ÉRIKA CAPELATO ATRATOR NO SENTIDO PULLBACK E TRAJETÓRIAS COMPLETAS EXTREMAS PARA PROBLEMAS GOVERNADOS PELO p-LAPLACIANO

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❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❊ ❙➹❖ ❈❆❘▲❖❙ ❈❊◆❚❘❖ ❉❊ ❈■✃◆❈■❆❙ ❊❳❆❚❆❙ ❊ ❉❊ ❚❊❈◆❖▲❖●■❆ P❘❖●❘❆▼❆ ❉❊ PÓ❙✲●❘❆❉❯❆➬➹❖ ❊▼ ▼❆❚❊▼➪❚■❈❆

❆tr❛t♦r ♥♦ s❡♥t✐❞♦ ♣✉❧❧❜❛❝❦ ❡ tr❛❥❡tór✐❛s ❝♦♠♣❧❡t❛s ❡①tr❡♠❛s ♣❛r❛ ♣r♦❜❧❡♠❛s ❣♦✈❡r♥❛❞♦s ♣❡❧♦ p✲▲❛♣❧❛❝✐❛♥♦

➱r✐❦❛ ❈❛♣❡❧❛t♦

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UNIVERSIDADE FEDERAL DE SÃO CARLOS

CENTRO DE CIÊNCIAS EXATAS E DE TECNOLOGIA

PROGRAMA DE PÓS-GRADUAÇÃO EM MATEMÁTICA

ÉRIKA CAPELATO

ATRATOR NO SENTIDO PULLBACK E TRAJETÓRIAS

COMPLETAS EXTREMAS PARA PROBLEMAS GOVERNADOS

PELO

p

-LAPLACIANO

Tese de doutorado apresentada ao Programa de Pós-Graduação em Matemática, para obtenção do título de doutor em Matemática.

Orientação:Profª.Drª.Cláudia Buttarello Gentile

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Ficha catalográfica elaborada pelo DePT da Biblioteca Comunitária/UFSCar

C238as

Capelato, Érika.

Atrator no sentido pullback e trajetórias completas extremas para problemas governados pelo p-Laplaciano /

Érika Capelato. -- São Carlos : UFSCar, 2011. 129 p.

Tese (Doutorado) -- Universidade Federal de São Carlos, 2011.

1. Matemática. 2. Equações diferenciais. 3. Processos multívocos. I. Título.

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GRADECIMENTOS

▼✉✐t❛s ♣❡ss♦❛s ♣❛ss❛r❛♠ ♣♦r ♠✐♥❤❛ ✈✐❞❛ ❡ ❝♦♥tr✐❜✉ír❛♠ ❞❡ ❛❧❣✉♠❛ ❢♦r♠❛ ♣❛r❛ ♠❡✉s ❡st✉❞♦s ❡ ❛q✉✐ ❢❛ç♦ ♠❡✉s ❛❣r❛❞❡❝✐♠❡♥t♦s ❛ ❡❧❛s✳ ❉❡s❞❡ ❥á ♣❡ç♦ ❞❡s❝✉❧♣❛s ❛♦s ♥ã♦ ❝✐t❛❞♦s✱ ♠❛s q✉❡ ♠❡r❡❝❡♠ ♠✐♥❤❛ ❣r❛t✐❞ã♦✳

➚ ❉❡✉s q✉❡ ♠❡ ❞❡✉ ♦ ❞♦♠ ❞❛ ✈✐❞❛✱ ♠❡ ✐♥s♣✐r♦✉ s❛❜❡❞♦r✐❛ ♣❛r❛ ❡✉ ❢❛③❡r ❛s ❡s❝♦❧❤❛s ❝❡rt❛s ❡ ♠❡ ♣r❡s❡♥t❡♦✉ ❝♦♠ ♦ ❞♦♠ ❞❛ ❝✐ê♥❝✐❛ ❞✉r❛♥t❡ t♦❞♦s ♦s ❞✐❛s ✈✐✈✐❞♦s ♥❛ ♣ós✲❣r❛❞✉❛çã♦✳

❆♦s ♠❡✉s ♣❛✐s✱ ♣❡❧❛ ❡❞✉❝❛çã♦ ❡ ❛♠♦r q✉❡ ❞❡❧❡s r❡❝❡❜✐✱ ♣❡❧❛ ♦♣♦rt✉♥✐❞❛❞❡ q✉❡ ♠❡ ❞❡r❛♠ ❞❡ ❧❡✈❛r ❛❞✐❛♥t❡ ♠❡✉s ❡st✉❞♦s ❡ ♣❡❧♦ ❛♣♦✐♦ ♥♦s ♠♦♠❡♥t♦s ❞❡ ♣r❡♦❝✉♣❛çõ❡s✳

❆ ♠✐♥❤❛ ♦r✐❡♥t❛❞♦r❛ ❈❧á✉❞✐❛ ❇✉t❛r❡❧❧♦ ●❡♥t✐❧❡✱ q✉❡ s❡♠♣r❡ ♠❡ ❛t❡♥❞❡✉ ❝♦♠ ❜♦♠ ❤✉♠♦r✱ ❡s❝♦❧❤❡✉ ❝♦♠ ♠✉✐t♦ t❛❧❡♥t♦ ♦ t❡♠❛ ❞❡st❡ ♣r♦❥❡t♦ ✭❝♦♠ ♦ q✉❛❧ t✐✈❡ ♠✉✐t♦ ♣r❛③❡r ❡♠ tr❛❜❛❧❤❛r✮ ❡ ❝♦♥tr✐❜✉✐✉ ✐♠❡♥s❛♠❡♥t❡ ♣❛r❛ ♦ s❡✉ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❡ ♣❛r❛ ♠✐♥❤❛ ❢♦r♠❛çã♦ ❝♦♠♦ ♣❡sq✉✐s❛❞♦r❛✳ ❆ ❈❆P❊❙ ♣❡❧♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦✱ ✐♥❞✐s♣❡♥sá✈❡❧ ♣❛r❛ ♠✐♥❤❛ ♣❡r♠❛♥ê♥❝✐❛ ❡♠ ❙ã♦ ❈❛r❧♦s✳

❆♦s ♣r♦❢❡ss♦r❡s ❞♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥❡s♣ ❞❡ ❘✐♦ ❈❧❛r♦ r❡s♣♦♥✲ sá✈❡✐s ♣❡❧♦ ✐♥í❝✐♦ ❞❡ ♠✐♥❤❛ ❢♦r♠❛çã♦ ❡ ♣❡❧♦ ♣r❛③❡r q✉❡ t❡♥❤♦ ❡♠ ❡st✉❞❛r ♠❛t❡♠át✐❝❛✱ ❛♦s ♣r♦❢❡ss♦r❡s ❞♦ ■❈▼❈ ✲ ❯❙P ❥✉♥t♦ ❛♦s q✉❛✐s ❛❣r❡❣✉❡✐ ♠❛✐s ❛♣r❡♥❞✐③❛❞♦ ❡ ♦❜t✐✈❡ ♦ tít✉❧♦ ❞❡ ♠❡str❡ ❡ ✜♥❛❧♠❡♥t❡ ❛♦s ♣r♦❢❡ss♦r❡s ❞♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❯❋❙❈❛r ❝♦♠ ♦s q✉❛✐s ❝♦♥✈✐✈✐ ❛❣r❛❞❛✈❡❧♠❡♥t❡ ❞✉r❛♥t❡ ❡st❡s q✉❛tr♦ ❛♥♦s✳

❆♦s ❛♠✐❣♦s ❞❛ ❣r❛❞✉❛çã♦ ❡ ❞❛ ♣ós✲❣r❛❞✉❛çã♦✱ ♣❡❧❛ tr♦❝❛ ❞❡ ❝♦♥❤❡❝✐♠❡♥t♦s ❡ r✐s❛❞❛s ❡ ❛♦s ❛♠✐❣♦s q✉❡ ♥ã♦ sã♦ ♠❛t❡♠át✐❝♦s✱ ♣❡❧♦ ❛♣♦✐♦ ❡ ❝♦♠♣❛♥❤❡✐r✐s♠♦✳

❆♦s ❝✐t❛❞♦s ❛❝✐♠❛✱ ▼✉✐t♦ ❖❜r✐❣❛❞❛✦

➱r✐❦❛ ❈❛♣❡❧❛t♦

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BSTRACT

❲❡ st✉❞② ♥♦♥✲❛✉t♦♥♦♠♦✉s ♣r♦❜❧❡♠s ✇✐t❤ t❤❡ ♠❛✐♥ ♣❛rt ❣✐✈❡♥ ❜② t❤❡ p✲ ▲❛♣❧❛❝✐❛♥ ❛♥❞ ❡st❛❜❧✐s❤ ❡①✐st❡♥❝❡ r❡s✉❧ts ♦❢ ♣✉❧❧❜❛❝❦ ❛ttr❛❝t♦r✱ ❝♦♠♣❛r✐s♦♥ ♦❢ s♦❧✉t✐♦♥s ❛♥❞ ❡①✐st❡♥❝❡ ♦❢ ❛ ♠❛①✐♠❛❧ ❛♥❞ ❛ ♠✐♥✐♠❛❧ ❝♦♠♣❧❡t❡ tr❛❥❡❝t♦✲ r✐❡s✳ ■♥ ❜♦✉♥❞❡❞ s♠♦♦t❤ ❞♦♠❛✐♥s ✇❡ ❛ss✉♠❡ r❛t❤❡r ❣❡♥❡r❛❧ ❝♦♥❞✐t✐♦♥s ♦♥ t❤❡ ♣❡rt✉r❜❛t✐✈❡ ♦♣❡r❛t♦r ❛♥❞ ✇❡ ❛❞♠✐t t❤❡ ♣r♦❝❡ss ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡ ♣r♦❜❧❡♠ ❜❡✐♥❣ ♠✉❧t✐✈❛❧✉❡❞✳ ❲❡ ♦❜t❛✐♥ ❛♥❛❧♦❣♦✉s r❡s✉❧ts ❢♦r s②st❡♠s ✐♥Rn✇✐t❤ ❢✉rt❤❡r r❡str✐❝t✐♦♥s ✐♥ ♣❡rt✉r❜❛t✐✈❡ t❡r♠s ✐♥ s✉❝❤ ✇❛② t❤❛t ✇❡ ❞❡❛❧ ✇✐t❤ ❛ ✉♥✐✈♦❝❛❧ ♣r♦❝❡ss✳

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R

ESUMO

◆❡st❡ tr❛❜❛❧❤♦ ❡st✉❞❛♠♦s ♣r♦❜❧❡♠❛s ♥ã♦ ❛✉tô♥♦♠♦s ❝♦♠ ♣❛rt❡ ♣r✐♥❝✐♣❛❧ ❞❛❞❛ ♣❡❧♦ p✲❧❛♣❧❛❝✐❛♥♦✱ ❡ ❡st❛❜❡❧❡❝❡♠♦s r❡s✉❧t❛❞♦s ❞❡ ❡①✐stê♥❝✐❛ ❞❡ ❛tr❛t♦r ♥♦ s❡♥✲ t✐❞♦ ♣✉❧❧❜❛❝❦✱ ❝♦♠♣❛r❛çã♦ ❞❡ s♦❧✉çõ❡s ❡ ❛ ❡①✐stê♥❝✐❛ ❞❡ ❞✉❛s tr❛❥❡tór✐❛s ❝♦♠✲ ♣❧❡t❛s✱ ✉♠❛ ♠❛①✐♠❛❧ ❡ ♦✉tr❛ ♠✐♥✐♠❛❧✳ ❊♠ ❞♦♠í♥✐♦s ❧✐♠✐t❛❞♦s ❡ s✉❛✈❡s ❝♦♥✲ s✐❞❡r❛♠♦s t❡r♠♦s ♣❡rt✉r❜❛t✐✈♦s ❜❛st❛♥t❡ ❣❡r❛✐s ❡ ❛❞♠✐t✐♠♦s q✉❡ ♦ ♣r♦❝❡ss♦ ❛ss♦❝✐❛❞♦ ❛♦ ♣r♦❜❧❡♠❛ s❡❥❛ ♠✉❧tí✈♦❝♦✳ ❖❜t✐✈❡♠♦s r❡s✉❧t❛❞♦s ❛♥á❧♦❣♦s ♣❛r❛ s✐st❡♠❛s ❞❡✜♥✐❞♦s ❡♠ t♦❞♦ ♦ Rn❝♦♠ r❡str✐çõ❡s ❛❞✐❝✐♦♥❛✐s ♥♦ t❡r♠♦ ♣❡rt✉r❜❛✲ t✐✈♦ ❞❡ ❢♦r♠❛ ❛ ❧✐❞❛r♠♦s ❝♦♠ ✉♠ ♣r♦❝❡ss♦ ✉♥í✈♦❝♦✳

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NTRODUÇÃO

◗✉❛♥❞♦ ✉♠ s✐st❡♠❛ ❞✐♥â♠✐❝♦ é ❞❡t❡r♠✐♥❛❞♦ ♣♦r ✉♠❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ❛✉tô ✲ ♥♦♠❛ ♣♦❞❡♠♦s ❛ss♦❝✐❛r ❛ ❡❧❡ ✉♠ s❡♠✐❣r✉♣♦ ❡ ❛♥❛❧✐s❛r ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❛ss✐♥tót✐❝♦ ❞♦ ♣r♦❜❧❡♠❛ ❝♦♥s✐❞❡r❛♥❞♦ ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠ ❛tr❛t♦r ❣❧♦❜❛❧ ✭✉♠ ❝♦♥❥✉♥t♦ ❝♦♠♣❛❝t♦ ✐♥✈❛r✐✲ ❛♥t❡ ♠❛①✐♠❛❧ q✉❡ ❛tr❛✐ ❛s ór❜✐t❛s✮ q✉❡✱ ✉♠❛ ✈❡③ ♣r♦✈❛❞❛✱ ♣❡r♠✐t❡ q✉❡ ❢❛ç❛♠♦s ✐♥ú♠❡r❛s r❡str✐çõ❡s ❡♠ ♥♦ss❛ ❛♥á❧✐s❡ s❡♠ ♣❡r❞❡r ❛ ❣❡♥❡r❛❧✐❞❛❞❡ ❞♦ ♣r♦❜❧❡♠❛✳ ❊①✐st❡ ✉♠❛ t❡♦r✐❛ ❜❛st❛♥t❡ ❡st❛❜❡❧❡❝✐❞❛ ♣❛r❛ ♦ tr❛t❛♠❡♥t♦ ❞❡ t❛✐s s✐st❡♠❛s ♥♦ ❝❛s♦ ✉♥í✈♦❝♦ ❬✹✷✱ ✺✷✱ ✼✸❪✳

❙✐st❡♠❛s ❞✐♥â♠✐❝♦s ♠✉❧tí✈♦❝♦s s✉r❣❡♠ q✉❛♥❞♦ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤② ❞❡ ✉♠❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ♥ã♦ ♣♦ss✉✐ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❛ ✉♥✐❝✐❞❛❞❡✳ ❍á ♣♦✉❝♦ ♠❛✐s ❞❡ ✈✐♥t❡ ❛♥♦s✱ ❛ ❛✉sê♥❝✐❛ ❞❡ ✉♥✐❝✐❞❛❞❡ ♣❛r❡❝✐❛ s❡r ✉♠ ♦❜stá❝✉❧♦ ✐♥❝♦♥t♦r♥á✈❡❧ ♣❛r❛ ♦ ❡st✉❞♦ ❞♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❛ss✐♥tót✐❝♦ ❞❡ ✉♠ ♣r♦❜❧❡♠❛ ♦✉✱ ♠❛✐s ❡s♣❡❝✐✜❝❛♠❡♥t❡✱ ♣❛r❛ ❝♦♥❝❧✉✐r ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠ ❛tr❛t♦r✳ ◆❡st❡s ❝❛s♦s✱ ♥ã♦ ♣♦❞❡♠♦s ❛ss♦❝✐❛r ❛♦s ♣r♦❜❧❡♠❛s ✉♠ s❡♠✐❣r✉♣♦ ❞❡ ♦♣❡r❛❞♦r❡s ❞❡ ❢♦r♠❛ q✉❡ ♥ã♦ s❡ ❛♣❧✐❝❛ ❛ ❡❧❡s ♦ ❡❧❡♥❝♦ ❞❡ r❡s✉❧t❛❞♦s q✉❡ s❡ ❡♥❝♦♥tr❛♠ ♥❛ t❡♦r✐❛ ❝❧áss✐❝❛✳

◆❛ ✈❡r❞❛❞❡✱ q✉❛♥❞♦ ❡st✉❞❛♠♦s ✉♠ ♣r♦❜❧❡♠❛ ❞❡ ✈❛❧♦r ✐♥✐❝✐❛❧✱ ❛♥t❡s ❞❡ t✉❞♦ é ✉s✉❛❧ ♥♦s q✉❡st✐♦♥❛r♠♦s s♦❜r❡ s✉❛ ❜♦❛ ❝♦❧♦❝❛çã♦✳ ❊st❛♠♦s ❛❝♦st✉♠❛❞♦s ❛ ❢❛③❡r três ♣❡r❣✉♥t❛s ❜ás✐❝❛s✿

✶✳ ❉❛❞❛ ✉♠❛ ❝♦♥❞✐çã♦ ✐♥✐❝✐❛❧✱ ❡①✐st❡ ✉♠❛ s♦❧✉çã♦ ❣❧♦❜❛❧❄ ✷✳ ❙❡ ❛ss✐♠ ❢♦r✱ é ú♥✐❝❛❄

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①✐✐ ■♥tr♦❞✉çã♦

❍♦❥❡ é s❛❜✐❞♦ q✉❡ ❛ ❛✉sê♥❝✐❛ ❞❡ ✉♥✐❝✐❞❛❞❡ ♥ã♦ ❝♦♥st✐t✉✐ ✉♠ ♦❜stá❝✉❧♦ ♣❛r❛ s❡ ♦❜t❡r ❛ ❡①✐stê♥❝✐❛ ❞❡ ❛tr❛t♦r❡s ❣❧♦❜❛✐s ♣❛r❛ ♣r♦❜❧❡♠❛s ❛✉tô♥♦♠♦s✳ ❉❡ ❢❛t♦✱ ♣❛r❛ s❡ ❝♦♥✲ ❝❧✉✐r ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠ ❛tr❛t♦r✱ sã♦ ♥❡❝❡ssár✐❛s ❜❛s✐❝❛♠❡♥t❡ ♣r♦♣r✐❡❞❛❞❡s ❞❡ ❞✐ss✐♣❛t✐✈✐✲ ❞❛❞❡ ❡ ❝♦♠♣❛❝✐❞❛❞❡ ❛ss✐♥tót✐❝❛✱ ❡ ✉♠❛ ❛♠♣❧❛ ❝❧❛ss❡ ❞❡ ♣r♦❜❧❡♠❛s ❞❡ ❈❛✉❝❤② ✉s✉❛❧♠❡♥t❡ ❞❡♥♦♠✐♥❛❞♦s ❝♦♠♦ ♠❛❧✲♣♦st♦s s❛t✐s❢❛③❡♠ ❡st❛s ❝♦♥❞✐çõ❡s✳ ❱ár✐♦s ❛✉t♦r❡s tê♠ ❧✐❞❛❞♦ ❝♦♠ ♣r♦❜❧❡♠❛s ❛✉tô♥♦♠♦s ♠✉❧tí✈♦❝♦s ❡ ❛❧❣✉♥s ❡s❢♦rç♦s ♥❡ss❡ s❡♥t✐❞♦ s✉r❣✐r❛♠ ❥á ♥❛ ♣r✐♠❡✐r❛ ♠❡t❛❞❡ ❞♦ sé❝✉❧♦ ✷✵✿ ✶✾✹✽✲❇❛r❜❛➨✐♥ ❬✶✵❪✱ ✶✾✹✽✲▼✐♥❦❡✈✐↔ ❬✺✼❪✱ ✶✾✺✷✲❇✉❞❛❦ ❬✶✽❪✱ ✶✾✻✷✲ ❇r♦♥➨t❡✞✙♥ ❬✶✺❪✱ ✶✾✻✺✲❘♦①✐♥ ❬✻✹✱ ✻✺❪✱ ✶✾✻✾✲❇r✐❞❣❧❛♥❞ ❬✶✹❪✱ ✶✾✻✾✲❙③❡❣ö ❛♥❞ ❚r❡❝❝❛♥✐ ❬✼✷❪✱ ✶✾✼✸✲❙❡❧❧ ❬✻✽❪✱ ✶✾✼✽✲❑❧♦❡❞❡♥ ❬✹✼❪✱ ✶✾✽✺✲❇❛❜✐♥ ❛♥❞ ❱✐s❤✐❦ ❬✻❪✱ ✶✾✾✺✲❈❤❡♣②③❤♦✈ ❛♥❞ ❱✐s❤✐❦ ❬✸✺❪✱ ✶✾✾✺✲▼❡❧♥✐❦ ❬✺✸❪✱ ✶✾✾✼✲❇❛❧❧ ❬✾❪✱ ✶✾✾✽✲❑❛♣✉st②❛♥ ❛♥❞ ▼❡❧♥✐❦ ❬✹✸❪✱ ✶✾✾✽✲▼❡❧♥✐❦ ❛♥❞ ❱❛❧❡r♦ ❬✺✹❪✱ ✶✾✾✾✲❈❛r✈❛❧❤♦ ❛♥❞ ●❡♥t✐❧❡ ❬✷✽❪✱ ✷✵✵✽✲❙✐♠s❡♥ ❛♥❞ ●❡♥t✐❧❡ ❬✻✾❪✳

P♦❞❡♠♦s ❝❧❛ss✐✜❝❛r ❡st❡s tê①t♦s ❡♠ três ❣r✉♣♦s ❞❡ ❛❜♦r❞❛❣❡♥s✿ ✉♠❛ ❞❡❧❛s ❧✐❞❛ ❝♦♠ ❛ ❢❛❧t❛ ❞❡ ✉♥✐❝✐❞❛❞❡ tr❛❜❛❧❤❛♥❞♦ ❡♠ ✉♠ ❡s♣❛ç♦ ❞❡ tr❛❥❡tór✐❛s ϕ : [0,+∞) → X ❡ ❞❡✜♥✐♥❞♦ ♥❡ss❡s ❡s♣❛ç♦s s❡♠✐❣r✉♣♦s ❞❡ tr❛♥s❧❛çõ❡s T(·)✱ ♣♦r T(t)ϕ = ϕt✱ ❬✸✺✱ ✻✽❪✳ ❯♠ s❡❣✉♥❞♦ ❣r✉♣♦✱ q✉❡ ✐♥❝❧✉✐ ❛ ♠❛✐♦r✐❛ ❞♦s tr❛❜❛❧❤♦s ❛❝✐♠❛✱ ❝♦♥s✐❞❡r❛ ✉♠❛ tr❛❥❡tór✐❛ ♠✉❧tí✈♦❝❛t 7→T(t)z q✉❡ é ❝♦♠♣♦st❛ ♣♦r t♦❞♦s ♦s ♣♦ssí✈❡✐s ♣♦♥t♦s ❛❧❝❛♥ç❛❞♦s ♥♦ t❡♠♣♦t ♣♦r s♦❧✉çõ❡s ❝♦♠ ♦s ❞❛❞♦s ✐♥✐❝✐❛✐s ❡♠z✭♣♦r ❡①❡♠♣❧♦ ❬✺✹❪✮ ✳ ❖ t❡r❝❡✐r♦ ❣r✉♣♦✱ r❡♣r❡s❡♥t❛❞♦ ♣♦r ❬✾❪✱ ❝♦♥s✐❞❡r❛ ❛ s♦❧✉çõ❡s ♣r♦♣r✐❛♠❡♥t❡ ❝♦♠♦ ♦s ♦❜❥❡t♦s ♣r✐♠✐t✐✈♦s ❞❛ t❡♦r✐❛✱ ❡ ❝♦♥s✐❞❡r❛ ❝♦♥❥✉♥t♦s ❞❡ ❢✉♥çõ❡s q✉❡ ❝♦♠♣õ❡♠ ✉♠ s❡♠✐❣r✉♣♦ ❣❡♥❡r❛❧✐③❛❞♦✳ ❆❝r❡❞✐t❛♠♦s q✉❡ ✉♠❛ ❛❜♦r❞❛❣❡♠ ❛❞❡q✉❛❞❛ ♣❛r❛ ❡st✉❞❛r ♣r♦❜❧❡♠❛s ❞✐❢❡r❡♥❝✐❛✐s ❞❡ ✈❛❧♦r❡s ♠✉❧tí✈♦❝♦s é ♦❜t✐❞❛ ♣❡❧❛ ♠✐st✉r❛ ❞♦s ♠ét♦❞♦s ✉t✐❧✐③❛❞♦s ♣❡❧♦s s❡❣✉♥❞♦ ❡ t❡r❝❡✐r♦ ❣r✉♣♦s ❛❝✐♠❛ ♠❡♥❝✐♦♥❛❞♦s✱ ❝♦♠♦ é ❢❡✐t♦ ❡♠ ❬✻✾❪✱ ♦♥❞❡ ♦s s❡♠✐❣r✉♣♦s ♠✉❧tí✈♦❝♦s sã♦ ❞❡✜♥✐❞♦s ❞❡ ❢♦r♠❛ ❛ ♥ã♦ ♣❡r❞❡r ❞❡ ✈✐st❛ ❝❛❞❛ ♣♦ssí✈❡❧ s♦❧✉çã♦✳

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■♥tr♦❞✉çã♦ ①✐✐✐

♣r♦✈❛r✮ q✉❡ ❛ s♦❧✉çã♦ ❝♦rr❡s♣♦♥❞❡♥t❡ ❛ ❝❛❞❛ ❞❛❞♦ ✐♥✐❝✐❛❧ é ú♥✐❝❛ ✐♥tr♦❞✉③✐♠♦s ♦ ❝♦♥❝❡✐t♦ ❞❡ ♣r♦❝❡ss♦ ♠✉❧tí✈♦❝♦✱ ♣❛r❛ ♦ q✉❛❧ ❥á s❡ ❡♥❝♦♥tr❛♠ ♥❛ ❧✐t❡r❛t✉r❛ ❣❡♥❡r❛❧✐③❛çõ❡s ❞❡ ❞✐✈❡rs♦s r❡s✉❧t❛❞♦s ♦r✐❣✐♥❛❧♠❡♥t❡ ♣r♦✈❛❞♦s ♣❛r❛ ♣r♦❜❧❡♠❛s ❜❡♠ ♣♦st♦s✱ ✭✈❡❥❛ ♣♦r ❡①❡♠♣❧♦ ❬✶✾❪✮✳

◆❡st❡ tr❛❜❛❧❤♦ ❡st❛♠♦s ❝♦♥s✐❞❡r❛♥❞♦ ♣r♦❜❧❡♠❛s ♥ã♦ ❛✉tô♥♦♠♦s ❣♦✈❡r♥❛❞♦s ♣❡❧♦ p✲❧❛♣❧❛❝✐❛♥♦✱ p > max{2, n/2}✱ ❛♦s q✉❛✐s s❡ ❛ss♦❝✐❛♠ ♣r♦❝❡ss♦s ♠✉❧tí✈♦❝♦s ❡♠ ❞♦♠í♥✐♦s

❧✐♠✐t❛❞♦s ❡ ♣r♦❝❡ss♦s ✉♥í✈♦❝♦s ❡♠ Rn✳ P❛r❛ ❡st❛s ❝❧❛ss❡s ❞❡ ♣r♦❜❧❡♠❛s✱ s♦❜ ❝♦♥❞✐çõ❡s ❛♣r♦♣r✐❛❞❛s✱ ❡①✐❜✐♠♦s ❛ ❡①✐stê♥❝✐❛ ❞❡ ❛tr❛t♦r ♥♦ s❡♥t✐❞♦ ♣✉❧❧❜❛❝❦✱ ❡st✐♠❛t✐✈❛s ❡♠ L∞

❛tr❛✈és ❞❡ r❡s✉❧t❛❞♦s ❞❡ ❝♦♠♣❛r❛çã♦ ❡ ♦❜t✐✈❡♠♦s ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠❛ tr❛❥❡tór✐❛ ❝♦♠✲ ♣❧❡t❛ ♠❛①✐♠❛❧ ❡ ✉♠❛ tr❛❥❡tór✐❛ ❝♦♠♣❧❡t❛ ♠✐♥✐♠❛❧ ❞❡❧✐♠✐t❛♥❞♦ ❡♠ ✉♠ ❝❡rt♦ s❡♥t✐❞♦ t❛✐s ❛tr❛t♦r❡s✳

❊①✐st❡ ✉♠ ♥ú♠❡r♦ ❡①♣r❡ss✐✈♦ ❞❡ tr❛❜❛❧❤♦s ♥❛ ❧✐t❡r❛t✉r❛ s♦❜r❡ ♣r♦❜❧❡♠❛s ♣❛r❛❜ó❧✐✲ ❝♦s ❡♥✈♦❧✈❡♥❞♦ ♦ ♦♣❡r❛❞♦rp✲❧❛♣❧❛❝✐❛♥♦✳ ❱ár✐♦s ❞❡❧❡s ❡st❛❜❡❧❡❝❡♥❞♦ ❛ ❡①✐stê♥❝✐❛ ❞❡ ❛tr❛t♦r ♣❛r❛ ♣r♦❜❧❡♠❛s ❛✉tô♥♦♠♦s ❡♠ ❞♦♠í♥✐♦s ❧✐♠✐t❛❞♦s✱ ✈❡❥❛ ♣♦r ❡①❡♠♣❧♦ ❬✼✱ ✷✺✱ ✷✽✱ ✹✶✱ ✺✹✱ ✼✵✱ ✼✾❪ ❡ r❡❢❡rê♥❝✐❛s ♥❡❧❡s ❝♦♥t✐❞❛s✳ P❛r❛ ♣r♦❜❧❡♠❛s ♥ã♦ ❛✉tô♥♦♠♦s✱ ❛ ❡①✐stê♥❝✐❛ ❞❡ ❛tr❛t♦r ♥♦ s❡♥t✐❞♦ ♣✉❧❧❜❛❝❦ ❥á ❤❛✈✐❛ s✐❞♦ ♦❜t✐❞❛ ❡♠ ❬✶✾❪✳ ❊♠ ❬✸✷❪ ♦ ❛✉t♦r ❧✐❞❛ ❝♦♠ ♣r♦❜❧❡♠❛s ♥ã♦ ❛✉tô♥♦♠♦s ♠❛s ♦❜té♠ ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠ ♦✉tr♦ t✐♣♦ ❞❡ ❛tr❛t♦r✱ ♦ ❛tr❛t♦r ✉♥✐❢♦r♠❡✱ ✐♥tr♦❞✉③✐❞♦ ♣♦r ❈❤❡♣②③❤♦✈ ❡ ❱✐s❤✐❦ ❡♠ ❬✸✻❪✱ q✉❡ ❛ ❣r♦ss♦ ♠♦❞♦ ❛tr❛✐ t♦❞❛s ❛s ór❜✐t❛s ✐♥✲ ❞❡♣❡♥❞❡♥t❡♠❡♥t❡ ❞❡ q✉❛♥❞♦ ❡ ♦♥❞❡ ❡❧❛s ❝♦♠❡ç❛r❛♠✳ ◆♦ss♦ ❛❝rés❝✐♠♦ ♥❡st❡ tr❛❜❛❧❤♦ ❢♦✐✱ ♥♦ q✉❡ ❞✐③ r❡s♣❡✐t♦ ❛♦s ♣r♦❜❧❡♠❛s ❡♠ ❞♦♠í♥✐♦s ❧✐♠✐t❛❞♦s✱ ♦❜t❡r ❝♦♠ ❞✐❢❡r❡♥t❡s ❝♦♥❞✐çõ❡s ✐♠♣♦st❛s ❛♦ t❡r♠♦ ♣❡rt✉r❜❛t✐✈♦✱ r❡s✉❧t❛❞♦s q✉❡ ♣❡r♠✐t❡♠ ❝♦♠♣❛r❛r s♦❧✉çõ❡s ❡ ❞❡♠♦♥str❛r ❛ ❡①✐stê♥❝✐❛ ❞❡ tr❛❥❡tór✐❛s ❝♦♠♣❧❡t❛s ❡①tr❡♠❛s✳

❆ ❡①✐stê♥❝✐❛ ❞❡ t❛✐s tr❛❥❡tór✐❛s ❢♦✐ ♦❜t✐❞❛ ❡♠ ❬✻✸❪ ♣❛r❛ ♣r♦❜❧❡♠❛s ♥ã♦ ❛✉tô♥♦♠♦s s❡♠✐❧✐♥❡❛r❡s✳ ❊♠ s✐st❡♠❛s q✉❛s✐❧✐♥❡❛r❡s✱ ❛ ♣r✐♥❝✐♣❛❧ ❞✐✜❝✉❧❞❛❞❡ ❡♠ s❡ ♣r♦✈❛r ❛ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çõ❡s ❣❧♦❜❛✐s ❡①tr❡♠❛s s❡ ❞á ♣♦r q✉❡ ❛s ❞❡♠♦♥str❛çõ❡s ❡①✐st❡♥t❡s ♣❛r❛ ❡st❡ ❢❛t♦ ❛♣♦✐❛♠✲s❡ ❡♠ ❢♦rt❡s r❡s✉❧t❛❞♦s ❞❡ ❝♦♠♣❛r❛çã♦ ❞❡ s♦❧✉çõ❡s✱ ♦s q✉❛✐s ♥ã♦ s❡ ❛♣❧✐❝❛♠✱ ✈✐❛ ❞❡ r❡❣r❛✱ ♣❛r❛ ♣r♦❜❧❡♠❛s ♠✉❧tí✈♦❝♦s✳ ▼❛✐s ❡s♣❡❝✐✜❝❛♠❡♥t❡✱ ❡♠ ❬✷✶❪✱ ♦♥❞❡ ♣r✐♠❡✐r❛♠❡♥t❡ s❡ ♦❜té♠ ✉♠ r❡s✉❧t❛❞♦ ❛❜str❛t♦ q✉❡ ❣❛r❛♥t❡ ❛ ❡①✐stê♥❝✐❛ ❞❡ ❡q✉✐❧í❜r✐♦s ❡①tr❡♠♦s ❡♠ ♣r♦❜✲ ❧❡♠❛s ❛✉tô♥♦♠♦s✱ s✉♣õ❡✲s❡ q✉❡ ♦ s❡♠✐❣r✉♣♦ ♠✉❧tí✈♦❝♦ ❛ss♦❝✐❛❞♦ ❛♦ s✐st❡♠❛ s❛t✐s❢❛③ ✉♠❛ ❝♦♥❞✐çã♦ ❞❡ ♠♦♥♦t♦♥✐❝✐❞❛❞❡ ❜❛st❛♥t❡ ❡①✐❣❡♥t❡✱ ❛ s❛❜❡r✱ s❡ ❞♦✐s ❞❛❞♦s ✐♥✐❝✐❛✐s s❛t✐s❢❛③❡♠ u0(x) ≤ v0(x) ✭❡♠ ✉♠ s❡♥t✐❞♦ q✉❡ s❡rá ❛♣r♦♣r✐❛❞❛♠❡♥t❡ ❞❡s❝r✐t♦ ❛♦ ❧♦♥❣♦ ❞❡st❡ tê①t♦✮✱

❡♥tã♦ ❞❡✈❡✲s❡ t❡r q✉❡ t♦❞❛ s♦❧✉çã♦ q✉❡ s❡ ✐♥✐❝✐❛ ❡♠u0 ♣❡r♠❛♥❡ç❛ ♠❡♥♦r q✉❡ t♦❞❛ s♦❧✉çã♦

q✉❡ s❡ ✐♥✐❝✐❛ ❡♠ v0✳ ◆♦ ❡♥t❛♥t♦✱ ❛ ❧✐t❡r❛t✉r❛ ❡♥✈♦❧✈❡♥❞♦ r❡s✉❧t❛❞♦s ❞❡ ❝♦♠♣❛r❛çã♦ ♣❛r❛

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①✐✈ ■♥tr♦❞✉çã♦

◆❡st❡ tê①t♦ ❡st❡♥❞❡♠♦s ✉♠ r❡s✉❧t❛❞♦ ❞❡ ❝♦♠♣❛r❛çã♦ ♦r✐❣✐♥❛❧♠❡♥t❡ ♣r♦✈❛❞♦ ♣❛r❛ ♣r♦❜✲ ❧❡♠❛s ❛✉tô♥♦♠♦s ❡♠ ❬✷✻❪✱ ♥♦ q✉❛❧ ❣❛r❛♥t❡✲s❡ q✉❡✱ ♣❛r❛ ✉♠❛ ❞❡t❡r♠✐♥❛❞❛ ❝❛t❡❣♦r✐❛ ❞❡ ♣r♦❜❧❡♠❛s ✭❡♠ q✉❡ s❡ ❡♥q✉❛❞r❛♠ ♦s s✐st❡♠❛s ❡♠ ❞♦♠í♥✐♦s ❧✐♠✐t❛❞♦s ❝♦♥s✐❞❡r❛❞♦s ♥❡st❡ tr❛❜❛❧❤♦✮✱ s❡ u0(x) ≤ v0(x)✱ ❡♥tã♦ ❞❛❞❛ ✉♠❛ s♦❧✉çã♦ u ❝♦♠ ❞❛❞♦ ✐♥✐❝✐❛❧ ❡♠ u0✱ ❡①✐st❡

✉♠❛ s♦❧✉çã♦ v q✉❡ s❡ ✐♥✐❝✐❛ ❡♠ v0 ❡ t❛❧ q✉❡ u(t) ≤ v(t) ♣❛r❛ t♦❞♦ t❡♠♣♦ t > 0✳ ❈♦♠♦

❡ss❡♥❝✐❛❧♠❡♥t❡ ♦ ♠❡❝❛♥✐s♠♦ q✉❡ s❡ ✉s❛ ♣❛r❛ ❣❛r❛♥t✐r ❛ ❡①✐stê♥❝✐❛ ❞❡ t❛✐s tr❛❥❡tór✐❛s ❡①✲ tr❡♠❛s ❡♠ s✐st❡♠❛s ♥ã♦ ❛✉tô♥♦♠♦s é ♦ ♠❡s♠♦ ❡♥✉♥❝✐❛❞♦ ❡♠ ❬✷✶❪✱ ❛ ❛✉sê♥❝✐❛ ❞❡ ✉♥✐❝✐❞❛❞❡ t♦r♥❛✲s❡ ✉♠ ❢♦rt❡ ♦❜stá❝✉❧♦✳ ◆❡st❡ tr❛❜❛❧❤♦ ❡st❛ ❞✐✜❝✉❧❞❛❞❡ ❢♦✐ ❝♦♥t♦r♥❛❞❛ ♣r♦✈❛♥❞♦✲s❡ q✉❡✱ q✉❛♥❞♦ r❡str✐t♦ ❛♦ ❛tr❛t♦r✱ ♦ ✢✉①♦ é ❞❡ ❢❛t♦ ✉♥í✈♦❝♦ ❡ ❞❡st❛ ❢♦r♠❛✱ ❛ ❡①✐stê♥❝✐❛ ❞❛s s♦❧✉çõ❡s ❡①tr❡♠❛s ♣♦❞❡ s❡r ❞❡♠♦♥str❛❞❛✳

❆ ✉♥✐❝✐❞❛❞❡ ❞❡ s♦❧✉çã♦ ❞❡♥tr♦ ❞♦ ❛tr❛t♦r ✭q✉❡ ❝❤❛♠❛♠♦s ❞❡ ✉♥✐❝✐❞❛❞❡ ❡♠ t❡♠♣♦ ❣r❛♥❞❡ ♦✉ ✉♥✐❝✐❞❛❞❡ ❡✈❡♥t✉❛❧✮ é ❣❛r❛♥t✐❞❛ t❛♠❜é♠ ❛tr❛✈és ❞❡ r❡s✉❧t❛❞♦s ❞❡ ❝♦♠♣❛r❛çã♦✳ ◆♦r♠❛❧♠❡♥t❡✱ ♣r♦❜❧❡♠❛s ❣♦✈❡r♥❛❞♦s ♣♦r ♦♣❡r❛❞♦r❡s ❞♦ t✐♣♦ ♠❛①✐♠❛❧ ♠♦♥ót♦♥♦ ❣♦③❛♠ ❞❡ ✉♥✐❝✐❞❛❞❡ s❡ ♦ t❡r♠♦ ♣❡rt✉r❜❛t✐✈♦ ❢♦r ❣❧♦❜❛❧♠❡♥t❡ ▲✐♣s❝❤✐t③✐❛♥♦✳ P♦ré♠✱ ❞❡ ❛❝ôr❞♦ ❝♦♠ ♦ tr❛❜❛❧❤♦ ❬✷✽❪✱ ♣♦❞❡✲s❡ ❛❢r♦✉①❛r ❡st❛ ❝♦♥❞✐çã♦ tr❛❜❛❧❤❛♥❞♦✲s❡ ❝♦♠ ♣❡rt✉r❜❛çõ❡s q✉❡ sã♦ ❧♦❝❛❧♠❡♥t❡ ▲✐♣s❝❤✐t③✐❛♥❛s✱ ♠❛s ❡♠ ❝♦♥tr❛♣❛rt✐❞❛ ♣♦❞❡♠ s❡r ❡st✐♠❛❞❛s ✭❡♠ t❡♠♣♦ ❣r❛♥❞❡✮ ♣♦r t❡r♠♦s ❞❡ ❝r❡s❝✐♠❡♥t♦ ❧✐♥❡❛r✳ ❆ ❡str❛té❣✐❛ ✉s❛❞❛ ♥❡st❡ ❝❛s♦ é ♦❜t❡r ❡st✐✲ ♠❛t✐✈❛s ❡♠ L∞ ❛tr❛✈és ❞❡ ❝♦♠♣❛r❛çã♦ ❞❡ s♦❧✉çõ❡s ❞❡ ❢♦r♠❛ q✉❡ ❛ ❛♥á❧✐s❡ ❛ss✐♥tót✐❝❛

♣♦ss❛ s❡r ❢❡✐t❛ ❡♠ ✉♠ ❝♦♥❥✉♥t♦ ♥♦ q✉❛❧ ♦ t❡r♠♦ ♣❡rt✉r❜❛t✐✈♦ é ❡ss❡♥❝✐❛❧♠❡♥t❡ ▲✐♣❝❤✐t③✳ ❆ ♣r✐♠❡✐r❛ ♣❛rt❡ ❞❡st❡ tr❛❜❛❧❤♦ tr❛♥s❝r❡✈❡ ❡st❛s ✐❞é✐❛s ♣❛r❛ ♣r♦❜❧❡♠❛s ♥ã♦ ❛✉tô♥♦♠♦s✳ ▼✉✐t♦ ❡♠❜♦r❛ ❛ ❛❞❛♣t❛çã♦ t❡♥❤❛ s✐❞♦ ❢❡✐t❛ ❞❡ ✉♠❛ ❢♦r♠❛ ❜❛st❛♥t❡ ♥❛t✉r❛❧✱ ❢♦✐ ♥❡❝❡ssár✐♦ ❛❥✉st❛r ❛ ❣r❛♥❞❡ ♠❛✐♦r✐❛ ❞♦s r❡s✉❧t❛❞♦s ✉s❛❞♦s✱ ♦r✐❣✐♥❛❧♠❡♥t❡ ❞❡s❡♥✈♦❧✈✐❞♦s ♣❛r❛ ♣r♦❜✲ ❧❡♠❛s ❛✉tô♥♦♠♦s✳ ❊♠ ❡s♣❡❝✐❛❧✱ ♦s r❡s✉❧t❛❞♦s ❞❡ ❝♦♠♣❛r❛çã♦ ❢♦r❛♠ r❡❢♦r♠✉❧❛❞♦s ♣❛r❛ ❝♦♥t❡♠♣❧❛r ♥♦ss❛s ♥❡❝❡ss✐❞❛❞❡s ❡✱ ❛❧é♠ ❞❡st❡s✱ ❢♦r❛♠ ❢❡✐t❛s ❛❧❣✉♠❛s ❡①t❡♥sõ❡s ❛♦ ❝♦♥✲ t❡①t♦ ❞♦s ♣r♦❝❡ss♦s ♠✉❧tí✈♦❝♦s ❞❡ ❢❛t♦s ❛♥t❡r✐♦r♠❡♥t❡ ♦❜s❡r✈❛❞♦s ♥❛ ❧✐t❡r❛t✉r❛ ❛♣❡♥❛s ♣❛r❛ ♣r♦❝❡ss♦s ✉♥í✈♦❝♦s✳

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■♥tr♦❞✉çã♦ ①✈

♥♦s ♣❡r♠✐t✐✉ ❣❛r❛♥t✐r ❛ ✐♥✈❛r✐â♥❝✐❛ ❞♦s ❛tr❛t♦r❡s ❛✐♥❞❛ q✉❡ t❡♥❤❛♠♦s ❛♣❡♥❛s ❛ s❡♠✐❝♦♥✲ t✐♥✉✐❞❛❞❡ s✉♣❡r✐♦r ❞♦ ✢✉①♦ ❝♦♠ r❡❧❛çã♦ ❛♦s ❞❛❞♦s ✐♥✐❝✐❛✐s✳ ◆♦ss❛ ❞❡♠♦♥str❛çã♦ ❞❡st❡ ❢❛t♦ r❡❢♦rç❛ ♥♦ss❛ ❝r❡♥ç❛ ❞❡ q✉❡✱ ♣❛r❛ tr❛t❛r ♣r♦❜❧❡♠❛s ♠✉❧tí✈♦❝♦s✱ s❡❥❛♠ ❡❧❡s ❛✉tô♥♦♠♦s ♦✉ ♥ã♦ ❛✉tô♥♦♠♦s✱ ❛s té❝♥✐❝❛s q✉❡ ❧✐❞❛♠ ❝♦♠ ♦♣❡r❛❞♦r❡s ♠✉❧tí✈♦❝♦s ❞❡ ❡✈♦❧✉çã♦ ❝♦♥str✉í✲ ❞♦s ❛ ♣❛rt✐r ❞❛s tr❛❥❡tór✐❛s✱ ♥♦s ♠❡s♠♦s ♠♦❧❞❡s ❞❡ ❬✻✾❪✱ r❡✈❡❧❛♠✲s❡ ❢❡rr❛♠❡♥t❛s ♣♦❞❡r♦s❛s ♣❛r❛ ❛ ❛♥á❧✐s❡ ❛ss✐♥tót✐❝❛✳

❆ s❡❣✉♥❞❛ ♣❛rt❡ ❞❡st❡ tr❛❜❛❧❤♦ ❞❡st✐♥❛✲s❡ ❛ ❛♥á❧✐s❡ ❞❡ ♣r♦❜❧❡♠❛s ❞❡✜♥✐❞♦s ❡♠ t♦❞♦ ♦ Rn✳ ❊♠ ❞♦♠í♥✐♦s ✐❧✐♠✐t❛❞♦s ❛ ♠❛✐♦r ❞✐✜❝✉❧❞❛❞❡✱ ❡♠ ❣❡r❛❧✱ s❡ ❞á ♣❡❧❛ ♣❡r❞❛ ❞❛s ✐♠❡rsõ❡s ❝♦♠♣❛❝t❛s ❞♦s ❡s♣❛ç♦s ❞❡ ❢✉♥çõ❡s ❡✱ ♣❛r❛ ❝♦♥t♦r♥❛r ❡st❡ ♣r♦❜❧❡♠❛✱ ❞✐✈❡rs♦s ♠ét♦❞♦s tê♠ s✐❞♦ ❞❡s❡♥✈♦❧✈✐❞♦s✳ ❆❧❣✉♥s ❛✉t♦r❡s ❛♠❜✐❡♥t❛♠ s❡✉s s✐st❡♠❛s ❡♠ ❡s♣❛ç♦s ❝✉❥❛ ♥♦r♠❛ é ❞❡✜♥✐❞❛ ❝♦♠ ✉♠ ♣❡s♦✱ ❞❡ ❢♦r♠❛ q✉❡ s❡ ♣♦ss❛ r❡s❣❛t❛r ♣r♦♣r✐❡❞❛❞❡s ❞❡ ❝♦♠✲ ♣❛❝✐❞❛❞❡✱ ❬✶✱ ✼✱ ✼✶❪✳ ❖✉tr♦ r❡❝✉rs♦ ❝♦♠✉♠❡♥t❡ ✉t✐❧✐③❛❞♦ ❝♦♥s✐st❡ ❡♠ ❞❡❝♦♠♣♦r ♦ ❡s♣❛ç♦Rn ❡♠ ✉♠❛ ❜♦❧❛ ❧✐♠✐t❛❞❛ ❡ s❡✉ ❝♦♠♣❧❡♠❡♥t♦✱ ❡ ❡♥tã♦ ♠♦str❛r q✉❡ ♦ ✢✉①♦ é ❛ss✐♥t♦t✐❝❛♠❡♥t❡ t♦t❛❧♠❡♥t❡ ❧✐♠✐t❛❞♦ ❛tr❛✈és ❞❡ ❡st✐♠❛t✐✈❛s ❞❛s s♦❧✉çõ❡s ❡♠ ✈❛r✐á✈❡✐s ❡s♣❛❝✐❛✐s ❝♦♠ ♥♦r♠❛ ❣r❛♥❞❡✱ ❛❧✐❛❞❛s às ✐♠❡rsõ❡s ❝♦♠♣❛❝t❛s ✉s✉❛✐s ❡♠ ❞♦♠í♥✐♦s ❧✐♠✐t❛❞♦s ❬✶✶✱ ✹✺✱ ✼✻❪✳ ◆♦s tr❛✲ ❜❛❧❤♦s ❬✸✾✱ ✺✽❪ ♦s ❛✉t♦r❡s✱ s❡❣✉✐♥❞♦ ✉♠❛ ❞✐r❡çã♦ ❞✐❢❡r❡♥t❡ ❞♦s ❛♥t❡r✐♦r❡s✱ ✉t✐❧✐③❛♠ ❡s♣❛ç♦s ❞❡ ❢✉♥çõ❡s ❧✐♠✐t❛❞❛s ❡ ✉♥✐❢♦r♠❡♠❡♥t❡ ❝♦♥tí♥✉❛s ❡♠ Rn ❝♦♠♦ ❡s♣❛ç♦ ❜❛s❡ ❡✱ s♦❜ ❤✐♣ót❡✲ s❡s ❝♦♥✈❡♥✐❡♥t❡s ♥♦ t❡r♠♦ ♣❡rt✉r❜❛t✐✈♦✱ ❡①✐❜❡♠ ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠❛ s♦❧✉çã♦ ❡st❛❝✐♦♥ár✐❛ ♠❛①✐♠❛❧ ❡ ❛ ❝♦♠♣❛❝✐❞❛❞❡ ❛ss✐♥tót✐❝❛ ❞♦ s✐st❡♠❛ é ♦❜t✐❞❛ ❛tr❛✈és ❞❡ ✉♠ ❛r❣✉♠❡♥t♦ ❞❡ ❝♦♠♣❛r❛çã♦✳ ❯♠❛ ❡①❝❡❧❡♥t❡ ❞✐s❝✉ssã♦ s♦❜r❡ ♣r♦❜❧❡♠❛s ❡♠ ❞♦♠í♥✐♦s ✐❧✐♠✐t❛❞♦s ♣♦❞❡ s❡r ❡♥❝♦♥t❛❞❛ ❡♠ ❬✷❪✱ ♦♥❞❡ ♦s ❛✉t♦r❡s ❢❛③❡♠ ✉♠❛ ❞❡s❝r✐çã♦ ❜❛st❛♥t❡ ❞❡t❛❧❤❛❞❛ ❞❛s ♣r✐♥❝✐♣❛✐s té❝♥✐❝❛s ✉s❛❞❛s ❡♠ ♣r♦❜❧❡♠❛s ❛✉tô♥♦♠♦s✳

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①✈✐ ■♥tr♦❞✉çã♦

♥❛ ❞❡✜♥✐çã♦ ❞♦ ♦♣❡r❛❞♦r ♣r✐♥❝✐♣❛❧✱ ❞❡ ❢♦r♠❛ q✉❡ ♣✉❞éss❡♠♦s ❧❛♥ç❛r ♠ã♦ ❞♦s r❡s✉❧t❛❞♦s ❞❡ ❝♦♠♣❛r❛çã♦ ♣r❡✈✐❛♠❡♥t❡ ❞❡s❡♥✈♦❧✈✐❞♦s ♥❡st❡ tê①t♦✳ ■ss♦ ❢♦✐ ❢❡✐t♦ ♥♦ ú❧t✐♠♦ ❝❛♣ít✉❧♦ s❡❣✉✐♥❞♦ ✉♠ ❡sq✉❡♠❛ ❛♥t❡r✐♦r♠❡♥t❡ ✉t✐❧✐③❛❞♦ ❡♠ ✉♠ ♣r♦❜❧❡♠❛ s❡♠✐❧✐♥❡❛r ♣♦r ▼♦r✐❧❧❛s ❡ ❱❛❧❡r♦ ❡♠ ❬✺✾❪ q✉❡ ♥♦s ♣❡r♠✐t✐✉ ❡st❛❜❡❧❡❝❡r ✉♠ r❡s✉❧t❛❞♦ ❞❡ ❝♦♠♣❛r❛çã♦ q✉❡ ❣❛r❛♥t❡ ❛ ❡①✐stê♥❝✐❛ ❞❡ tr❛❥❡tór✐❛s ❝♦♠♣❧❡t❛s ❡①tr❡♠❛s ❞ês❞❡ q✉❡ ♦ s✐st❡♠❛ s❡❥❛ ✉♥í✈♦❝♦✳ ❉❡ ❢❛t♦✱ ♣♦❞❡♠♦s ❛✜r♠❛r q✉❡✱ s❡ u0(x) ≤ v0(x) ❡♥tã♦ ❡①✐st❡ ✉♠❛ s♦❧✉çã♦ u ❝♦♠ ❞❛❞♦ ✐♥✐❝✐❛❧ ❡♠

u0 ❡ ❡①✐st❡ ✉♠❛ s♦❧✉çã♦ v q✉❡ s❡ ✐♥✐❝✐❛ ❡♠ v0 s❛t✐s❢❛③❡♥❞♦ u(t) ≤ v(t) ♣❛r❛ t♦❞♦ t❡♠♣♦

t >0✳ ❆ ♥❡❝❡ss✐❞❛❞❡ ❞❡ s❡ ❝♦♥s✐❞❡r❛r ♣r♦❜❧❡♠❛s q✉❡ tê♠ ✉♠❛ ú♥✐❝❛ s♦❧✉çã♦ ❛♣❛r❡❝❡ ❥á ♥❛ té❝♥✐❝❛ q✉❡ ✉t✐❧✐③❛♠♦s ♣❛r❛ ❞❡♠♦♥str❛r ❛ ❝♦♠♣❛❝✐❞❛❞❡ ❛ss✐♥tót✐❝❛ ♥♦ s❡♥t✐❞♦ ♣✉❧❧❜❛❝❦✱ q✉❡ r❡q✉❡r q✉❡ ♦ ♣r♦❝❡ss♦ s❡❥❛ ❢r❛❝❛♠❡♥t❡ ❢❡❝❤❛❞♦✱ ♣r♦♣r✐❡❞❛❞❡ q✉❡ só ♦❜t✐✈❡♠♦s ♣❛r❛ ♣r♦❝❡ss♦s ✉♥í✈♦❝♦s ❡ ✈❡♠ s❡r r❡❢♦rç❛❞❛ ♣❡❧❛s ❝❛r❛❝t❡ríst✐❝❛s ❞♦ r❡s✉❧t❛❞♦ ❞❡ ❝♦♠♣❛r❛çã♦ q✉❡ ❞❡♠♦♥str❛♠♦s✳

❊st❡ tê①t♦ ❡stá ♦r❣❛♥✐③❛❞♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ ♥♦ ♣r✐♠❡✐r♦ ❝❛♣ít✉❧♦ ❛♣r❡s❡♥t❛♠♦s ❞✐✈❡rs♦s r❡s✉❧t❛❞♦s ♣r❡❧✐♠✐♥❛r❡s q✉❡ s❡rã♦ ✉t✐❧✐③❛❞♦s ❛♦ ❧♦♥❣♦ ❞♦ tr❛❜❛❧❤♦✳ ◆♦s ❈❛♣ít✉❧♦s ✷ ❡ ✸ tr❛t❛♠♦s ♦s ♣r♦❜❧❡♠❛s ❡♠ ❞♦♠í♥✐♦s ❧✐♠✐t❛❞♦s s❡♥❞♦ q✉❡✱ ♥♦ ❈❛♣ít✉❧♦ ✷ ❡♥❢♦❝❛♠♦s ♦s r❡s✉❧t❛❞♦s ❞❡ ❝♦♠♣❛r❛çã♦ ❞❡ s♦❧✉çõ❡s ❡ ♦ ❈❛♣ít✉❧♦ ✸ é ❞❡❞✐❝❛❞♦ à ❡①✐stê♥❝✐❛ ❞❡ ❛tr❛t♦r ♥♦ s❡♥t✐❞♦ ♣✉❧❧❜❛❝❦ ❡ à ❡①✐stê♥❝✐❛ ❞❡ tr❛❥❡tór✐❛s ❡①tr❡♠❛s ❝♦♠♣❧❡t❛s✳ ◆♦ ❈❛♣ít✉❧♦ ✹ ❡①✐❜✐♠♦s ❛ ❡①✐stê♥❝✐❛ ❞❡ ❛tr❛t♦r ♥♦ s❡♥t✐❞♦ ♣✉❧❧❜❛❝❦ ❡ tr❛❥❡tór✐❛s ❡①tr❡♠❛s ❡♠Rn✉t✐❧✐③❛♥❞♦ ❡s♣❛ç♦s ❞❡✜♥✐❞♦s ❝♦♠ ♣❡s♦ ♥❛ ♥♦r♠❛ ❡✱ ♥♦ ❈❛♣ít✉❧♦ ✺ tr❛t❛♠♦s ♦ ♠❡s♠♦ ♣r♦❜❧❡♠❛ ❡♠Rnr❡t✐r❛♥❞♦ ❞❛ ❞❡✜♥✐çã♦ ❞♦ ♦♣❡r❛❞♦r ♣r✐♥❝✐♣❛❧ ❛s ❢✉♥çõ❡s ♣❡s♦ ❛✉①✐❧✐❛r❡s✳

❱❛❧❡ ❛ ♣❡♥❛ ♦❜s❡r✈❛r q✉❡ ♥ã♦ t✐✈❡♠♦s ❛ ♣r❡♦❝✉♣❛çã♦ ❞❡ s✉♣♦r ❝♦♥❞✐çõ❡s ót✐♠❛s ♥❛s ♣❡rt✉r❜❛çõ❡s q✉❡ ❝♦♥s✐❞❡r❛♠♦s✳ ❉❡ ❢❛t♦✱ q✉❛s❡ t♦❞❛s ❛s ❝♦♥❞✐çõ❡s ✐♠♣♦st❛s ❛♦ t❡r♠♦ ♣❡rt✉r❜❛t✐✈♦ ♥♦s ❈❛♣ít✉❧♦s ✹ ❡ ✺ ♣♦❞❡r✐❛♠ s❡r ❝♦♥s✐❞❡r❛✈❡❧♠❡♥t❡ r❡❧❛①❛❞❛s✱ ♥♦ ❡♥t❛♥t♦ ❛s ❡st✐♠❛t✐✈❛s s❡ t♦r♥❛r✐❛♠ ❜❛st❛♥t❡ ♠❛✐s ❝♦♠♣❧✐❝❛❞❛s✱ ❞❡ ❢♦r♠❛ q✉❡ ♦♣t❛♠♦s ♣♦r ✉♠❛ ❡①♣♦s✐çã♦ ♠❛✐s s✐♠♣❧✐✜❝❛❞❛ q✉❡ ♥ã♦ ♦❜s❝✉r❡❝❡ss❡ ♦ ♦❜❥❡t✐✈♦ ♣r✐♥❝✐♣❛❧✳ ❆✐♥❞❛ ❛ss✐♠✱ ❛s ❝♦♥❞✐çõ❡s q✉❡ ❝♦♥s✐❞❡r❛♠♦s sã♦ ❜❛st❛♥t❡ ❝♦♠♣❛tí✈❡✐s ❝♦♠ ❛s ✐♠♣♦st❛s ❡♠ ❬✹✺✱ ✹✻✱ ✽✵❪ ♥♦s q✉❛✐s sã♦ ❝♦♥s✐❞❡r❛❞❛s ❛♣❡♥❛s ❛ ♣❛rt❡ ♠♦♥ót♦♥❛ ❞❛ ♣❡rt✉r❜❛çã♦✱ ❡ ❡♠ ❬✼✼❪✱ ♦♥❞❡ B =f(x, u) +g(t, x)✱ ♦✉ s❡❥❛✱ ♥ã♦ ❤á ✉♠ t❡r♠♦ q✉❡ ❞❡♣❡♥❞❛ s✐♠✉❧t❛♥❡❛♠❡♥t❡ ❞❡t ❡ ❞❡ u✳ ❆ ❡①✐❣ê♥❝✐❛ p > n/2 ❡stá r❡❧❛❝✐♦♥❛❞❛ ❝♦♠ ❛s ❡st✐♠❛t✐✈❛s ❡♠ L∞ q✉❡ sã♦ ❡ss❡♥❝✐❛✐s

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C

ONTEÚDO

AGRADECIMENTOS V

ABSTRACT VII

RESUMO IX

INTRODUÇÃO XI

1 PRELIMINARES 3

✶✳✶ Pr✐♥❝✐♣❛✐s ❉❡s✐❣✉❛❧❞❛❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✷ ❖♣❡r❛❞♦r ▼❛①✐♠❛❧ ▼♦♥ót♦♥♦ ❡♠ ❊s♣❛ç♦s ❞❡ ❍✐❧❜❡rt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✷✳✶ ❖ ❖♣❡r❛❞♦r ♣✲▲❛♣❧❛❝✐❛♥♦ ❡♠ ❉♦♠í♥✐♦s ▲✐♠✐t❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✸ ❚❡♦r❡♠❛s ❞❡ ❈♦♠♣❛❝✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✹ ❆tr❛t♦r P✉❧❧❜❛❝❦ ❡ Pr♦❝❡ss♦ ❉✐♥â♠✐❝♦ ▼✉❧tí✈♦❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶

2 EXISTÊNCIA ECOMPARAÇÃO DE SOLUÇÕES 21

✷✳✶ ❊①✐stê♥❝✐❛ ❧♦❝❛❧ ❞❡ s♦❧✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✷ ❊①✐stê♥❝✐❛ ●❧♦❜❛❧ ❞❡ ❙♦❧✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✷✳✸ ❈♦♠♣❛r❛çã♦ ❞❡ ❙♦❧✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ 3 ATRATORPULLBACK ETRAJETÓRIASEXTREMAS EMDOMÍNIOSLIMITADOS 37 ✸✳✶ ❊st✐♠❛t✐✈❛s ♣❛r❛ ❛s ❙♦❧✉çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✸✳✷ ❚r❛❥❡tór✐❛s ❡①tr❡♠❛s ♣❛r❛ ♦ ❛tr❛t♦r ♣✉❧❧❜❛❝❦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✸✳✷✳✶ ❖ Pr♦❝❡ss♦ ▼✉❧tí✈♦❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✸✳✷✳✷ ❊①✐stê♥❝✐❛ ❞♦ ❛tr❛t♦r ♣✉❧❧❜❛❝❦ ❡♠ L2(Ω) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹

✸✳✷✳✸ ■♥✈❛r✐â♥❝✐❛ ❞♦ ❆tr❛t♦r P✉❧❧❜❛❝❦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✸✳✷✳✹ ❚r❛❥❡tór✐❛s ❝♦♠♣❧❡t❛s ❡①tr❡♠❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼

4 SISTEMAS GLOBALMENTELIPSCHITZ EMESPAÇOS COMPESO 61

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✹✳✷ ❊①✐stê♥❝✐❛ ❞❡ s♦❧✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✼ ✹✳✸ ❆tr❛t♦r ♣✉❧❧❜❛❝❦ ❡ tr❛❥❡tór✐❛s ❡①tr❡♠❛s ❡♠ L2(Rn) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✹ ✹✳✸✳✶ ❊st✐♠❛t✐✈❛ ❡♠ L2(Rn), ❡♠ E ❡ ❡①✐stê♥❝✐❛ ❞♦ ❛tr❛t♦r ♣✉❧❧❜❛❝❦ ✳ ✳ ✳ ✼✹ ✹✳✸✳✷ ❚r❛❥❡tór✐❛s ❝♦♠♣❧❡t❛s ❡①tr❡♠❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✽

5 ATRATOR PULLBACK ETRAJETÓRIAS EXTREMAS EML2(Rn) 79

✺✳✵✳✸ ❊st✐♠❛t✐✈❛s ❡♠ ❞♦♠í♥✐♦s ❧✐♠✐t❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✺ ✺✳✵✳✹ ❊①✐stê♥❝✐❛ ❞❡ s♦❧✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✹ ✺✳✶ ❊①✐stê♥❝✐❛ ❡ ❙❡♠✐✲❝♦♥t✐♥✉✐❞❛❞❡ s✉♣❡r✐♦r ❞♦ Pr♦❝❡ss♦ ▼✉❧tí✈♦❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✺ ✺✳✷ ❊①✐stê♥❝✐❛ ❞♦ ❆tr❛t♦r P✉❧❧❜❛❝❦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✽ ✺✳✸ ❚r❛❥❡tór✐❛s ❈♦♠♣❧❡t❛s ❊①tr❡♠❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✾

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❈❛♣ít✉❧♦ ✶

P

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◆❡st❡ ❝❛♣ít✉❧♦ ❛♣r❡s❡♥t❛♠♦s ❞❡✜♥✐çõ❡s✱ ♥♦t❛çõ❡s ❡ r❡s✉❧t❛❞♦s✱ ❛❧❣✉♥s ❛❞❛♣t❛❞♦s ❞❛ ❧✐t❡r❛t✉r❛✱ q✉❡ ❢♦r❛♠ ❢✉♥❞❛♠❡♥t❛✐s ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ t❡s❡✱ ✉♠❛ ✈❡③ q✉❡ ❢♦r❛♠ ✉t✐❧✐③❛❞♦s ❡♠ ❞❡♠♦♥str❛çõ❡s ❞❡ t❡♦r❡♠❛s ✐♠♣♦rt❛♥t❡s✳ ❊st❡ ❝❛♣ít✉❧♦ ❡stá ❞✐✈✐❞✐❞♦ ❡♠ ✈ár✐❛s s❡çõ❡s✱ ♦♥❞❡ ❝❛❞❛ ✉♠❛ tr❛t❛ ❞❡ ✉♠ t❡♠❛ ❡s♣❡❝í✜❝♦✳ ◆❛ ❙❡çã♦ ✶✳✶ ❞❡♠♦♥str❛♠♦s ❞✉❛s ❞❡s✐❣✉❛❧❞❛❞❡s q✉❡ sã♦ ♥❛ ✈❡r❞❛❞❡ ♣❡q✉❡♥❛s ❛❞❛♣t❛çõ❡s ❞♦ ▲❡♠❛ ✺✳✶ ❡ ▲❡♠❛ ❯♥✐❢♦r♠❡ ❞❡ ●r♦♥✇❛❧❧✱ ❬✼✸❪✳ ◆❛ ❙❡çã♦ ✶✳✷ ❛♣r❡s❡♥t❛♠♦s ♦s ♣r✐♥❝✐♣❛✐s r❡s✉❧t❛❞♦s q✉❡ ✉s❛r❡♠♦s ❞❛ ❚❡♦r✐❛ ❞❡ ❖♣❡r❛❞♦r❡s ▼♦♥ót♦♥♦s✳ ❆ ❙❡çã♦ ✶✳✸ é ❞❡❞✐❝❛❞❛ ❛ ♣r♦♣r✐❡❞❛❞❡s ❞❡ ❝♦♠♣❛❝✐❞❛❞❡ ❞❡ s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s ❞✐❢❡r❡♥❝✐❛✐s ❡ ♥❛ ❙❡çã♦ ✶✳✹ ✐♥tr♦❞✉③✐♠♦s ♦s ❝♦♥❝❡✐t♦s ❞❡ ❛tr❛çã♦ ♣✉❧❧❜❛❝❦ ♣❛r❛ ♣r♦❝❡ss♦s ♠✉❧tí✈♦❝♦s✳

✶✳✶ Pr✐♥❝✐♣❛✐s ❉❡s✐❣✉❛❧❞❛❞❡s

❉❡s✐❣✉❛❧❞❛❞❡ ✶✳✶✳✶ ✭❉❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❚❛rt❛r✱ ♣✳✷ ❬✼✺❪✮ ❙❡❥❛ p ≥ 2✳ ❊♥tã♦ ♣❛r❛ t♦❞♦ a✱b ∈Rn✱n N t❡♠♦s

|a|p−2a− |b|p−2b✱ab γ0|a−b|p✱

♦♥❞❡ γ0 é ♣♦s✐t✐✈♦ ❡ ❞❡♣❡♥❞❡ ❛♣❡♥❛s ❞❡ p ❡ ❞❡ n✳ ❙❡ 1< p <2 ❡♥tã♦ ♣❛r❛ t♦❞♦ a✱b ∈Rn

t❡♠♦s

|a|p−2a− |b|p−2b✱ab γ1|a−b|p✱

♦♥❞❡ γ1 ❞❡♣❡♥❞❡ ❛♣❡♥❛s ❞❡ p ❡ ❞❡ n✳

❉❡s✐❣✉❛❧❞❛❞❡ ✶✳✶✳✷ ✭❉❡s✐❣✉❛❧❞❛❞❡ ❞❡ ●r♦♥✇❛❧❧✲❇❡❧❧♠❛♥ ♣✳✶✺✻ ❬✶✷❪✮ ❙❡❥❛♠ τ, T ∈ R, τ < T. ❙❡❥❛♠ m ∈ L1(τ, T,R) t❛❧ q✉❡ m 0 q✳t✳♣✳ ❡♠ (τ, T) a 0 ✉♠❛ ❝♦♥st❛♥t❡✳

❙❡❥❛ φ: [τ✱T]→R ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛ q✉❡ ✈❡r✐✜❝❛

φ(t)a+

Z t

τ

m(s)φ(s)ds✱ t [τ✱T].

(25)

✹ ❈❛♣ít✉❧♦ ✶ ✕ Pr❡❧✐♠✐♥❛r❡s

❊♥tã♦

φ(t)≤aeRτtm(s)ds✱ ∀ t ∈[τ✱T].

❉❡s✐❣✉❛❧❞❛❞❡ ✶✳✶✳✸ ✭❉❡s✐❣✉❛❧❞❛❞❡ ❞❡ ●r♦♥✇❛❧❧ ♣✳ ✶✺✼ ❬✶✷❪✮ ❙❡❥❛♠ τ, T ∈ R, τ < T. ❙❡❥❛♠ m ∈ L1(τ, T,R) t❛❧ q✉❡ m 0 q✳t✳♣✳ ❡♠ (τ, T) a 0 ✉♠❛ ❝♦♥st❛♥t❡✳ ❙❡❥❛

φ: [τ✱T]→R ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛ ✈❡r✐✜❝❛♥❞♦ 1

2φ 2(t)

≤ 12a2+

Z t

τ

m(s)φ(s)ds✱ t [τ✱T]. ❊♥tã♦

|φ(t)| ≤a+

Z t

τ

m(s)ds✱ t[τ✱T].

❖s ♣ró①✐♠♦s ❞♦✐s ❧❡♠❛s sã♦ ❧✐❣❡✐r❛s ❛❞❛♣t❛çõ❡s ❞♦ ▲❡♠❛ ✺✳✶ ❡ ▲❡♠❛ ❯♥✐❢♦r♠❡ ❞❡ ●r♦♥✇❛❧❧ ❡♥❝♦♥tr❛❞♦s ❡♠ ❬✼✸❪✳

▲❡♠❛ ✶✳✶✳✶ ✭▲❡♠❛ ✺✳✶✱ ❬✼✸❪✮ ❙❡❥❛ y : [τ,) R ✉♠❛ ❢✉♥çã♦ ♣♦s✐t✐✈❛ ❛❜s♦❧✉t❛♠❡♥t❡ ❝♦♥tí♥✉❛ ♦♥❞❡ τ R✱ ❛ q✉❛❧ s❛t✐s❢❛③ ♣❛r❛ ❝❛❞❛ tτ

˙

y+γyRδ,

❝♦♠ R > 1✱γ > 0 ❡ δ : [τ,∞) → R ✉♠❛ ❢✉♥çã♦ ♣♦s✐t✐✈❛ ❛❜s♦❧✉t❛♠❡♥t❡ ❝♦♥tí♥✉❛ ❡ ♥ã♦ ❞❡❝r❡s❝❡♥t❡ ❧✐♠✐t❛❞❛ ❡♠ ❧✐♠✐t❛❞♦s✳ ❊♥tã♦ ♣❛r❛ ❝❛❞❛ t ≥τ t❡♠♦s

y(t)

δ(t)

γ 1/R

+ 1

(γ(R−1)(t−τ))1/(R−1)

❉❡♠♦♥str❛çã♦✿ ❙❡ y(τ) δ(γτ)1/R ❡♥tã♦ ♣❛r❛ ❝❛❞❛ t τ✱ y(t) δ(γt)1/R. ❉❡ ❢❛t♦✱ s✉♣♦♥❤❛ q✉❡ ♣❛r❛ ❛❧❣✉♠t τ✱ y(t)>δ(γt)1/R. ❊♥tã♦ t❡r❡♠♦s q✉❡ ♣❛r❛ ❛❧❣✉♠ t0 ∈[τ, t)✱ y(t0) =

δ(t0)

γ 1/R

✳ ▲♦❣♦

˙

y(t0) +γ

" δ(t0)

γ

1/R#R

≤δ(t0)⇒y˙(t0)≤0

❆ss✐♠✱ y ❛t✐♥❣❡ ❛ ❢✉♥çã♦ δγ1/R ❞❡❝r❡s❝❡♥❞♦✳ ❈♦♠♦ δ é ♥ã♦ ❞❡❝r❡s❝❡♥t❡✱ ♣❛r❛ ❝❛❞❛ t≥τ t❡♠♦s q✉❡y(t)≤δ(γt)

1/R

+(γ(R1)(t1τ))1/(R−1)

❙❡y(τ)>δ(γτ)1/R, s✉♣♦♥❤❛ q✉❡ y(t)>δ(γt)1/R ♣❛r❛ ❛❧❣✉♠ t0 > τ ❡♥tã♦

δ(t)y˙(t) +γyR(t)y˙(t) +γ "

δ(t) γ

1/R#R

(26)

✶✳✶✳ Pr✐♥❝✐♣❛✐s ❉❡s✐❣✉❛❧❞❛❞❡s ✺

✐st♦ ✐♠♣❧✐❝❛ q✉❡ y˙(t) ≤ 0 ♣❛r❛ ❝❛❞❛ t = t0✳ ❈♦♠♦ δ é ♥ã♦ ❞❡❝r❡s❝❡♥t❡ ❛ ♣❛rt✐r ❞❡ t0,

y(t)≤δ(γt) 1/R

❡ ❡♥tã♦ y(t)≤δ(γt) 1/R

+(γ(R1)(t1τ))1/(R−1) ♣❛r❛ t≥t0✳

P❛r❛ ❝❛❞❛ t[τ✱t0]❡s❝r❡✈❛z(t) = y(t)−

δ(t)

γ 1/R

≥0✳ ❈♦♠♦ aR+bR(a+b)R s❡a✱b sã♦ ♣♦s✐t✐✈♦s ❡R ≥1,t❡♠♦s

y(t)R= z(t) +

δ(t)

γ

1/R!R

≥z(t)R+ δ(t) γ . P♦rt❛♥t♦

˙

z(t) +γz(t)R y˙(t)δ(t)(1/R)−1

Rγ1/R δ˙(t) +γy(t)

Rδ(t)

❈♦♠♦ ♣♦r ❤✐♣ót❡s❡ y˙(t) +γy(t)Rδ(t) 0 δ é ✉♠❛ ❢✉♥çã♦ ♣♦s✐t✐✈❛ ❡ ♥ã♦ ❞❡❝r❡s❝❡♥t❡ t❡♠♦s q✉❡

˙

z(t) +γz(t)R0 ■♥t❡❣r❛♥❞♦ ❞❡ τ ❛ t t❡♠♦s

Z t

τ

z(s)−Rz˙(s)ds ≤

Z t

τ − γds P♦rt❛♥t♦✱

z(t)1−R

1R −

z(τ)1−R

1R ≤ −γ(t−τ) z(t)1−R z(τ)1−R+γ(R1)(tτ) z(t)R−1 1

γ(R−1)(t−τ)

z(t) 1

[γ(R1)(tτ)]1/(R−1)

P♦rt❛♥t♦ t❡♠♦s ♣❛r❛ ❝❛❞❛ t ∈[τ✱t0]

y(t) = z(t) +

δ(t)

γ 1/R

δ(t)

γ 1/R

+ 1

[γ(R1)(tτ)]1/(R−1)

■st♦ ❝♦♥❝❧✉✐ ❛ ❞❡♠♦♥str❛çã♦✳

▲❡♠❛ ✶✳✶✳✷ ✭▲❡♠❛ ❯♥✐❢♦r♠❡ ❞❡ ●r♦♥✇❛❧❧✱ ❬✼✸❪✮ ❙❡❥❛♠g✱h✱y❢✉♥çõ❡s ♣♦s✐t✐✈❛s ❝♦♠y✱g✱h˙ ❡ y ❧♦❝❛❧♠❡♥t❡ ✐♥t❡❣rá✈❡✐s ❡♠ [t0✱∞] ❝♦♠

dy

(27)

✻ ❈❛♣ít✉❧♦ ✶ ✕ Pr❡❧✐♠✐♥❛r❡s

❙❡❥❛ R >0 ✜①♦ ❡ s✉♣♦♥❤❛ q✉❡ ♣❛r❛ ❝❛❞❛ t≥t0 t❡♥❤❛♠♦s

Z t+R

t

g(s)dsa1(t)✱

Z t+R

t

h(s)dsa2(t)✱

Z t+R

t

y(s)dsa3(t).

❊♥tã♦ ♣❛r❛ ❝❛❞❛ t≥t0 t❡♠♦s

y(t+R)

a3(t)

R +a2(t)

ea1(t)

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ t0 ≤t ≤θ≤s ≤τ ≤t+R.

▼✉❧t✐♣❧✐❝❛♥❞♦ dy

dτ −gy≤h ♣♦re−

s g(σ)dσ ♦❜t❡♠♦s d

y(τ)e−Rsτg(σ)dσ

≤h(τ)e−Rsτg(σ)dσ ≤h(τ) ■♥t❡❣r❛♥❞♦ ❡♠ τ ❞❡θ ❛ t+R

y(t+R)e−Rst+Rg(σ)dσ−y(θ)e−

s g(σ)dσ ≤ Z t+R

θ

h(τ)dτ Z t+R

t

h(τ)dτ a2(t)

y(t+R)y(θ)e−Rsθg(σ)dσ+

Rt+R

s g(σ)dσ+a2(t)e

Rt+R s g(σ)dσ

y(t+R)(y(θ) +a2(t))e

Rt+R θ g(σ)dσ

y(t+R)≤(y(θ) +a2(t))ea1(t)

■♥t❡❣r❛♥❞♦ ❡♠ θ ❞❡t ❛ t+R Z t+R

t

y(t+R)dθ Z t+R

t

y(θ)ea1(t)+ Z t+R

t

a2(t)ea1(t)dθ

Ry(t+R)ea1(t) Z t+R

t

y(θ)dθ+a2(t)ea1(t)

Z t+R

t

ea1(t)a

3(t) +a2(t)ea1(t)R

P♦rt❛♥t♦ ♣❛r❛ ❝❛❞❛t ≥t0 t❡♠♦s

y(t+R)

a3(t)

R +a2(t)

ea1(t)

❖❜s❡r✈❛çã♦ ✶✳✶✳✶ ❈♦♠ ❛s ♠❡s♠❛s ❤✐♣ót❡s❡s ❞♦ ▲❡♠❛ ✶✳✶✳✷✱ ♠❛s ❝♦♥s✐❞❡r❛♥❞♦ ❛❣♦r❛ g 0 t❡♠♦s ♣❛r❛ R > 0 ✜①♦ y(t+R) a3(t)

(28)

✶✳✷✳ ❖♣❡r❛❞♦r ▼❛①✐♠❛❧ ▼♦♥ót♦♥♦ ❡♠ ❊s♣❛ç♦s ❞❡ ❍✐❧❜❡rt ✼

✶✳✷ ❖♣❡r❛❞♦r ▼❛①✐♠❛❧ ▼♦♥ót♦♥♦ ❡♠ ❊s♣❛ç♦s ❞❡ ❍✐❧❜❡rt

◆❡st❛ s❡çã♦ ✈❛♠♦s ❡①✐❜✐r ❛❧❣✉♠❛s ❞❡✜♥✐çõ❡s ❡ r❡s✉❧t❛❞♦s q✉❡ ♣♦❞❡♠ s❡r ❡♥❝♦♥✲ tr❛❞♦s ❡♠ ❬✶✷❪✳

❙❡❥❛ H ✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt s♦❜r❡ R ♠✉♥✐❞♦ ❞♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ,·i. ❯♠ ♦♣❡r❛❞♦r ♠✉❧tí✈♦❝♦✱ A, ❡♠ H é ✉♠❛ ❛♣❧✐❝❛çã♦ ❞❡ H ❡♠ ℘(H), ❝♦♥❥✉♥t♦ ❞❛s ♣❛rt❡s ❞❡ H. ❙❡ ♣❛r❛ t♦❞♦ x ∈ H ♦ ❝♦♥❥✉♥t♦ Ax ❝♦♥té♠ ♥♦ ♠á①✐♠♦ ✉♠ ❡❧❡♠❡♥t♦ ❞✐③❡♠♦s q✉❡ A é ✉♥í✈♦❝♦✳ ❖ ❞♦♠í♥✐♦ ❞❡A é ♦ ❝♦♥❥✉♥t♦ D(A) = {xH;Ax6=∅} ❡ ❛ ✐♠❛❣❡♠ ❞❡ A é ♦ ❝♦♥❥✉♥t♦ R(A) =x∈HAx.

■❞❡♥t✐✜❝❛r❡♠♦s A ❝♦♠ s❡✉ ❣rá✜❝♦ ❡♠ H ×H, ✐st♦ é✱ A = {(x, y);yAx}. ❖ ♦♣❡r❛❞♦r A−1 é ❛q✉❡❧❡ ❝✉❥♦ ❣rá✜❝♦ é s✐♠étr✐❝♦ ❛♦ ❞❡ A, ✐st♦ é✱ y A−1x ⇐⇒ x Ay;

❡✈✐❞❡♥t❡♠❡♥t❡D(A−1) = R(A).

❖ ❝♦♥❥✉♥t♦ ❞♦s ♦♣❡r❛❞♦r❡s ❞❡ H é ♣❛r❝✐❛❧♠❡♥t❡ ♦r❞❡♥❛❞♦ ♣❡❧❛ ✐♥❝❧✉sã♦ ❞♦s ❣rá✲ ✜❝♦s✿ A ⊂B s❡ ❡ s♦♠❡♥t❡ s❡ ♣❛r❛ t♦❞♦ x∈H, Ax⊂Bx.

❉❡✜♥✐çã♦ ✶✳✷✳✶ ❉✐③❡♠♦s q✉❡ ✉♠ ♦♣❡r❛❞♦r A ❡♠ H é ♠♦♥ót♦♥♦ s❡ ♣❛r❛ t♦❞♦ x1, x2 ∈

D(A), hAx1−Ax2, x1−x2i ≥ 0 ♦✉✱ ♠❛✐s ♣r❡❝✐s❛♠❡♥t❡✱ ♣❛r❛ t♦❞♦ y1 ∈ Ax1 ❡ ♣❛r❛ t♦❞♦

y2 ∈Ax2, hy1−y2, x1−x2i ≥0.

❊①❡♠♣❧♦ ✶✳✷✳✶ ✭❊①❡♠♣❧♦ ✷✳✶✳✹ ♣✳ ✷✶ ❬✶✷❪✮ ❙❡❥❛ ϕ ✉♠❛ ❢✉♥çã♦ ❝♦♥✈❡①❛ ❡ ♣ró♣r✐❛ s♦❜r❡ H, ♦✉ s❡❥❛✱ ✉♠❛ ❛♣❧✐❝❛çã♦ ❞❡ H ❡♠ ]− ∞,+], t❛❧ q✉❡ ϕ6≡+

ϕ(tx+ (1−t)y)≤tϕ(x) + (1−t)ϕ(y)

♣❛r❛ t♦❞♦ x, y H ❡ ♣❛r❛ t♦❞♦ t (0,1). ❖ ❞♦♠í♥✐♦ ❞❛ ❢✉♥çã♦ ϕ,D(ϕ), é ❞❛❞♦ ♣♦r

D(ϕ) = {xH;ϕ(x)<+∞}. ❆ s✉❜❞✐❢❡r❡♥❝✐❛❧ ∂ϕ ❞❡ ϕ, ❞❡✜♥✐❞❛ ♣♦r y∈∂ϕ(x)⇐⇒♣❛r❛ t♦❞♦ ξ ∈H, ϕ(ξ)≥ϕ(x) +hy, ξ−xi, é ♠♦♥ót♦♥❛ ❡♠ H.

❉❡✜♥✐çã♦ ✶✳✷✳✷ ❖ ♦♣❡r❛❞♦r ♠♦♥ót♦♥♦ A ❞❡ H é ♠❛①✐♠❛❧ ♠♦♥ót♦♥♦ s❡ ❡❧❡ ♥ã♦ ❡stá ♣r♦♣r✐❛♠❡♥t❡ ❝♦♥t✐❞♦ ❡♠ q✉❛❧q✉❡r ♦✉tr♦ ♦♣❡r❛❞♦r ♠♦♥ót♦♥♦ ❞❡ H.

❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✿ A é ♠❛①✐♠❛❧ ♠♦♥ót♦♥♦ s❡ ❡ s♦♠❡♥t❡ s❡ A é ♠♦♥ót♦♥♦ ❡✱ s❡ (x, y)∈H×H ❢♦r t❛❧ q✉❡

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✽ ❈❛♣ít✉❧♦ ✶ ✕ Pr❡❧✐♠✐♥❛r❡s

♣❛r❛ t♦❞♦ ξ ∈ D(A) ✭♦✉ ♠❛✐s ♣r❡❝✐s❛♠❡♥t❡✱ hy−η, x−ξi ≥ 0, ♣❛r❛ t♦❞♦ (ξ, η) ∈ A), ❡♥tã♦ y ∈Ax.

❖✉tr❛ ❝❛r❛❝t❡r✐③❛çã♦ ♣❛r❛ ♦♣❡r❛❞♦r❡s ♠❛①✐♠❛✐s ♠♦♥ót♦♥♦s é ❛ s❡❣✉✐♥t❡ ♣r♦♣♦s✐çã♦✿

Pr♦♣♦s✐çã♦ ✶✳✷✳✶ ✭Pr♦♣♦s✐çã♦ ✷✳✷ ♣✳✷✸ ❬✶✷❪✮ ❙❡❥❛ A ✉♠ ♦♣❡r❛❞♦r ❞❡ H. ❆s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s sã♦ ❡q✉✐✈❛❧❡♥t❡s✿

✶✳ A é ♠❛①✐♠❛❧ ♠♦♥ót♦♥♦❀

✷✳ A é ♠♦♥ót♦♥♦ ❡ R(I+A) = H;

✸✳ P❛r❛ t♦❞♦ λ >0, (I+λA)−1 é ✉♠❛ ❝♦♥tr❛çã♦ ❞❡✜♥✐❞❛ s♦❜r❡ t♦❞♦ H.

❊①❡♠♣❧♦ ✶✳✷✳✷ ✭❊①❡♠♣❧♦ ✷✳✸✳✸ ♣✳✷✺✱ ❬✶✷❪✮ ❙❡❥❛ ϕ ✉♠❛ ❢✉♥çã♦ ❝♦♥✈❡①❛ ❡ ♣ró♣r✐❛ s♦❜r❡ H. ❙❡ ϕ é s❡♠✐❝♦♥tí♥✉❛ ✐♥❢❡r✐♦r♠❡♥t❡ ❡♥tã♦ ∂ϕ é ♠❛①✐♠❛❧ ♠♦♥ót♦♥♦✳

❊①❡♠♣❧♦ ✶✳✷✳✸ ✭❊①❡♠♣❧♦ ✷✳✸✳✼ ♣✳✷✻✱ ❬✶✷❪✮ ❙❡❥❛V ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ r❡✢❡①✐✈♦ ❡V∗

s❡✉ ❞✉❛❧ t❛❧ q✉❡V H V∗ ✐♥❥❡çõ❡s ❝♦♥tí♥✉❛s ❡ ❞❡♥s❛s✳ ❙❡❥❛ A:V V✉♠ ♦♣❡r❛❞♦r

✉♥í✈♦❝♦ ❞❡✜♥✐❞♦ ❡♠ t♦❞♦ V, ❤❡♠✐❝♦♥tí♥✉♦ ❡ ❝♦❡r❝✐✈♦✳ ❊♥tã♦ ♦ ♦♣❡r❛❞♦r AH r❡str✐çã♦ ❞❡ A ❛ H ❞❡✜♥✐❞♦ ♣♦r

D(AH) ={x∈V;Ax∈H} ❡ AH =A é ✉♠ ♦♣❡r❛❞♦r ♠❛①✐♠❛❧ ♠♦♥ót♦♥♦ ❡♠ H.

P❛r❛ ♦s ♣ró①✐♠♦s r❡s✉❧t❛❞♦s ♣r❡❝✐s❛♠♦s ❞❛ s❡❣✉✐♥t❡ ❞❡✜♥✐çã♦✿

❉❡✜♥✐çã♦ ✶✳✷✳✸ ✭❉❡✜♥✐çã♦ ✸✳✶ ♣✳ ✻✹✱ ❬✶✷❪✮ ❙❡❥❛A ✉♠ ♦♣❡r❛❞♦r ❞❡H ❡f ∈L1(0, T;H).

❯♠❛ s♦❧✉çã♦ ❢♦rt❡ ❞❛ ❡q✉❛çã♦ du

dt +Au ∋f é t♦❞❛ ❢✉♥çã♦u∈C([0, T];H),❛❜s♦❧✉t❛♠❡♥t❡ ❝♦♥tí♥✉❛ ❡♠ t♦❞♦ ❝♦♠♣❛❝t♦ ❞❡ (0, T) ✈❡r✐✜❝❛♥❞♦✿ u(t) ∈ D(A) ❡ du

dt(t) +Au(t) ∋ f(t), q✳t✳♣ t(0, T).

❉✐③❡♠♦s q✉❡ u C([0, T];H), é ✉♠❛ s♦❧✉çã♦ ❢r❛❝❛ ❞❛ ❡q✉❛çã♦ du

dt +Au ∋ f s❡ ❡①✐st❡♠ s❡q✉ê♥❝✐❛s fn ∈ L1(0, T;H) ❡ un ∈ C([0, T];H) t❛❧ q✉❡ un é ✉♠❛ s♦❧✉çã♦ ❢♦rt❡ ❞❛ ❡q✉❛çã♦ dun

dt +Aun ∋fn, fn →f ❡♠ L

1(0, T;H) u

n →u ✉♥✐❢♦r♠❡♠❡♥t❡ ❡♠[0, T]. ▲❡♠❛ ✶✳✷✳✶ ✭❚❡♦r❡♠❛ ✸✳✹ ♣✳✻✺✱ ❬✶✷❪✮ ❙❡❥❛ A ✉♠ ♦♣❡r❛❞♦r ♠❛①✐♠❛❧ ♠♦♥ót♦♥♦ ❞❡ H. P❛r❛ t♦❞❛ f L1(0, T;H) ❡ t♦❞♦ u

0 ∈ D(A), ❡①✐st❡ ✉♠❛ ú♥✐❝❛ s♦❧✉çã♦ ❢r❛❝❛ ❞❛ ❡q✉❛çã♦

du

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✶✳✷✳ ❖♣❡r❛❞♦r ▼❛①✐♠❛❧ ▼♦♥ót♦♥♦ ❡♠ ❊s♣❛ç♦s ❞❡ ❍✐❧❜❡rt ✾

Pr♦♣♦s✐çã♦ ✶✳✷✳✷ ✭Pr♦♣♦s✐çã♦ ✸✳✻ ♣✳✼✵✱ ❬✶✷❪✮ ❙❡❥❛ A ✉♠ ♦♣❡r❛❞♦r ♠❛①✐♠❛❧ ♠♦♥ót♦♥♦✱ u∈C([0, T];H) ❡ f ∈L1(0, T;H). ❊♥tã♦ u é ✉♠❛ s♦❧✉çã♦ ❢r❛❝❛ ❞❛ ❡q✉❛çã♦ du

dt +Au∋f s❡ ❡ s♦♠❡♥t❡ s❡ u ✈❡r✐✜❝❛

1

2ku(t)−xk 2

H ≤

1

2ku(s)−xk 2

H + Z t

s h

f(σ)y, u(σ)xidσ,

∀ [x, y]∈A, ∀ 0≤s≤t ≤T.

❚❡♦r❡♠❛ ✶✳✷✳✶ ✭❚❡♦r❡♠❛ ✸✳✻ ♣✳✼✷✱ ❬✶✷❪✮ ❙✉♣♦♥❤❛ q✉❡ ♦ ♦♣❡r❛❞♦r A s❡❥❛ s✉❜❞✐❢❡r❡♥❝✐❛❧ ❞❡ ✉♠❛ ❢✉♥çã♦ϕ♣ró♣r✐❛✱ ❝♦♥✈❡①❛ ❡ s❡♠✐❝♦♥tí♥✉❛ ✐♥❢❡r✐♦r♠❡♥t❡✳ ❙❡f ∈L2(τ, T;H)❡♥tã♦ t♦❞❛ s♦❧✉çã♦ ❢r❛❝❛ ❞❛ ❡q✉❛çã♦ du

dt +Au ∋ f é ✉♠❛ s♦❧✉çã♦ ❢♦rt❡ ❡

tdu dt ∈ L

2(τ, T;H).

❆❧é♠ ❞✐ss♦✱ s❡ ♦ ❞❛❞♦ ✐♥✐❝✐❛❧✱ u0 ∈ D(ϕ), ❡♥tã♦ dudt ∈L2(τ, T;H).

▲❡♠❛ ✶✳✷✳✷ ✭▲❡♠❛ ✸✳✸ ♣✳✼✸✱ ❬✶✷❪✮ ❙❡❥❛ϕ❝♦♥✈❡①❛✱ ♣ró♣r✐❛ ❡ s❡♠✐❝♦♥tí♥✉❛ ✐♥❢❡r✐♦r♠❡♥t❡ ❡♠ H. ❙❡❥❛ u W1,2(0, T;H) t❛❧ q✉❡ u(t) D(∂ϕ) q✳t✳♣✳ ❡♠ (0, T). ❙✉♣♦♥❤❛ q✉❡

❡①✐st❛ g L2(0, T;H) t❛❧ q✉❡ g(t) ∂ϕ(u(t)) q✳t✳♣✳ ❡♠ (0, T). ❊♥tã♦ t 7→ ϕ(u(t)) é

❛❜s♦❧✉t❛♠❡♥t❡ ❝♦♥tí♥✉❛ ❡♠ [0, T]. ❆❧é♠ ❞✐ss♦✱ s❡❥❛ L ♦ ❝♦♥❥✉♥t♦ ❞♦s ♣♦♥t♦s t ∈ (0, T) t❛✐s q✉❡ u(t)∈D(∂ϕ) ❡ u(·) ❡ ϕ(u(·)) sã♦ ❞❡r✐✈á✈❡✐s ❡♠ t ❡♥tã♦ ♣❛r❛ t♦❞♦ t∈ L t❡♠♦s

d

dtϕ(u(t)) =

h, d dtu(t)

,h∂ϕ(u(t)).

Pr♦♣♦s✐çã♦ ✶✳✷✳✸ ✭Pr♦♣♦s✐çã♦ ✸✳✶✸ ♣✳✶✵✼✱ ❬✶✷❪✮ ❙❡❥❛A✉♠ ♦♣❡r❛❞♦r ♠❛①✐♠❛❧ ♠♦♥ót♦♥♦ t❛❧ q✉❡ Int(D(A))6= ❡ s❡❥❛ ❛ ❛♣❧✐❝❛çã♦ B : [0, T]× D(A)H ✈❡r✐✜❝❛♥❞♦

• ❊①✐st❡ω ≥0t❛❧ q✉❡kB(t1, x1)−B(t2, x2)kH ≤ωkx1−x2kH, ∀t∈[0, T],∀x1, x2 ∈

D(A);

• P❛r❛ t♦❞♦ x∈ D(A) ❛ ❛♣❧✐❝❛çã♦ t7→B(t, x) ♣❡rt❡♥❝❡ ❛ L∞(0, T;H).

❊♥tã♦ ♣❛r❛ t♦❞♦ u0 ∈ D(A) ❡①✐st❡ ✉♠❛ ú♥✐❝❛ s♦❧✉çã♦ u ∈ W1,1(0, T;H) ❞❛ ❡q✉❛çã♦

du

dt(t) +Au(t)−B(t, u(t))∋0.

❯♠❛ ✈❡③ q✉❡ ♦ ♦♣❡r❛❞♦r é ❞♦ t✐♣♦ s✉❜❞✐❢❡r❡♥❝✐❛❧ ❞❡ ✉♠❛ ❢✉♥çã♦ ❝♦♥✈❡①❛✱ ♣ró♣r✐❛ ❡ s❡♠✐❝♦♥tí♥✉❛ ✐♥❢❡r✐♦r♠❡♥t❡✱ ♦ s❡♠✐❣r✉♣♦ ❣❡r❛❞♦ ♣♦r ❡st❡ ♦♣❡r❛❞♦r ❡①❡r❝❡ ✉♠ ❡❢❡✐t♦ r❡❣✉❧❛r✐③❛♥t❡ s♦❜r❡ ♦ ❞❛❞♦ ✐♥✐❝✐❛❧✿

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✶✵ ❈❛♣ít✉❧♦ ✶ ✕ Pr❡❧✐♠✐♥❛r❡s

✶✳✷✳✶ ❖ ❖♣❡r❛❞♦r ♣✲▲❛♣❧❛❝✐❛♥♦ ❡♠ ❉♦♠í♥✐♦s ▲✐♠✐t❛❞♦s

◆❡st❡ tr❛❜❛❧❤♦ ♦s ♣r♦❜❧❡♠❛s ❞❡ ❡✈♦❧✉çã♦ q✉❡ ❝♦♥s✐❞❡r❛r❡♠♦s tê♠ ❝♦♠♦ ♣❛rt❡ ♣r✐♥❝✐♣❛❧ ♦ ♦♣❡r❛❞♦rp✲▲❛♣❧❛❝✐♥♦✱∆p,❝♦♠ p >2.❙❡❣✉♥❞♦ ♦s ❛✉t♦r❡s ❬✶✻❪ ❡ ❬✻✵❪ s❡V é ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ r❡✢❡①✐✈♦ ❡V′

♦ s❡✉ ❞✉❛❧✱ ❡H é ✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt✱ ❝♦♠V H V′

❝♦♠ ✐♠❡rsõ❡s ❝♦♥tí♥✉❛s ❡ ❞❡♥s❛s✱A:V →V′ é ✉♠ ♦♣❡r❛❞♦r ♠♦♥ót♦♥♦✱ ✉♥í✈♦❝♦✱ ❞❡✜♥✐❞♦ ❡♠ t♦❞♦ V, ❤❡♠✐❝♦♥tí♥✉♦ ❡ ❝♦❡r❝✐✈♦✱ ❡♥tã♦ ♦ ♦♣❡r❛❞♦r AH, r❡str✐çã♦ ❞❡ A à H, ❞❡✜♥✐❞♦ ♣♦r

D(AH) ={u∈V;Au∈H}, AH(u) =A(u), u∈ D(AH) é ♠❛①✐♠❛❧ ♠♦♥ót♦♥♦ ❡♠ H.

❯s❛♥❞♦ ❡st❡ r❡s✉❧t❛❞♦ é ♣♦ssí✈❡❧ ♠♦str❛r q✉❡ q✉❛♥❞♦ ♠✉❧t✐♣❧✐❝❛❞♦ ♣♦r ✭✲✶✮ ♦ ♦♣❡r❛❞♦r p✲▲❛♣❧❛❝✐❛♥♦✱ A:W01,p(Ω) W01,p(Ω)

❞❡✜♥✐❞♦ ♣♦r Au=❞✐✈(|∇u|p−2u),

é t❛❧ q✉❡ ❛ s✉❛ r❡❛❧✐③❛çã♦ AH, H =L2(Ω) é ♠❛①✐♠❛❧ ♠♦♥ót♦♥♦ ❡♠ L2(Ω), ♦♥❞❡ Ω é ✉♠ ❞♦♠í♥✐♦ ❧✐♠✐t❛❞♦ ❝♦♠ ❢r♦♥t❡✐r❛ r❡❣✉❧❛r ❡ p >2. ❱❛♠♦s ❞❡♥♦t❛r ♣♦r−∆p ♦ ♦♣❡r❛❞♦r AH. ❆❧é♠ ❞✐ss♦✱ ♦ ♦♣❡r❛❞♦r AH é ❞♦ t✐♣♦ s✉❜❞✐❢❡r❡♥❝✐❛❧✳ ❈♦♥s✐❞❡r❡ ϕ : L2(Ω) → R∪ {+∞}❞❡✜♥✐❞❛ ♣♦r

✭✶✳✶✮ ϕ(u) =

(

1

p R

Ω|∇u|

pdx, uW1,p

0 (Ω) +, ❝❛s♦ ❝♦♥trár✐♦

❊st❛ é ✉♠❛ ❛♣❧✐❝❛çã♦ ❝♦♥✈❡①❛✱ ♣ró♣r✐❛ ❡ s❡♠✐❝♦♥tí♥✉❛ ✐♥❢❡r✐♦r♠❡♥t❡✳ ❊♥tã♦ é ♣♦ssí✈❡❧ ♠♦str❛r q✉❡ ∂ϕ é ♠❛①✐♠❛❧ ♠♦♥ót♦♥♦✱ ♣❡❧♦ ❊①❡♠♣❧♦ ✶✳✷✳✷✱ D(∂ϕ) ⊂ D(ϕ) ⊂ D(ϕ) =

D(∂ϕ), ❡✱ ❛❧é♠ ❞✐ss♦✱∂ϕ=−∆p.

❆ss✐♠✱ ♦ s❡♠✐❣r✉♣♦ ❣❡r❛❞♦ ♣❡❧♦ ♦♣❡r❛❞♦r−∆p ♣♦ss✉✐ ❛ ♣r♦♣r✐❡❞❛❞❡ r❡❣✉❧❛r✐③❛♥t❡ ❡♥✉♥❝✐❛❞❛ ♥♦ ❚❡♦r❡♠❛ ✶✳✷✳✷ ❡ ❡st❡ s❡♠✐❣r✉♣♦ é ❝♦♠♣❛❝t♦✱ ♣❛r❛ ❡st❛ ❞❡♠♦♥str❛çã♦ é ♥❡❝❡ssár✐♦ ❢❛③❡r ✉♠❛ ❛❞❛♣t❛çã♦ ❞❛ ❖❜s❡r✈❛çã♦ ✷✳✶✳✶✱ ♣✳✹✼✱ ❬✼✺❪✳

P❛r❛ ♦❜t❡r♠♦s ✐♥❢♦r♠❛çõ❡s s♦❜r❡ ♦ ❞♦♠í♥✐♦ ❞♦ ♦♣❡r❛❞♦r p✲▲❛♣❧❛❝✐❛♥♦ ♣♦❞❡♠♦s ❡♥❝♦♥tr❛r ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✿

❈♦♥s✐❞❡r❡♠♦s ♦ s❡❣✉✐♥t❡ ♣r♦❜❧❡♠❛ ✭✶✳✷✮

(

−❞✐✈(|∇u|p−2u) +g(u) = f(x), ❡♠

u= 0✱ ❡♠ ∂Ω,

♦♥❞❡ hg(u), ui ≥0 q✉❛❧q✉❡r q✉❡ s❡❥❛ u∈ D(∆p).

❚❡♦r❡♠❛ ✶✳✷✳✸ ✭❚❡♦r❡♠❛ ❆✸✱ ♣✳✷✼✶✱ ❬✷✽❪✮ ❙❡ f (Ω) ❝♦♠ υ > n/p u é s♦❧✉çã♦ ❞❡ ✭✶✳✷✮ ❡♥tã♦ uL∞(Ω) kuk

(32)

✶✳✸✳ ❚❡♦r❡♠❛s ❞❡ ❈♦♠♣❛❝✐❞❛❞❡ ✶✶

✶✳✸ ❚❡♦r❡♠❛s ❞❡ ❈♦♠♣❛❝✐❞❛❞❡

❖ ♦❜❥❡t✐✈♦ ❞❡st❛ s❡çã♦ é ❡♥✉♥❝✐❛r r❡s✉❧t❛❞♦s ❞❡ ❝♦♠♣❛❝✐❞❛❞❡ ♣❛r❛ ♦ ❝♦♥❥✉♥t♦ ❞❡ s♦❧✉çõ❡s ❞❡ ✉♠❛ ❢❛♠í❧✐❛ ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s✳

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

✭✶✳✸✮

( dun

dt +Aun∋fn un(τ) =u0n ∈H

♦♥❞❡ A é ♠❛①✐♠❛❧ ♠♦♥ót♦♥♦ ❡♠ ✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt H, u0n ∈ H ❡ fn ∈L

1(τ, T;H).

◗✉❛♥❞♦ ✈❛r✐❛♠♦s fn ❡ u0n ♦❜t❡♠♦s ✉♠❛ ❢❛♠í❧✐❛ ❞❡ ♣r♦❜❧❡♠❛s ❡ ❡♥tã♦ ✉♠❛ ❢❛♠í❧✐❛ ❞❡ s♦❧✉çõ❡s✳

❉❡✜♥✐çã♦ ✶✳✸✳✶ ❯♠ s✉❜❝♦♥❥✉♥t♦ K L1(a, b;H) é ✉♥✐❢♦r♠❡♠❡♥t❡ ✐♥t❡❣rá✈❡❧ s❡✱ ❞❛❞♦

ǫ >0, ❡①✐st❡ δ >0 t❛❧ q✉❡ Z

Ek

f(t)kHdt < ǫ

♣❛r❛ ❝❛❞❛ s✉❜❝♦♥❥✉♥t♦ ♠❡♥s✉rá✈❡❧ E [a, b] ❝✉❥❛ ♠❡❞✐❞❛ ❞❡ ▲❡❜❡s❣✉❡✱ m(E) < δ(ǫ), ✉♥✐❢♦r♠❡♠❡♥t❡ ♣❛r❛ f ∈K.

❉❡✜♥✐♠♦sM(K) = {un;un é ❛ ú♥✐❝❛ s♦❧✉çã♦ ❢r❛❝❛ ❞❡ ✭✶✳✸); (fn, u0n)∈K×H}. ❖ ♣ró①✐♠♦ r❡s✉❧t❛❞♦ é ✉♠❛ ❛❞❛♣t❛çã♦ ❞♦ ❚❡♦r❡♠❛ ❞❡ ❇❛r❛s ♣✳ ✹✼✱ ❬✼✺❪✳

❚❡♦r❡♠❛ ✶✳✸✳✶ ✭❚❡♦r❡♠❛ ✷✳✺ ❬✼✵❪✮ ❙❡❥❛♠ H ✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt r❡❛❧✱ A : D(A) H ℘(H) ✉♠ ♦♣❡r❛❞♦r ♠❛①✐♠❛❧ ♠♦♥ót♦♥♦ ❡♠ H ♦ q✉❛❧ ❣❡r❛ ✉♠ s❡♠✐❣r✉♣♦ ❝♦♠♣❛❝t♦✱

{u0n} ⊂ D(A) ❝♦♠ u0n → u0 ❡♠ H, ❡ K = {fn :n ∈N} ✉♠ s✉❜❝♦♥❥✉♥t♦ ✉♥✐❢♦r♠❡✲ ♠❡♥t❡ ✐♥t❡❣rá✈❡❧ ❡♠ L1(τ, T;H) ❡♥tã♦ ♦ ❝♦♥❥✉♥t♦ M(K) é r❡❧❛t✐✈❛♠❡♥t❡ ❝♦♠♣❛❝t♦ ❡♠

C([τ, T];H).

❚❡♦r❡♠❛ ✶✳✸✳✷ ✭❬✻✷❪ ♣✳ ✷✶✹✮❙❡❥❛♠ X, Y ❡ H ❡s♣❛ç♦s ❞❡ ❇❛♥❛❝❤✱ ❝♦♠X ⊂⊂H Y, ❡ X r❡✢❡①✐✈♦✳ ❙✉♣♦♥❤❛ q✉❡ {un} é ✉♠❛ s❡q✉ê♥❝✐❛ ✉♥✐❢♦r♠❡♠❡♥t❡ ❧✐♠✐t❛❞❛ ❡♠ L2(τ, T;X) ❡ d

dtun ✉♥✐❢♦r♠❡♠❡♥t❡ ❧✐♠✐t❛❞❛ ❡♠ L

p(τ, T;Y), ♣❛r❛ ❛❧❣✉♠ p > 1. ❊♥tã♦ ❡①✐st❡ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛ ❞❡ {un} q✉❡ ❝♦♥✈❡r❣❡ ❢♦rt❡♠❡♥t❡ ❡♠ L2(τ, T;H).

✶✳✹ ❆tr❛t♦r P✉❧❧❜❛❝❦ ❡ Pr♦❝❡ss♦ ❉✐♥â♠✐❝♦ ▼✉❧tí✈♦❝♦

Figure

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