♦st♦ é ❞❡ ❡①tr❡♠❛ ✐♠♣♦rtâ♥❝✐❛ ❞❡✈✐❞♦ à ❞✐✜❝✉❧❞❛❞❡ ♥✉♠ér✐❝❛ ❛ s❡r ❝♦♥t♦r♥❛❞❛ ✭❛ss♦❝✐❛❞❛ ❛♦ ♠❛❧ ❝♦♥✲ ❞✐❝✐♦♥❛♠❡♥t♦ ❞♦ ♣r♦❜❧❡♠❛✮✳ ❊♥tã♦✱ ♣❛r❛ ❝♦♥t♦r♥❛r ❡ss❛s ❞✐✜❝✉❧❞❛❞❡s ♣r♦✈❡♥✐❡♥t❡s ❞❡ ♣r♦❜❧❡♠❛s ♠❛❧ ♣♦st♦s✱ ✉t✐❧✐③❛✲s❡ ♠ét♦❞♦s ❞❡ r❡❣✉❧❛r✐✲ ③❛çã♦ ❝♦♠ ♦ ✐♥t✉✐t♦ ❞❡ ♦❜t❡r ✉♠❛ ❛♣r♦①✐♠❛çã♦ ♣❛r❛ ❛ s♦❧✉çã♦ ❞❡s❡❥❛❞❛✱ ♥ã♦ só ❝♦♥✈❡r❣❡♥t❡✱ ♠❛s✱ t❛♠❜é♠✱ ❡stá✈❡❧✳

  ✶✺ ❞❡ ✉♠ ♣r♦❜❧❡♠❛ ♠❛❧ ♣♦st♦ é ✉♠ ♠ét♦❞♦ ❞❡ r❡❣✉❧❛r✐③❛çã♦✳ ❆ ❞✐✜❝✉❧❞❛❞❡

  ∗

  ❡♠ ❡♥❝♦♥tr❛r ✉♠❛ s♦❧✉çã♦ ϕ ❞❡ F (ϕ) = g ✐♥✈❡rt❡♥❞♦ ♦ ♦♣❡r❛❞♦r F ❡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 q✉❡ ❞❡♣❡♥❞❡ ❝♦♥t✐♥✉❛♠❡♥t❡ ❞♦s ❞❛❞♦s

  k ∗

  ♣❡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✈❡❧✳

  δ k

  ❆♣ós ❣❛r❛♥t✐r ❡st❛❜✐❧✐❞❛❞❡ ϕ → ϕ ✱ s❡ δ → 0✱ ❡ ❛ ❝♦♥✈❡r❣ê♥❝✐❛

  k δ ∗

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

  k

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

  δ ∗

  ϕ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ❞♦ ♠ét♦❞♦✱ ♦✉ s❡❥❛✱ ❛ t❛①❛ ❝♦♠ q✉❡ − ϕ → 0✱

  k

  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✳

  ❖ t❡①t♦ ❡stá ❞✐✈✐❞✐❞♦ ❡♠ três ❝❛♣ít✉❧♦s✳ ❖ ♣r✐♠❡✐r♦ ❞❡s❝r❡✈❡ ♦ ♣r♦✲ ❜❧❡♠❛ ❞❡ ❈❛✉❝❤② ❡❧í♣t✐❝♦ ❡ ❛ss♦❝✐❛ ♦ ♣r♦❜❧❡♠❛ ✐♥✈❡rs♦❀ ❛✐♥❞❛✱ ❡♠ s❡çã♦✱ ❛❜♦r❞❛ q✉❛✐s ♦s ❛s♣❡❝t♦s ❞❡ ❍❛❞❛♠❛r❞ ♣❛r❛ s❡ ❞✐③❡r q✉❡ ♦ ♣r♦❜❧❡♠❛ é ♠❛❧ ♣♦st♦✳ ❖ s❡❣✉♥❞♦ ❝❛♣ít✉❧♦ ❡stá ❞✐✈✐❞✐❞♦ ❡♠ ❞✉❛s s❡çõ❡s✱ ❛ ♣r✐♠❡✐r❛ ❞❡s❝r❡✈❡ ♦ ♠ét♦❞♦ ❞❡ ▼❛③✬✐❛ ♠♦str❛♥❞♦ ❛ ❛♥á❧✐s❡ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ s♦❜ ❞❛❞♦s ❡①❛t♦s❀ ❛ s❡❣✉♥❞❛✱ tr❛③ ♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r ❧✐♥❡❛r ❡ ❢❛③ ❛ ❛♥á❧✐s❡ ❝♦♠♣❧❡t❛ ❞♦ ♠ét♦❞♦✳ ◆♦ t❡r❝❡✐r♦ ❝❛♣ít✉❧♦✱ ❝♦♥❝❧✉✐✲s❡ ❛ ❛♥á❧✐s❡ ❞♦ ♠ét♦❞♦ ❞❡ ▼❛③✬✐❛ s♦❜ ❞❛❞♦s ❝♦♠ r✉í❞♦✱ ✐st♦ é✱ q✉❡ é ✉♠ ♠ét♦❞♦

  ✶✻ ♠❡♥t❡ ♣♦ssí✈❡❧ ❛♣ós ♦❜t❡r ✉♠❛ r❡❧❛çã♦ ❢✉♥❞❛♠❡♥t❛❧ ❡♥tr❡ ♦s ♠ét♦❞♦s✱ ❛ ✐❣✉❛❧❞❛❞❡ ❡♥tr❡ ❛s ✐t❡r❛çõ❡s✳ ◆♦ ❆♣ê♥❞✐❝❡ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞♦ ♦s ❞❡♠❛✐s r❡s✉❧t❛❞♦s q✉❡ ❢✉♥❞❛♠❡♥t❛♠ ❡ss❛ ❞✐ss❡rt❛çã♦✳

  ❈❛♣ít✉❧♦ ✶ ❖ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤② ❡❧í♣t✐❝♦

  2

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

  1

  2

  1 2 = ∅

  1 2 = ∂Ω

  ♥❡♥t❡s ❛❜❡rt❛s ❡ ❝♦♥❡①❛s Γ ✱ Γ t❛✐s q✉❡ Γ ∩ Γ ❡ Γ ∪ Γ ✳ ❖ ♣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 ❡♠ Γ

  1 ⊂ ∂Ω✳

  ❆ss✐♠✱ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤② ❡❧í♣t✐❝♦ ❝♦♥s✐❞❡r❛❞♦ é ♣❛r❛ ♦ ♦♣❡r❛❞♦r ❞❡ ▲❛♣❧❛❝❡✱ ❞❛❞♦ ♣♦r 1 1

  Γ = f ; u ν Γ = g;

  (CP ) ∆u = 0, Ω; u | | ♦♥❞❡ (f, g) é ❝❤❛♠❛❞♦ ❞❡ ♣❛r ❞❡ ❞❛❞♦s ❞❡ ❈❛✉❝❤②✳

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

  1

  (Ω) ❜✉✐çõ❡s u ∈ H q✉❡ r❡s♦❧✈❡♠ ♦ ♣r♦❜❧❡♠❛ ♥❛ ❢♦r♠✉❧❛çã♦ ❢r❛❝❛ ❡ t❛♠❜é♠ s❛t✐s❢❛③❡♠ ♦s ❞❛❞♦s ❞❡ ❈❛✉❝❤② (f, g) s♦❜r❡ Γ

  1 ♥♦ s❡♥t✐❞♦ ❞♦

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

  1/2 1/2 ′

  (Γ

  1 (Γ 1 )

  ♥♦ ❡s♣❛ç♦ H ) × H ❬✹❪✳

  00

  ❖ ♣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✱ ❛✈❛❧✐❛r u ν s♦❜r❡ Γ

  2 ✳

  ◆♦t❡ q✉❡ r❡s♦❧✈❡r ❡st❡ ♣r♦❜❧❡♠❛ ✭❛✈❛❧✐❛r ♦ tr❛ç♦ ❞❡ ✉♠❛ s♦❧✉çã♦ s♦❜r❡ Γ

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

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

  2 ✱ ❡♥tã♦

  ♣❛r❛ ❡♥❝♦♥tr❛r ❡st❛ s♦❧✉çã♦ ❞❡ (CP )✱ ❜❛st❛ r❡s♦❧✈❡r ♦ s❡❣✉✐♥t❡ ♣r♦❜❧❡♠❛

  ✶✽

  

1

2 Γ = f ; u ν Γ = ϕ.

  ∆u = 0, Ω; u | | ❆q✉✐✱ ϕ é ♦ 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 ❡

  ∞

  2

  g (Ω)

  ❞❡ ❝❧❛ss❡ C ✳ ❙❡ f ≡ 0 ❡ ❡①✐st❡ s♦❧✉çã♦ ♣❛r❛ (CP ) ❡♠ C ✱ ❡♥tã♦ g é ✉♠❛ ❢✉♥çã♦ ❛♥❛❧ít✐❝❛✳

  2

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

  ∞

  (Γ

  1 )

  ❝❤② (f, g)✱ ♦♥❞❡ f ≡ 0 ❡ g é ✉♠❛ ❢✉♥çã♦ ❞❡ ❝❧❛ss❡ C ✳ ❊♥tã♦✱ ❝♦♠♦ ∆u = 0

  ✱ u t❡♠ ❛ s❡❣✉✐♥t❡ r❡♣r❡s❡♥t❛çã♦ ❞❡ ●r❡❡♥ ❬♣✳ ✶✽✱ ✸❪✿ Z u(y) = (u(x)Γ

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

  1 ♦♥❞❡ Γ(x − y) = ln |x − y|✱ ♦♥❞❡ x ∈ ∂Ω ❡ y ∈ Ω✳

  2π ❈♦♠♦ ♦ ✐♥t❡❣r❛♥❞♦ é ❛♥❛❧ít✐❝♦ ❝♦♠ r❡s♣❡✐t♦ ❛ y✱ ♣♦✐s ❝♦♠♦ y ∈ Ω

  ❡ x ∈ ∂Ω ❡ Ω é ❛❜❡rt♦✱ t❡♠✲s❡ q✉❡ x 6= y✱ ♦✉ s❡❥❛✱ ❛ s✐♥❣✉❧❛r✐❞❛❞❡ ❞❡ Γ(x − y) ♥ã♦ ♦❝♦rr❡ ❡✱ ❛ss✐♠✱ ❝♦♥❝❧✉✐✲s❡ q✉❡ u é ❛♥❛❧ít✐❝❛ ❡♠ Ω✳

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

  1 n n ≥ 0;

  U (x , . . . , x ) =

  1 n

  u (x ) , x < 0;

  1 , . . . , −x n n 1 (U )

  s❛t✐s❢❛③ ∆U = 0 ❡✱ ❛❧é♠ ❞✐ss♦✱ é ❛♥❛❧ít✐❝❛✳ ❊♥tã♦✱ γ N,Γ é ❛♥❛❧ít✐❝❛ ❬♣✳✹✾✹✱ ✻❪✱ ♦✉ s❡❥❛✱ g é ❛♥❛❧ít✐❝❛✳

  ✶✾ s♦❧✉çã♦ ♣❛r❛ (CP )✳

  ❉❛s ❝♦♥❞✐çõ❡s ❞❡ ❍❛❞❛♠❛r❞✱ ❛ ✉♥✐❝✐❞❛❞❡ ❞❛ s♦❧✉çã♦ ❞❡ (CP ) é ❛ ú♥✐❝❛ ♣♦ssí✈❡❧ ❞❡ s❡r ❞❡♠♦♥str❛❞❛✱ ❝♦♠♦ s❡❣✉❡ ❛❜❛✐①♦✳

  ❝♦♠ ❢r♦♥t❡✐r❛ ❛♥❛❧ít✐❝❛ ❡ ❝♦♥s✐❞❡r❡ Γ ⊂ ∂Ω ❛❜❡rt♦ ❡ s✐♠♣❧❡s♠❡♥t❡ ❝♦✲ ♥❡①♦✳ ❊♥tã♦✱ ♦ ♣r♦❜❧❡♠❛ (CP ) ❝♦♠ ♣❛r ❞❡ ❞❛❞♦s ❞❡ ❈❛✉❝❤② (f, g) ∈

  1/2 ′ 1/2

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

  00

  1

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

  1/2

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

  1/2 ′

  H (Γ) ❝♦♥s✐st❡♥t❡✳ ❊♥tã♦✱ u := w − v é s♦❧✉çã♦ ❞❡

  00 = 0; u = 0.

  ∆u = 0, Ω; u | Γ ν | Γ

  ∞

  (Ω) ❈♦♠♦ ∆u = 0✱ t❡♠✲s❡ q✉❡ u ∈ C ✱ ♦✉ s❡❥❛✱ u é ✉♠❛ 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

  λ 0≤λ≤1 = Γ 1 λ

  {Γ } t❛❧ q✉❡ Γ ❡ Γ = ∂Ω \ Γ ❡ Γ t❡♠ ♦s ♠❡s♠♦s ♣♦♥t♦s

  λ 0≤λ≤1

  ✜♥❛✐s✳ ▲♦❣♦✱ s❡ ✉t✐❧✐③❛r ❡ss❛ ❢❛♠í❧✐❛ {Γ } ♥❛ ❞❡♠♦♥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✿

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

  

k x ∈ (0, 1);

  ∂u

  1 k (x, 0) = sin (πkx) , x ∈ (0, 1);

  ∂y πk ♣❛r❛ k ∈ N✳

  Pr✐♠❡✐r❛♠❡♥t❡✱ ♥♦t❡ q✉❡

  1

  1

  1

  1 sin (πkx) . 0 ≤ ≤ ||sin (πkx)|| ≤

  πk πk π k

  1 sin(πkx) ❈♦♠♦ ♣❛r❛ k → ∞✱ k 1/k k→ 0✱ ❡♥tã♦ ❝♦♥✈❡r❣❡

  πk ✉♥✐❢♦r♠❡♠❡♥t❡ ♣❛r❛ ③❡r♦✱ ♣❛r❛ k s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡✳ ❆ss✐♠✱ ♣❛r❛

  ✷✵ ❛♣❡♥❛s ❛ s♦❧✉çã♦ tr✐✈✐❛❧✳

  1

  k (x, y) = sinh (πky) sin (πkx)

  ❆❣♦r❛✱ ♥♦t❡ q✉❡ ❛s s♦❧✉çõ❡s u

  2

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

  1 sin (πkx) sinh (πky) sin (πkx) ; ≤

  2

  2

  (πk) 2 (πk) ♦✉ s❡❥❛✱ u k t♦r♥❛✲s❡ ✐❧✐♠✐t❛❞♦ ❡ ♦s❝✐❧❛ ❝❛❞❛ ✈❡③ ♠❛✐s à ♠❡❞✐❞❛ q✉❡

  9 k → ∞ ❡✱ ♣♦rt❛♥t♦✱ u k ❡♠ ♥❡♥❤✉♠❛ ♥♦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❛r f✱ ❡♥tã♦ ♦ ❝♦♥❥✉♥t♦ ❞❡ ❞❛❞♦s g ♣❛r❛ ♦ q✉❛❧ ❡①✐st❡

  1/2 ′

  1 ❀ ♦ ♠❡s♠♦ ✈❛❧❡

  00

  (Γ ) s♦❧✉çã♦ ❞❡ (CP ) é ❞❡♥s♦ ♥♦ ❡s♣❛ç♦ ❞❡ ❙♦❜♦❧❡✈ H

  s❡ ✜①❛r g✱ ♦✉ s❡❥❛✱ ♦ ❝♦♥❥✉♥t♦ ❞❡ ❞❛❞♦s f ♣❛r❛ ♦ q✉❛❧ ❡①✐st❡ s♦❧✉çã♦ ❞❡

  1/2

  (CP ) (Γ ) é ❞❡♥s♦ ♥♦ ❡s♣❛ç♦ ❞❡ ❙♦❜♦❧❡✈ H

  1 ✳ ❊st❡ r❡s✉❧t❛❞♦ é ♦❜t✐❞♦

  ❞❡ ❬✶❪✳

  1/2

  (Γ

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

  H (Γ )

  1 ✱ tê♠✲s❡ q✉❡

  00 1/2 ′ 1/2

  (Γ

  1 ) (Γ 1 )

  ✶✳ ♣❛r❛ f ∈ H ✜①♦✱ ♦ ❝♦♥❥✉♥t♦ ❞❡ ❞❛❞♦s g ∈ H

  00 ♣❛r❛

  1

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

  1/2 ′

  (Γ ) é ❞❡♥s♦ ❡♠ H

  1 ❀

  00 1/2 ′ 1/2

  (Γ ) (Γ ) ✷✳ ♣❛r❛ g ∈ H

  1 ✜①♦✱ ♦ ❝♦♥❥✉♥t♦ ❞❡ ❞❛❞♦s f ∈ H 1 ♣❛r❛

  00

  1

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

  1/2

  (Γ

  1 )

  é ❞❡♥s♦ ❡♠ H ✳ ❉❡♠♦♥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✳

  1

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

  1/2

  1

  (Γ ) (Ω) ❝♦♥❥✉♥t♦ ❞❡ ❞❛❞♦s ❞❡ g ∈ H

  1 t❛❧ q✉❡ ❡①✐st❡ u ∈ H s❛t✐s❢❛✲

  00 1/2 ′

  (Γ ) ③❡♥❞♦ (CP ) ♥ã♦ é ❞❡♥s♦ ❡♠ H

  1 ✳

  ✷✶

  1/2 ′

  ❙❡❥❛ G ⊆ H ♦ ❝♦♥❥✉♥t♦ ❞❡ ❞❛❞♦s ❞❡ g t❛❧ q✉❡ ♦ ♣r♦❜❧❡♠❛

  00

  ❞❡ ✈❛❧♦r ✐♥✐❝✐❛❧ 1 = 0; u 1 = g; ∆u = 0, Ω; u | Γ ν | Γ t❡♥❤❛ s♦❧✉çã♦✳

  1/2 ′

  (Γ

  1 )

  ◆♦t❡ q✉❡ G 6= H

  00 ✳ ❊♥tã♦✱ ❡①✐st❡ ✉♠❛ ❢♦r♠❛ ❧✐♥❡❛r ❝♦♥tí♥✉❛ 1/2 ′ 1/2

  (Γ

  1 ) (Γ 1 )

  ♥ã♦ ♥✉❧❛ l ∈ H ❡ l ∈ G ⊆ H t❛❧ q✉❡ hl, gi = 0, ∀g ∈ G✳

  00

  ❈♦♥s✐❞❡r❡ ♦ s❡❣✉✐♥t❡ ♣r♦❜❧❡♠❛ ❞❡ ✈❛❧♦r ❞❡ ❝♦♥t♦r♥♦ ♠✐st♦ 1 = l; v 2 = 0; ∆v = 0, Ω; v | Γ ν | Γ

  ♥♦t❡ q✉❡ ❡st❡ ♣r♦❜❧❡♠❛ é ❜❡♠ ♣♦st♦ ❬❆✳✶❪✳ ❆❣♦r❛✱ ♣❡❧❛ s❡❣✉♥❞❛ ❢ór✲ R (vu ) d∂Ω

  ♠✉❧❛ ❞❡ ●r❡❡♥ R (v∆u − u∆v) dx = ν − uv ν ❬♣✳ ✶✼✱ 1 2

  Ω Γ ⊔ Γ

  (vu )dx = 0; ✸❪✱ t❡♠✲s❡ q✉❡ R ν − uv ν ❛✐♥❞❛ ♠❛✐s✱ ♣❡❧❛s ❝♦♥❞✐çõ❡s 1 ⊔ 2

  Γ Γ

  ❞❡ ❢r♦♥t❡✐r❛✱ Z vu dx = 0. 2

ν

Γ

  ∞

  (Γ ) ❈♦♥s✐❞❡r❡ t❛♠❜é♠ h ∈ C

  2 ❡ ♦ s❡❣✉✐♥t❡ ♣r♦❜❧❡♠❛ ❞❡ ✈❛❧♦r ❞❡

  ❝♦♥t♦r♥♦ ♠✐st♦ ✭❜❡♠ ♣♦st♦ ❬❆✳✶❪✮ 1 2

  Γ = 0; w ν Γ = h;

  ∆w = 0, Ω; w | |

  1

  (Ω) ❝✉❥❛ s♦❧✉çã♦ é ú♥✐❝❛ ❡♠ H ✳ ❆ss✐♠✱ ❛♥❛❧♦❣❛♠❡♥t❡✱ ♣❡❧❛ ❢ór♠✉❧❛ ❞❡ ●r❡❡♥ ❛❝✐♠❛ ♣❛r❛ v ❡ w ❡ ✉t✐❧✐③❛♥❞♦ ❛s ❝♦♥❞✐çõ❡s ❞❡ ❢r♦♥t❡✐r❛✱ t❡♠✲s❡

  ∞

  (Γ ) q✉❡ R vhdx = 0, ∀h ∈ C 2 2 ✳ Γ ▲♦❣♦✱ v s❛t✐s❢❛③ ♦ ♣r♦❜❧❡♠❛ ❤♦♠♦❣ê♥❡♦ 2 = 0; v 2 = 0;

  ∆v = 0, Ω; v | Γ ν | Γ ♦✉ s❡❥❛✱ v ≡ 0 ♣❡❧♦ t❡♦r❡♠❛ ❞❡ ❍♦❧♠❣r❡♥ ❬♣✳✶✻✻✱ ✼❪ ❡✱ ❛ss✐♠✱ l ≡ 0✱ ♦ q✉❡ é ✉♠❛ ❝♦♥tr❛❞✐çã♦✳

  1/2 ′

  (Γ ) P♦rt❛♥t♦✱ ♦ ❝♦♥❥✉♥t♦ ❞❡ ❞❛❞♦s g ∈ H

  2 ♣❛r❛ ♦ q✉❛❧ ❡①✐st❡

  00 1/2 ′

  1

  (Ω) (Γ ) u ∈ H s❛t✐s❢❛③❡♥❞♦ (CP ) é ❞❡♥s♦ ❡♠ H

  1 ✳

  00

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

  2 ✭♦♥❞❡ ♥❡♥❤✉♠ ❞❛❞♦

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

  1/2 ′ 1/2

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

  1 ) × H 1 ❡

  00 1/2 1/2 ′ ′

  F : H (Γ ) (Γ )

  2 −→ H 1 é ✉♠ ♦♣❡r❛❞♦r ❡♥tr❡ ❡s♣❛ç♦s ❞❡ ❍✐❧❜❡rt

  00

  00

  ✭♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦ q✉❡ ♦s t♦r♥❛ ❡s♣❛ç♦s ❞❡ ❍✐❧❜❡rt s❡rá ❛♣r❡s❡♥t❛❞♦ ♥♦ ♣ró①✐♠♦ ❝❛♣ít✉❧♦✮✱ ❝♦♥s✐❞❡r❡ ♦s s❡❣✉✐♥t❡s ♦♣❡r❛❞♦r❡s✿

  ✷✷

  1/2 ′

  1

  ❼ U −→ H ✱ ♦♥❞❡ U é s♦❧✉çã♦ ❞❡

  00 1 2 Γ = f ; u ν Γ = ϕ;

  ∆u = 0, Ω; u | | ♥♦t❡ q✉❡ ❡st❡ ♦♣❡r❛❞♦r ❡stá ❜❡♠ ❞❡✜♥✐❞♦✱ ♣♦✐s ♦ ♣r♦❜❧❡♠❛ ❞❡ ✈❛❧♦r ❞❡ ❝♦♥t♦r♥♦ ♠✐st♦ ❛ s❡r r❡s♦❧✈✐❞♦ é ❜❡♠ ♣♦st♦ ❬❆✳✶❪✳

  1/2

1 : H (Γ )

  1 1 (u) := u 1

  ❼ γ N,Γ (Ω) −→ H

  1 ✱ ♦♥❞❡ γ N,Γ ν | Γ ✱ ❡st❡ é ♦

  00

  ♦♣❡r❛❞♦r tr❛ç♦ ❞❡ ◆❡✉♠❛♥♥ s♦❜r❡ Γ

  1 ✳ 1 1/2 ′ 1/2 ′ N,Γ f : H (Γ 2 ) (Γ 1 )

  ❆ss✐♠✱ ❞❡✜♥❛ F := γ ◦ U

  00 −→ H 00 ♣♦r

  F (ϕ) := γ 1 (U (ϕ)) ;

  N,Γ f

  ♥♦t❡ q✉❡ γ 1 (U (ϕ)) = γ 1 (u) = u 1 .

  

N,Γ f N,Γ ν | Γ

1/2

1/2 ′

  (Γ

  

1 (Γ

1 )

  ▲♦❣♦✱ ❞❛❞♦ (f, g) ∈ H ) × H

  00 ✉♠ ♣❛r ❞❡ ❞❛❞♦s ❞❡ ❈❛✉✲

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

  2 ✮ ♣♦❞❡ s❡r ✈✐st♦ ♣❡❧♦ s❡❣✉✐♥t❡ ♣r♦❜❧❡♠❛ ✐♥✈❡rs♦

  ♠❛❧ ♣♦st♦✱ F (ϕ) = g✳ (ϕ) + F

  ❖❜s❡r✈❡ q✉❡ F é ✉♠ ♦♣❡r❛❞♦r ❛✜♠✱ ✐st♦ é✱ F (ϕ) := F l f ❀ (ϕ) := v 1 := w 1

  ♦♥❞❡ F l ν | Γ é ❛ ♣❛rt❡ ❧✐♥❡❛r ❡ F f ν | Γ é ❛ ♣❛rt❡ ❛✜♠✱ ❝♦♠ v s❡♥❞♦ ❛ s♦❧✉çã♦ ❞❡ 1 = 0; v 2 = ϕ;

  ∆v = 0, Ω; v | Γ ν | Γ ❡ w✱ ❛ s♦❧✉çã♦ ❞❡ 1 = f ; w 2 = 0.

  ∆w = 0, Ω; w | Γ ν | Γ ❖ ♣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♦ ❞❡

  1

  (Ω) ◆❡✉♠❛♥♥ ❞❡ u ∈ H s♦❜r❡ Γ

  2 ✱ ♦♥❞❡ u é s♦❧✉çã♦ ❞❡

1

1 = f ; u = g.

  ∆u = 0, Ω; u | Γ ν | Γ ❖ ♠ét♦❞♦ ❞❡ ▼❛③✬✐❛ ❡ ♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r✳

  ❈❛♣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♦♣♦❧♦❣✐❛ ❞✐✲

  1/2 ′

  (Γ

  2 )

  ❢❡r❡♥t❡ ♣❛r❛ ♦ ❡s♣❛ç♦ ❞❡ ❙♦❜♦❧❡✈ H ✱ t♦r♥❛♥❞♦✲♦s ❡s♣❛ç♦s ❞❡

  00

  ❍✐❧❜❡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✳

  1/2 ′ 1/2

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

  1 ) × H

  1

  00

  ❝♦♥s✐st❡♥t❡✱ ♦ ♦❜❥❡t✐✈♦ ❞♦s ♠ét♦❞♦s ❛ s❡❣✉✐r é ♦ ❞❡ ❝❛❧❝✉❧❛r ♦ tr❛ç♦ ❞❡

  ✷✹

  1

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

  ❡♥❝♦♥tr❛r u ν | Γ ✳ ❊♠ 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♦ ✐♥✐❝✐❛❧

  1/2 ′

  1

  ϕ (Γ ) 2 (Ω) ∈ H

  2 ♣❛r❛ u ν | Γ ✱ ♦♥❞❡ u ∈ H é s♦❧✉çã♦ ❞❡ (CP )✱ ❣❡r❛r

  00

  ❛ s❡❣✉✐♥t❡ ✐t❡r❛çã♦ {ϕ k } k∈N ✿

  1

  (Ω) resolve : 1 = f ; w 2 = ϕ ; w ∈ H ∆w = 0, Ω; w | Γ ν | Γ k 2 ψ ;

  k := w | Γ

  1 1 2  v ∈ H ν | Γ = g; v | Γ k (Ω) resolve : ∆v = 0, Ω; v = ψ ; 2

  ϕ k+1 := v ν Γ ; |

  1/2 ′ 1/2

  (Γ

  1 (Γ 1 )

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

  00 ❝♦♥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❡ Γ

  2 ❀ ❞❡✜♥✐❞❛s ❛♦ ❧♦♥❣♦

  1

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

  1/2 ′ 1/2

  H (Γ ) (Γ )

  2 ❬❆✳✸❪ ❡ {ψ k } k∈N ⊆ H 2 ❬✹❪✳

  00

  ❈♦♠ ♦ ♦❜❥❡t✐✈♦ ❞❡ ❢❛③❡r ❛ ❛♥á❧✐s❡ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ❞❡st❡ ♠ét♦❞♦✱ ♦❜s❡r✈❡ q✉❡ ❛ ✐t❡r❛çã♦ ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞❛ ♣♦r ♣♦tê♥❝✐❛s ❞❡ ✉♠ ♦♣❡✲

  1/2 1/2 ′ ′

  2 −→ H 2 ✱ ❝♦♥str✉✐♥❞♦ ❡ss❡ ♦♣❡r❛❞♦r T ❞❛

  00

  00

  (Γ ) (Γ ) r❛❞♦r T : H

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

  1/2 ′

  1

  : H (Γ ) (Ω), L (ϕ) := w, ❼ L N

  2 −→ H N ♦♥❞❡ w é s♦❧✉çã♦ ❞❡

  00 1 = f ; w 2 = ϕ;

  ∆w = 0, Ω; w | Γ ν | Γ

  1/2

  1

  : H (Γ (Ω), L (ψ) := v, ❼ L D

  2 ) −→ H D ♦♥❞❡ v é s♦❧✉çã♦ ❞❡ 1 2

  ∆v = 0, Ω; v = ψ;

  ν | Γ = g; v | Γ

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

  ❆ss✐♠✱ ❛ ✐t❡r❛çã♦ ❞♦ ♠ét♦❞♦ ❞❡ ▼❛③✬✐❛ ♣♦❞❡ s❡r r❡❡s❝r✐t❛ ♣♦r w = L (ϕ ); ψ = γ 2 (w);

  

N k k D,Γ

2

  v = L (ψ ); ϕ = γ (v);

  D k k+1 N,Γ

  ♦♥❞❡

  

1/2 ′

2

  1 2 2

  γ N,Γ : H (Γ

  2 ) , γ N,Γ (u) := u ν Γ ;

  (Ω) −→ H |

  00

  é ♦ tr❛ç♦ ❞❡ ◆❡✉♠❛♥♥ s♦❜r❡ Γ

  2 ❡ 1 1/2

  γ 2 : H (Γ ), γ 2 2 ;

  D,Γ (Ω) −→ H

  2 D,Γ (u) := u | Γ

  é ♦ tr❛ç♦ ❞❡ ❉✐r✐❝❤❧❡t s♦❜r❡ Γ

  2 ✳

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

  1/2 1/2 ′ ′

  T : H (Γ ) (Γ )

  2 −→ H 2 ✱ ❜❛st❛ ❞❡✜♥í✲❧♦ ♣♦r

  00

  00 2 2 T := γ N,Γ D D,Γ N .

  ◦ L ◦ γ ◦ L ◆♦t❡ q✉❡ T s❛t✐s❢❛③✱ ♣♦r ❞❡✜♥✐çã♦✱

  k+1

  ϕ k+1 = T (ϕ k ) = T (ϕ ) , k ∈ N;

  ♦✉ s❡❥❛✱ ♦ ♠ét♦❞♦ ❞❡ ▼❛③✬✐❛ é ✉♠❛ ✐t❡r❛çã♦ ❞❡ ♣♦♥t♦ ✜①♦ q✉❡ ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞❛ ♣♦r ♣♦tê♥❝✐❛s ❞❡ ✉♠ ♦♣❡r❛❞♦r ❛✜♠ ✭♣♦✐s L D ❡ L N ❝❧❛r❛♠❡♥t❡ sã♦ ♦♣❡r❛❞♦r❡s ❛✜♥s✮✳

  = T (ϕ ) := T (ϕ ) + T ❈♦♠♦ T é ✉♠ ♦♣❡r❛❞♦r ❛✜♠✱ ❡♥tã♦ ϕ k+1 k l k ✱ 2 l l 2

  (ϕ ) = γ (ϕ ) k ∈ N✱ ♦♥❞❡ T l k N,Γ ◦ L ◦ γ D,Γ ◦ L k é ❛ ♣❛rt❡ ❧✐♥❡❛r✱

  D N l l D N

  ❝♦♠ L ❡ L s❡♥❞♦ ❛s ♣❛rt❡s ❧✐♥❡❛r❡s ❞❡ L ❡ L ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡❀ ❡

  D N 2 l 2 2 T = γ N,Γ D,Γ (w f ) + γ N,Γ (v g ) f

  ◦ L ◦ γ s❡♥❞♦ ❛ ♣❛rt❡ ❛✜♠✱ ❝♦♠ w

  D

  1 g (Ω) N D

  ❡ v ❡♠ H s❡♥❞♦ ❛s ♣❛rt❡s ❛✜♥s ❞❡ L ❡ L ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❡

  f g

  ❝♦♠ w ❞❡♣❡♥❞❡♥❞♦ ❛♣❡♥❛s ❞❡ f ❡ v ❞❡♣❡♥❞❡♥❞♦ ❛♣❡♥❛s ❞❡ g✳ ❆ss✐♠✱ r❡❝✉rs✐✈❛♠❡♥t❡✱ t❡♠✲s❡ q✉❡ X k k+1 j

  ϕ = T (ϕ ) + T (T

k+1 ) , k ∈ N.

  l l j=0 2 (u)

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

  1/2 1/2 ′

  (Γ

  1 (Γ 1 )

  ❝♦♠ ❞❛❞♦s ❞❡ ❈❛✉❝❤② (f, g) ∈ H ) × H

  00 ❝♦♥s✐st❡♥t❡✱ ❡♥tã♦

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

  ❛s ❢✉♥çõ❡s w ❡ v tê♠ ♦s ♠❡s♠♦s tr❛ç♦s s♦❜r❡ Γ

  2 ✳ ▲♦❣♦✱ ♣❡❧❛ ✉♥✐❝✐❞❛❞❡

  ❞❛ s♦❧✉çã♦✱ w = v é s♦❧✉çã♦ ❞❡ 1 = f ; u 1 = g.

  ∆u = 0, Ω; u | Γ ν | Γ ◆❛ s✉❜s❡çã♦ ❡stá ❛♣r❡s❡♥t❛❞♦ ❛s ♣r♦♣r✐❡❞❛❞❡s s♦❜r❡ ♦ ♦♣❡r❛❞♦r T l

  ❡ ❛ ❛♥á❧✐s❡ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ❞♦ ♠ét♦❞♦ ❞❡ ▼❛③✬✐❛ s♦❜ ❞❛❞♦s ❡①❛t♦s✳

  ✷✻

  ✷✳✶✳✶ ❆♥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♥❝✐❛

  k

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

  ϕ ❝♦♥s✐st❡♥t❡ ❡ ϕ := lim k→∞ k 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❛❞♦r T l ✳ ❆ ♣r✐♠❡✐r❛ ♣r♦♣♦s✐çã♦ ❞á ✐♥❢♦r♠❛çõ❡s ❛ r❡s♣❡✐t♦ ❞♦ ♦♣❡r❛❞♦r T l ✱ ❛

  ♣♦s✐t✐✈✐❞❛❞❡ ❡ ❛ ✐♥❥❡t✐✈✐❞❛❞❡ ❞♦ ♦♣❡r❛❞♦r T l ✱ 1 ♥ã♦ é ❛✉t♦✈❛❧♦r ❞❡ T l ❡ ♦ ♦♣❡r❛❞♦r T l é ❛✉t♦❛❞❥✉♥t♦✳

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

  1/2 ′

  1

  (Γ ) (Ω) ❉❡✜♥❛ ♦ ♦♣❡r❛❞♦r W : H

  2 −→ H ♣♦r

  00 l l

  W (ϕ) := L 2 (ϕ); ◦ γ D,Γ ◦ L

  D N l l 2

  ❡♠ q✉❡ L ✱ L ❡ γ D,Γ sã♦ ♦s ♦♣❡r❛❞♦r❡s ❞❡✜♥✐❞♦s ❛♥t❡r✐♦r♠❡♥t❡ ♥❡st❛

  D N

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

  ❬❆✳✹❪ Z Z Z

  t

  ∆uvdx = u 1 ⊔ 2 ν vd∂Ω − (∇v) (∇u) dx.

  Ω Γ Γ Ω 1/2 ′

  (Γ ) ❆ss✐♠✱ ❝♦♠ ❡ss❛ ❢ór♠✉❧❛ ❡ ❝♦♥s✐❞❡r❛♥❞♦ ϕ, ψ ∈ H

  2 ✱ t❡♠✲s❡

  00

  q✉❡ Z

  t

l l

  T l (ψ)dx = ∇L N (ϕ) − ∇W (ϕ) ∇L N Z

l l

  L T L (ψ)d∂Ω

  

l (ϕ) − W (ϕ)

N N

1 ⊔ 2 ν Γ Γ Z l l

  ∆ L T L (ψ)dx − l (ϕ) − W (ϕ)

  N N Ω l l

  (ϕ) (ψ) ❆❣♦r❛✱ s❡ ✉s❛r ❛ ♥♦t❛çã♦ w := L ✱ w := L ❡ v := W (ϕ)✱

  N N

  ❝♦♥❝❧✉✐✲s❡ q✉❡ Z

  t l l

  T (ψ)dx = 0; ∇L l (ϕ) − ∇W (ϕ) ∇L

  N N

  1 = 0 2 = ✷✼ 2 T (ϕ) = L (v l l

2 )

v | Γ ❥á q✉❡ L l ν | Γ ❡ t❛♥t♦ w q✉❛♥t♦ v s❛t✐s❢❛③❡♠ ❛

  N N

  ❡q✉❛çã♦ ❞❡ ▲❛♣❧❛❝❡ ❡♠ Ω✳ ▲♦❣♦✱ ✈❛❧❡ ❛ ✐❣✉❛❧❞❛❞❡ Z Z

  t t

l l l

  T l (ϕ) (ψ)dx = (ψ)dx; ∇L N ∇L N (∇W (ϕ)) ∇L N

  Ω Ω

  ❡✱ ♣♦rt❛♥t♦✱

  l l l ∗ T

  hT l (ϕ), ψi = h∇L l (ϕ), ∇L (ψ)i = h∇W (ϕ), ∇L (ψ)i;

  N N N

  2

  (Ω) ♦♥❞❡ h., .i é ♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ✐♥t❡r♥♦ s♦❜r❡ L ❡ Z

  t l l ∗ := (ϕ) (ϕ)dx;

  h., .i ∇L ∇L

  N N Ω 1/2

  (Γ

  

2 )

  é ♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦ s♦❜r❡ H

  00 ✳

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

  l

  h∇L N (ϕ), ∇W (ψ)i = h∇W (ϕ), ∇W (ψ)i; ✐st♦ é✱ Z

  t l

  (ϕ) ∇W (ϕ) − ∇L ∇W (ψ)dx = 0;

  N Ω 1 1 = 0 2 =

  ♣♦✐s✱ ♣❡❧❛ ❞❡✜♥✐çã♦ ❞❡ v ❡ ❞❡ w✱ tê♠✲s❡ q✉❡ v ν | Γ = w | Γ ✱ v | Γ 2 w | Γ ❡ v ❡ w s❛t✐s❢❛③❡♠ ❛ ❡q✉❛çã♦ ❞❡ ▲❛♣❧❛❝❡✳ ❆❧é♠ ❞✐ss♦✱ ✈❛❧❡ ❛s s❡❣✉✐♥t❡s ✐❣✉❛❧❞❛❞❡s✿

  l l l

  T h∇L N (ϕ), ∇L N l (ψ)i = h∇L N (ϕ), ∇W (ψ)i; ❡

  l

  h∇W (ϕ), ∇W (ψ)i = h∇W (ϕ), ∇L N (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❡

  

1/2 1/2

′ ′

  (Γ ) (Γ ) ❛ ♥♦r♠❛ ✉s✉❛❧ ❞❡ H

  

2 ✳ ❆❧é♠ ❞✐ss♦✱ H

2 é ✉♠ ❡s♣❛ç♦ ❞❡

  00

  00

  ❍✐❧❜❡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❛❞❛

  1/2 ′

  2 é ♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦ h., .i ✱ ❛✜♠

  00

  (Γ ) ∗ s♦❜r❡ ♦ ❡s♣❛ç♦ ❞❡ ❙♦❜♦❧❡✈ H

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

  ∗

  ❈♦♠ ❡ss❛s ✐❞❡♥t✐❞❛❞❡s✱ ❛ ❞❡♠♦♥str❛çã♦ ❛ s❡❣✉✐r é ❢❛❝✐❧✐t❛❞❛✳

  ✷✽

  1/2 ′

  Pr♦♣♦s✐çã♦ ✷✳✶ ❙❡❥❛ T ∈ L(H ♦ ♦♣❡r❛❞♦r ❞❡✜♥✐❞♦ ❝♦♠♦ ♥❛

  00 l

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

  l

  ❛❞❥✉♥t♦✱ 1 ♥ã♦ é ❛✉t♦✈❛❧♦r ❞❡ T ❡ ✐♥❥❡t✐✈♦✳ ❉❡♠♦♥str❛çã♦✿ P❛rt❡ ✶ ✲ T l é ♣♦s✐t✐✈♦✿ ❈♦♠ ❛s ✐❞❡♥t✐❞❛❞❡s ❡st❛✲ ❜❡❧❡❝✐❞❛s ❛♥t❡r✐♦r♠❡♥t❡ à ❞❡♠♦♥str❛çã♦✱ ❝♦♥❝❧✉✐✲s❡ ❢❛❝✐❧♠❡♥t❡ q✉❡

  l l ∗ = T

  hT l (ϕ), ϕi h∇L l (ϕ), ∇L (ϕ)i

  N N l

  = h∇W (ϕ), ∇L (ϕ)i

  N

  = h∇W (ϕ), ∇W (ϕ)i

  2 1

  ; ≥ c ||W (ϕ)||

  H (Ω) 1/2 ′

  (Γ ) ∀ϕ ∈ H

  2 ❡ ❛❧❣✉♠❛ ❝♦♥st❛♥t❡ c ∈ R✳

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

  

1/2

  1 (Γ ) (ϕ) = ϕ é ❛✉t♦✈❛❧♦r ❞❡ T l ✱ ✐st♦ é✱ ∃ϕ ∈ H

  2 ♥ã♦ ♥✉❧♦ t❛❧ q✉❡ T l ✳

  

00

  ❈♦♠ ❛ ♠❡s♠❛ ♥♦t❛çã♦ ♣r❡❝❡❞❡♥t❡ à ♣r♦♣♦s✐çã♦✱ ❧❡♠❜r❡ q✉❡ w s❛t✐s❢❛③ 1 = 0; w 2 = ϕ; ∆w = 0, Ω; w | Γ ν | Γ

  ❡ q✉❡ v s❛t✐s❢❛③ 1 2 2 ∆v = 0, Ω; v .

  ν | Γ = 0; v | Γ = w | Γ

  ❆❧é♠ ❞✐ss♦✱ ♦❜s❡r✈❡ q✉❡ ❛ ❞✐❢❡r❡♥ç❛ (v − w) é s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛ ❤♦♠♦❣ê♥❡♦ 2 2 = 0; 2 ∆(v − w) = 0, Ω; (v − w) | Γ = 0; (v − w) ν | Γ 2

  ♣♦✐s v | Γ = w | Γ ❀ ♦✉ s❡❥❛✱ ♣❡❧❛ ✉♥✐❝✐❞❛❞❡ ❞❛ s♦❧✉çã♦✱ ❞❡✈❡♠♦s t❡r v = w ❀ ♣❡❧♦ ❢❛t♦ ❞❡ q✉❡ ❛ ú♥✐❝❛ s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛ ❤♦♠♦❣ê♥❡♦ é ❛ 1 = 0 s♦❧✉çã♦ tr✐✈✐❛❧✳ P❡❧❛ ❞❡✜♥✐çã♦ ❞❡ v ❡ ❞❡ w✱ tê♠✲s❡ w | Γ ❡ 0 = v 1 = w 1

  

ν | Γ ν | Γ ❀ ♣♦✐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 ❞❡ T l ✳

  1/2 ′

  (Γ ) P❛rt❡ ✸ ✲ T l é ❛✉t♦❛❞❥✉♥t♦✿ P❛r❛ ϕ, ψ ∈ H

  2 ✱ ♣❡❧❛s ✐❞❡♥✲

  00

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

  l l l

  T h∇L (ϕ), ∇L l (ψ)i = h∇L (ϕ), ∇W (ψ)i;

  ✷✾

  l

  h∇W (ϕ), ∇W (ψ)i = h∇W (ϕ), ∇L (ψ)i;

  N l ∗ l ∗

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

  l

  P♦rt❛♥t♦✱ T é ❛✉t♦❛❞❥✉♥t♦✳

  1/2 ′

  , ϕ (Γ ) P❛rt❡ ✹ ✲ T l é ✐♥❥❡t✐✈❛✿ ❙❡❥❛♠ ϕ

  1 2 ∈ H 2 t❛✐s q✉❡

  00

l

  T (ϕ ) = T (ϕ ) (ϕ ) )

  l 1 l 2 ❡ ❞❡✜♥❛ w := L 1 − ϕ 2 ❡ v := W (ϕ 1 − ϕ 2 ✳

N

2

  (ϕ ) = 0 = 0 ◆♦t❡ q✉❡ ❛ ❤✐♣ót❡s❡ T l 2 1 − ϕ 2 2 ✐♠♣❧✐❝❛ ❡♠ v ν | Γ ✱ ♣♦✐s

  T (ϕ ) = γ (v) = v

  l 1 − ϕ

  ∆v = 0, Ω; v 1 2 2 .

  ν | Γ = 0; v | Γ = w | Γ

  ❈♦♠♦ v é ❝♦♥st❛♥t❡✱ ♣♦✐s s❛t✐s❢❛③ ∆v = 0, Ω; v = 0, ∂Ω;

  ν 1/2 ′

  (Γ ) ❞❡✈❡✲s❡ t❡r q✉❡ v ≡ 0✱ ♣♦✐s ❛ ú♥✐❝❛ ❢✉♥çã♦ ❝♦♥st❛♥t❡ ❡♠ H

  2

  00

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

  1 2 ✱ ♣♦✐s

  0 = w 2 = ϕ

  ν | Γ 1 − ϕ 2 ✳

  P♦rt❛♥t♦✱ T l é ✐♥❥❡t✐✈❛✳ ◆❛ ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r ❢♦✐ ♣r♦✈❛❞♦ q✉❡ 1 ♥ã♦ é ❛✉t♦✈❛❧♦r ❞❡ T l ✳

  ❆❣♦r❛✱ ❛ ♣r♦♣♦s✐çã♦ ❛ s❡❣✉✐r ❞✐③ q✉❡✱ ❞❡ ❢❛t♦✱ ♦ ♦♣❡r❛❞♦r é ♥ã♦ ❡①♣❛♥✲

  1/2 ′

  2

  00

  (Γ ) s✐✈♦ ❡✱ ❛❧é♠ ❞✐ss♦✱ ❛ ♣r♦✈❛ ❞❡ q✉❡ T l é r❡❣✉❧❛r ❛ss✐♥tót✐❝♦ ❡♠ H

  é ❛♣r❡s❡♥t❛❞❛✳

  1/2 ′

  (Γ ) ) Pr♦♣♦s✐çã♦ ✷✳✷ ❙❡❥❛ T l ∈ L(H

  2 ♦ ♦♣❡r❛❞♦r ❞❡✜♥✐❞♦ ❝♦♠♦ ♥❛

  00

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

  1/2 ′

  (Γ

  

2 )

  tót✐❝♦ ❡ ♥ã♦ ❡①♣❛♥s✐✈♦ ❡♠ H

  00 ✳

  ❉❡♠♦♥str❛çã♦✿ P❛rt❡ ✶ ✲ T l é r❡❣✉❧❛r ❛ss✐♥tót✐❝♦✿ P❛r❛ ♣r♦✈❛r q✉❡ T l é r❡❣✉❧❛r ❛ss✐♥tót✐❝♦✱ ❞❡✈❡✲s❡ ♠♦str❛r q✉❡

  k+1 k

  lim T (ϕ) = 0; (ϕ) − T l

  l ∗ k→∞ 1/2 ′

  (Γ ) ∀ϕ ∈ H

  2 ✳

  00 k

  (ϕ ❆ss✐♠✱ ❜❛st❛ ✈❡r✐✜❝❛r q✉❡ T ) → 0, ∀ϕ ∈ R(T l − I)✱ ♣♦✐s ✈❛❧❡

  l

  ❛ ✐❞❡♥t✐❞❛❞❡

  k+1

k k

  T (ϕ) = T (T (ϕ) − T l − I) (ϕ).

  

l l

l 1/2 ′

  (Γ ) ❉❡ ❢❛t♦✱ s❡❥❛ ψ ∈ H

  2 t❛❧ q✉❡ (T l − I) (ψ) = ϕ✳ P❛r❛ ❡ss❛

  00 k

  (ψ) ❞❡♠♦♥str❛çã♦✱ ♦❜s❡r✈❡ q✉❡ ❛ ✐t❡r❛çã♦ T ♣♦❞❡ s❡r r❡❡s❝r✐t❛ ♣♦r

  l l

2

  w k (ψ) = L (γ N,Γ (v k−1 (ψ))), k ≥ 1;

  N l

2

  ✸✵

  l

  (ψ) = L (ψ)

  N

  ❆ss✐♠✱ ♣♦r ❡ss❛ ❞❡s❝r✐çã♦✱ s❡❣✉❡ q✉❡ ❝♦♠♦ w k s❛t✐s❢❛③ ∆w = 0, Ω; w 1 = 0; (w ) 2 = (v ) 2 ;

  

k k | Γ k ν | Γ k−1 ν | Γ

  ❡ v k é s♦❧✉çã♦ ❞❡ ∆v = 0, Ω; (v ) 1 = 0; v 2 = w 2 ; 1

k k ν | Γ k | Γ k | Γ

1 k Γ = (v k ) ν Γ = 0

  2 k = w k k ) ν = (v k−1 ) ν

  tê♠✲s❡ q✉❡ w | | ❡ ❡♠ Γ ✱ v ❡ (w ✳ ❈♦♠ ✐ss♦ ❡ ✉t✐❧✐③❛♥❞♦ ❛ ❢ór♠✉❧❛ ❞❡ ●r❡❡♥ ❬❆✳✹❪✱ ❝♦♥❝❧✉✐✲s❡ ❛s s❡✲

  ❣✉✐♥t❡s ✐❣✉❛❧❞❛❞❡s✿ Z Z

  t

  ) ) dx = v (v ) d∂Ω; (∇v k (∇v k k k ν 2 Z Z

  Ω Γ t

  ) ) dx = w (w ) d∂Ω; (∇w k (∇w k k k ν 2 Z Z

  Ω Γ t

  ) ) dx = w (v ) d∂Ω; (∇v k (∇w k k k ν 2 Z Z

  Ω Γ t

  ) ) dx = w (v ) d∂Ω; (∇v k−1 (∇w k k k−1 ν 2

  Ω Γ

  ❝♦♠ ❛ ♠❡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

  t k k−1 ) k k−1 ) dx =

  ∇ (w − v ∇ (w − v Z

t t

  k−1 ) k−1 k ) k ) dx;

  (∇v (∇v ) − (∇w (∇w

  Ω

  ❡ Z

  t

  ) ) dx = ∇ (v k − w k ∇ (v k − w k Z

t t

  ) ) ) dx. (∇w k (∇w k ) − (∇v k (∇v k

  Ω

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

  ✈❡r ✐ss♦ ❜❛st❛ ♦❜s❡r✈❛r q✉❡

  k k

  T (ϕ) = T (T l − I) (ψ) . l (ϕ)i 1/2 ∗ ;

  hT l (ϕ), T

  l N

  l N

  L

  D,Γ 2 (v)) ν

  (γ

  Γ 1 ⊔ Γ 2 L l N

  = Z

  D,Γ 2 (v)) dx

  (γ

  l N

  ∇L

  

D,Γ

2 (v)) t

  (γ

  ∇L

  D,Γ 2 (v)) d∂Ω

  l (ϕ)i ∗ = Z

  (ϕ), T

  ❡ ♣❡❧❛ ❢ór♠✉❧❛ ❞❡ ●r❡❡♥ ❬❆✳✹❪✱ t❡♠✲s❡ q✉❡ hT l

  N,Γ 2 (v)

  ❆❣♦r❛✱ ❝♦♠♦ T l (ϕ) = γ

  (ϕ) ✳

  

l

N

  ◦ γ D,Γ 2 ◦ L

  l D

  (ϕ) ❡ v := L

  l N

  ✱ w := L

  (γ

  Ω

  )

  (γ

  1/2

  (∇v) dx

  t

  Z (∇v)

  (γ D,Γ 2 (v)) dx ≤

  

l

N

  ∇L

  t

  (∇v)

  Ω

  = Z

  D,Γ 2 (v)) dx

  l N

  ∆ L

  ∆(v)L

  Ω

  D,Γ 2 (v)) d∂Ω

  (γ

  l N

  L

  ν

  v

  Γ 1 ⊔ Γ 2

  (γ D,Γ 2 (v)) dx = Z

  l N

  (γ D,Γ 2 (v)) L

  l N

  ′

  2

  ✸✶ Z

  t

  − T

  

2

  (ψ)

  k l

  ≤ 2 T

  2 ∗

  (ϕ)

  k l

  ♦✉ s❡❥❛✱ ✈❛❧❡ q✉❡ T

  (∇w k+1 (ψ)) dx;

  t

  (∇w k (ψ)) − (∇w k+1 (ψ))

  (∇w k (ψ))

  (ψ)

  Ω

  = 2 Z

  ∇ (w k+1 (ψ) − w k (ψ)) dx

  t

  ∇ (w k+1 (ψ) − w k (ψ))

  Ω

  ≤ 2 Z

  k (ϕ)dx

  ∇w

  k (ϕ))

t

  (∇w

  Ω

  k+1 l

  2 ∗

  (Γ

  (ψ)

  00

  1/2

  ≤ 1✱ ❞❡✜♥❛ ♣❛r❛ ϕ ∈ H

  L (H 1 /2 00 (Γ2 ))

  ❡①♣❛♥s✐✈♦✱ ✐st♦ é✱ ||T l ||

  P♦rt❛♥t♦✱ T l é r❡❣✉❧❛r ❛ss✐♥tót✐❝♦✳ P❛rt❡ ✷ ✲ T l é ♥ã♦ ❡①♣❛♥s✐✈♦✿ P❛r❛ ❞❡♠♦♥str❛r q✉❡ T l é ♥ã♦

  ∗ = 0.

  (ϕ)

  k l

  T

  ♥ã♦ é ❝r❡s❝❡♥t❡✳ ▲♦❣♦✱ lim k→∞

  ∗ k∈N

  k l

  , ∀k ∈ N; ♣❡❧❛ ❞❡✜♥✐çã♦ w k

  T

  ≥ 0✱ ∀k ∈ N✱ ✐st♦ é✱ ❛ s❡q✉ê♥❝✐❛

  2 ∗

  (ψ)

  k+1 l

  − T

  2 ∗

  (ψ)

  k l

  ❆❧é♠ ❞✐ss♦✱ ♦❜s❡r✈❡ q✉❡ T

  (.) ✳

  l N

  (.) := L

  • Z
  • Z

  ✸✷ ♦✉ s❡❥❛✱ ✈❛❧❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ Z

  t

  (ϕ), T ∗ hT l l (ϕ)i ≤ (∇v) (∇v) dx.

  Ω

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

  t

  v vd∂Ω (∇v) (∇v) dx = ν 2

  Ω Γ Z

  = v ν wd∂Ω 1 ⊔ 2 Z

  Γ Γ t

  = (∇v) (∇w) dx

  Ω 1/2 1/2

  Z Z

  t t

  ; ≤ (∇v) (∇v) dx (∇w) (∇w) dx

  

Ω Ω

  ✐st♦ é✱ Z Z

  

t t

(∇v) (∇v) dx ≤ (∇w) (∇w) dx. Ω Ω

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

  1/2

  Z

  t 1/2 ∗ ;

  ||T l (ϕ)|| ∗ ≤ (∇w) (∇w) dx = hϕ, ϕi = ||ϕ|| ∗

  Ω l

  (ϕ) ♣♦✐s w := L ✳

  N

  P♦rt❛♥t♦✱ T l é ♥ã♦ ❡①♣❛♥s✐✈♦✳

  l

  ❆♣ós ❡ss❛ ❛♥á❧✐s❡ s♦❜r❡ ♦ ♦♣❡r❛❞♦r T ✱ ♦ tr❛❜❛❧❤♦ ❞❡ ❞❡♠♦♥str❛r ❛ ❛♥á❧✐s❡ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ♣❛r❛ ♦ ♠ét♦❞♦ ❞❡ ▼❛③✬✐❛ é ❢❛❝✐❧✐t❛❞♦❀ ❝♦♠♦ ❛❜❛✐①♦ ♦ t❡♦r❡♠❛ ❞❡st❛ s❡çã♦✳

  l

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

  1/2 ′ 1/2

  (Γ (Γ ) ❞♦ ♠ét♦❞♦ ❞❡ ▼❛③✬✐❛✳ ❙❡ (f, g) ∈ H

  1 )×H 1 é ✉♠ ♣❛r ❞❡ ❞❛✲

  00 k

  (ϕ ❞♦s ❞❡ ❈❛✉❝❤② ❝♦♥s✐st❡♥t❡✱ ❡♥tã♦ ❛ s❡q✉ê♥❝✐❛ {ϕ k } k∈N = {T )} k∈N ❝♦♥✈❡r❣❡ ♣❛r❛ ♦ tr❛ç♦ ❞❡ ◆❡✉♠❛♥♥ ❞❛ s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤②✱

  1/2 2 (Γ )

  ✐st♦ é✱ ♣❛r❛ u ν | Γ ✱ ♣❛r❛ t♦❞♦ ϕ ∈ H

  2 ✳

  ✸✸

  k

  T (ϕ ) ❝♦♥✈❡r❣❡ ♣❛r❛ ♦ ♣♦♥t♦ ✜①♦ ϕ ❞♦ ♦♣❡r❛❞♦r T ✱ ♣❡❧❛ ♦❜s❡r✈❛çã♦

  ❢❡✐t❛ ♥❡st❛ s❡çã♦✳ = ϕ

  P❛r❛ ✐ss♦✱ ❞❡✜♥❛✱ ♣❛r❛ ❝❛❞❛ k ∈ N✱ ε k k − ϕ✳ ❊♥tã♦✱ t❡♠✲s❡ q✉❡ ε = ϕ

  

k+1 k+1 − ϕ = T (ϕ k ) − T (ϕ)

= T l (ϕ k ) + T l (ϕ) = T l (ϕ k l (ϕ) = T l (ε k ) .

  − T − T ) − T

  1/2 ′

  (Γ )

  ∗

  ❈♦♠♦ H

  2 é ✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt ✭❝♦♠ ♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦ h., .i ✮✱

  00

  t❡♠✲s❡ q✉❡ ε k → 0✱ ♣❡❧❛s ♣r♦♣♦s✐çõ❡s ❛♥t❡r✐♦r❡s✳ ❉❡ ❢❛t♦✱ T l s❡r r❡❣✉❧❛r ❛ss✐♥tót✐❝♦ ✐♠♣❧✐❝❛ ❡♠ s❡r ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❛✉❝❤② ❡✱ ❝♦♠♦ ♦ ❡s♣❛ç♦ é ✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt✱ ❛ s❡q✉ê♥❝✐❛ é ❝♦♥✈❡r❣❡♥t❡✳ P♦rt❛♥t♦✱ ❝♦♠♦ 1 ♥ã♦ é ❛✉t♦ ✈❛❧♦r ❞❡ T l ✱ t❡♠✲s❡ q✉❡ ϕ k → ϕ✱ s❡ k → ∞✳

  ❖ ú❧t✐♠♦ r❡s✉❧t❛❞♦ ❛ r❡s♣❡✐t♦ ❞❡ss❡ ♠ét♦❞♦ é ❛ r❡❝✐♣r♦❝✐❞❛❞❡ ❞♦ t❡♦r❡♠❛ ❞❛ ❛♥á❧✐s❡ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛✳ ❆ ♣r♦♣♦s✐çã♦ ❛ s❡❣✉✐r t❛♠❜é♠ ♣♦❞❡ s❡r ❡♥t❡♥❞✐❞❛ ❝♦♠♦ ✉♠ ❝r✐tér✐♦ ♣❛r❛ ❛ ❡①✐stê♥❝✐❛ ❞♦ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤② ❡❧í♣t✐❝♦✳

  1/2 ′

1/2

  (Γ

  1 (Γ 1 )

  Pr♦♣♦s✐çã♦ ✷✳✸ ❙❡❥❛ (f, g) ∈ H ) × H

  00 ✉♠ ♣❛r ❞❡ ❞❛❞♦s k

  (ϕ

  k } k∈N = {T )} k∈N

  ❞❡ ❈❛✉❝❤② ❝♦♥s✐st❡♥t❡✳ ❙❡ ❛ s❡q✉ê♥❝✐❛ {ϕ ❝♦♥✲

  1/2 ′

  (Γ ) ✈❡r❣❡ ❡♠ H

  2 ✱ ❡♥tã♦ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤② ❡❧í♣t✐❝♦ (CP ) t❡♠

  00

  1

  (Ω) 2 = lim ϕ ✉♠❛ s♦❧✉çã♦ u ∈ H ❡ u ν | Γ k→∞ k ✳

  1/2 ′ k→∞ ϕ k (Γ 2 )

  ❉❡♠♦♥str❛çã♦✿ ❉❡✜♥❛ ϕ = lim ∈ H

  00 ✱ ❡♥tã♦ T (ϕ) = T lim ϕ = lim ϕ = ϕ. k k+1

k→∞ k→∞

  P♦rt❛♥t♦✱ ϕ é ♣♦♥t♦ ✜①♦ ❞❡ T ❡✱ ❛ss✐♠✱ ❛ ♦❜s❡r✈❛çã♦ ❢❡✐t❛ ♥❡st❛ s❡çã♦ ❣❛r❛♥t❡ ❛ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çã♦ ♣❛r❛ (CP )✳ ❆ ♣ró①✐♠❛ s❡çã♦ tr❛③ ♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r ❝♦♠ ✉♠❛ s✉❜s❡çã♦

  ❛❜♦r❞❛♥❞♦ ❛ ❛♥á❧✐s❡ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛✱ ❡st❛❜✐❧✐❞❛❞❡ ❞♦ ♠ét♦❞♦ ❡ ❛ t❛①❛ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ✈✐❛ ❛ t❡♦r✐❛ ❝❧áss✐❝❛ ❞❡ r❡❣✉❧❛r✐③❛çã♦✳

  ✷✳✷ ▼ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r 1/2 ′

  1/2

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

  1 ) × H

  1

  00

  ❝♦♥s✐st❡♥t❡✱ s❛❜❡✲s❡ q✉❡ ♦ ♦♣❡r❛❞♦r F ❞♦ ♣r♦❜❧❡♠❛ ✐♥✈❡rs♦ F (ϕ) = g é ✉♠ ♦♣❡r❛❞♦r ❛✜♠ ❡✱ ♣♦rt❛♥t♦✱ F (ϕ) = g ♣♦❞❡ s❡r r❡❡s❝r✐t♦ ♣♦r

  1/2 ′

  F (ϕ) = h (Γ )

  l ✱ ♦♥❞❡ h = h(f, g) := g −F f ✳ ◆♦t❡ q✉❡ ❝♦♠♦ F f ∈ H 1 ✱

  00 1/2 ′

  (Γ ) h ∈ H

  1 t❛♠❜é♠✳

  ✸✹ ♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r ❛♣r❡s❡♥t❛❞♦ ❛q✉✐ é ❛♣❡♥❛s ♣❛r❛ ♣r♦❜❧❡♠❛s ❧✐♥❡❛r❡s✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ♣❛r❛ r❡s♦❧✈❡r ♦ ♣r♦❜❧❡♠❛ ✐♥✈❡rs♦ ❛ss♦❝✐❛❞♦ ❛♦

  (ϕ) = h ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤② ❡❧í♣t✐❝♦✱ F l ✳

  ❖ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r t❛♠❜é♠ ❝♦♥s✐st❡ ❡♠ ✉♠❛ ✐t❡r❛çã♦ ❞❡ ♣♦♥t♦

  1/2 1/2 ′ ′

  = L (ϕ ) (Γ ) (Γ ) ✜①♦✱ ϕ k+1 k ✱ k ∈ N✱ ❝♦♠ L : H

  2 −→ H 2 ❞❡✜♥✐❞❛

  

00

  00

  ♣♦r

  ∗

  L (ϕ) := ϕ + F (ϕ)); (h − F l

  l

  ❡st❛ ✐t❡r❛çã♦ ❞❡ ♣♦♥t♦ ✜①♦ ♣r♦✈é♠ ❞❛ ❝♦♥❞✐çã♦ ❞❡ ♦t✐♠❛❧✐❞❛❞❡ ❞❡ ♣r✐✲ ♠❡✐r❛ ♦r❞❡♠ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ❞❡ ♠í♥✐♠♦s q✉❛❞r❛❞♦s ♥ã♦ ❧✐♥❡❛r

  1

  2

  min ; 1 ||h − F l (ϕ)||

  /2 2 ′

  2

  h∈H (Γ ) 00

  q✉❡ é ❛ ❡q✉❛çã♦ ♥♦r♠❛❧

  ∗ ∗ F F l (ϕ) = F (ϕ) h. l l

  ❈♦♠ ♦ ✐♥t✉✐t♦ ❞❡ ❛♥❛❧✐s❛r ♦ ♠ét♦❞♦ s♦❜ ❞❛❞♦s ♣❡rt✉r❜❛❞♦s✱ ✐st♦ é✱ ❛✜♠ ❞❡ ❞❡♠♦♥str❛r q✉❡ é ✉♠ ♠ét♦❞♦ ❞❡ r❡❣✉❧❛r✐③❛çã♦ ✭❛♥á❧✐s❡ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ❡ ❡st❛❜✐❧✐❞❛❞❡✮ ❡ ♦❜t❡r t❛①❛ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛✱ ❝♦♥s✐❞❡r❡

  1/2 ′ 1/2

  (Γ (Γ ) (f, g) ∈ H

  1 ) × H 1 ✉♠ ♣❛r ❞❡ ❞❛❞♦s ❞❡ ❈❛✉❝❤② ❝♦♥s✐st❡♥t❡

  00 1/2 ′ 1/2

  (Γ (Γ ) ❡ ✉♠ ♦✉tr♦ ♣❛r f, g ∈ H

  1 ) × H 1 ❝♦♥s✐st❡♥t❡ ♦✉ ♥ã♦✳

  00

  ❈♦♥s✐❞❡r❡ t❛♠❜é♠ h ❝♦♠♦ ❛♥t❡r✐♦r♠❡♥t❡✱ ❡♥tã♦✱ ❛♥❛❧♦❣❛♠❡♥t❡✱ ♣♦❞❡✲ s❡ ❝❛❧❝✉❧❛r h✱ ❛ s❛❜❡r h := g − F ✳

  f 1/2

δ δ

2 ′

  , g (Γ (Γ ) ❊♥tã♦✱ ❞❛❞♦ ✉♠ ♣❛r ❞❡ ❞❛❞♦s f ∈ L

  1 ) × H 1 t❛✐s

  00

  q✉❡

  δ δ

  f g 2 /2 1 − f − g ′ ≤ δ. 1

  • 1

  L (Γ ) H 00 (Γ ) δ

  ❆ ♣❡r❣✉♥t❛ ❛ s❡r ❢❡✐t❛ é✿ ♣♦❞❡✲s❡ ❣❛r❛♥t✐r q✉❡ ❡①✐st❡ ✉♠ t❡r♠♦ h t❛❧

  δ 1

  h /2 q✉❡ − h ≤ δ ❄ 1

  H 00 (Γ )

  P❛r❛ ✐ss♦ é ♥❡❝❡ssár✐♦ ❛ ❤✐♣ót❡s❡ ❛ ♣r✐♦r✐ s♦❜r❡ ♦ ♣❛r ❞❡ ❞❛❞♦s ❞❡

  r

  (Γ ) ❈❛✉❝❤② ❡①❛t♦s✿ f ∈ H

  1 ✱ ♣❛r❛ r ≥ 1/2 ❬✺❪✳ ❖ s❡❣✉✐♥t❡ ❧❡♠❛ ❬✺❪

  ❢♦r♥❡❝❡ ❛ r❡s♣♦st❛ ❛ ❡ss❛ ♣❡r❣✉♥t❛✳

  r δ

  2

  (Γ ) (Γ ) ▲❡♠❛ ✷✳✶ ❙❡❥❛♠ f ∈ H

  

1 ✱ r > s > 0 ❡ f ∈ L

1 ❝♦♠ δ 2

  f − f ≤ δ; 1 L (Γ )

  2

  (Γ ♣❛r❛ ❛❧❣✉♠ δ > 0 ✳ ❊♥tã♦✱ ❡①✐st❡♠ ✉♠ ♦♣❡r❛❞♦r s✉❛✈✐③❛♥t❡ S : L

  1 ) −→ s δ

  H (Γ ) γ (x) = 0

  1 ❡ ✉♠❛ ❢✉♥çã♦ ♣♦s✐t✐✈❛ γ ❝♦♠ lim x↓0 t❛✐s q✉❡ f − Sf s ≤ 1 H (Γ )

  γ (δ) ✳

  ✸✺

  2 1/2 δ 1/2 δ

  L (Γ (Γ ) := Sf (Γ )

  1 ) −→ H 1 ♣❛r❛ ❣❡r❛r ❛ ❢✉♥çã♦ f ∈ H 1 t❛❧ q✉❡ ′ ′

  δ

  > 0 f − f 1 ≤ δ ✱ ♣❛r❛ ❛❧❣✉♠ ♦✉tr♦ δ ✳ ❆ss✐♠✱ ❛♣ós s✉❛✈✐③❛r

  /2 1 H (Γ ) 1/2 ′

  δ 2 δ δ

  f (Γ ) := g (Γ ) ∈ L

  1 ✱ ♣♦❞❡✲s❡ ❝❛❧❝✉❧❛r ♦ t❡r♠♦ h − F δ ∈ H

  1

  00 f ′

  δ 1 /2

  h t❛❧ q✉❡ − h ′ ≤ δ ❛ ♣❛rt✐r ❞♦ ♣❛r ❞❡ ❞❛❞♦s ❞❡ ❈❛✉❝❤② 1 H 00 (Γ ) δ

  δ

  f , g ✱ ♣♦✐s ♦ t❡r♠♦ h := g − F f ❞❡♣❡♥❞❡ ❝♦♥t✐♥✉❛♠❡♥t❡ ❞♦ ♣❛r ❞❡

  ❞❛❞♦s (f, g)✳ ❊ss❛ ❛♥á❧✐s❡ é ♥❡❝❡ssár✐❛ ♣❛r❛ ♦❜t❡r ✉♠❛ ❡st✐♠❛t✐✈❛ ❞♦ ❡rr♦ ❞❡ ♣r♦♣❛❣❛çã♦ ❞♦ ❛❧❣♦rít♠♦ ❞❡ ▲❛♥❞✇❡❜❡r✱ ❢❡✐t❛ ♥❛ s✉❜s❡çã♦✳

  δ l (ϕ) = h

  ▲♦❣♦✱ ♣♦❞❡✲s❡ ♣❡♥s❛r ❡♠ r❡s♦❧✈❡r ♦ ♣r♦❜❧❡♠❛ ✐♥✈❡rs♦ F

  δ 1 /2

  t❛❧ q✉❡ h − h ≤ δ ❬✺❪✳ 1

  H 00 (Γ )

  ❆ s✉❜s❡çã♦ ❛ s❡❣✉✐r✱ ❛❧é♠ ❞❡ ❛❜♦r❞❛r ❛ ❛♥á❧✐s❡ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛✱ t❛♠❜é♠ ❛♣r❡s❡♥t❛ ✉♠ t❡♦r❡♠❛ s♦❜r❡ ❛ t❛①❛ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ❞♦ ❛❧❣♦✲ rít♠♦ ❞❡ ▲❛♥❞✇❡❜❡r ♣❛r❛ ❞❛❞♦s ♣❡rt✉r❜❛❞♦s✳

  ✷✳✷✳✶ ❆♥á❧✐s❡ ❞♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r

  ❖s r❡s✉❧t❛❞♦s ❝❧áss✐❝♦s ♣❛r❛ ♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r ❧✐♥❡❛r sã♦ ❛♣r❡✲ s❡♥t❛❞♦s ♥❡st❛ s✉❜s❡çã♦✳ ❖ ♦❜❥❡t✐✈♦ ❞❡st❛ s✉❜s❡çã♦ é ♦ ❞❡ ❛♣r❡s❡♥t❛r ❛s ❞❡♠♦♥str❛çõ❡s r❡❢❡✲ r❡♥t❡ ❛ ❛♥á❧✐s❡ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ❡ ❡st❛❜✐❧✐❞❛❞❡ ✭r❡❣✉❧❛r✐③❛çã♦✮ ❡ t❛①❛ ❞❡

  ❝♦♥✈❡r❣ê♥❝✐❛ ♣❛r❛ ♦ ♠ét♦❞♦✳ ❆♣❡♥❛s ❛ ❞❡♠♦♥str❛çã♦ ❞❡ q✉❡ ♦ ♠ét♦❞♦ é ❞❡ ♦r❞❡♠ ót✐♠❛ ♥ã♦ ❡stá ♣r❡s❡♥t❡✳

  ❈♦♠♦ ❛ t❡♦r✐❛ ❞❡ r❡❣✉❧❛r✐③❛çã♦ ♣❛r❛ ♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r é ❢❡✐t❛ ♣❛r❛ ♦♣❡r❛❞♦r❡s ♥ã♦ ❡①♣❛♥s✐✈♦s✱ ♣♦rt❛♥t♦✱ ♦ ♣r✐♠❡✐r♦ r❡s✉❧t❛❞♦ ❣❛r❛♥t❡

  (ϕ) = h q✉❡ ❡st❛ t❡♦r✐❛ ♣♦❞❡ s❡r ❛♣❧✐❝❛❞❛ ❛♦ ♣r♦❜❧❡♠❛ ✐♥✈❡rs♦ F l ✳

  1/2 1/2 ′ ′

  (Γ ) , H (Γ ) ) Pr♦♣♦s✐çã♦ ✷✳✹ ❙❡❥❛ F l ∈ L(H

  2 1 ♦ ♦♣❡r❛❞♦r ❞❡✜✲

  

00

  00

  ♥✐❞♦ ❝♦♠♦ ♥❛ ❞❡s❝r✐çã♦ ❞♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r✳ ❊♥tã♦✱ t❡♠✲s❡ q✉❡ F

  l é ♥ã♦ ❡①♣❛♥s✐✈♦✳ 1/2 ′

  1

  (Γ ) (Ω) ❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ ϕ ∈ H

  2 ✳ ❙❡❥❛♠ v, w, q ∈ H s♦❧✉çõ❡s

  00

  ❞♦s s❡❣✉✐♥t❡s ♣r♦❜❧❡♠❛s ❞❡ ✈❛❧♦r ❞❡ ❝♦♥t♦r♥♦ 1 = 0; v 2 = ϕ; ∆v = 0, Ω; v | Γ ν | Γ 1 2 2

  ∆w = 0, Ω; w ;

  ν | Γ = 0; w | Γ = v | Γ 1 1 2

  ∆q = 0, Ω; q ν Γ = v ν Γ Γ = 0; | | ; q |

  ✸✻ R

  2

  2

  (ϕ) = v 1 = q 1 = dx ❊♥tã♦✱ tê♠✲s❡ q✉❡ F l ν | Γ ν | Γ , ||ϕ|| |∇v| ❡

  ∗ R

  2 1

  2 /2

  = dx ||F l (ϕ)|| |∇q| ✳ 1 Ω

  H 00 (Γ )

  ❆ss✐♠✱ r := v − w é s♦❧✉çã♦ ❞❡ ∆r = 0, Ω; r 1 = v 1 2 = 0;

  ν | Γ ν | Γ ; r | Γ

  ♦✉ s❡❥❛✱ t❡♠✲s❡ q✉❡ q = v − w✳ Z Z Z Z ❆❧é♠ ❞✐ss♦✱

  2 t

  dx = ww d∂Ω = vw d∂Ω = |∇w| ν ν (∇v) (∇w) dx. 1 2 1 2

  ⊔ ⊔ Ω Γ Γ Γ Γ Ω

  ▲♦❣♦✱ Z

  2 1

  2 /2 l = dx

  ||F (ϕ)|| ′ |∇q| 1 H 00 (Γ ) Z

  2

  = dx |∇ (v − w)| Z

  

2

2 t

  = dx |∇v| + |∇w| − 2 (∇v) (∇w) Z

  

2

  2

  = dx |∇v| − |∇w|

  Ω

Z

  2

  2 dx.

  = ||ϕ|| − |∇w|

  ∗

1/2 1

  /2

  (Γ ) P♦rt❛♥t♦✱ ||F l (ϕ)|| ≤ ||ϕ|| , ∀ϕ ∈ H

1 ′ ∗

2 ✱ ♦✉ s❡❥❛✱

  00 1 H 00 (Γ ) /2

  ||F l || ′ ≤ 1 ❡✱ ❛ss✐♠✱ F l é ♥ã♦ ❡①♣❛♥s✐✈♦✳ 1 H 00 (Γ ) 1

  /2

  ❱❛❧❡ r❡ss❛❧t❛r q✉❡ ♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦ h., .i ′ ❞❡✜♥✐❞♦ s♦❜r❡ ♦ 1 H 00 (Γ ) 1/2

  ′

  (Γ ) ❡s♣❛ç♦ ❞❡ ❙♦❜♦❧❡✈ H

  1 ❞❡ ♠♦❞♦ ❛ t♦r♥á✲❧♦ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt é

  00 1/2 ′ ∗ (Γ )

  ❛♥á❧♦❣♦ ❛♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦ h., .i s♦❜r❡ ♦ ❡s♣❛ç♦ H

  2 ✱ ✐st♦ é✱ ❞❛❞♦s

  00 1/2 ′

  (Γ ) ϕ, ψ ∈ H

  1 ✱ ❡♥tã♦

  00 Z 1 t /2

  := hϕ, ψi ′ (∇v) (∇w) dx; 1 H 00 (Γ )

  1

  (Ω) ♦♥❞❡ v, w ∈ H sã♦ s♦❧✉çõ❡s ❞♦s s❡❣✉✐♥t❡s ♣r♦❜❧❡♠❛s ❞❡ ✈❛❧♦r ❞❡ ❝♦♥t♦r♥♦

  ✸✼ ∆v = 0, Ω; v 1 2 = 0;

  ν | Γ = ϕ; v | Γ

  ∆w = 0, Ω; w 1 2 = 0;

  ν | Γ = ψ; w | Γ 1 /2

  r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❆ ♥♦r♠❛ ❞❡♥♦t❛❞❛ ♣♦r ||.|| é ❛ ♥♦r♠❛ ♣r♦✈❡✲ 1 ′

  H 00 (Γ )

  ♥✐❡♥t❡ ❞❡ss❡ ♣r♦❞✉t♦ ✐♥t❡r♥♦✳ ❆ ♣r♦♣♦s✐çã♦ ❛ s❡❣✉✐r ❣❛r❛♥t❡ q✉❡ ❛ s❡q✉ê♥❝✐❛ {ϕ k } k∈N ❝♦♥✈❡r❣❡

  (ϕ) = h ♣❛r❛ ❛ s♦❧✉çã♦ ❞❡ ♠í♥✐♠♦s q✉❛❞r❛❞♦s ❞❡ F l ✱ s❡ ♦ ♣❛r ❞❡ ❞❛❞♦s

  

1/2

1/2 ′

  (Γ (Γ ) ❞❡ ❈❛✉❝❤② (f, g) ∈ H

  1 ) × H 1 ❢♦r ❝♦♥s✐st❡♥t❡✳ ❊♠ ♦✉tr❛s

  

00

  ♣❛❧❛✈r❛s✱ ❝❛s♦ s❡❥❛ ❢♦r♥❡❝✐❞♦ ✉♠ ♣❛r ❞❡ ❞❛❞♦s ❞❡ ❈❛✉❝❤② ❡①❛t♦s ✭❡ ❝♦♥s✐st❡♥t❡✮✱ t❡♠✲s❡ ❛ ❝♦♥✈❡r❣ê♥❝✐❛ ❞♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r✳

  1/2 1/2 ′

  (Γ (Γ ) Pr♦♣♦s✐çã♦ ✷✳✺ ❙❡ (f, g) ∈ H

  1 ) × H 1 é ✉♠ ♣❛r ❞❡ ❞❛❞♦s

  00 †

  (h) ❞❡ ❈❛✉❝❤② ❝♦♥s✐st❡♥t❡✱ ❡♥tã♦ ϕ k → F ✱ s❡ k → ∞✳

  l 1/2

  (Γ ❉❡♠♦♥str❛çã♦✿ Pr✐♠❡✐r❛♠❡♥t❡✱ ♦❜s❡r✈❡ q✉❡ ❝♦♠♦ (f, g) ∈ H

  1 )× 1/2 ′

  H (Γ )

  1 é ✉♠ ♣❛r ❞❡ ❞❛❞♦s ❞❡ ❈❛✉❝❤② ❝♦♥s✐st❡♥t❡✱ ♦ ♣r♦❜❧❡♠❛ ❞❡

  00

  1

  (Ω) ❈❛✉❝❤② (CP ) ❛❞♠✐t❡ ✉♠❛ ú♥✐❝❛ s♦❧✉çã♦ u ∈ H ✳ ❊♥tã♦✱ F (ϕ) = g

  ) ❛❞♠✐t❡ s♦❧✉çã♦✳ ▲♦❣♦✱ g ∈ R(F ) ❡✱ ♣♦rt❛♥t♦✱ h ∈ R(F l ✳

  ❆ ✐❞❡✐❛ ❞❡st❛ ❞❡♠♦♥str❛çã♦ é ❞❡✜♥✐r ✉♠❛ ❢✉♥çã♦ g k ❞❡ ♠♦❞♦ q✉❡ s❛t✐s❢❛ç❛ ❛s ❤✐♣ót❡s❡s ❞❡ ❬❆✳✺❪ ♣❛r❛ ❛♣❧✐❝á✲❧♦ ❡ ❝♦♥❝❧✉✐r ❛ ❞❡♠♦♥str❛çã♦✳

  

  = ϕ + F (ϕ )) , P❛r❛ k ∈ N✱ ❝♦♠♦ ϕ k+1 k (h − F l k ❡♥tã♦ ϕ k ♣♦❞❡ s❡r

  

l

  r❡❡s❝r✐t♦ ♣♦r ✉♠❛ ❢♦r♠❛ ♥ã♦ r❡❝✉rs✐✈❛ X k−1 j

  ∗ ∗

  ϕ = F ) F (h) ;

  k (I − F l l l j=0 ∗ ∗ k F l ) (ϕ k−1 ) + F (h)

  ♣❛r❛ ✈❡r ✐ss♦ ❜❛st❛ ❡s❝r❡✈❡r ϕ = (I − F ❡ ✉t✐❧✐③❛r

  l l

  = 0 ❡ss❛ ❡①♣r❡ssã♦ r❡❝✉rs✐✈❛♠❡♥t❡ ❞❡ k − 1 ❛té k = 0✱ s❡ ϕ ✳

  † 1/2 ∗ ∗ † † ′

  (h) = F F ϕ = F (Γ ) ❊♥tã♦✱ t❡♠✲s❡ q✉❡ F l ✱ ♦♥❞❡ ϕ (h) ∈ H

  2 ✳ l l l

  00

  ▲♦❣♦✱ ✉t✐❧✐③❛♥❞♦ ❛ ❢♦r♠❛ ♥ã♦ r❡❝✉rs✐✈❛ ♣❛r❛ ϕ k ✱

X

k−1

j

  † † ∗ ∗ † ϕ = ϕ F F ) ϕ .

  − ϕ k − F l (I − F l

  l l P

j=0

∗ k−1 ∗ j ∗ k

  F F ) F ) ❈♦♠ ❛ ✐❣✉❛❧❞❛❞❡ F l (I − F l = I − (I − F l ✱ ✈❛❧❡

  l l l j=0

  q✉❡

  

† ∗ k †

ϕ F ) ϕ .

  − ϕ k = (I − F l

  ✸✽ P k−1 j k (λ) = .

  j=0

  q✉❡

  ∗ ∗

  ϕ = g (F F ) F (h) ;

  k k l l l

  ❡ q✉❡

  

† ∗ †

ϕ = r (F F ) ϕ .

  − ϕ k k l

  l k k (λ)

  ❈♦♥s✐❞❡r❡ λ ∈ (0, 1]✳ ◆♦t❡ q✉❡ ♥❡st❡ ✐♥t❡r✈❛❧♦ λg (λ) = 1 − r é ✉♥✐❢♦r♠❡♠❡♥t❡ ❧✐♠✐t❛❞❛ ❡ g k (λ) → 1/λ✱ s❡ k → ∞✱ ❥á q✉❡ r k (λ) → 0✱ s❡ k → ∞✳

  P♦rt❛♥t♦✱ ♣♦❞❡✲s❡ ❛♣❧✐❝❛r ♦ r❡s✉❧t❛❞♦ ❬❆✳✺❪ ♣❛r❛ ❝♦♥❝❧✉✐r ❛ ❞❡♠♦♥s✲ tr❛çã♦✳ ❆ ♣ró①✐♠❛ ♣r♦♣♦s✐çã♦ ❞á ✉♠❛ s✐♠♣❧❡s ❡st✐♠❛t✐✈❛ ❞♦ ❡rr♦ ❞❡ ♣r♦♣❛✲

  ❣❛çã♦ ❞♦ ❛❧❣♦rít♠♦ ❞❡ ▲❛♥❞✇❡❜❡r ♥❛ ♣r❡s❡♥ç❛ ❞❡ ❞❛❞♦s ♣❡rt✉r❜❛❞♦s

  δ δ

  f , g ✱ ✐st♦ é✱ ✉♠❛ ❡st✐♠❛t✐✈❛ ♣❛r❛ s❛❜❡r ❞❡ q✉❡ ♠♦❞♦ ♦ ❡rr♦ ♥♦s ❞❛✲

  δ k∈N

  ❞♦s ❞❡t❡r✐♦r❛ ❛ ✐t❡r❛çã♦ {ϕ } ✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ é ❞❡♠♦♥str❛❞❛

  k

  ❛ ❡st❛❜✐❧✐❞❛❞❡ ❞♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r✳ ◆♦t❡ q✉❡ s❡ ❢♦r♥❡❝✐❞♦s ❞❛❞♦s

  δ δ

  , g ♣❡rt✉r❜❛❞♦s f t❛✐s q✉❡

  δ δ 2 /2 1 f − f g − g ≤ δ;

  • 1
  • 1 L (Γ ) H 00 (Γ )

    δ δ δ

    = h f , g

      é ♣♦ssí✈❡❧ ❝❛❧❝✉❧❛r ♦ t❡r♠♦ h t❛❧ q✉❡

      δ

    1

    /2

    h − h ≤ δ. 1 H 00 (Γ )

    1/2

      ′ δ

      (Γ ) Pr♦♣♦s✐çã♦ ✷✳✻ ❙❡❥❛♠ h, h ∈ H

      1 ❞❛❞♦s t❛✐s q✉❡

      

    00

    δ

    1

    /2

      h − h ≤ δ, 1

      H 00 (Γ ) δ k ϕ

      ❡ ❝♦♥s✐❞❡r❡ ❛s ✐t❡r❛çõ❡s ❞❡ ▲❛♥❞✇❡❜❡r ❝♦rr❡s♣♦♥❞❡♥t❡s✱ {ϕ } ❡

      k

      ✭❝♦♠ ❞❛❞♦s ❡①❛t♦s ❡ ✐♥❡①❛t♦s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✮✳ ❊♥tã♦✱ t❡♠✲s❡ q✉❡ √

      δ

      ϕ

    k − ϕ ≤ kδ, k ≥ 0.

      k ∗ k−1 j ∗ ∗

      δ δ

      := P F ) F ❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ R k h − h (I − F l h − h ✳

      j=0 l l δ δ

      = R ❊♥tã♦✱ ♣❡❧❛ ❞❡♠♦♥str❛çã♦ ❞❛ ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r✱ ϕ k −ϕ k h − h ✳ P k ∗ k−1 ∗ j

      F F ) = ❆ss✐♠✱ ✉t✐❧✐③❛♥❞♦ ♥♦✈❛♠❡♥t❡ ❛ ✐❣✉❛❧❞❛❞❡ F l (I − F l

      l l j=0 k ∗

      F l ) I − (I − F ✱ s❡❣✉❡ q✉❡

      l 2 ∗

      R ||R k || = ||R k ||

      ✸✾ X k−1 ∗ j ∗ k = F l ) F l )

      (I − F l I − (I − F l

      j=0 X k−1

    ∗ j

      F l ) ; ≤ (I − F l

      j=0 ∗

      F ♣♦✐s I − F l é ✉♠ ♦♣❡r❛❞♦r s❡♠✐❞❡✜♥✐❞♦ ♣♦s✐t✐✈♦ ♥ã♦ ❡①♣❛♥s✐✈♦✳

      l

      ❈♦♠♦ ♦ ❧❛❞♦ ❞✐r❡✐t♦ ❞❛ ❡①♣r❡ssã♦ ❛❝✐♠❛ é ❧✐♠✐t❛❞♦ ♣♦r k✱ t❡♠✲s❡ q✉❡ √

      δ δ 1 /2 ϕ kδ.

    k − ϕ k ≤ ||R k || h − h ′ ≤

    1 H 00 (Γ )

      √

      δ

      k ∗

      ϕ P♦rt❛♥t♦✱ k − ϕ ≤ kδ, k ≥ 0.

      ❖ ❝r✐tér✐♦ ❞❡ ♣❛r❛❞❛ ✉t✐❧✐③❛❞♦ ♣❛r❛ ♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r é ♦ ♣r✐♥❝í♣✐♦ ❞❛ ❞✐s❝r❡♣â♥❝✐❛✱ ✉♠ ❝r✐tér✐♦ ❛ ♣♦st❡r✐♦r✐✱ ❞❡✈✐❞♦ ❛ ▼♦r♦③♦✈ ❬♣✳✽✸✱ ✷❪✳ ❖ ♣r✐♥❝í♣✐♦ ❞❛ ❞✐s❝r❡♣â♥❝✐❛ é ✉♠ ❝r✐tér✐♦ q✉❡ ❞❡t❡r♠✐♥❛ ❛

      

    δ

      ♣❛r❛❞❛ ❞❛ ✐t❡r❛çã♦ ❡♠ ¯k = k δ, h q✉❛♥❞♦✱ ♣❡❧❛ ♣r✐♠❡✐r❛ ✈❡③✱

      δ δ 1 /2

      h ϕ − F l ¯ ≤ τδ; 1 ′

      k H 00 (Γ )

      ♣❛r❛ τ > 1 ✜①♦✳ ❆ ♠♦t✐✈❛çã♦ ❞♦ ✉s♦ ❞♦ ♣r✐♥❝í♣✐♦ ❞❛ ❞✐s❝r❡♣â♥❝✐❛ ❝♦♠♦ ❝r✐tér✐♦ ❞❡

      ♣❛r❛❞❛ ♣❛r❛ ♦ ❛❧❣♦rít♠♦✱ ❛❧é♠ ❞♦ r❡s✉❧t❛❞♦ ❛❜❛✐①♦✱ é ♦ ❞❡ ❣❛r❛♥t✐r ✉♠ í♥❞✐❝❡ ❞❡ ♣❛r❛❞❛ ✜♥✐t♦✱ ❢❛❝✐❧✐t❛♥❞♦✱ ❛ss✐♠✱ ❞❡♠♦♥str❛r t❛①❛ ❞❡ ❝♦♥✈❡r✲ ❣ê♥❝✐❛ ♣❛r❛ ♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r✳

      ❆❧é♠ ❞✐ss♦✱ t❡♠ ❝♦♠♦ ♦❜❥❡t✐✈♦ ♦ ❞❡ t♦r♥á✲❧♦ ✉♠ ♠ét♦❞♦ ❞❡ r❡❣✉✲ ❧❛r✐③❛çã♦ ❞❡ ♦r❞❡♠ ót✐♠❛✳ ❊♠ ❜r❡✈❡s ♣❛❧❛✈r❛s✱ ✐ss♦ s✐❣♥✐✜❝❛ q✉❡ s❡ ♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r s♦❜ ❞❛❞♦s ♣❡rt✉r❜❛❞♦s ❝♦♠ ❡st❡ ❝r✐tér✐♦ ❞❡ ♣❛✲

      † 1/2 ′

      1 ✱ ❡♥tã♦ ♦ ❡rr♦

      00 l

      (Γ ) r❛❞❛ é ✉s❛❞♦ s♦❜ ❛ ❤✐♣ót❡s❡ ❛ ♣r✐♦r✐ F (h) ∈ H

      ❞♦ ♣✐♦r ❝❛s♦ ♣❛r❛ ❡st❡ ♠ét♦❞♦ é ♦ í♥✜♠♦ ❞♦s ❡rr♦s ❞♦s ♣✐♦r❡s ❝❛s♦s ♣❛r❛ ♦ ♠ét♦❞♦ ❝♦♠ ♦✉tr♦s ❝r✐tér✐♦s ❞❡ ♣❛r❛❞❛✳

      1/2 ′ 1/2

      (Γ (Γ ) Pr♦♣♦s✐çã♦ ✷✳✼ ❙❡❥❛♠ (f, g) ∈ H

      1 ) × H 1 ✉♠ ♣❛r ❞❡ ❞❛✲

      00 1/2 ′

      (Γ ) (ϕ) = h ❞♦s ❞❡ ❈❛✉❝❤② ❝♦♥s✐st❡♥t❡ ❡ ϕ ∈ H

      2 ✉♠❛ s♦❧✉çã♦ ❞❡ F l

      

    00

    δ δ δ

    1 /2

      k 1 ′ k+1 H 00 (Γ ) † δ

      h ϕ > 2δ q✉❛❧q✉❡r✳ ❙❡ − F l ✱ ❡♥tã♦ ϕ é ✉♠❛ ❛♣r♦①✐✲

      ♠❛çã♦ ♠❡❧❤♦r ♣❛r❛ ϕ ❞♦ q✉❡ ϕ ✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ ❛ ✐t❡r❛çã♦ ♥ã♦

      k

      t❡r♠✐♥❛ ❛♥t❡s ❞❡

      δ δ 1 /2

      h ϕ − F l ¯ ≤ 2δ. 1 ′

      

    k H

    00 (Γ )

      ✹✵

      δ k+1

      q✉❡

      2

      2 † δ † δ ∗ δ δ

      ϕ = ϕ h ϕ − ϕ − ϕ − F − F l

      k+1 k l k ∗ ∗ 2 † † ∗

      δ δ δ δ

      = ϕ , F h ϕ ∗ − ϕ − 2hϕ − ϕ − F l i

      k k l k ∗

    δ δ δ δ 1

      /2

      ϕ , F F h ϕ

    • hh − F l l − F l i

      k l k 1 ′ H 00 (Γ )

      2

    † δ δ δ δ

    1 /2

      = ϕ , h ϕ − ϕ − 2hh − h − F l i

      k k 1 ′ ∗ H 00 (Γ )

      2 δ δ

    1

    /2

      h ϕ − − F l

      k 1 ′ H 00 (Γ ) δ δ ∗ δ δ 1 /2 ϕ , (F F ϕ .

    • hh − F l l − I) h − F l i

      k l k 1 ′ H (Γ ) 00 ∗ l l F

      ❈♦♠♦ F é ✉♠ ♦♣❡r❛❞♦r ♥ã♦ ❡①♣❛♥s✐✈♦✱ t❡♠✲s❡ q✉❡ F − I é

      l

      s❡♠✐❞❡✜♥✐❞♦ ♥❡❣❛t✐✈♦✱ ❡♥tã♦

      2

      2 † †

    δ δ

      ϕ ϕ − ϕ k − − ϕ k+1

      ∗ ∗ δ δ δ δ 1 1 /2 /2 h ϕ h ϕ .

      ≥ − F l − F l − 2δ

      k

    1 ′ k

    1 ′ H

    00 (Γ ) H

    00 (Γ ) δ δ 1

      h ϕ /2 > 2δ P♦r ❤✐♣ót❡s❡✱ − F l ✱ ❡♥tã♦ ♦ ❧❛❞♦ ❞✐r❡✐t♦ ❞❛

      ′ k 1 H

    00 (Γ )

      ❞❡s✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛ é ♣♦s✐t✐✈♦✳

      2

      2 † † δ δ

      ϕ ϕ > 0 P♦rt❛♥t♦✱ − ϕ − − ϕ ✱ ♦✉ s❡❥❛✱

      k ∗ k+1 ∗ † †

    δ δ

      ϕ > ϕ ; − ϕ − ϕ

      

    k k+1

    ∗ ∗ † δ

      δ

      ❡✱ ❛ss✐♠✱ ϕ é ✉♠❛ ♠❡❧❤♦r ❛♣r♦①✐♠❛çã♦ ♣❛r❛ ϕ ❞♦ q✉❡ ϕ ✳

      k+1 k

      ❖ t❡♦r❡♠❛ ❞❡st❛ s❡çã♦ ❣❛r❛♥t❡ q✉❡ ♦ ♣r✐♥❝í♣✐♦ ❞❛ ❞✐s❝r❡♣â♥❝✐❛ ❞❡✲ t❡r♠✐♥❛ ✉♠ í♥❞✐❝❡ ❞❡ ♣❛r❛❞❛ ✜♥✐t♦ ❡✱ ❛✐♥❞❛ ♠❛✐s✱ ❢♦r♥❡❝❡ ❛ t❛①❛ ❞❡

      − δ

      2

      ❝♦♥✈❡r❣ê♥❝✐❛ k δ, h = δ ✳ ❚❡♦r❡♠❛ ✷✳✷ ❙❡❥❛ τ > 1 ✜①♦✳ ❊♥tã♦✱ ♦ ♣r✐♥❝í♣✐♦ ❞❛ ❞✐s❝r❡♣â♥❝✐❛ ❞❡✲

      δ

      ) t❡r♠✐♥❛ ✉♠ í♥❞✐❝❡ ❞❡ ♣❛r❛❞❛ ✜♥✐t♦ k(δ, h ♣❛r❛ ❛ ✐t❡r❛çã♦ ❞❡ ▲❛♥❞✇❡✲

      − δ

      2

      ❜❡r ❝♦♠ k δ, h = δ ✳ ❉❡♠♦♥str❛çã♦✿ ❈♦♥s✐❞❡r❡ ♠♦♠❡♥t❛♥❡❛♠❡♥t❡ {ϕ k } ❛ ✐t❡r❛çã♦ ❞❡ ▲❛♥❞✇❡✲ ❜❡r ❝♦♠ ❞❛❞♦ ❡①❛t♦ h ♣♦r ❢❛❝✐❧✐❞❛❞❡ ♥❛ ♥♦t❛çã♦✳

      P❡❧❛ ❞❡♠♦♥str❛çã♦ ❛♥t❡r✐♦r✱ t❡♠✲s❡ q✉❡

      2

      2

      2 † † 1 /2 ϕ ϕ (ϕ .

      − ϕ j − − ϕ j+1 ≥ ||h − F l j )|| 1 ′

      2

      H 1 /2 00 (Γ 1

      l

      ) ′

      H 1 /2 00 (Γ 1

      (h − F l (ϕ ))

      k

      )

      ∗ l

      F

      l

      ≤ (I − F

      ) ′

      − F l (ϕ )

      ∗ l

      δ

      h

      k

      )

      ∗ l

      = (I − F l F

      ′

      /2 00 (Γ 1 )

      δ k H 1

      − F l ϕ

      δ

      . ❆ss✐♠✱ h

      F

      )

      − ϕ

      − ϕ

      −

      δ

      2

      1

      − ϕ

      †

      ϕ

      2

      −

      < τ δ ✱ ❛♣❡♥❛s s❡ k > (τ − 1)

      1 ∗

      †

      k

      ϕ

      − 1/2

      . P♦rt❛♥t♦✱ t❡♠✲s❡ q✉❡ δ + k

      1 ∗

      − ϕ

      †

      ϕ

      

    1/2

      ≤ δ + k

      ) ′

      δ H 1 /2 00 (Γ 1

      h − h

      1 ∗

      †

      ✹✶ ϕ

      ) ′

      F

      k−1 ) − F l

      (ϕ

      k ) = h − F l

      ◆♦t❡ q✉❡ h − F l (ϕ

      ; ♦♥❞❡ ✈❛❧❡ ❛ ú❧t✐♠❛ ❞❡s✐❣✉❛❧❞❛❞❡ ♣❡❧❛ ♠♦♥♦t♦♥✐❝✐❞❛❞❡ ❞❛s ♥♦r♠❛s ❞♦s r❡sí❞✉♦s✳

      ) ′

      2 H 1 /2 00 (Γ 1

      )||

      l (ϕ k

      ≥ k ||h − F

      2 H 1 /2 00 (Γ 1

      (h − F l (ϕ

      j )||

      ||h − F l (ϕ

      k X j=1

      ≥

      

    2

      − ϕ k+1

      †

      − ϕ

      2 ∗

      1

      − ϕ

      †

      ∗ l

      k−1

      ϕ

      l

      − 1/2

      ≤ k

      ) ′

      k )|| H 1 /2 00 (Γ 1

      = ||h − F l (ϕ

      ) ′

      H

    1

    /2

    00 (Γ

    1

      (h − F l (ϕ ))

      k

      )

      ∗ l

      F

      ✳ ❊♥tã♦✱ ♣❡❧♦ q✉❡ ❢♦✐ ❢❡✐t♦ ❛❝✐♠❛✱ (I − F

      )) = (I − F l

      l (ϕ ))

      (h − F

      k

      )

      l F ∗ l

      ) = (I − F

      l (ϕ k

      )) ; ❡♥tã♦✱ ♣♦r ✐♥❞✉çã♦✱ t❡♠✲s❡ q✉❡ h−F

      k−1

      ) (h − F l (ϕ

      ∗ l

      F

    • (I − F

      ✹✷

      − δ

      2

      ❖ ú❧t✐♠♦ r❡s✉❧t❛❞♦ ❞♦ ❝❛♣ít✉❧♦ ❞✐③ q✉❡ ♦ ❛❧❣♦rít✐♠♦ ❞❡ ▲❛♥❞✇❡❜❡r ❝♦♠ ♦ ♣r✐♥❝í♣✐♦ ❞❛ ❞✐s❝r❡♣â♥❝✐❛ ♣❛r❛ τ > 1 ✜①♦ é✱ ❞❡ ❢❛t♦✱ ✉♠ ♠ét♦❞♦ ❞❡ r❡❣✉❧❛r✐③❛çã♦ ❞❡ ♦r❞❡♠ ót✐♠❛✳

      1/2 ′ 1/2

      (Γ (Γ ) Pr♦♣♦s✐çã♦ ✷✳✽ ❙❡ (f, g) ∈ H

      1 ) × H 1 é ✉♠ ♣❛r ❞❡ ❞❛❞♦s

      00

      ❞❡ ❈❛✉❝❤② ❝♦♥s✐st❡♥t❡✱ ❡♥tã♦ ❛ ✐t❡r❛çã♦ ❞❡ ▲❛♥❞✇❡❜❡r ❝♦♠ ♦ ♣r✐♥❝í♣✐♦ ❞❛ ❞✐s❝r❡♣â♥❝✐❛ ❝♦♠ τ > 1 ✜①♦ é ✉♠ ♠ét♦❞♦ ❞❡ r❡❣✉❧❛r✐③❛çã♦ ❞❡ ♦r❞❡♠ ót✐♠❛✳

      ❊st❛ ❞❡♠♦♥str❛çã♦ ❡♥❝♦♥tr❛✲s❡ ❡♠ ❬✷❪✳ P♦rt❛♥t♦✱ ♥❡st❡ ❝❛♣ít✉❧♦✱ ❢♦✐ ❞❡♠♦♥str❛❞♦ ❛ ❛♥á❧✐s❡ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛

      ♣❛r❛ ♦ ♠ét♦❞♦ ❞❡ ▼❛③✬✐❛ s♦❜ ❞❛❞♦s ❡①❛t♦s ❡ q✉❡ ♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r ❧✐♥❡❛r é ✉♠ ♠ét♦❞♦ ❞❡ r❡❣✉❧❛r✐③❛çã♦✳ ❆❧é♠ ❞✐ss♦✱ ❢♦✐ ♦❜t✐❞♦ ✉♠❛ t❛①❛ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ♣❛r❛ ♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r ❡ ✈✐st♦ q✉❡ é ✉♠ ♠ét♦❞♦ ❞❡ ♦r❞❡♠ ót✐♠❛✳

      ❖ ♣ró①✐♠♦ ❝❛♣ít✉❧♦ r❡❧❛❝✐♦♥❛ ♦s ♠ét♦❞♦s ❛♣r❡s❡♥t❛❞♦s✳ ▼♦str❛✲ s❡ q✉❡ ❛s ✐t❡r❛çõ❡s ❞♦s ♠ét♦❞♦s sã♦ ✐❞ê♥t✐❝❛s ❡ ❝♦♠ ✐ss♦✱ ✉♠ ❝r✐tér✐♦ ❞❡ ♣❛r❛❞❛ ❡ t❛①❛ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ♣❛r❛ ♦ ♠ét♦❞♦ ❞❡ ▼❛③✬✐❛ ♣♦❞❡ s❡r ❝♦♥s✐❞❡r❛❞♦ s♦❜ ❞❛❞♦s ♣❡rt✉r❜❛❞♦s✱ ❥á q✉❡ ❛ ❛♥á❧✐s❡ ❢♦✐ ❢❡✐t❛ ❛♣❡♥❛s ♣❛r❛ ❞❛❞♦s ❡①❛t♦s✳ ▲♦❣♦✱ ❝♦♥❝❧✉✐✲s❡ q✉❡ ♦ ♠ét♦❞♦ ❞❡ ▼❛③✬✐❛ é ✉♠ ♠ét♦❞♦ ❞❡ r❡❣✉❧❛r✐③❛çã♦ ❛♣ós ♠♦str❛r ❛ ✐❣✉❛❧❞❛❞❡ ❡♥tr❡ ❛s ✐t❡r❛çõ❡s ❞♦s ♠ét♦❞♦s✳

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

      ❆ r❡❧❛çã♦ ❡♥tr❡ ♦s ♠ét♦❞♦s ❞❡ ▼❛③✬✐❛ ❡ ❞❡ ▲❛♥❞✇❡❜❡r ❡♥❝♦♥tr❛❞❛ é ❛ ❞❡ q✉❡ ❛s ✐t❡r❛çõ❡s ❝❛❧❝✉❧❛❞❛s ❞❡ss❡s ❛❧❣♦rít♠♦s ❞❡ ♣♦♥t♦ ✜①♦ sã♦

      1/2 ′

      (Γ ) ✐❞ê♥t✐❝❛s✱ ✐st♦ é✱ L(ϕ) = T (ϕ)✱ ∀ϕ ∈ H

      2 ✱ ❝♦♠ L ❡ T s❡♥❞♦

      00

      ♦s ♦♣❡r❛❞♦r❡s ❞♦ ❛❧❣♦rít♠♦ ❞❡ ▲❛♥❞✇❡❜❡r ❡ ▼❛③✬✐❛✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳

      1/2 ′

      (Γ ) ❉❡♠♦♥str❛r ❛ ✐❣✉❛❧❞❛❞❡ L(ϕ) = T (ϕ)✱ ∀ϕ ∈ H

      2 ✱ s✐❣♥✐✜❝❛ ♣r♦✈❛r

      00

      ❞❡ ❢❛t♦ q✉❡

      ∗ 1/2 ′

      ϕ + F (ϕ)) = T (ϕ) + T (Γ ) ; (h − F l l , ∀ϕ ∈ H

      2 l

      00

      ♦♥❞❡ h = g − F f ✳ ❈♦♠♦ ❝♦♥❝❧✉sã♦ ❛♣ós ♣r♦✈❛r ❡ss❛ ✐❣✉❛❧❞❛❞❡✱ t❡♠✲s❡ q✉❡ ♣❛r❛ ❞❛❞♦s

      δ δ

      , g ❝♦♠ r✉í❞♦s f t❛✐s q✉❡

      δ δ 2 /2 1 f − f g − g ≤ δ, 1 1 ′

    • L (Γ ) H

      00 (Γ )

      ♣♦❞❡✲s❡ ♦❜s❡r✈❛r ❛ ❡st❛❜✐❧✐❞❛❞❡ ❞♦ ♠ét♦❞♦ ❡✱ ❛ss✐♠✱ ✉s❛r ❝♦♠♦ ❝r✐tér✐♦ ❞❡ ♣❛r❛❞❛ ♦ ♣r✐♥❝í♣✐♦ ❞❛ ❞✐s❝r❡♣â♥❝✐❛ ♣❛r❛ ✈❡r q✉❡ ♦ ♠ét♦❞♦ ❞❡ ▼❛③✬✐❛ é ✉♠ ♠ét♦❞♦ ❞❡ r❡❣✉❧❛r✐③❛çã♦ ❡ t❡♠ ❛ ♠❡s♠❛ t❛①❛ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ q✉❡ ♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r✳

      P❛r❛ ❡st❛❜❡❧❡❝❡r ❛ ✐❣✉❛❧❞❛❞❡ ❞❛s ✐t❡r❛çõ❡s ♦s três ❧❡♠❛s ❛ s❡❣✉✐r s❡

      1/2 ∗ ′

      (ψ) (Γ ) ❢❛③❡♠ ♥❡❝❡ssár✐♦s✳ ❖ ♣r✐♠❡✐r♦ ❞❡❧❡s ❝❛❧❝✉❧❛ F ✱ ∀ψ ∈ H

      1 ✳

      ✹✹

      1/2 ′

      1

      ▲❡♠❛ ✸✳✶ ❙❡❥❛♠ ψ ∈ H ❡ q ∈ H s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛ ❞❡

      00

      ✈❛❧♦r ❞❡ ❝♦♥t♦r♥♦ ✭❜❡♠ ♣♦st♦ ❬❆✳✶❪✮ 2 1 Γ = 0; q ν Γ = ψ.

      ∆q = 0, Ω; q | |

      ∗ 2

      (ψ) = q ν Γ ❊♥tã♦✱ t❡♠✲s❡ q✉❡ F | ✳

      l 1/2 1/2 ′ ′

      (Γ ) (Γ ) ❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ (ψ, ϕ) ∈ H

      1 ×H 2 ✳ ❙❡❥❛♠ v, w, q, r ∈

      00

      00

      s♦❧✉çõ❡s ❞♦s s❡❣✉✐♥t❡s ♣r♦❜❧❡♠❛s ❞❡ ✈❛❧♦r ❞❡ ❝♦♥t♦r♥♦ 1 2 = 0; v = ϕ;

      ∆v = 0, Ω; v | Γ ν | Γ 1 1 2 ∆w = 0, Ω; w = v = 0;

      ν | Γ ν | Γ ; w | Γ 1 2

      ∆q = 0, Ω; q ν Γ Γ = 0; | = ψ; q | 1 2 2

      Γ = 0; r ν Γ = q ν Γ ;

      ∆r = 0, Ω; r | | | r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❊♥tã♦✱ ✉t✐❧✐③❛♥❞♦ ❛ ❢ór♠✉❧❛ ❞❡ ●r❡❡♥ ❬❆✳✹❪ Z Z Z

      t

      ∆uvdx = u ν 1 ⊔ 2 vd∂Ω − (∇u) (∇v) dx;

      Ω Γ Γ Ω

      ❡ ❛s ❝♦♥❞✐çõ❡s ❞❡ ❢r♦♥t❡✐r❛ ❞♦s ♣r♦❜❧❡♠❛s ❞❡ ✈❛❧♦r ❞❡ ❝♦♥t♦r♥♦ ❛❝✐♠❛✱ tê♠✲s❡ q✉❡ Z 1 t

      /2

      = hF l (ϕ), ψi (∇w) (∇q) dx 1 ′

      H 00 (Γ ) Z

      = w qd∂Ω 1 ⊔ 2 ν Z Γ Γ = w qd∂Ω 1 ν Z Γ

      = v qd∂Ω 1 ν Z Γ = v ν qd∂Ω 1 ⊔ 2 Z

      Γ Γ t

      = (∇v) (∇q) dx;

      Ω

      ✹✺ Z 1 t 2 = /2 hϕ, q ν | Γ i (∇v) (∇r) dx 2 ′

      H 00 (Γ ) Z

      = vr ν d∂Ω 1 ⊔ 2 Z

      Γ Γ

      = vr d∂Ω 2 ν Z Γ = vq d∂Ω 2 ν Z Γ

      = vq d∂Ω 1 ⊔ 2 ν Z Γ Γ t = (∇v) (∇q) dx. 11

      /2 2 /2

    l ν Γ

      ▲♦❣♦✱ ✈❛❧❡ q✉❡ hF (ϕ), ψi ′ = hϕ, q | i ′ ✱ ♦✉ s❡❥❛✱ 1 2 H 00 (Γ ) H 00 (Γ ) ∗ F (ψ) = q 2

      ν | Γ ✳ l

      ❖s ♣ró①✐♠♦s ❞♦✐s ❧❡♠❛s ✈✐s❛♠ ❢❛❝✐❧✐t❛r ❛ ❞❡♠♦♥str❛çã♦ ❞❛ ✐❣✉❛❧❞❛❞❡ ❡♥tr❡ ❛s ✐t❡r❛çõ❡s ❞❡ ♣♦♥t♦ ✜①♦ ❞❡ ▼❛③✬✐❛ ❡ ❞❡ ▲❛♥❞✇❡❜❡r✳

      1/2 ′ ∗

      (Γ

      2 ) F l l (ϕ)

      ▲❡♠❛ ✸✳✷ P❛r❛ ϕ ∈ H ✱ t❡♠✲s❡ q✉❡ F (ϕ) = ϕ − T ✳

      

    00 l

    1/2 ′

      1

      (Γ ) (Ω) ❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ ϕ ∈ H

      2 ✳ ❙❡❥❛♠ v, w ∈ H s♦❧✉çõ❡s

      00

      ❞♦s s❡❣✉✐♥t❡s ♣r♦❜❧❡♠❛s ❞❡ ✈❛❧♦r ❞❡ ❝♦♥t♦r♥♦ 1 = 0; v 2 = ϕ; ∆v = 0, Ω; v | Γ ν | Γ ∆w = 0, Ω; w 1 2 2 ;

      ν | Γ = 0; w | Γ = v | Γ

      r❡s♣❡❝t✐✈❛♠❡♥t❡✳ (ϕ) = v 1 (ϕ) = w 2

      ❊♥tã♦✱ tê♠✲s❡ q✉❡ F l ν | Γ ❡ q✉❡ T l ν | Γ ✱ ♣♦r ❞❡✜♥✐çã♦ ❞♦s ♦♣❡r❛❞♦r❡s ❧✐♥❡❛r❡s F l ❡ T l ✳ ❆❧é♠ ❞✐ss♦✱ ❞❡✜♥✐♥❞♦ q := v − w✱ ♦✉

      1

      (Ω) s❡❥❛✱ q ∈ H é s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛ ❞❡ ✈❛❧♦r ❞❡ ❝♦♥t♦r♥♦ ♠✐st♦ ∆q = 0, Ω; q 1 = v 1 2 = 0;

      ν | Γ ν | Γ ; q | Γ ∗

      (v 1 ) = q 2 t❡♠✲s❡ q✉❡ F ν | Γ ν | Γ ✱ ♣❡❧♦ ❧❡♠❛ ❛♥t❡r✐♦r✳

      l 2 2 = v 2 2 (ϕ)

      P♦rt❛♥t♦✱ ❝♦♠♦ q ν | Γ = (v − w) ν | Γ ν | Γ − w ν | Γ = ϕ − T l ✱

      ∗

      l

      F (ϕ) t❡♠✲s❡ q✉❡ F l (ϕ) = ϕ − T l ✳

      ❆❣♦r❛✱ ♦ ú❧t✐♠♦ ❧❡♠❛ ❛♥t❡s ❞♦ t❡♦r❡♠❛ ♣r✐♥❝✐♣❛❧ ❞❡st❡ ❝❛♣ít✉❧♦✳

      ✹✻

      1/2 1/2 ′

      ▲❡♠❛ ✸✳✸ P❛r❛ ♦ ♣❛r ❞❡ ❞❛❞♦s ❞❡ ❈❛✉❝❤② (f, g) ∈ H )×H

      00 ∗

      (h) = T f ❝♦♥s✐st❡♥t❡✱ t❡♠✲s❡ q✉❡ F ✱ ❡♠ q✉❡ h = g − F ✳

      l

      

    1

      (Ω) ❉❡♠♦♥str❛çã♦✿ ❙❡❥❛♠ v, w ∈ H s♦❧✉çõ❡s ❞♦s s❡❣✉✐♥t❡s ♣r♦❜❧❡✲ ♠❛s ❞❡ ✈❛❧♦r ❞❡ ❝♦♥t♦r♥♦ 1 = f ; v 2 = 0;

      ∆v = 0, Ω; v | Γ ν | Γ ∆w = 0, Ω; w 1 2 2 ;

      ν | Γ = g; w | Γ = v | Γ

      r❡s♣❡❝t✐✈❛♠❡♥t❡✳ = v 1 = w 2

      ❊♥tã♦✱ tê♠✲s❡ q✉❡ F f ν | Γ ❡ q✉❡ T ν | Γ ✱ ♣♦r ❞❡✜♥✐çã♦✳

      1

      (Ω) ◆♦t❡ q✉❡✱ ❞❡✜♥✐♥❞♦ r := w − v✱ r ∈ H é s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛ ❞❡ ✈❛❧♦r ❞❡ ❝♦♥t♦r♥♦ ♠✐st♦ ✭❜❡♠ ♣♦st♦ ❬❆✳✶❪✮

      ∆r = 0, Ω; r 1 1 2 = 0;

      ν | Γ = g − v ν | Γ ; r | Γ ∗

      (h) = r 2 2 = w 2 ❛ss✐♠✱ t❡♠✲s❡ q✉❡ F ν | Γ = (w − v) ν | Γ ν | Γ

      l ∗

      (h) = T P♦rt❛♥t♦✱ F ✳

      l

      ❆ss✐♠✱ ❝♦♠ ❡ss❛s ❢❡rr❛♠❡♥t❛s✱ ♦ t❡♦r❡♠❛ ❛ s❡❣✉✐r é ❢❛❝✐❧♠❡♥t❡ ❞❡✲ ♠♦♥str❛❞♦✳

      1/2

      (Γ ) ❚❡♦r❡♠❛ ✸✳✶ P❛r❛ ϕ ∈ H

      2 ✱ t❡♠✲s❡ q✉❡ L(ϕ) = T (ϕ)✳

      00 ∗ ∗

      (ϕ)) = ϕ + F ❉❡♠♦♥str❛çã♦✿ ◆♦t❡ q✉❡ L(ϕ) := ϕ + F (h − F l (h) −

      l l ∗ ∗ ∗

      F F (ϕ) F (ϕ) (h)+T (ϕ)

      l ✳ ❈♦♠♦ F l (ϕ) = ϕ−T l ✱ t❡♠✲s❡ q✉❡ L(ϕ) = F l ✳ l l l ∗

      (h) = T (ϕ) + T ▼❛s F ✱ ❛ss✐♠✱ L(ϕ) = T l ✳ ▲♦❣♦✱ ❝♦♠♦ T (ϕ) =

      l 1/2 ′

      T (ϕ) + T (Γ )

      l ✱ ❝♦♥❝❧✉✐✲s❡ q✉❡ L(ϕ) = T (ϕ)✱ ∀ϕ ∈ H 2 ✳

      00

      ❊♥tã♦✱ ❝♦♥❝❧✉✐✲s❡ ❛ss✐♠✱ ❛❧é♠ ❞❛ ❡st❛❜✐❧✐❞❛❞❡ ❞♦ ♠ét♦❞♦ ❞❡ ▼❛③✬✐❛✱ q✉❡ ❞❡✈❡✲s❡ ✉t✐❧✐③❛r ♦ ♣r✐♥❝í♣✐♦ ❞❛ ❞✐s❝r❡♣â♥❝✐❛ ❝♦♠♦ ❝r✐tér✐♦ ❞❡ ♣❛r❛❞❛ ♣❛r❛ ♦ ❛❧❣♦rít♠♦ ❞❡ ▼❛③✬✐❛✱ s❡ ❢♦r♥❡❝✐❞♦ s♦♠❡♥t❡ ❞❛❞♦s ❝♦♠ r✉í❞♦s✳ P❛r❛ ❛ ❡st❛❜✐❧✐❞❛❞❡✱ ❜❛st❛ ♦❜s❡r✈❛r ✉s❛r ❛ ✐❣✉❛❧❞❛❞❡ ❡♥tr❡ ❛s ✐t❡r❛çõ❡s ❡ s❡❣✉✐r ❛s ❧✐♥❤❛s ❞❛ ❞❡♠♦♥str❛çã♦ ♣❛r❛ ♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r✳

      δ δ

      , g ❉❡ ❢❛t♦✱ ♥♦t❡ q✉❡ s❡ f é ✉♠ ♣❛r ❞❡ ❞❛❞♦s t❛✐s q✉❡

      δ δ 2 /2 1 f − f 1 g − g ≤ δ, 1

    • L (Γ ) H (Γ )
    • 00

        ♣❛r❛ (f, g) ✉♠ ♣❛r ❞❡ ❞❛❞♦s ❝♦♥s✐st❡♥t❡✱ ❡♥tã♦ ♣♦❞❡✲s❡ ❝❛❧❝✉❧❛r ♦ t❡r♠♦

        δ δ δ

        h = g − F f t❛❧ q✉❡

        δ

      1

      /2

      h − h ≤ δ. 1 ′

        ✹✼

        δ δ δ δ

        ϕ = T ϕ − ϕ − ϕ

        k+1 k k k

      ∗ ∗

      δ δ

        = L ϕ − ϕ

        k k ∗ ∗ δ δ

        = F h ϕ − F l

        l k ∗ ∗ δ δ 1

        /2

        h ϕ ≤ ||F || − F l

        l k 1 ′ H 00 (Γ ) δ δ 1 /2 h ϕ .

        ≤ − F l

        k 1 ′ H 00 (Γ )

        ❆ss✐♠✱ ❝♦♠♦ ❛s ✐t❡r❛çõ❡s ❞♦s ❛❧❣♦rít♠♦s sã♦ ❞❡ ♣♦♥t♦ ✜①♦ ❡ sã♦ ✐❣✉❛✐s✱ ♦✉ s❡❥❛✱

        δ δ δ

        ϕ = T ϕ = L ϕ , k ∈ N,

        k+1 k k

        ❡♥tã♦✱ ❝♦♠♦ í♥❞✐❝❡ ❞❡ ♣❛r❛❞❛ ♣❛r❛ ♦ ♠ét♦❞♦ ❞❡ ▼❛③✬✐❛✱ ❜❛st❛ ❡s❝♦❧❤❡r

        δ

        ¯ k = k δ, h ✐❣✉❛❧ ❛♦ ¯k ❞❛❞♦ ♣❡❧♦ ♣r✐♥❝í♣✐♦ ❞❛ ❞✐s❝r❡♣â♥❝✐❛ ✉t✐❧✐③❛❞♦

        δ

        (ϕ) = h ♥♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r ✭❛♣❧✐❝❛❞♦ ❛♦ ♣r♦❜❧❡♠❛ ✐♥✈❡rs♦ F l ✮

        − δ

        2

        ❡ ❝♦♥❝❧✉✐r ❛ ♠❡s♠❛ t❛①❛ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ k δ, h = δ ♣❛r❛ ♦ ♠ét♦❞♦ ❞❡ ▼❛③✬✐❛✳ P❛r❛ ✐ss♦✱ ❜❛st❛ s❡❣✉✐r ♦s ♣❛ss♦s ❞❛ ❞❡♠♦♥str❛çã♦ ❞♦ t❡♦r❡♠❛ s♦❜r❡ ♦ í♥❞✐❝❡ ❞❡ ♣❛r❛❞❛ ✜♥✐t♦ ❡ t❛①❛ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ♣❛r❛ ♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r✳

        P♦rt❛♥t♦✱ t❡♠✲s❡ q✉❡ ♦ ♠ét♦❞♦ ❞❡ ▼❛③✬✐❛ é t❛♠❜é♠ ✉♠ ♠ét♦❞♦ ❞❡ r❡❣✉❧❛r✐③❛çã♦ ❡ ❛ t❛①❛ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ❝♦♥❝❧✉í❞❛ é ❛ ♠❡s♠❛ ❞♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r✳

        ❱❛❧❡ ♥♦t❛r q✉❡ ❛s ❞❡♠♦♥str❛çõ❡s ♦♠✐t✐❞❛s ♥❡ss❛ ú❧t✐♠❛ ❞✐s❝✉ssã♦ s❡❣✉❡♠ ❡①❛t❛♠❡♥t❡ ❛s ♠❡s♠❛s ❧✐♥❤❛s✱ ♦❜s❡r✈❛♥❞♦ ❛♣❡♥❛s ❛ ✐❣✉❛❧❞❛❞❡ ❡♥tr❡ ❛s ✐t❡r❛çõ❡s✳

        ❆ss✐♠✱ ❛ ✐❣✉❛❧❞❛❞❡ ❡♥tr❡ ❛s ✐t❡r❛çõ❡s ❞♦s ♠ét♦❞♦s ❢♦✐ ❢✉♥❞❛♠❡♥t❛❧ ♣❛r❛ t❡r♠✐♥❛r ❛ ❛♥á❧✐s❡ ❞♦ ♠ét♦❞♦ ❞❡ ▼❛③✬✐❛✳

        ❈♦♥s✐❞❡r❛çõ❡s ❋✐♥❛✐s

        ◆❡st❡ tr❛❜❛❧❤♦ ❢♦✐ ❛♣r❡s❡♥t❛❞♦ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤② ❡❧í♣t✐❝♦ ❡✱ ❛♥❛❧✐s❛❞♦✱ s❡❣✉♥❞♦ ❍❛❞❛♠❛r❞✱ q✉❡ ♦ ♠❡s♠♦ é ✉♠ ♣r♦❜❧❡♠❛ ♠❛❧ ♣♦st♦✳ ❖ ♣r♦❜❧❡♠❛ ❛♥❛❧✐s❛❞♦ ❢♦✐ ♦ ❞❛ r❡❝♦♥str✉çã♦ ❞♦ tr❛ç♦ ❞❡ ✉♠❛ s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤② s♦❜r❡ ❛ ♣❛rt❡ ❞❛ ❢r♦♥t❡✐r❛ ♦♥❞❡ ♥ã♦ s❡ t✐♥❤❛ ❞❛❞♦s❀ ♣❛r❛ ❡st❡ ♣r♦❜❧❡♠❛ ❢♦r❛♠ ❛♣r❡s❡♥t❛❞♦s ❡ ❛♥❛❧✐s❛❞♦s ❞♦✐s ♠ét♦❞♦s ❝❛✲ ♣❛③❡s ❞❡ r❡s♦❧✈ê✲❧♦✱ ♦ ♠ét♦❞♦ ❞❡ ▼❛③✬✐❛ ❡ ♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r✳ ◆❛ ❛♥á❧✐s❡ ❞♦ ♠ét♦❞♦ ❞❡ ▼❛③✬✐❛✱ ❢♦✐ ❞❡♠♦♥str❛❞♦ ❛ ❛♥á❧✐s❡ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ♣❛r❛ ❞❛❞♦s ❡①❛t♦s ✉t✐❧✐③❛♥❞♦ ✉♠❛ t♦♣♦❧♦❣✐❛ ♥ã♦ ✉s✉❛❧ s♦❜r❡ ♦ ❡s♣❛ç♦ ❡♠ q✉❡stã♦✳ ❆♣ós ❛ ❛♥á❧✐s❡ ❝♦♠♣❧❡t❛ ❞♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r✱ ♦✉ s❡❥❛✱ ✉♠❛ ✈❡③ ❞❡♠♦♥str❛❞♦ q✉❡ é ✉♠ ♠ét♦❞♦ ❞❡ r❡❣✉❧❛r✐③❛çã♦ ❡ ♦❜t✐❞♦ ❛ t❛①❛ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛✱ ❢♦✐ ❡♥❝♦♥tr❛❞❛ ✉♠❛ r❡❧❛çã♦ ❡♥tr❡ ♦s ♠ét♦❞♦s ❞❡ ▼❛③✬✐❛ ❡ ❞❡ ▲❛♥❞✇❡❜❡r✳ ❙❡♥❞♦✱ ❛ss✐♠✱ ♣♦ssí✈❡❧ ❝♦♠♣❧❡t❛r ❛ ❛♥á❧✐s❡ ❞♦ ♠ét♦❞♦ ❞❡ ▼❛③✬✐❛✱ ✐st♦ é✱ s♦❜ ❞❛❞♦s ❝♦♠ r✉í❞♦s✳

        ❉❡ss❛ ❢♦r♠❛✱ ♣♦❞❡✲s❡ ❝♦♥s✐❞❡r❛r q✉❡ ❛ ♣❛rt❡ t❡ór✐❝❛ ❡stá ❝♦♠♣❧❡t❛❀ ♣♦rt❛♥t♦✱ ♣❛r❛ ❝♦♠♣❧❡♠❡♥t❛r✱ ❜✉s❝❛✲s❡ ✐♠♣❧❡♠❡♥t❛r ❞❡✈✐❞❛♠❡♥t❡ ❡ss❡s ♠ét♦❞♦s ❡ ❢❛③❡r ❡①♣❡r✐♠❡♥t♦s ♥✉♠ér✐❝♦s✳ P❛r❛ ❛❧é♠ ❞✐ss♦✱ ❡♠ tr❛❜❛❧❤♦s ❢✉t✉r♦s✱ ❛❜♦r❞❛r ♦s ♠ét♦❞♦s ✐t❡r❛t✐✈♦s ❞❡ r❡❣✉❧❛r✐③❛çã♦ t✐♣♦ ❑❛❝③♠❛r③ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤② ❡❧í♣t✐❝♦✳

        ✺✵

        ❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s

        ❬✶❪ ❆◆❉❘■❊❯❳✱ ❙✳❀ ❇❆❘❆◆●❊❘✱ ❚✳ ◆✳❀ ❆❇❉❆✱ ❆✳ ❇✳✳ ❙♦❧✈✐♥❣ ❈❛✉✲ ❝❤② ♣r♦❜❧❡♠s ❜② ♠✐♥✐♠✐③✐♥❣ ❛♥ ❡♥❡r❣②✲❧✐❦❡ ❢✉♥❝t✐♦♥❛❧✳ ■♥✈❡rs❡ Pr♦❜❧❡♠s✳ ✈✳ ✷✷✱ ♣✳ ✶✶✺✲✶✸✸✱ ✷✵✵✻✳

        ❬✷❪ ❊◆●▲✱ ❍✳ ❲✳❀ ❍❆◆❑❊✱ ▼✳❀ ◆❊❯❇❆❯❊❘✱ ❆✳✳ ❘❡❣✉❧❛r✐③❛t✐♦♥ ♦❢ ■♥✈❡rs❡ Pr♦❜❧❡♠s✳ ▼❛t❤❡♠❛t✐❝s ❛♥❞ ■ts ❆♣♣❧✐❝❛t✐♦♥s✳ ✈✳ ✸✼✺✱ ❑❧✉✇❡r ❆❝❛❞❡♠✐❝ P✉❜❧✐s❤❡rs✱ ❉♦r❞r❡❝❤t✱ ✶✾✾✻✳

        ❬✸❪ ●■▲❇❆❘●✱ ❉✳❀ ❚❘❯❉■◆●❊❘✱ ◆✳ ❙✳✳ ❊❧❧✐♣t✐❝ P❛rt✐❛❧ ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s ♦❢ ❙❡❝♦♥❞ ❖r❞❡r✳ ●r✉♥❞❧❡❤r❡♥ ❞❡r ♠❛t❤❡♠❛t✐s✲ ❝❤❡♥ ❲✐ss❡♥s❝❤❛❢t❡♥✳ ✈✳ ✷✷✹✱ ✷ ❡❞✳✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ❇❡r❧✐♥ ❡ ◆❡✇ ❨♦r❦✱ ✶✾✽✸✳

        ❬✹❪ ▲❊■❚➹❖✱ ❆✳✳ ❆♥ ✐t❡r❛t✐✈❡ ♠❡t❤♦❞ ❢♦r s♦❧✈✐♥❣ ❡❧❧✐♣t✐❝ ❈❛✉❝❤② ♣r♦✲ ❜❧❡♠✳ ◆✉♠❡r✐❝❛❧ ❋✉♥❝t✐♦♥❛❧ ❆♥❛❧②s✐s ❛♥❞ ❖♣t✐♠✐③❛t✐♦♥✳ ✈✳ ✷✶✱ ♥✳ ✺✲✻✱ ♣✳ ✼✶✺✲✼✹✷✱ ✷✵✵✵✳

        ❬✺❪ ▲❊■❚➹❖✱ ❆✳❀ ❇❆❯▼❊■❙❚❊❘✱ ❏✳✳ ❆♥ ✐t❡r❛t✐✈❡ ♠❡t❤♦❞s ❢♦r s♦❧✲ ✈✐♥❣ ✐❧❧✲♣♦s❡❞ ♣r♦❜❧❡♠s ♠♦❞❡❧❡❞ ❜② ♣❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✳ ❏♦✉r♥❛❧ ♦❢ ■♥✈❡rs❡ ❛♥❞ ■❧❧✲P♦s❡❞ Pr♦❜❧❡♠s✳ ✈✳ ✾✱ ♥✳ ✶✱ ♣✳ ✶✲✶✼✱ ✷✵✵✶✳

        ❬✻❪ ▲■❖◆✱ ❏✳ ▲✳❀ ❉❆❯❚❘❆❨✱ ❘✳✳ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❛♥❞ ◆✉♠❡✲ r✐❝❛❧ ▼❡t❤♦❞s ❢♦r ❙❝✐❡♥❝❡ ❛♥❞ ❚❡❝❤♥♦❧♦❣②✳ ❋✉♥❝t✐♦♥❛❧ ❛♥❞ ❱❛✲ r✐❛t✐♦♥❛❧ ▼❡t❤♦❞s✳ ✈✳ ✶✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ❇❡r❧✐♥ ❡ ❍❡✐❞❡❧❜❡r❣✱ ✶✾✽✺✳

        ❬✼❪ ▲■❖◆✱ ❏✳ ▲✳❀ ❉❆❯❚❘❆❨✱ ❘✳✳ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❛♥❞ ◆✉♠❡✲ r✐❝❛❧ ▼❡t❤♦❞s ❢♦r ❙❝✐❡♥❝❡ ❛♥❞ ❚❡❝❤♥♦❧♦❣②✳ ❋✉♥❝t✐♦♥❛❧ ❛♥❞ ❱❛✲ r✐❛t✐♦♥❛❧ ▼❡t❤♦❞s✳ ✈✳ ✷✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ❇❡r❧✐♥ ❡ ❍❡✐❞❡❧❜❡r❣✱ ✶✾✽✽✳

        ✺✷ t✐✈❡ ♠❡t❤♦❞ ❢♦r s♦❧✈✐♥❣ t❤❡ ❈❛✉❝❤② ♣r♦❜❧❡♠ ❢♦r ❡❧❧✐♣t✐❝ ❡q✉❛t✐♦♥s✳ ❈♦♠♣✉t✳ ▼❛t❤✳ P❤②s✳✳ ✈✳ ✸✶✱ ♥✳ ✶✱ ♣✳ ✹✺✲✺✷✱ ✶✾✾✶✳

        ❆♣ê♥❞✐❝❡ ❆ ❘❡s✉❧t❛❞♦s ❆✉①✐❧✐❛r❡s

        ◆❡st❛ ♣❛rt❡ ❞♦ tr❛❜❛❧❤♦✱ ❡stá ❛♣r❡s❡♥t❛❞♦ ♦s ❡♥✉♥❝✐❛❞♦s ❞♦s r❡s✉❧✲ t❛❞♦s ✉t✐❧✐③❛❞♦s ♦s q✉❛✐s ♥ã♦ ❝♦♥✈ê♠ ❡♥✉♥❝✐á✲❧♦s ♦✉ ❞❡♠♦♥strá✲❧♦s ❛♦ ❧♦♥❣♦ ❞♦ t❡①t♦✳

        ❆ ❡①✐stê♥❝✐❛✱ ✉♥✐❝✐❞❛❞❡ ❡ ❛ ❞❡♣❡♥❞ê♥❝✐❛ ❝♦♥tí♥✉❛ ❞♦s ❞❛❞♦s ❞♦ s❡✲ ❣✉✐♥t❡ ♣r♦❜❧❡♠❛ ❞❡ ✈❛❧♦r ❞❡ ❝♦♥t♦r♥♦ ♠✐st♦ 1 = f ; u 2 = g;

        ∆u = 0, Ω; u | Γ ν | Γ

        1/2 ′ 1/2

        (Γ (Γ ) ♣❛r❛ ♦ ♣❛r ❞❡ ❞❛❞♦s (f, g) ∈ H

        1 ) × H 2 ✭❝♦♠ Ω✱ Γ 1 ❡ Γ

        2

        00

        ❝♦♠♦ ✉t✐❧✐③❛❞♦s ❞✉r❛♥t❡ ♦ t❡①t♦✮✱ s❡❣✉❡ ❞❡ ✉♠ t❡♦r❡♠❛ ❞♦ t✐♣♦ ▲❛①✲ ▼✐❧❣r❛♠♠✱ ❝✉❥❛ ❞❡♠♦♥str❛çã♦ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ❛tr❛✈és ❞❡ ❬✹❪✳

        1/2 ′ 1/2

        (Γ (Γ ) Pr♦♣♦s✐çã♦ ❆✳✶ P❛r❛ ❝❛❞❛ ♣❛r ❞❡ ❞❛❞♦s (f, g) ∈ H

        1 )×H 2 ✱

        00

        ♦ ♣r♦❜❧❡♠❛ ❞❡ ✈❛❧♦r ❞❡ ❝♦♥t♦r♥♦ ♠✐st♦ 1 2

        Γ = f ; u ν Γ = g;

        ∆u = 0, Ω; u | |

        1

        (Ω) t❡♠ s♦❧✉çã♦ ú♥✐❝❛ u ∈ H ✳ ❆❧é♠ ❞✐ss♦✱ ✈❛❧❡ q✉❡ 1 1 1

        /2 /2 .

        ||u|| ≤ c ||f|| 1 + ||g||

        H (Ω) H (Γ ) 2 ′ H 00 (Γ )

        P❛r❛ ❞❡♠♦♥str❛r ❛ ♣r♦♣♦s✐çã♦ ✶✳✶ é ♥❡❝❡ssár✐♦ ♦ ♣r✐♥❝í♣✐♦ ❞❛ r❡✲ ✢❡①ã♦ ❞❡ ❙❝❤✇❛r③ ❬✸❪✿

      • n t❡♥❞♦

        > 0 Pr♦♣♦s✐çã♦ ❆✳✷ ❙❡❥❛ Ω ✉♠ s✉❜❞♦♠í♥✐♦ ❞♦ s❡♠✐❡s♣❛ç♦ x

        = 0 ❝♦♠♦ ♣❛rt❡ ❞❛ ❢r♦♥t❡✐r❛ ✉♠❛ s❡çã♦ ❛❜❡rt❛ T ❞♦ ❤✐♣❡r♣❧❛♥♦ x n ✳

        ✺✹

        ❝♦♠ u = 0 ❡♠ T ✳ ❊♥tã♦✱ ❛ ❢✉♥çã♦ u (x

        1 , . . . , x n ) , x n

        ≥ 0; U (x

        1 , . . . , x n ) =

        u (x

        1 n ) , x n < 0;

        , . . . , −x

        − −

        é ❤❛r♠ô♥✐❝❛ ♥♦ ❞♦♠í♥✐♦ Ω ✱ ♦♥❞❡ Ω é ❛ r❡✢❡①ã♦ ❞❡ Ω ∪ T ∪ Ω

        − n n

        1 n ) ∈ R 1 , . . . , −x n ) ∈ R }✳

        = 0 , . . . , x ; (x s♦❜r❡ x n ✱ ✐st♦ é✱ Ω := {(x

        ❯♠ r❡s✉❧t❛❞♦ ✐♠♣♦rt❛♥t❡ ♥❛ ❞❡✜♥✐çã♦ ❞♦s ♦♣❡r❛❞♦r❡s ✐♥✈♦❧✈✐❞♦s ♥❛ ❞❡✜♥✐çã♦ ❞♦s ♠ét♦❞♦s ❞❡ ▼❛③✬✐❛ ❡ ▲❛♥❞✇❡❜❡r é ♦ t❡♦r❡♠❛ ❞♦ tr❛ç♦ ❬✹❪✿ Pr♦♣♦s✐çã♦ ❆✳✸ ❖ ♦♣❡r❛❞♦r ❞♦ tr❛ç♦ ❞❡ ◆❡✉♠❛♥♥ ❞❡✜♥✐❞♦ s♦❜r❡ ♦

        ∞ ∞

        (Ω) (Γ j ) ❡s♣❛ç♦ C ❝♦♠ ✐♠❛❣❡♠ ♥♦ ❡s♣❛ç♦ C ✱ i = 1, 2 t❡♠ ✉♠❛ ú♥✐❝❛ ❡①t❡♥sã♦ ❞❡✜♥✐❞❛ ♣♦r

        1/2 1 ′

        γ : H (Γ ) ;

        

      N,Γ i (Ω) −→ H i

        00

        ♣❛r❛ i = 1, 2✳ ❆ ♣r♦♣♦s✐çã♦ s❡❣✉✐♥t❡ ❢♦✐ ❧❛r❣❛♠❡♥t❡ ✉t✐❧✐③❛❞❛ ♥❛s ❞❡♠♦♥str❛çõ❡s

        ❞❡ t♦❞♦ ♦ tr❛❜❛❧❤♦✱ ❛ s❡❣✉✐♥t❡ ❢ór♠✉❧❛ ❞❡ ●r❡❡♥✱ ❝✉❥❛ ❞❡♠♦♥str❛çã♦ ♣♦❞❡ s❡r ♦❜t✐❞❛ ❛tr❛✈és ❞❡ ❬✹❪✿

        1

        (Ω) Pr♦♣♦s✐çã♦ ❆✳✹ ❉❛❞♦s u ∈ H ❡ n o

        1/2

        1

        (Ω); γ (Γ ); i = 1, 2 ; v ∈ v ∈ H N,Γ i (v) ∈ H i

        00

        ❡♥tã♦ ✈❛❧❡ q✉❡ Z Z Z

        t

        ∆uvdx = u 1 2 ν vd∂Ω − (∇v) (∇u) dx.

        Ω Γ ⊔ Γ Ω

        ❖ ú❧t✐♠♦ r❡s✉❧t❛❞♦ ♥❡st❡ ❛♣ê♥❞✐❝❡ é ❡ss❡♥❝✐❛❧ ♣❛r❛ ❛ ❛♥á❧✐s❡ ❞❡ ❝♦♥✲ ✈❡r❣ê♥❝✐❛ ❞♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r✱ ♥❛ ♣r♦♣♦s✐çã♦ ✷✳✺✱ ❛ ❞❡♠♦♥str❛✲ çã♦ ❡♥❝♦♥tr❛✲s❡ ❡♠ ❬♣✳ ✼✷✱ ✷❪✳ Pr♦♣♦s✐çã♦ ❆✳✺ ❙❡❥❛✱ ♣❛r❛ t♦❞♦ α > 0 ❡ ❛❧❣✉♠ ǫ > 0✱ ❛ ❢✉♥çã♦ ❞❡✜✲

        2

        ♥✐❞❛ ♣♦r g α : [0, ||T || ] −→ R t❛❧ q✉❡ s❛t✐s❢❛ç❛ ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s✿ g

        α é ❝♦♥tí♥✉❛ ♣♦r ♣❛rt❡s❀ ∃c > 0 t❛❧ q✉❡ α

        |λg (λ)| ≤ c; lim g α (λ) = 1/λ;

        α→0 2 †

        ] ) ♣❛r❛ t♦❞♦ λ ∈ (0, ||T || ✳ ❊♥tã♦✱ ♣❛r❛ t♦❞♦ y ∈ R(T ✱ t❡♠✲s❡

        ∗ ∗ †

        lim g α (T T ) T y = x ;

        α→0 † † † ∗ ∗

        = T (y) ) α→0 α (T T ) T ♦♥❞❡ x ✳ ❙❡ y /∈ D(T ✱ ❡♥tã♦ lim ||g y|| = +∞✳

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