❊st✉❞♦s ❞❡ ❚é❝♥✐❝❛s ❞❡ ❘❡✐♥✐❝✐❛❧✐③❛çã♦ ❡♠
▼ét♦❞♦s ▲❡✈❡❧✲❙❡t ❡ s✉❛s ❆♣❧✐❝❛çõ❡s ❡♠
❊s❝♦❛♠❡♥t♦s ❇✐❢ás✐❝♦s
❆♠❛✉r✐ ●❛❜r✐❡❧ ❞❡ ❏❡s✉s ❏✉♥✐♦r
❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❈❛ss✐♦ ▼❛❝❤✐❛✈❡❧✐ ❖✐s❤✐Pr♦❣r❛♠❛✿ ▼❛t❡♠át✐❝❛ ❆♣❧✐❝❛❞❛ ❡ ❈♦♠♣✉t❛❝✐♦♥❛❧
❯◆■❱❊❘❙■❉❆❉❊ ❊❙❚❆❉❯❆▲ P❆❯▲■❙❚❆
❋❛❝✉❧❞❛❞❡ ❞❡ ❈✐ê♥❝✐❛s ❡ ❚❡❝♥♦❧♦❣✐❛ ❞❡ Pr❡s✐❞❡♥t❡ Pr✉❞❡♥t❡
Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❆♣❧✐❝❛❞❛ ❡ ❈♦♠♣✉t❛❝✐♦♥❛❧
❊st✉❞♦s ❞❡ ❚é❝♥✐❝❛s ❞❡ ❘❡✐♥✐❝✐❛❧✐③❛çã♦ ❡♠
▼ét♦❞♦s ▲❡✈❡❧✲❙❡t ❡ s✉❛s ❆♣❧✐❝❛çõ❡s ❡♠
❊s❝♦❛♠❡♥t♦s ❇✐❢ás✐❝♦s
❆♠❛✉r✐ ●❛❜r✐❡❧ ❞❡ ❏❡s✉s ❏✉♥✐♦r
❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❈❛ss✐♦ ▼❛❝❤✐❛✈❡❧✐ ❖✐s❤✐❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❆♣❧✐❝❛❞❛ ❡ ❈♦♠♣✉t❛❝✐♦♥❛❧ ❞❛ ❋❛❝✉❧❞❛❞❡ ❞❡ ❈✐ê♥❝✐❛s ❡ ❚❡❝♥♦❧♦❣✐❛ ❞❛ ❯◆❊❙P ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tí✲ t✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛ ❆♣❧✐❝❛❞❛ ❡ ❈♦♠♣✉t❛❝✐♦♥❛❧✳
FICHA CATALOGRÁFICA
Jesus Junior, Amauri Gabriel de.
J56e Estudos de técnicas de reinicialização em métodos level-set e suas aplicações em escoamentos bifásicos / Amauri Gabriel de Jesus Junior. - Presidente Prudente : [s.n.], 2016
104 f.
Orientador: Cássio Machiaveli Oishi
Dissertação (mestrado) - Universidade Estadual Paulista, Faculdade de Ciências e Tecnologia
Inclui bibliografia
❆❣r❛❞❡❝✐♠❡♥t♦s
❘❡s✉♠♦
❊ss❡ tr❛❜❛❧❤♦ ❛♣r❡s❡♥t❛ ✉♠ ❡st✉❞♦ s♦❜r❡ té❝♥✐❝❛s ❞❡ r❡✐♥✐❝✐❛❧✐③❛çã♦ ♣❛r❛ ♠ét♦❞♦s ❧❡✈❡❧✲ s❡t ❝♦♠ ❛♣❧✐❝❛çã♦ ❡♠ ❡s❝♦❛♠❡♥t♦s ❞❡ ✢✉✐❞♦s✳ ❖ tr❛♥s♣♦rt❡ ❞❛ ✐♥t❡r❢❛❝❡ é ❢❡✐t♦ ❝♦♠ ✉♠ ♠ét♦❞♦ ❲❊◆❖ ❞❡ q✉✐♥t❛ ♦r❞❡♠ ♣❛r❛ ❛s ❞❡r✐✈❛❞❛s ❡s♣❛❝✐❛✐s✱ ❡ ✉♠ ❘✉♥❣❡✲❑✉tt❛ ❞❡ t❡r❝❡✐r❛ ♦r❞❡♠ ♣❛r❛ ❛ ❞❡r✐✈❛❞❛ t❡♠♣♦r❛❧✳ ❯t✐❧✐③❛♠♦s ♠❛❧❤❛ ✉♥✐❢♦r♠❡✱ ♥♦ ❝♦♥t❡①t♦ ❞❡ ❞✐❢❡r❡♥ç❛s ✜♥✐t❛s✱ ❡ ♥♦s ❧✐♠✐t❛♠♦s ❛ ❡st✉❞❛r ❝❛s♦s ❜✐❞✐♠❡♥s✐♦♥❛✐s✳ ❆♣ós ♦ ❡st✉❞♦ ❞❛ ❛❞✈❡❝çã♦ ❞❛ ✐♥t❡r❢❛❝❡✱ ❛♣r❡s❡♥t❛♠♦s ❞✉❛s té❝♥✐❝❛s ❞❡ r❡✐♥✐❝✐❛❧✐③❛çã♦ ♣❛r❛ r❡st❛✉r❛r ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ ❢✉♥çã♦ ❞✐stâ♥❝✐❛ ❞❛ ❢✉♥çã♦ ❧❡✈❡❧✲s❡t s❡♠✱ q✉❡ ❤❛❥❛ ♣❡r❞❛ ❞❡ ♠❛ss❛✳ ❆ ♣r✐♠❡✐r❛ ❞❛s té❝♥✐❝❛s ❝♦♥s✐st❡ ♥❛ r❡s♦❧✉çã♦ ❞❡ ✉♠❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ♣❛r❝✐❛❧✱ ❡ ❛ ♦✉tr❛ é ♣✉r❛♠❡♥t❡ ❣❡♦♠étr✐❝❛✳ ❯t✐❧✐③❛♥❞♦ ❛s ❊q✉❛çõ❡s ❞❡ ◆❛✈✐❡r✲❙t♦❦❡s✱ ❡st✉❞❛♠♦s ❞✉❛s té❝♥✐❝❛s ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❛ ❢♦rç❛ ❞❡✈✐❞♦ à t❡♥sã♦ s✉♣❡r✜❝✐❛❧✿ ❈❙❋ ❡ ❙❙❋✳ ❆♣r❡s❡♥t❛♠♦s t❡st❡s ♥✉♠ér✐❝♦s ❝♦♠ ✈❡❧♦❝✐❞❛❞❡ ♣r❡s❝r✐t❛ ♣❛r❛ ✈❡r✐✜❝❛r ❛s ❞✉❛s té❝♥✐❝❛s ❞❡ r❡✐♥✐❝✐❛❧✐③❛çã♦✱ ❡ r❡s♦❧✈❡♠♦s ♣r♦❜❧❡♠❛s ❞❡ ❡s❝♦❛♠❡♥t♦s ❜✐❢ás✐❝♦s ❝♦♠ t❡♥sã♦ s✉♣❡r✜❝✐❛❧ ♣❛r❛ ✈❡r✐✜❝❛r ❝♦♠♦ ❛s ❞✉❛s té❝♥✐❝❛s ❞❡ r❡✐♥✐❝✐❛❧✐③❛çã♦ s❡ ❝♦♠♣♦rt❛♠ ♥❡ss❡ t✐♣♦ ❞❡ ♣r♦❜❧❡♠❛✳
❆❜str❛❝t
❚❤✐s ✇♦r❦ ♣r❡s❡♥ts ❛ st✉❞② ❛❜♦✉t r❡❞✐st❛♥❝✐♥❣ s❝❤❡♠❡s ❢♦r ❧❡✈❡❧✲s❡t ♠❡t❤♦❞s ✇✐t❤ ❛♣♣❧✐❝❛t✐♦♥s ✐♥ ✢✉✐❞ ✢♦✇✳ ❚❤❡ tr❛♥s♣♦rt ♦❢ t❤❡ ✐♥t❡r❢❛❝❡ ✐s ♣❡r❢♦r♠❡❞ ✇✐t❤ ❛ ✜❢t❤ ♦r❞❡r ❲❊◆❖ s❝❤❡♠❡ ❝♦♠❜✐♥❡❞ ✇✐t❤ ❛ t❤✐r❞ ♦r❞❡r ❛❝❝✉r❛t❡ ❘✉♥❣❡✲❑✉tt❛ ♠❡t❤♦❞✳ ❆❢t❡r t❤❡ ❛❞✈❡❝t✐♦♥ ♦❢ t❤❡ ✐♥t❡r❢❛❝❡✱ ✇❡ ❛♥❛❧✐s❡ t✇♦ r❡❞✐st❛♥❝✐♥❣ ♠❡t❤♦❞s t♦ r❡st♦r❡ t❤❡ ❞✐st❛♥❝❡ ❢✉♥❝t✐♦♥ ♣r♦♣❡rt② ♦❢ t❤❡ ❧❡✈❡❧✲s❡t ❢✉♥❝t✐♦♥✳ ❚❤❡ ✜rst ♠❡t❤♦❞ ❝♦♥s✐sts ✐♥ t❤❡ s♦❧✉t✐♦♥ ♦❢ ❛ ♣❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥✱ ✇❤✐❧❡ t❤❡ ♦t❤❡r ✐s ❢✉❧❧② ❣❡♦♠❡tr✐❝✳ ❯s✐♥❣ t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s✱ ✇❡ st✉❞② t✇♦ ❞✐✛❡r❡♥t t❡❝❤♥✐q✉❡s ❢♦r t❤❡ ❝❛❧❝✉❧❛t✐♦♥ ♦❢ t❤❡ s✉r❢❛❝❡ t❡♥s✐♦♥ ❢♦r❝❡✿ ❈❙❋ ❛♥❞ ❙❙❋✳ ■♥ ♦r❞❡r t♦ ♥✉♠❡r✐❝❛❧❧② ✈❡r✐❢② t❤❡ t✇♦ r❡❞✐st❛♥❝✐♥❣ s❝❤❡♠❡s✱ ✇❡ ♣r❡s❡♥t t❡sts ✇✐t❤ ♣r❡s❝r✐❜❡❞ ✈❡❧♦❝✐t② ✜❡❧❞✳ ❋✐♥❛❧❧②✱ ✇❡ ❛♣♣❧② t❤❡ r❡❞✐st❛♥❝✐♥❣ t❡❝❤♥✐q✉❡s ❢♦r s♦❧✈✐♥❣ ❜✐♣❤❛s❡ ✢✉✐❞ ✢♦✇s ✐♥ t❤❡ ♣r❡s❡♥❝❡ ♦❢ t❤❡ s✉r❢❛❝❡ t❡♥s✐♦♥✳
▲✐st❛ ❞❡ ❋✐❣✉r❛s
✷✳✶ ❖ ❝♦♥t♦r♥♦ ❞❛ ❡str❡❧❛ ✈❡r♠❡❧❤❛ r❡♣r❡s❡♥t❛ ❛ ✐♥t❡r❢❛❝❡∂Ω✱ ♦ ❝ír❝✉❧♦ ❛❧❛r❛♥✲
❥❛❞♦ ♣♦ss✉✐ r❛✐♦|x−xc|❡ ♦ ❝ír❝✉❧♦ ❡♠ ❛♠❛r❡❧♦ ♣♦ss✉✐ r❛✐♦|y−xc|✳ ❙❡xc é
♦ ♣♦♥t♦ ❞❛ ✐♥t❡r❢❛❝❡ ♠❛✐s ♣ró①✐♠♦ ❞❡ x✱ ❡❧❡ t❛♠❜é♠ é ♦ ♣♦♥t♦ ❞❛ ✐♥t❡r❢❛❝❡
♠❛✐s ♣ró①✐♠♦ ❞❡ q✉❛❧q✉❡r ♣♦♥t♦ q✉❡ ♣❡rt❡♥ç❛ ❛♦ s❡❣♠❡♥t♦ ❞❡ r❡t❛ q✉❡ ✈❛✐ ❞❡ x❛ xc✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸
✷✳✷ ❉✉❛s r❡❣✐õ❡s ❞❡ ✉♠ ❞♦♠í♥✐♦ s❡♣❛r❛❞❛s ♣♦r ✉♠❛ ✐♥t❡r❢❛❝❡✳ ❆ r❡❣✐ã♦ ✐♥t❡r♥❛ ✭Ω+✮ ❡stá r❡♣r❡s❡♥t❛❞❛ ❡♠ ✈❡r♠❡❧❤♦✱ ❛ r❡❣✐ã♦ ❡①t❡r♥❛ ✭Ω−✮ ❡♠ ❛③✉❧ ❡ ❛
❧✐♥❤❛ ♣r❡t❛ q✉❡ ❛s s❡♣❛r❛ ✭∂Ω✮ é ❛ ✐♥t❡r❢❛❝❡✳ né ♦ ✈❡t♦r ♥♦r♠❛❧ á ✐♥t❡r❢❛❝❡✳ ✷✹
✷✳✸ ▼❛❧❤❛ ✉t✐❧✐③❛❞❛ ♥❡ss❡ tr❛❜❛❧❤♦✳ ❛✮ ■❞❡♥t✐✜❝❛çã♦ ❞♦s ♥ós ❞❛ ♠❛❧❤❛✳ ❖ ♥ó ❡♠ ✈❡r♠❡❧❤♦ ❞❛ ❝é❧✉❧❛ ❝❡♥tr❛❧ é ✐❞❡♥t✐✜❝❛❞♦ ♣❡❧❛ ♣♦s✐çã♦ (i, j)✱ ❡♥q✉❛♥t♦
q✉❡ ♦ ♥ó ❞❛ q✉✐♥❛ s✉♣❡r✐♦r ❞✐r❡✐t❛ ❞❛ ♠❡s♠❛ ❝é❧✉❧❛ ♣♦ss✉✐ ❝♦♦r❞❡♥❛❞❛s (i+12, j+12)✳ ❜✮ ♥♦ ❝❡♥tr♦ ❞❛ ❝é❧✉❧❛ sã♦ ❛r♠❛③❡♥❛❞❛s ❛ ❢✉♥çã♦φ❡ ❛ ♣r❡ssã♦ p ✭q✉❡ s❡rá ✉t✐❧✐③❛❞❛ ♥♦ ❈❛♣ít✉❧♦ ✹✮❀ ❛ ❝♦♠♣♦♥❡♥t❡ u ❞♦ ✈❡t♦r ✈❡❧♦❝✐❞❛❞❡
u = (u, v) é ❛r♠❛③❡♥❛❞❛ ♥❛s ❢❛❝❡s ✈❡rt✐❝❛✐s ❞❛ ❝é❧✉❧❛✱ ❡ ❛ ❝♦♠♣♦♥❡♥t❡ v é
❛r♠❛③❡♥❛❞❛ ♥❛s ❢❛❝❡s ❤♦r✐③♦♥t❛✐s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✷✳✹ ❈✉r✈❛s ❞❡ ♥í✈❡❧ ❞❛ ❝✉r✈❛t✉r❛ ❞❡ φ q✉❛♥❞♦ ✉♠ ❝ír❝✉❧♦ é tr❛♥s❧❛❞❛❞♦ ✈❡rt✐✲
❝❛❧♠❡♥t❡ ♣❛r❛ ❝✐♠❛✳ ❊♠ ❛✮ ❛ tr❛♥s❧❛çã♦ é ❢❡✐t❛ ❝♦♠ ♦ ♠ét♦❞♦ ❲❊◆❖ ❞❡ q✉✐♥t❛ ♦r❞❡♠✳ ❊♠ ❜✮✱ é ❢❡✐t❛ ❝♦♠ ✉♠ ❡sq✉❡♠❛ ❈❯❇■❙❚❆ ❬✷❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✷✳✺ ❛✮ ▼♦❧é❝✉❧❛ ❝♦♠♣✉t❛❝✐♦♥❛❧ à ❡sq✉❡r❞❛✱ ✉t✐❧✐③❛❞♦ ♣❛r❛ ❛♣r♦①✐♠❛r φ−x,i❀ ❜✮
▼♦❧é❝✉❧❛ ❝♦♠♣✉t❛❝✐♦♥❛❧ à ❞✐r❡✐t❛✱ ✉t✐❧✐③❛❞♦ ♣❛r❛ ❛♣r♦①✐♠❛r φ+x,i✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾
✸✳✶ ❆❞✈❡❝çã♦ ❞❛ ✐♥t❡r❢❛❝❡ s♦❜ ✉♠ ❝❛♠♣♦ ❞❡ ✈❡❧♦❝✐❞❛❞❡ ♥♦r♠❛❧ ❛ ❡❧❛✳ ❆ ❧✐♥❤❛ tr❛❝❡❥❛❞❛ r❡♣r❡s❡♥t❛ ❛ ✐♥t❡r❢❛❝❡ ♥♦ ✐♥st❛♥t❡ n✱ ❡ ❛ ❧✐♥❤❛ ❝♦♥tí♥✉❛ ❛ r❡♣r❡✲
s❡♥t❛ ♥♦ ✐♥st❛♥t❡ n+ 1✳ ❖s ♣♦♥t♦s ❞♦ ✐s♦❝♦♥t♦r♥♦ φ = 0 ♥♦ ✐♥st❛♥t❡ n
❡stã♦ ♥❛ r❡❣✐ã♦ ✐♥t❡r✐♦r à ✐♥t❡r❢❛❝❡ ♥♦ ✐♥st❛♥t❡n+ 1✱ ❡ ❛ ✉♠❛ ❞✐stâ♥❝✐❛ ❞❡ a∆t ❞♦ ♥♦✈♦ ✐s♦❝♦♥t♦r♥♦ ♥✉❧♦ ❞❡ φ✳ ▲♦❣♦✱ φ(n+1) =a∆t ♥❡ss❡s ♣♦♥t♦s✳ ✳ ✳ ✳ ✸✹
✸✳✷ ❚rês ❝✉r✈❛s ❞❡ ♥í✈❡❧ ❞✐❢❡r❡♥t❡s ❞❡ φ✳ ❊♠ ❛③✉❧✱ ♥❛ r❡❣✐ã♦ ✐♥t❡r♥❛ à ✐♥t❡r✲
❢❛❝❡✱ ✉♠❛ ❝✉r✈❛ ❞❡ ♥í✈❡❧ ♣♦s✐t✐✈❛❀ ❡♠ ✈❡r♠❡❧❤♦✱ ❛ ❝✉r✈❛ ❞❡ ♥í✈❡❧ ③❡r♦ q✉❡ r❡♣r❡s❡♥t❛ ❛ ✐♥t❡r❢❛❝❡❀ ❡♠ ♣r❡t♦✱ ♥❛ r❡❣✐ã♦ ❡①t❡r♥❛✱ ✉♠❛ ❝✉r✈❛ ❞❡ ♥í✈❡❧ ♥❡❣❛t✐✈❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✸✳✸ ❈é❧✉❧❛s ✉t✐❧✐③❛❞❛s ❡♠ ✭✸✳✷✶✮ ♣❛r❛ ❛ ❛♣r♦①✐♠❛çã♦ ❞♦ ❝á❧❝✉❧♦ ❞❛ ✐♥t❡❣r❛❧✳
❚❡♠♦s m(1,1) = 3 ✭❤á três ❝é❧✉❧❛s ✈✐③✐♥❤❛s à ❝é❧✉❧❛ (1,1) ♥♦ ❞♦♠í♥✐♦✮✱
m(i1,1) = 5 ❡ m(i2,2) = 8✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾
✸✳✹ ❆♦ ❢✉♥❞♦✱ ❞❡ ❝♦r ❛✈❡r♠❡❧❤❛❞❛✱ é r❡♣r❡s❡♥t❛❞❛ ❛ ♠❛❧❤❛ ❞❡ ❛❞✈❡❝çã♦✱ ❡ à ❢r❡♥t❡✱ ❡♠ ❝♦r ❛❧❛r❛♥❥❛❞❛✱ ❛ ♠❛❧❤❛ tr✐❛♥❣✉❧❛r✳ ❖s ♣♦♥t♦s ❡♠ ✈❡r♠❡❧❤♦ sã♦ ♦s ❝❡♥tr♦s ❞❛s ❝é❧✉❧❛s ❞❡ ❛❞✈❡❝çã♦✱ ♥♦s q✉❛✐s ♦ ✈❛❧♦r ❞❛ ❢✉♥çã♦ ❧❡✈❡❧✲s❡t é ❝♦♥❤❡❝✐❞♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✸✳✺ ∂φé ♦ ❝♦♥t♦r♥♦ ❞❛ ✐♥t❡r❢❛❝❡ r❡♣r❡s❡♥t❛❞❛ ♣♦r φ✳ P1 é ♦ ♣♦♥t♦ ❞❛ ✐♥t❡r❢❛❝❡
♠❛✐s ♣ró①✐♠♦ ❛♦s ♣♦♥t♦s x1 ❡ ρi,1✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵
▲■❙❚❆ ❉❊ ❋■●❯❘❆❙ ✶✵ ✸✳✻ ▼❛❧❤❛ tr✐❛♥❣✉❧❛r ❡♠ q✉❡ ❛s ❤✐♣♦t❡♥✉s❛s ❞♦s tr✐â♥❣✉❧♦s ♥ã♦ ♣♦ss✉❡♠ t♦❞❛s
❛s ♠❡s♠❛s ❞✐r❡çõ❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✸✳✼ ■❞❡♥t✐✜❝❛çã♦ ❞♦s ❡❧❡♠❡♥t♦s ✉t✐❧✐③❛❞♦s ♥❛ r❡✐♥✐❝✐❛❧✐③❛çã♦ ❣❡♦♠étr✐❝❛✳ ✳ ✳ ✳ ✳ ✹✷ ✸✳✽ ∂φ¯ ✐♥t❡rs❡❝t❛ ♥♦ ♣♦♥t♦P1 ❛ ❛r❡st❛ q✉❡ ✉♥❡ ♦s ✈ért✐❝❡s ρi,j1 ❡ ρi,j2✳ ❈♦♠♦
∂φ¯ é ♦ ✐s♦❝♦♥t♦r♥♦ ♥✉❧♦ ❞❡ φ¯✱ ♦✉ φ¯é ♥❡❣❛t✐✈♦ ❡♠ t♦❞♦ ♦ ♣♦❧í❣♦♥♦ ❛❜❛✐①♦
❞❡∂φ¯♦✉ ❡♠ t♦❞♦ ♦ ♣♦❧í❣♦♥♦ ❛❝✐♠❛ ❞❡ss❛ r❡t❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸
✸✳✾ ∂φ é ♦ ❝♦♥t♦r♥♦ ❞❛ ✐♥t❡r❢❛❝❡ r❡♣r❡s❡♥t❛❞❛ ♣♦r φ✱ ❡ ∂φ¯ é ♦ ❝♦♥t♦r♥♦ ❞❛
✐♥t❡r❢❛❝❡ r❡♣r❡s❡♥t❛❞❛ ♣♦r φ¯✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✸✳✶✵ P1 P2 é ❛ ♣❛rt❡ ❞❛ ✐♥t❡r❢❛❝❡ q✉❡ s❡ ❡♥❝♦♥tr❛ ❞❡♥tr♦ ❞♦ tr✐â♥❣✉❧♦ τ¯k✳ ✳ ✳ ✳ ✳ ✹✹
✸✳✶✶ ■♥t❡r❢❛❝❡s r❡♣r❡s❡♥t❛❞❛s ♣♦rφ¯✭✈❡r♠❡❧❤♦✮ ❡φ∗✭♣r❡t♦✮✳ ❊♠ ❧❛r❛♥❥❛✱ ♠♦str❛♠♦s
❛ ár❡❛ ❞❛ r❡❣✐ã♦ ✐♥t❡r♥❛ à ✐♥t❡r❢❛❝❡ r❡♣r❡s❡♥t❛❞❛ ♣♦r φ¯✱ ❡ ❡♠ ❛③✉❧ ❛ ár❡❛ ✐♥t❡r♥❛ às ❞✉❛s ✐♥t❡r❢❛❝❡s✳ ❆ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ❛s ❞✉❛s r❡t❛s ❢♦✐ ✐♥t❡♥❝✐♦♥❛❧✲ ♠❡♥t❡ ❡①❛❣❡r❛❞❛ ♣❛r❛ ❢❛❝✐❧✐t❛r ❛ ✈✐s✉❛❧✐③❛çã♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✸✳✶✷ ❚✐♣♦s ❞❡ tr✐â♥❣✉❧♦s ✉t✐❧✐③❛❞♦s ♥❡ss❡ tr❛❜❛❧❤♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✸✳✶✸ P♦ssí✈❡✐s ❝❛s♦s ❛♦ ❝❛❧❝✉❧❛r ❛ ár❡❛ ❞❛ r❡❣✐ã♦ ✐♥t❡r♥❛ à ✐♥t❡r❢❛❝❡ ✭r❡❣✐ã♦
❛❧❛r❛♥❥❛❞❛✮✳ φ¯ é ♣♦s✐t✐✈♦ ♥♦s ✈ért✐❝❡s ❞♦ tr✐â♥❣✉❧♦ q✉❡ ❡stã♦ ❞❡♥tr♦ ❞❛ r❡❣✐ã♦ ❛❧❛r❛♥❥❛❞❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✸✳✶✹ ■♥t❡r❢❛❝❡s r❡♣r❡s❡♥t❛❞❛s ♣♦r φ¯✭✈❡r♠❡❧❤♦✮ ❡ φ∗+η ✭♠❛rr♦♠✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽
✸✳✶✺ ◆❛ ❋✐❣✉r❛✱ φ¯(P2) < φ¯(P1) < 0 < φ¯(P3)✳ ❊♠ ♠❛rr♦♠✱ ár❡❛ ✐♥t❡r♥❛ ❞❛ ✐♥t❡r❢❛❝❡ r❡♣r❡s❡♥t❛❞❛ ♣♦r φ∗+η(0)✳ ❊♠ ✈❡r♠❡❧❤♦✱ ár❡❛ ❞❛ r❡❣✐ã♦ ✐♥t❡r♥❛
às ✐♥t❡r❢❛❝❡s r❡♣r❡s❡♥t❛❞❛s ♣♦r φ∗+η(0) ❡ ♣♦r φ¯✳ ❊♠ ❛③✉❧✱ ár❡❛ ❞❛ r❡❣✐ã♦
✐♥t❡r♥❛ às três ✐♥t❡r❢❛❝❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✸✳✶✻ ∂d˜é ♦ ✐s♦❝♦♥t♦r♥♦ ♥✉❧♦ ❞❡d˜✱ ❛♣ós s❡r ❝❛❧❝✉❧❛❞❛ ♣♦r ✭✸✳✹✾✮✳ r1 é ✉♠ ✈ért✐❝❡
❛❞❥❛❝❡♥t❡ à ✐♥t❡r❢❛❝❡ ✭♣♦✐s ♣❡rt❡♥❝❡ ❛ ✉♠ tr✐â♥❣✉❧♦ ✐♥t❡rs❡❝t❛❞♦ ♣♦r ∂d˜✮✱
❞❡ ♠♦❞♦ q✉❡ s✉❛ ❞✐stâ♥❝✐❛ ❛té ∂d˜❥á ❢♦✐ ♦❜t✐❞❛ ✉s❛♥❞♦ ✭✸✳✹✾✮✱ ❡ r1✱ r2 ❡
r3 sã♦ ✈ért✐❝❡s ❞❡ V+✳ ❊♠ ✈❡r❞❡ é ♠♦str❛❞♦ ♦ s❡❣♠❡♥t♦ ❞❡ r❡t❛ ❞❡ ♠❡♥♦r
❝♦♠♣r✐♠❡♥t♦ q✉❡ ✈❛✐ ❞❡r1 à ✐♥t❡r❢❛❝❡✳ ❖ ❝♦♠♣r✐♠❡♥t♦ ❞❡ss❡ s❡❣♠❡♥t♦ ❞❡
r❡t❛✱ ❝♦♠♣❧❡♠❡♥t❛❞♦ ♣❡❧♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦ ❝❛♠✐♥❤♦ ❡♠ ❛③✉❧✱ é ♦ ✈❛❧♦r ❞❡ ˜
d(r3) q✉❡ s❡rá ♦❜t✐❞♦ ♣♦r ✭✸✳✺✵✮ q✉❛♥❞♦ d˜❞❡✐①❛r ❞❡ ✈❛r✐❛r✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶
✸✳✶✼ Fi é ❛ ❢❛❝❡ ♦♣♦st❛ ❛♦ ✈ért✐❝❡ ri✱ ❞❡❧✐♠✐t❛❞❛ ♣❡❧♦s ✈ért✐❝❡s Fi,1 ❡ Fi,2✳ x é
✉♠ ♣♦♥t♦ q✉❛❧q✉❡r ❞❡ss❛ ❢❛❝❡✱ ❞✐st❛♥t❡h ❞♦ ♣♦♥t♦ Fi,1✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷
✸✳✶✽ ψ ❡ ψ′ ❡♠ ❢✉♥çã♦ ❞❡ h✳ ◆❡ss❡s ❣rá✜❝♦s✱ ❢♦r❛♠ ❝♦♥s✐❞❡r❛❞♦s d˜(F
i,2) = 0.8✱
˜
d(Fi,1) = 0.5✱ Fi,2 = (0,∆x)✱ Fi,1 = (0,0) ❡ ri = (∆x,0)✱ ♣❛r❛ ∆x = 0.5✳
◆♦t❡ q✉❡ ψ′ ♥ã♦ ♣♦ss✉✐ r❛í③ ♥❡ss❡ ✐♥t❡r✈❛❧♦✱ ❞❡ ♠♦❞♦ q✉❡ ♦ ♠á①✐♠♦ ❞❡ψ′
♦❝♦rr❡ ❡♠ ✉♠ ❞♦s ❡①tr❡♠♦s ❞♦ ✐♥t❡r✈❛❧♦ ✭♥❡ss❡ ❝❛s♦✱ ♥♦ ❡①tr❡♠♦ ❞✐r❡✐t♦✮✳ ✳ ✺✸ ✸✳✶✾ ❘❡♣r❡s❡♥t❛çã♦ ❞♦ t❡st❡ ❞❡ ❛❞✈❡❝çã♦ s♦❜ ✉♠ ❝❛♠♣♦ ❞❡ ✈❡❧♦❝✐❞❛❞❡ ❝✐s❛❧❤❛♥t❡
r❡✈❡rsí✈❡❧✳ ❆ ♠❛❧❤❛ ✉t✐❧✐③❛❞❛ ❢♦✐ ❞❡ 128 × 128✳ ❊♠ ❛③✉❧✱ ✉t✐❧✐③❛♠♦s ❛ r❡✐♥✐❝✐❛❧✐③❛çã♦ ♣♦r ❊❉P✱ ❡ ❡♠ ✈❡r♠❡❧❤♦ ❛ r❡✐♥✐❝✐❛❧✐③❛çã♦ ❣❡♦♠étr✐❝❛✳ ❊♠ ❛✮ ❛♣r❡s❡♥t❛♠♦s ❛ s✐♠✉❧❛çã♦ ♥♦s ✐♥st❛♥t❡st = 0✭❝ír❝✉❧♦ ♣❡r❢❡✐t♦✮✱t=T /10 ❡ T =T /2 ✭❝ír❝✉❧♦ ♠❛✐s ❞❡❢♦r♠❛❞♦✮✳ ❊♠ ❜✮✱ ❛♣r❡s❡♥t❛♠♦s ❛s s✐♠✉❧❛çõ❡s ♥♦s ✐♥st❛♥t❡s t=T /2✱t = 9T /10 ❡t =T ✭❝ír❝✉❧♦ q✉❛s❡ ♣❡r❢❡✐t♦✮✳ ✳ ✳ ✳ ✳ ✳ ✺✺
▲■❙❚❆ ❉❊ ❋■●❯❘❆❙ ✶✶ ✸✳✷✶ ❈♦♠♣❛r❛çã♦ ❡♥tr❡ ♦s ❡st❛❞♦s ✐♥✐❝✐❛✐s ❡ ✜♥❛✐s ❞❛ ✐♥t❡r❢❛❝❡ ♥♦ t❡st❡ ❞♦ ❝ír✲
❝✉❧♦ ❞❡ ❩❛❧❡s❛❦✳ ❆ ♠❛❧❤❛ ✉t✐❧✐③❛❞❛ ❢♦✐ ❞❡128×128✱ ❡♠ ❛③✉❧✱ ♠♦str❛♠♦s ♦ r❡s✉❧t❛❞♦ ✜♥❛❧ ❞❛ s✐♠✉❧❛çã♦ ✉t✐❧✐③❛♥❞♦ ❛ r❡✐♥✐❝✐❛❧✐③❛çã♦ ♣♦r ❊❉P✱ ❡♠ ✈❡r✲ ♠❡❧❤♦ ✉t✐❧✐③❛♥❞♦ ❛ r❡✐♥✐❝✐❛❧✐③❛çã♦ ❣❡♦♠étr✐❝❛✱ ❡ ❡♠ ♣r❡t♦ ♦ ❡st❛❞♦ ✐♥✐❝✐❛❧ ❞❛ s✐♠✉❧❛çã♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼ ✸✳✷✷ ❘❡♣r❡s❡♥t❛çã♦ ❞♦ t❡st❡ ❞♦ ❝ír❝✉❧♦ ❞❡ ❩❛❧❡s❛❦✳ ❆ ♠❛❧❤❛ ✉t✐❧✐③❛❞❛ ❢♦✐ ❞❡
128×128✳ ❊♠ ❛③✉❧✱ r❡♣r❡s❡♥t❛♠♦s ❛ s♦❧✉çã♦ ✉t✐❧✐③❛♥❞♦ r❡✐♥✐❝✐❛❧✐③❛çã♦ ♣♦r ❊❉P✱ ❡ ❡♠ ✈❡r♠❡❧❤♦ ✉t✐❧✐③❛♥❞♦ ❛ r❡✐♥✐❝✐❛❧✐③❛çã♦ ❣❡♦♠étr✐❝❛✳ ❆ s✐♠✉❧❛çã♦ ♥♦ t❡♠♣♦ t = 0 é ❛ q✉❡ s❡ ❡♥❝♦♥tr❛ ♠❛✐s ❛❝✐♠❛ ❞♦ ❞♦♠í♥✐♦✳ ❆ ♣❛rt✐r ❞❛í✱ ❡①✐❜✐♠♦s ♥♦ s❡♥t✐❞♦ ❛♥t✐❤♦rár✐♦ ♦s ❡st❛❞♦s ❞❛s s✐♠✉❧❛çõ❡s ♥♦s ✐♥st❛♥t❡s
t = 13T /50,✱ t= 26T /50❡ t= 39T /50✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼
✹✳✶ ❯♠ ✜♥♦ ❡❧❡♠❡♥t♦ ❞❡ ✈♦❧✉♠❡dV✱ ❝♦♠ ❢r♦♥t❡✐r❛sdS ❝♦♥t❡♥❞♦ ✉♠❛ ✐♥t❡r❢❛❝❡ S✳ ❙❡ dV ❢♦r s✉✜❝✐❡♥t❡♠❡♥t❡ ✜♥♦✱ ♥ã♦ ♣♦❞❡ ❤❛✈❡r ❛❝ú♠✉❧♦ ❞❡ ♠❛ss❛ ❡♠
s❡✉ ✐♥t❡r✐♦r✳ ■♠❛❣❡♠ r❡t✐r❛❞❛ ❞❡ [✹✷]✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷ ✹✳✷ ❖ ✈❡t♦r p é ♣❡r♣❡♥❞✐❝✉❧❛r ❛♦ ❝♦♥t♦r♥♦ ❞❛ ✐♥t❡r❢❛❝❡✱ ♠❛s é ❞✐❢❡r❡♥t❡ ❞♦
✈❡t♦r n ❬✹✷❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✸
✹✳✸ ❱♦❧✉♠❡ ❞❡ ❝♦♥tr♦❧❡V✱ ❝♦♠ ❞✐♠❡♥sõ❡s∆x×∆y✳ ❆ ✐♥t❡r❢❛❝❡ é r❡♣r❡s❡♥t❛❞❛
♣❡❧❛ ❧✐♥❤❛ ✈❡r♠❡❧❤❛✳ ❖ ❝♦♠♣r✐♠❡♥t♦ ❞❛ ♣❛rt❡ ❞❛ ✐♥t❡r❢❛❝❡ ✐♥t❡r♥❛ ❛ V é Li✱ ❡ né ✉♠ ✈❡t♦r ✉♥✐tár✐♦ ♥♦r♠❛❧ à ✐♥t❡r❢❛❝❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺
✺✳✶ ❈❛♠♣♦ ❞❡ ✈❡❧♦❝✐❞❛❞❡ ✭❡♠ m/s✮ ❞♦ t❡st❡ ❞❛ ❣♦t❛ ❡stát✐❝❛✱ ❛♣ós ❞✉③❡♥t♦s
♣❛ss♦s ❞❡ t❡♠♣♦✱ ❝♦♠ ♠❛❧❤❛ ❞❡ 64×64✳ ❛✮ ▲❙✶ ❡ ❈❙❋❀ ❜✮ ▲❙✶ ❡ ❙❙❋❀ ❝✮ ▲❙✷ ❡ ❈❙❋❀ ❞✮ ▲❙✷ ❡ ❙❙❋❀ ❡✮ ▲❙✸ ❡ ❈❙❋❀ ❢✮ ▲❙✸ ❡ ❙❙❋❀ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✶ ✺✳✷ ❱❛r✐❛çã♦ ❞❛s ♥♦r♠❛sku¯k∞ ✭❛✮ ❡kuk2,1 ✭❜✮ ❡♠ ❢✉♥çã♦ ❞♦ t❡♠♣♦✱ ♥♦ t❡st❡
❞❛s ❝♦rr❡♥t❡s ♣❛r❛s✐t❛s✳ ❯t✐❧✐③❛♠♦s ✉♠❛ ♠❛❧❤❛ ❞❡128×128✱ s❡♠ r❡✐♥✐❝✐❛❧✲ ✐③❛çã♦ ❞❛ ❢✉♥çã♦ ❧❡✈❡❧✲s❡t✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✷ ✺✳✸ ❊✈♦❧✉çã♦ ❞♦ t❡st❡ ❞❛ ♦s❝✐❧❛çã♦ ❞❛ ❣♦t❛✱ ♥✉♠❛ ♠❛❧❤❛ ❞❡ 64 × 64✱ ❝♦♠
❝✉r✈❛t✉r❛ ▲❙✶✳ ❊♠ ❛③✉❧ ❡stá ❛ s♦❧✉çã♦ ❝♦♠ r❡✐♥✐❝✐❛❧✐③❛çã♦ ♣♦r ❊❉P✱ ❡ ❡♠ ✈❡r♠❡❧❤♦ ❝♦♠ r❡✐♥✐❝✐❛❧✐③❛çã♦ ❣❡♦♠étr✐❝❛✳ ❉❡ ❝✐♠❛ ♣❛r❛ ❜❛✐①♦✱ ❞❛ ❡s✲ q✉❡r❞❛ ♣❛r❛ ❛ ❞✐r❡✐t❛✱ ❛s ✐♠❛❣❡♥s sã♦ r❡❢❡r❡♥t❡s ❛♦s ✐♥st❛♥t❡s t = iT /50✱
i= 0, . . . ,5✱ ❝♦♠ T = 10s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✸
✺✳✹ ◆❡ss❡ ❣rá✜❝♦ é ❡①✐❜✐❞❛ ❛ ❝♦♦r❞❡♥❛❞❛x❞❛ ❝é❧✉❧❛ ❞❡ ✐♥t❡r❢❛❝❡ ♠❛✐s à ❞✐r❡✐t❛
❞♦ ❞♦♠í♥✐♦ ✭❛ q✉❡ ♣♦ss✉✐ ❛ ❝♦♦r❞❡♥❛❞❛ x ❝♦♠ ♠❛✐♦r ✈❛❧♦r✮✱ ❛ ❝❛❞❛ ♣❛ss♦
❞❡ t❡♠♣♦ ❞❛ s✐♠✉❧❛çã♦❀ ❛✮ ❙✐♠✉❧❛çã♦ ❢❡✐t❛ ♥✉♠❛ ♠❛❧❤❛ 128×128 ❡ s❡♠ r❡✐♥✐❝✐❛❧✐③❛çã♦ ❞❛ ❢✉♥çã♦ ❧❡✈❡❧✲s❡t❀ ❜✮ ❙♦❧✉çã♦ ❞❡ r❡❢❡rê♥❝✐❛ ❞❡ ❬✹✵❪ ♣❛r❛ três ♠❛❧❤❛s ❞✐❢❡r❡♥t❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✹ ✺✳✺ ❈♦♠♣♦♥❡♥t❡ x ❞❛ ✈❡❧♦❝✐❞❛❞❡ ❞❛ ❝é❧✉❧❛ ❝♦♠♣✉t❛❝✐♦♥❛❧ ❞❡ ✐♥t❡r❢❛❝❡ ♠❛✐s à
❞✐r❡✐t❛ ❞♦ ❞♦♠í♥✐♦ ♣❛r❛ ❞✐❢❡r❡♥t❡s ♠❛❧❤❛s✱ ❡ ✈❛r✐❛♥❞♦ ❛ té❝♥✐❝❛ ❞❡ r❡✐♥✐✲ ❝✐❛❧✐③❛çã♦ ❡♠♣r❡❣❛❞❛✳ ❙❘ s✐❣♥✐✜❝❛ ❙❡♠ ❘❡✐♥✐❝✐❛❧✐③❛çã♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✺ ✺✳✻ ❙✐♠✉❧❛çã♦ ❞♦ t❡st❡ ❞❛ ❛s❝❡♥sã♦ ❞❛ ❜♦❧❤❛✱ ❝♦♠ ❡s♣❛ç❛♠❡♥t♦ 1/h = 80✱
❝✉r✈❛t✉r❛ ▲❙✶ ❡ ✉t✐❧✐③❛♥❞♦ ❛ r❡✐♥✐❝✐❛❧✐③❛çã♦ ❞❡ φ ❡♠ t♦❞♦s ♦s ♣❛ss♦s ❞❡
t❡♠♣♦✳ ❆ t❡♥sã♦ s✉♣❡r✜❝✐❛❧ ❢♦✐ ❝❛❧❝✉❧❛❞❛ ♣❡❧♦ ♠ét♦❞♦ ❙❙❋✳ ❊♠ ❛✮ ❢♦✐ s✐♠✉❧❛❞♦ ♦ t❡st❡ ✶✱ ❡ ❡♠ ❜✮ ♦ t❡st❡ ✷✱ ❝♦♠ ♦s ♣❛râ♠❡tr♦s ❞❛ t❛❜❡❧❛ ✺✳✸✳ ❖s r❡s✉❧t❛❞♦s ❢♦r❛♠ ✐♠♣r❡ss♦s ♥♦s ✐♥st❛♥t❡st= 0✱t= 1.5s❡t= 3.0s✳ ❆ ❧✐♥❤❛
tr❛❝❡❥❛❞❛ ♥❛ ✜❣✉r❛ ❜✮ ❢♦✐ ✉t✐❧✐③❛❞❛ ❛♣❡♥❛s ♣❛r❛ ❢❛❝✐❧✐t❛r ❛ ✈✐s✉❛❧✐③❛çã♦✳ ❊♠ ❝✮ ❡ ❞✮ sã♦ ❡①✐❜✐❞❛s ❛s s♦❧✉çõ❡s ❞❡ r❡❢❡rê♥❝✐❛ ❞❡ ❬✶✺❪ ❡♠ t = 3.0s ♣❛r❛ ♦s
t❡st❡s ✶ ❡ ✷✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✽ ✺✳✼ ❙✐♠✉❧❛çã♦ ❞♦ t❡st❡ ✶ ❞❛ ❛s❝❡♥sã♦ ❞❛ ❜♦❧❤❛✱ ❝♦♠ ❡s♣❛ç❛♠❡♥t♦1/h= 80✱ ❝✉r✲
▲■❙❚❆ ❉❊ ❋■●❯❘❆❙ ✶✷ ✺✳✽ ❙✐♠✉❧❛çã♦ ❞♦ t❡st❡ ✷ ❞❛ ❛s❝❡♥sã♦ ❞❛ ❜♦❧❤❛✱ ❝♦♠ ❡s♣❛ç❛♠❡♥t♦1/h= 80✱ ❝✉r✲
✈❛t✉r❛ ▲❙✶ ❡ t❡♥sã♦ s✉♣❡r✜❝✐❛❧ ❝❛❧❝✉❧❛❞❛ ♣❡❧♦ ♠ét♦❞♦ ❙❙❋✳ ❛✮ ❈❡♥tr♦✐❞❡❀ ❜✮ ❱❡❧♦❝✐❞❛❞❡ ♠é❞✐❛❀ ❝✮ ❈✐r❝✉❧❛r✐❞❛❞❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✵ ❇✳✶ ❋❧✉①♦❣r❛♠❛ ❣❡r❛❧ ❞❡ ✉♠❛ s✐♠✉❧❛çã♦ ❝♦♠♣❧❡t❛ r❡s♦❧✈❡♥❞♦ ❛s ❡q✉❛çõ❡s ❞❡
◆❛✈✐❡r✲❙t♦❦❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✽ ❇✳✷ ❱✐sã♦ ❣❡r❛❧ ❞❛ r❡✐♥✐❝✐❛❧✐③❛çã♦ ❞❡ φ ♣♦r ❊❉P✳ ◆♦t❡ q✉❡ ♦ ♣s❡✉❞♦t❡♠♣♦ τ
❞♦ ♣r♦❝❡ss♦ ❞❡ r❡✐♥✐❝✐❛❧✐③❛çã♦ ♥ã♦ ♣♦ss✉✐ r❡❧❛çã♦ ❞✐r❡t❛ ❝♦♠ ♦ t❡♠♣♦t ❞❛
▲✐st❛ ❞❡ ❚❛❜❡❧❛s
✸✳✶ ❱❛r✐❛çã♦ ❞❡ ár❡❛ ❡ ❡rr♦ ❛♣ós ❛ ❛❞✈❡❝çã♦ ❞❡ φ♣♦r ✉♠ ❝❛♠♣♦ ❞❡ ✈❡❧♦❝✐❞❛❞❡
❝✐s❛❧❤❛♥t❡ r❡✈❡rsí✈❡❧✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✸✳✷ ❚❡♠♣♦ ❞❡ ❡①❡❝✉çã♦ ✭❡♠ s❡❣✉♥❞♦s✮ ❞❛s r♦t✐♥❛s ❞❡ ❛❞✈❡❝çã♦✱ r❡✐♥✐❝✐❛❧✐③❛çã♦
♣♦r ❊❉P ❡ r❡✐♥✐❝✐❛❧✐③❛çã♦ ❣❡♦♠étr✐❝❛✱ ♥♦ t❡st❡ ❞❡ ❛❞✈❡❝çã♦ ♣♦r ✉♠ ❝❛♠♣♦ ❞❡ ✈❡❧♦❝✐❞❛❞❡ ❝✐s❛❧❤❛♥t❡ r❡✈❡rsí❡❧✳ ❖ ♥ú♠❡r♦ ❈❋▲ ✉t✐❧✐③❛❞♦ ❢♦✐ 1.0✳ ✳ ✳ ✳ ✳ ✳ ✺✻ ✸✳✸ ❱❛r✐❛çã♦ ❞❡ ár❡❛ ❡ ❡rr♦ ❛♣ós ❛ ❛❞✈❡❝çã♦ ❞❡ φ♣♦r ✉♠ ❝❛♠♣♦ ❞❡ ✈❡❧♦❝✐❞❛❞❡
❝✐s❛❧❤❛♥t❡ r❡✈❡rsí✈❡❧✳ ❖ ♥ú♠❡r♦ ❈❋▲ ✉t✐❧✐③❛❞♦ ❢♦✐ ❞❡ 0.67✱ ❞❡ ♠♦❞♦ q✉❡✱
♥❛s ♠❛❧❤❛s ♠❛✐s ❣r♦ss❛s✱ ♦ ♣❛ss♦ t❡♠♣♦r❛❧ ❢♦✐ ❞❡0.010467✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽ ✺✳✶ ❙✐♠✉❧❛çã♦ ♣❛r❛ ♦ t❡st❡ ❞❛s ❝♦rr❡♥t❡s ♣❛r❛s✐t❛s✱ ♣❛r❛ ❞✐✈❡rs❛s ❝♦♠❜✐♥❛çõ❡s
❞❡ ♠❛❧❤❛s ❡ ♠ét♦❞♦s ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❛ ❝✉r✈❛t✉r❛ ❡ ❞❛ t❡♥sã♦ s✉♣❡r✜❝✐❛❧✳ ◆❡ss❡s t❡st❡s ♥ã♦ ❢♦✐ ❢❡✐t❛ r❡✐♥✐❝✐❛❧✐③❛çã♦ ❞❛ ❢✉♥çã♦ ❧❡✈❡❧✲s❡t✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✵ ✺✳✷ ❘❡s✉❧t❛❞♦s ♣❛r❛ ♦ t❡st❡ ❞❛ ♦s❝✐❧❛çã♦ ❞❛ ❜♦❧❤❛✱ ♣❛r❛ ❛s ♠❛❧❤❛s ❞❡64×64❡
128×128✱ ❞✐❢❡r❡♥t❡s t✐♣♦s ❞❡ ❝✉r✈❛t✉r❛✱ ✈❛r✐❛♥❞♦ ❛ té❝♥✐❝❛ ❞❡ r❡✐♥✐❝✐❛❧✐③❛çã♦ ✉t✐❧✐③❛❞❛ ❡ ❝❛❧❝✉❧❛♥❞♦ ❛ t❡♥sã♦ s✉♣❡r✜❝✐❛❧ ♣❡❧♦ ♠ét♦❞♦ ❙❙❋✳ P❡rí♦❞♦s r❡♣✲ r❡s❡♥t❛❞♦s ♣♦r ✲ ♥ã♦ ♣✉❞❡r❛♠ s❡r ❞❡t❡❝t❛❞♦s ♣♦rq✉❡ ❛ ♦s❝✐❧❛çã♦ s❡ t♦r♥♦✉ ♠✉✐t♦ ♣❡q✉❡♥❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✸ ✺✳✸ P❛râ♠❡tr♦s ❞❡ ❡♥tr❛❞❛ ♣❛r❛ ♦ t❡st❡ ❞❛ ❛s❝❡♥sã♦ ❞❛ ❜♦❧❤❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✹ ✺✳✹ ❆♥á❧✐s❡ q✉❛♥t✐t❛t✐✈❛ ❞♦ t❡st❡ ❛s❝❡♥sã♦ ❞❛ ❜♦❧❤❛ ✭t❡st❡ ✶✮✱ ❝♦♠♣❛r❛❞❛s ❝♦♠
❛ s♦❧✉çã♦ ❞❡ r❡❢❡rê♥❝✐❛ ❞❡ ❬✶✺❪ ✭❡♠ ♥❡❣r✐t♦✮✳ t1 s❡ r❡❢❡r❡ ❛♦ t❡♠♣♦ ❡♠ q✉❡ ❛
❝✐r❝✉❧❛r✐❞❛❞❡ ❛t✐♥❣❡ s❡✉ ✈❛❧♦r ♠í♥✐♠♦✱ ❡ t2 é ♦ t❡♠♣♦ ❡♠ q✉❡ ❛ ✈❡❧♦❝✐❞❛❞❡
♠é❞✐❛ ❞❛ ❜♦❧❤❛ ❛t✐♥❣❡ s❡✉ ✈❛❧♦r ♠á①✐♠♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✻ ✺✳✺ ❆♥á❧✐s❡ q✉❛♥t✐t❛t✐✈❛ ❞♦ t❡st❡ ❛s❝❡♥sã♦ ❞❛ ❜♦❧❤❛ ✭t❡st❡ ✷✮✱ ❝♦♠♣❛r❛❞❛s ❝♦♠
❛ s♦❧✉çã♦ ❞❡ r❡❢❡rê♥❝✐❛ ❞❡ ❬✶✺❪ ✭❡♠ ♥❡❣r✐t♦✮✳ t1 s❡ r❡❢❡r❡ ❛♦ t❡♠♣♦ ❡♠ q✉❡ ❛
❝✐r❝✉❧❛r✐❞❛❞❡ ❛t✐♥❣❡ s❡✉ ✈❛❧♦r ♠í♥✐♠♦✱ ❡ t2 é ♦ t❡♠♣♦ ❡♠ q✉❡ ❛ ✈❡❧♦❝✐❞❛❞❡
♠é❞✐❛ ❞❛ ❜♦❧❤❛ ❛t✐♥❣❡ s❡✉ ✈❛❧♦r ♠á①✐♠♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✼ ✺✳✻ ❚❡♠♣♦ ❞❡ ❡①❡❝✉çã♦ ✭❡♠ s❡❣✉♥❞♦s✮ ❞❛s té❝♥✐❝❛s ❞❡ r❡✐♥✐❝✐❛❧✐③❛çã♦ ♣♦r ❊❉P
❙✉♠ár✐♦
❘❡s✉♠♦ ✺
❆❜str❛❝t ✼
▲✐st❛ ❞❡ ❋✐❣✉r❛s ✽
▲✐st❛ ❞❡ ❚❛❜❡❧❛s ✶✷
✶ ■♥tr♦❞✉çã♦ ✶✼
✷ ❈♦♥❝❡✐t♦s ❇ás✐❝♦s ❡♠ ▼ét♦❞♦s ▲❡✈❡❧✲❙❡t ✷✶
✷✳✶ ❘❡♣r❡s❡♥t❛çã♦ ❞❡ ■♥t❡r❢❛❝❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✶✳✶ ❯♠❛ ❞✐♠❡♥sã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✶✳✷ ❉✉❛s ❉✐♠❡♥sõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✷ ❋✉♥çã♦ ❉✐stâ♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✸ ❈á❧❝✉❧♦ ◆✉♠ér✐❝♦ ❞❡ Pr♦♣r✐❡❞❛❞❡s ●❡♦♠étr✐❝❛s ❞❛ ■♥t❡r❢❛❝❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✸✳✶ ▼❛❧❤❛ ❯t✐❧✐③❛❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✸✳✷ ❱❡t♦r ◆♦r♠❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✸✳✸ ❈✉r✈❛t✉r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✷✳✹ ❆❞✈❡❝çã♦ ❞❛ ■♥t❡r❢❛❝❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✷✳✹✳✶ ❖ ▼ét♦❞♦ ❲❊◆❖ ❞❡ ◗✉✐♥t❛ ❖r❞❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✷✳✹✳✷ ■♥t❡❣r❛çã♦ ❚❡♠♣♦r❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶
✸ ❘❡✐♥✐❝✐❛❧✐③❛çã♦ ❞❛ ❋✉♥çã♦ ▲❡✈❡❧✲❙❡t ✸✸
✹✳✶✳✹ ❊s❝♦❛♠❡♥t♦s ■♥❝♦♠♣r❡ssí✈❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶ ✹✳✷ ❊s❝♦❛♠❡♥t♦s ▼✉❧t✐❢ás✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷ ✹✳✷✳✶ ❈♦♥❞✐çã♦ ❞❡ ❈♦♥t♦r♥♦ ♣❛r❛ ❛ ❊q✉❛çã♦ ❞❛ ❈♦♥t✐♥✉✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷ ✹✳✷✳✷ ❆ ❚❡♥sã♦ ❙✉♣❡r✜❝✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✸ ✹✳✸ ❙♦❧✉çã♦ ◆✉♠ér✐❝❛ ❞❛s ❊q✉❛çõ❡s ❞❡ ◆❛✈✐❡r✲❙t♦❦❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻ ✹✳✸✳✶ ❉✐s❝r❡t✐③❛çã♦ ❚❡♠♣♦r❛❧ ❡ ❊s♣❛❝✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻ ✹✳✸✳✷ ▼ét♦❞♦ ❞❡ Pr♦❥❡çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻ ✹✳✸✳✸ ❈♦♥❞✐çõ❡s ❞❡ ❈♦♥t♦r♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✽
✺ ❆♣❧✐❝❛çõ❡s ❡♠ ❊s❝♦❛♠❡♥t♦s ❇✐❢ás✐❝♦s ✻✾
✺✳✶ ❈♦rr❡♥t❡s P❛r❛s✐t❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✾ ✺✳✷ ❖s❝✐❧❛çã♦ ❞❛ ❜♦❧❤❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✵ ✺✳✸ ❆s❝❡♥sã♦ ❞❛ ❇♦❧❤❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✷
✻ ❈♦♥❝❧✉sã♦ ✽✶
❆ ❉✐s❝r❡t✐③❛çõ❡s ▲❙✷ ❡ ▲❙✸ ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❛ ❝✉r✈❛t✉r❛ ✽✸ ❆✳✶ ▼ét♦❞♦ ▲❙✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✸ ❆✳✷ ▼ét♦❞♦ ▲❙✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✹
❇ ❆❧❣♦r✐t♠♦s ❈♦♠♣✉t❛❝✐♦♥❛✐s ✽✼
❇✳✶ ❘❡✐♥✐❝✐❛❧✐③❛çã♦ ♣♦r ❊❉P ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✼ ❇✳✷ ❘❡✐♥✐❝✐❛❧✐③❛çã♦ ●❡♦♠étr✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✶
❈❛♣ít✉❧♦
✶
■♥tr♦❞✉çã♦
▼✉✐t♦s ♣r♦❜❧❡♠❛s ❡♠ ▼❡❝â♥✐❝❛ ❞♦s ❋❧✉✐❞♦s ❝♦♥t❛♠ ❝♦♠ ❛ ♣r❡s❡♥ç❛ ❞❡ ❞♦✐s ♦✉ ♠❛✐s ✢✉✐❞♦s✳ P❛r❛ s✐♠✉❧❛r ❝♦♠♣✉t❛❝✐♦♥❛❧♠❡♥t❡ ❡ss❡s ♣r♦❜❧❡♠❛s✱ é ♥❡❝❡ssár✐♦ r❡♣r❡s❡♥t❛r ❞❡ ❛❧❣✉♠❛ ❢♦r♠❛ ❛ r❡❣✐ã♦ ❞❡ ❢r♦♥t❡✐r❛ ✭✐♥t❡r❢❛❝❡✮ ❡♥tr❡ ♦s ❞✐❢❡r❡♥t❡s ✢✉✐❞♦s ❡♥✈♦❧✈✐❞♦s✳ ❊①✲ ❡♠♣❧♦s ❞❡ ♣r♦❜❧❡♠❛s q✉❡ ❡♥✈♦❧✈❡♠ ❞♦✐s ✢✉✐❞♦s sã♦ ♣r♦❜❧❡♠❛s ❝♦♠ s♣r❛②s ❡ ❛t♦♠✐③❛çã♦ ❬✶✾❪✳
❆ r❡♣r❡s❡♥t❛çã♦ ♠❛t❡♠át✐❝❛ ❞❡ ✐♥t❡r❢❛❝❡s ♣♦❞❡ s❡r ❢❡✐t❛ ❞❡ ❞✉❛s ❢♦r♠❛s ❞✐st✐♥t❛s✿ ❞❡ ♠❛♥❡✐r❛ ❡①♣❧í❝✐t❛ ❡ ❞❡ ♠❛♥❡✐r❛ ✐♠♣❧í❝✐t❛✳ ❆ r❡♣r❡s❡♥t❛çã♦ ❡①♣❧í❝✐t❛ ♣♦❞❡ s❡r ❢❡✐t❛✱ ♣♦r ❡①❡♠♣❧♦✱ ❝♦♠ ♠ét♦❞♦s ❞♦ t✐♣♦ ❢r♦♥t✲tr❛❝❦✐♥❣ ❬✹✸❪✱ ♥♦s q✉❛✐s ❛ ✐♥t❡r❢❛❝❡ é r❡♣r❡s❡♥t❛❞❛ ♣♦r ♣❛rtí❝✉❧❛s ❞❡ ♠❛ss❛ ❞❡s♣r❡③í✈❡❧✱ ❡ ♦ ❝♦♥t♦r♥♦ ✏r❡❛❧✑ ❞❛ ✐♥t❡r❢❛❝❡ é r❡❝♦♥str✉í❞♦ ❛♣❡♥❛s ❛tr❛✈és ❞❛ ♣♦s✐çã♦ ❞❡ss❛s ♣❛rtí❝✉❧❛s ✭♣♦r ❡①❡♠♣❧♦✱ ❢❛③❡♥❞♦ ❛❧❣✉♠❛ ✐♥t❡r♣♦❧❛çã♦✮✳ ❆❧❣✉♠❛s ❞❛s ✈❛♥t❛❣❡♥s ❡ ❞❡s✈❛♥t❛❣❡♥s ❞❛ r❡♣r❡s❡♥t❛çã♦ ❡①♣❧í❝✐t❛ sã♦ ❛♣r❡s❡♥t❛❞❛s ♥❛ s❡çã♦ ✷✳✶✳
◆❛ r❡♣r❡s❡♥t❛çã♦ ✐♠♣❧í❝✐t❛✱ ❛ ✐♥t❡r❢❛❝❡ é r❡♣r❡s❡♥t❛❞❛ s❡♠ ♦ ✉s♦ ❞❡ ♣❛rtí❝✉❧❛s ♠❛r✲ ❝❛❞♦r❛s✳ P♦r ❡①❡♠♣❧♦✱ ♥♦ ♠ét♦❞♦ ❱♦❧✉♠❡ ♦❢ ❋❧✉✐❞ ✭❱❖❋✮✱ ✉t✐❧✐③❛✲s❡ ❛ ❢✉♥çã♦ ❢r❛çã♦ ❞❡ ✈♦❧✉♠❡✱ q✉❡ ❞✐③ q✉❛❧ ♣♦rçã♦ ❞❡ ❝❛❞❛ ❝é❧✉❧❛ ❝♦♠♣✉t❛❝✐♦♥❛❧ ❡stá ♣r❡❡♥❝❤✐❞❛ ❝♦♠ ✉♠ ❞❡t❡r♠✐♥❛❞♦ t✐♣♦ ❞❡ ✢✉✐❞♦✳ ❏á ♠ét♦❞♦s ❧❡✈❡❧✲s❡t ✉t✐❧✐③❛♠ ✉♠❛ ❝✉r✈❛ ❞❡ ♥í✈❡❧ q✉❛❧q✉❡r ❞❡ ❛❧❣✉♠❛ ❢✉♥çã♦ ❡s❝❛❧❛r ♣❛r❛ ❢❛③❡r ❡ss❛ r❡♣r❡s❡♥t❛çã♦✳ ❚r❛❞✐❝✐♦♥❛❧♠❡♥t❡✱ ❡ss❛ ❢✉♥çã♦ é ✉♠❛ ❢✉♥çã♦ ❞✐stâ♥❝✐❛ s✐♥❛❧✐③❛❞❛✱ ❡ ❛ ❝✉r✈❛ ❞❡ ♥í✈❡❧ ❛❞♦t❛❞❛ é ♦ ✐s♦❝♦♥t♦r♥♦ ♥✉❧♦ ❬✸✾❪✳
❆ ✉t✐❧✐③❛çã♦ ❞❡ ♠ét♦❞♦s ❧❡✈❡❧✲s❡t ♣❛r❛ ❛ ♣r♦♣❛❣❛çã♦ ❞❡ ✐♥t❡r❢❛❝❡s ✈❡♠ ❞❡s❞❡ ✶✾✽✽✱ q✉❛♥❞♦ ❖s❤❡r ❡ ❙❡t❤✐❛♥ ❬✷✼❪ ❛♣r❡s❡♥t❛r❛♠ ❛ ♣r✐♠❡✐r❛ ❢♦r♠✉❧❛çã♦✳ ❊♠ ✶✾✾✵✱ ▼✉❧❞❡r ❡ ❖s❤❡r ❬✷✸❪ ❛♣r❡s❡♥t❛r❛♠ ❡st✉❞♦s s♦❜r❡ ❛ ✉t✐❧✐③❛çã♦ ❞❡ ♠ét♦❞♦s ❧❡✈❡❧✲s❡t ♣❛r❛ ❡s❝♦❛♠❡♥t♦s ❝♦♠♣r❡ssí✈❡✐s✱ ❡st✉❞❛♥❞♦ ✉♠ ❡s❝♦❛♠❡♥t♦ ❝♦♠ ❤é❧✐♦ ❡ ❛r✳
❆ s❡❣✉✐r✱ ❡♠ ✶✾✾✹✱ ❙✉ss♠❛♥ ❡t ❛❧✳ ❬✸✾❪ ♣r♦♣✉s❡r❛♠ ✉♠ ♠ét♦❞♦ ❧❡✈❡❧✲s❡t ♣❛r❛ s❡r ✉t✐❧✐③❛❞♦ ❝♦♠ ❡s❝♦❛♠❡♥t♦s ✐♥❝♦♠♣r❡ssí✈❡✐s ❜✐❢ás✐❝♦s✱ r❡❛❧✐③❛♥❞♦ s✐♠✉❧❛çõ❡s ❝♦♠ á❣✉❛ ❡ ❛r✳ ❉❡s❞❡ ❡♥tã♦✱ ❞✐✈❡rs♦s ❛✉t♦r❡s tê♠ ✉t✐❧✐③❛❞♦ ♠ét♦❞♦s ❧❡✈❡❧✲s❡t ♣❛r❛ ❡s❝♦❛♠❡♥t♦s ❜✐❢ás✐❝♦s ❬✺✱ ✶✵✱ ✶✽❪✳
❖ ♠ét♦❞♦ ♣r♦♣♦st♦ ♣♦r ❙✉ss♠❛♥ ❡t ❛❧✳ ✉t✐❧✐③❛ ✉♠❛ ❢✉♥çã♦ ❞✐stâ♥❝✐❛ s✐♥❛❧✐③❛❞❛ ♣❛r❛ r❡♣r❡s❡♥t❛r ❛ ✐♥t❡r❢❛❝❡✳ ❈♦♥t✉❞♦✱ ❝♦♥❢♦r♠❡ ❛ ✐♥t❡r❢❛❝❡ é ♠♦✈✐♠❡♥t❛❞❛✱ ♦❝♦rr❡ ♦ s✉r❣✐✲ ♠❡♥t♦ ❞❡ ❣r❛❞✐❡♥t❡s ♠✉✐t♦ ❛❧t♦s ✭♦✉ ♠✉✐t♦ ❜❛✐①♦s✮ ♥❛s ♣r♦①✐♠✐❞❛❞❡s ❞❛ ✐♥t❡r❢❛❝❡✱ ❢❛③❡♥❞♦ ❝♦♠ q✉❡ ❛ ❢✉♥çã♦ ❧❡✈❡❧✲s❡t ♣❡r❝❛ ❛ s✉❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ ❢✉♥çã♦ ❞✐stâ♥❝✐❛✳ ■ss♦ ❛❝❛rr❡t❛ ❡♠ ✐♠♣r❡❝✐sõ❡s ♥♦ ❝á❧❝✉❧♦ ❞❛ ❢♦rç❛ ❞❡✈✐❞♦ à t❡♥sã♦ s✉♣❡r✜❝✐❛❧✱ ❝♦♥❢♦r♠❡ ❞✐t♦ ♣♦r ❈r♦❝❡ ❡t ❛❧✳ ❬✶✵❪✳
❆❧é♠ ❞✐ss♦✱ ♠ét♦❞♦s ❧❡✈❡❧✲s❡t ❝♦st✉♠❛♠ ❛♣r❡s❡♥t❛r ❣r❛♥❞❡s ♣r♦❜❧❡♠❛s ❞❡ ♣❡r❞❛ ❞❡ ♠❛ss❛ ❬✷✶❪✱ ♦ q✉❡ ♣♦❞❡ ♣r♦✈♦❝❛r ❝♦♠♣♦rt❛♠❡♥t♦s ♥ã♦✲❢ís✐❝♦s ❞✉r❛♥t❡ ❛s s✐♠✉❧❛çõ❡s ♥✉♠ér✐✲ ❝❛s✳ ▲✉♦ ❡t ❛❧✳ ❬✷✶❪ ✜③❡r❛♠ ✉♠❛ ❞❡s❝r✐çã♦ ❞❡t❛❧❤❛❞❛ s♦❜r❡ ❢♦r♠❛s ❞❡ ♠❡❧❤♦r❛r ❛ ❝♦♥s❡r✲ ✈❛çã♦ ❞❡ ♠❛ss❛✳ ❊♥tr❡ ❡❧❛s✱ ♣♦❞❡♠♦s ❞❡st❛❝❛r✿
✶✳ ■♥tr♦❞✉çã♦ ✶✽ ❚r❛♥s♣♦rt❡ ❞❛ ❢✉♥çã♦ ❧❡✈❡❧✲s❡t ❝♦♠ ♠ét♦❞♦s ❞❡ ❛❧t❛ ♦r❞❡♠ ◆♦✉r❣❛❧✐❡✈ ❡t ❛❧✳ ❬✷✹❪
✈❡r✐✜❝❛r❛♠ q✉❡ ❛ ♣❡r❞❛ ❞❡ ♠❛ss❛ q✉❛♥❞♦ ❛ ❞✐s❝r❡t✐③❛çã♦ é ❢❡✐t❛ ❝♦♠ ✉♠ ♠ét♦❞♦ ❲❊◆❖ ❞❡ q✉✐♥t❛ ♦r❞❡♠ é ♠❡♥♦r ❞♦ q✉❡ q✉❛♥❞♦ é ❢❡✐t❛ ❝♦♠ ♠ét♦❞♦s ❞❡ ♦r❞❡♠ ♠❛✐s ❜❛✐①❛✳ ❙❛❧✐❤ ❡ ▼♦✉❧✐❝ ❬✸✶❪ ✜③❡r❛♠ ♦❜s❡r✈❛çã♦ s❡♠❡❧❤❛♥t❡✳
❯t✐❧✐③❛çã♦ ❞❡ ♠ét♦❞♦s ❤í❜r✐❞♦s ▼ét♦❞♦s ❱❖❋ sã♦ ❝♦♥s✐❞❡r❛✈❡❧♠❡♥t❡ ♠❡❧❤♦r❡s q✉❡ ♠ét♦❞♦s ❧❡✈❡❧✲s❡t q✉❛♥t♦ à ❝♦♥s❡r✈❛çã♦ ❞❡ ♠❛ss❛✱ ❡♠❜♦r❛ s❡❥❛♠ ♠❡♥♦s ♣r❡❝✐s♦s ♥♦ ❝á❧❝✉❧♦ ❞❛ ❝✉r✈❛t✉r❛ ❡ ❞♦ ✈❡t♦r ♥♦r♠❛❧✳ ❙❡♥❞♦ ❛ss✐♠✱ é ♥❛t✉r❛❧ ♣❡♥s❛r ❡♠ ❛❧❣✉♠❛ ❢♦r♠❛ ❞❡ ❝♦♠❜✐♥❛r ❝❛r❛❝t❡ríst✐❝❛s ❞♦s ❞♦✐s ♠ét♦❞♦s✳ ❊♥tr❡ ❡ss❡s ♠ét♦❞♦s✱ ♣♦❞❡♠♦s ❞❡st❛❝❛r ♦ ♠ét♦❞♦ ❱❖❙❊❚ ❬✸✹❪ ❡ ♦ ❈▲❙❱❖❋ ❬✸✽❪✳
▼ét♦❞♦s ❧❡✈❡❧✲s❡t ❛❞❛♣t❛t✐✈♦s ❈♦♠♦✱ ❡♠ ♣r♦❜❧❡♠❛s ❞❡ ✢✉✐❞♦s✱ ❛ s✐♠✉❧❛çã♦ ♣r❡❝✐s❛ ✜❝❛r ❜❡♠ ❝❛❧❝✉❧❛❞❛ ❛♣❡♥❛s ♥✉♠❛ r❡❣✐ã♦ ✈✐③✐♥❤❛ à ✐♥t❡r❢❛❝❡✱ s✉r❣✐r❛♠ ♠ét♦❞♦s ❧❡✈❡❧✲ s❡t ❛❞❛♣t❛t✐✈♦s ❬✷✵✱ ✷✹✱ ✸✻❪✱ q✉❡ ❜✉s❝❛♠ ♠❡❧❤♦r❛r ❛ s♦❧✉çã♦ ❛♣❡♥❛s ♥❡ss❛s r❡❣✐õ❡s✱ s♦❜ ❝✉st♦ ❝♦♠♣✉t❛❝✐♦♥❛❧ ❛❝❡✐tá✈❡❧✳
▼❡❧❤♦r✐❛ ❞❛ r❡✐♥✐❝✐❛❧✐③❛çã♦ ❆ r❡✐♥✐❝✐❛❧✐③❛çã♦ é ♥❡❝❡ssár✐❛ ♣❛r❛ r❡st❛✉r❛r ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ ❢✉♥çã♦ ❞✐stâ♥❝✐❛ ❞❛ ❢✉♥çã♦ ❧❡✈❡❧✲s❡t✳ ❚♦❞❛✈✐❛✱ ❡❧❛ t❛♠❜é♠ ♣♦❞❡ ❝♦♥tr✐❜✉✐r ♣❛r❛ ❛ ♣❡r❞❛ ❞❡ ♠❛ss❛✳ ❊♠ ✶✾✾✽✱ ❙✉ss♠❛♥ ❡t ❛❧✳ ❬✸✼❪ ❛❞✐❝✐♦♥❛r❛♠ ✉♠❛ ❝♦rr❡çã♦ à s✉❛ r❡✐♥✐❝✐❛❧✐③❛çã♦ ♣♦r ❊❉P✱ ❛ ✜♠ ❞❡ ♠❡❧❤♦r❛r ❛ ❝♦♥s❡r✈❛çã♦ ❞❡ ♠❛ss❛✳ ❆❧❣✉♠❛s ✈❛r✐✲ ❛çõ❡s ❞❡ss❛ r❡✐♥✐❝✐❛❧✐③❛çã♦ ❢♦r❛♠ ♣r♦♣♦st❛s ♥❛ ❧✐t❡r❛t✉r❛ ❬✶✵✱ ✹✺❪✳ ❆ r❡✐♥✐❝✐❛❧✐③❛çã♦ ❣❡♦♠étr✐❝❛ t❛♠❜é♠ ♣♦ss✉✐ ❛❧❣✉♥s ♣❛ss♦s q✉❡ ❞❡✈❡♠ s❡r s❡❣✉✐❞♦s ♣❛r❛ ♣r❡s❡r✈❛r ❛ ♠❛ss❛ ❞❛ ✐♥t❡r❢❛❝❡ ❬✹❪✳
P❛r❛ r❡st❛✉r❛r ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ ❢✉♥çã♦ ❞✐stâ♥❝✐❛ ❞❛ ❢✉♥çã♦ ❧❡✈❡❧✲s❡t✱ ♦♣t❛♠♦s ♣♦r ✉t✐✲ ❧✐③❛r ❞✉❛s r♦t✐♥❛s ❞❡ r❡✐♥✐❝✐❛❧✐③❛çã♦ ❞✐st✐♥t❛s✳ ❊ss❡r ❡t ❛❧✳ ❬✶✶❪ ❝❧❛ss✐✜❝♦✉ ❛s r❡✐♥✐❝✐❛❧✐③❛çõ❡s ❡①✐st❡♥t❡s ♥❛ ❧✐t❡r❛t✉r❛ ❡♠ ❞♦✐s ♣r✐♥❝✐♣❛✐s ❣r✉♣♦s✿ ❛s ❜❛s❡❛❞❛s ❡♠ ❊❉P ❡ ❛s ❣❡♦♠étr✐❝❛s✳ ◆❡ss❡ tr❛❜❛❧❤♦✱ ❡st✉❞❛♠♦s ✉♠❛ r❡✐♥✐❝✐❛❧✐③❛çã♦ ❞❡ ❝❛❞❛ ✉♠ ❞❡ss❡s ❣r✉♣♦s✳ ❆ ♣r✐♠❡✐r❛ ❞❡✲ ❧❛s✱ q✉❡ é ✉♠❛ r❡✐♥✐❝✐❛❧✐③❛çã♦ ♣♦r ❊❉P✱ ❢♦✐ ♣r♦♣♦st❛ ♣♦r ❙✉ss♠❛♥ ❡t ❛❧✳ ❬✸✾❪ ❥á ❡♠ ✶✾✾✹✱ ❡ ❝♦♥s✐st❡ ❞❡ r❡s♦❧✈❡r ✉♠❛ ♥♦✈❛ ❡q✉❛çã♦ ❞❡ ❛❞✈❡❝çã♦ ❛ ❝❛❞❛ ♣❛ss♦ ❞❡ t❡♠♣♦ ❞❛ ❡q✉❛çã♦ ❞❡ tr❛♥s♣♦rt❡ ❞❛ ✐♥t❡r❢❛❝❡✳ ❊ss❛ r❡✐♥✐❝✐❛❧✐③❛çã♦ s❡❣✉❡ ❜❡♠ ✉t✐❧✐③❛❞❛ ❛té ♦s ❞✐❛s ❞❡ ❤♦❥❡✳ ❊♠ ✷✵✶✵ ❡❧❛ ❢♦✐ ✉t✐❧✐③❛❞❛ ♣♦r ❈r♦❝❡ ❡t ❛❧✳ ❬✶✵❪✱ ❡ ❡♠ ✷✵✶✸ ❢♦✐ t❡st❛❞❛ ❞❡ ♠❛♥❡✐r❛ ♣❛r❛❧❡❧❛ ❡♠ ✉♠ ❝❧✉st❡r ●P❯ ♣♦r ❩❛s♣❡❧ ❡ ●r✐❡❜❡❧ ❬✹✻❪✳ ▲❛❧❛♥♥❡ ❡t ❛❧✳ ❬✶✽❪ t❛♠❜é♠ ✉t✐❧✐③❛r❛♠ ❛❜♦r❞❛❣❡♠ s❡♠❡❧❤❛♥t❡ ❡♠ ✷✵✶✺✳ ❆ r❡✐♥✐❝✐❛❧✐③❛çã♦ ❣❡♦♠étr✐❝❛ q✉❡ ❡s❝♦❧❤❡♠♦s ❢♦✐ ♣r♦♣♦st❛ ♣♦r ❆✉s❛s ❡t ❛❧✳ ❬✹❪✳ ◆ã♦ ❤á ♠✉✐t♦s tr❛❜❛❧❤♦s ♣✉❜❧✐❝❛❞♦s ✉t✐❧✐③❛♥❞♦ ❡ss❛ r❡✐♥✐❝✐❛❧✐③❛çã♦ ❣❡♦♠étr✐❝❛✱ ♠❛s ❆✉s❛s ❡t ❛❧✳ ❬✸❪ ❡ ❇✉s❝❛❣❧✐❛ ❡ ❆✉s❛s ❬✽❪ r❡s♦❧✈❡r❛♠ ❛❧❣✉♥s ♣r♦❜❧❡♠❛s ❝♦♠ ❡s❝♦❛♠❡♥t♦s ❜✐❢ás✐❝♦s ♥♦ ❝♦♥t❡①t♦ ❞❡ ❡❧❡♠❡♥t♦s ✜♥✐t♦s✳
❆❧é♠ ❞✐ss♦ tr❛❜❛❧❤♦ ♦♣t❛♠♦s ♣♦r ✉t✐❧✐③❛r ♠ét♦❞♦s ❞❡ ❛❧t❛ ♦r❞❡♠ ♣❛r❛ ❛ ❛♣r♦①✐♠❛çã♦ ❞❛ s♦❧✉çã♦ ❞❛s ❡q✉❛çõ❡s ❞❡ tr❛♥s♣♦rt❡ ❞❛ ❢✉♥çã♦ ❧❡✈❡❧✲s❡t✳ ❯t✐❧✐③❛♠♦s ❲❊◆❖ ❞❡ q✉✐♥t❛ ♦r❞❡♠ ♣❛r❛ ❛ ❛♣r♦①✐♠❛çã♦ ❞❛s ❞❡r✐✈❛❞❛s ❡s♣❛❝✐❛✐s✱ ❡ ❘✉♥❣❡✲❑✉tt❛ ❞❡ t❡r❝❡✐r❛ ♦r❞❡♠ ♣❛r❛ ❛s ❞❡r✐✈❛❞❛s t❡♠♣♦r❛✐s✳ ❆✐♥❞❛✱ ✜③❡♠♦s ✉♠❛ ✐♠♣❧❡♠❡♥t❛çã♦ ❝✉✐❞❛❞♦s❛ ❞♦s ♣❛ss♦s ❞❡ ❝♦♥s❡r✈❛çã♦ ❞❡ ♠❛ss❛ ♥❛s ❞✉❛s r♦t✐♥❛s ❞❡ r❡✐♥✐❝✐❛❧✐③❛çã♦ ❡st✉❞❛❞❛s✳
◆♦ ❈❛♣ít✉❧♦ ✷✱ ❞♦✐s ♣❛r❛❞✐❣♠❛s ♣❛r❛ ❛ r❡♣r❡s❡♥t❛çã♦ ❞❡ ✐♥t❡r❢❛❝❡s sã♦ ❞✐s❝✉t✐❞♦s✿ ❛ r❡♣r❡s❡♥t❛çã♦ ❡①♣❧í❝✐t❛ ❡ ❛ ✐♠♣❧í❝✐t❛✳ ❖ ♠ét♦❞♦ ❧❡✈❡❧✲s❡t✱ q✉❡ é ✉♠ ♠ét♦❞♦ ❞❡ r❡♣r❡s❡♥✲ t❛çã♦ ✐♠♣❧í❝✐t❛✱ é✱ ❡♥tã♦✱ ❛♣r❡s❡♥t❛❞♦✳ ◆❡ss❡ ♠❡s♠♦ ❈❛♣ít✉❧♦ ❛♣r❡s❡♥t❛♠♦s ❛s ❡q✉❛çõ❡s ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞♦ ✈❡t♦r ♥♦r♠❛❧ ❡ ❞❛ ❝✉r✈❛t✉r❛ ❞❛ ✐♥t❡r❢❛❝❡✳ ▼♦str❛♠♦s ❛ ♠❛❧❤❛ ❝♦♠✲ ♣✉t❛❝✐♦♥❛❧ ✉t✐❧✐③❛❞❛✱ ♣❛r❛ q✉❡ ♣♦ss❛♠♦s ❛♣r❡s❡♥t❛r ❛ ❞✐s❝r❡t✐③❛çã♦ ❞❛s ❡q✉❛çõ❡s✳ P♦r ✜♠✱ ❞✐s❝r❡t✐③❛♠♦s ❛ ❡q✉❛çã♦ ❞❡ tr❛♥s♣♦rt❡ ❞❛ ✐♥t❡r❢❛❝❡ ❝♦♠ ♦ ♠ét♦❞♦ ❲❊◆❖ ❞❡ q✉✐♥t❛ ♦r❞❡♠ ❡ ♦ ❘✉♥❣❡✲❑✉tt❛ ❞❡ t❡r❝❡✐r❛ ♦r❞❡♠ ♠❡♥❝✐♦♥❛❞♦s ❛♥t❡r✐♦r♠❡♥t❡✳
✶✳ ■♥tr♦❞✉çã♦ ✶✾ ♥❡ss❡ ❈❛♣ít✉❧♦✱ ❛♣r❡s❡♥t❛♠♦s r❡s✉❧t❛❞♦s ♥✉♠ér✐❝♦s q✉❡ ❝♦♠♣❛r❛♠ ❡ss❛s ❞✉❛s té❝♥✐❝❛s ❡♠ ♣r♦❜❧❡♠❛s ❝♦♠ ❝❛♠♣♦ ❞❡ ✈❡❧♦❝✐❞❛❞❡ ♣r❡s❝r✐t♦✳
❉❡♣♦✐s✱ ♥♦ ❈❛♣ít✉❧♦ ✹✱ é ❢❡✐t❛ ❛ ❞❡r✐✈❛çã♦ ❞❛s ❡q✉❛çõ❡s ❞❡ ◆❛✈✐❡r✲❙t♦❦❡s✱ q✉❡ ♠♦❞❡✲ ❧❛♠ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞♦s ✢✉✐❞♦s✳ ❊❧❛s sã♦ ❛♣r❡s❡♥t❛❞❛s ❡♠ s✉❛ ❢♦r♠❛ ♠❛✐s ❣❡r❛❧ ♣❛r❛ ❡s❝♦❛♠❡♥t♦s ♠✉❧t✐❢ás✐❝♦s✱ ❡ ❞❡♣♦✐s ♣❛rt✐❝✉❧❛r✐③❛❞❛s ♣❛r❛ ✢✉✐❞♦s ◆❡✇t♦♥✐❛♥♦s ❡ ✐♥❝♦♠✲ ♣r❡ssí✈❡✐s✳ ❆✐♥❞❛ ♥❡ss❡ ♠❡s♠♦ ❈❛♣ít✉❧♦✱ ♦s ♠ét♦❞♦s ✉t✐❧✐③❛❞♦s ♣❛r❛ ❛ r❡s♦❧✉çã♦ ♥✉♠ér✐❝❛ ❞❛s ❡q✉❛çõ❡s ❞❡ ◆❛✈✐❡r✲❙t♦❦❡s ❡ ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❛ t❡♥sã♦ s✉♣❡r✜❝✐❛❧ sã♦ ♠♦str❛❞♦s ❝♦♠ ❞❡t❛❧❤❡s✳ ❋♦✐ ✉t✐❧✐③❛❞♦ ✉♠ ♠ét♦❞♦ ✐♠♣❧í❝✐t♦ ♣❛r❛ ❛ ✐♥t❡❣r❛çã♦ t❡♠♣♦r❛❧✱ ❡ ✉♠ ♠ét♦❞♦ ❞❡ ♣r♦❥❡çã♦ ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❛ ✈❡❧♦❝✐❞❛❞❡ ❡ ❞❛ ♣r❡ssã♦✳ ❆ t❡♥sã♦ s✉♣❡r✜❝✐❛❧ ❢♦✐ ❝❛❧❝✉❧❛❞❛ ❛tr❛✈és ❞♦s ♠ét♦❞♦s ❙❤❛r♣ ❙✉r❢❛❝❡ t❡♥s✐♦♥ ❋♦r❝❡ ✭❙❙❋[✶✷]✮ ❡ ❈♦♥t✐♥✉♦✉s ❙✉r❢❛❝❡ t❡♥s✐♦♥ ❋♦r❝❡ ✭❈❙❋ ❬✼❪✮✳
❈❛♣ít✉❧♦
✷
❈♦♥❝❡✐t♦s ❇ás✐❝♦s ❡♠ ▼ét♦❞♦s
▲❡✈❡❧✲❙❡t
◆❡ss❡ ❈❛♣ít✉❧♦✱ ❢❛r❡♠♦s ✉♠ ❡st✉❞♦ t❡ór✐❝♦ s♦❜r❡ ♠ét♦❞♦s ❧❡✈❡❧✲s❡t✳ ■♥✐❝✐❛❧♠❡♥t❡✱ é ✐♥tr♦❞✉③✐❞♦ ♦ ❝♦♥❝❡✐t♦ ❞❡ r❡♣r❡s❡♥t❛çã♦ ✐♠♣❧í❝✐t❛ ❞❡ ✐♥t❡r❢❛❝❡s ❡ ♦ ❞❡ ❢✉♥çã♦ ❞✐stâ♥❝✐❛✳ ❉❡✲ ♣♦✐s✱ ♦ ♠ét♦❞♦ ❲❊◆❖ ❞❡ q✉✐♥t❛ ♦r❞❡♠ é ❛♣r❡s❡♥t❛❞♦ ♣❛r❛ ❛ ❛♣r♦①✐♠❛çã♦ ❞❛s ❞❡r✐✈❛❞❛s ❡s♣❛❝✐❛✐s ♥❛ ❡q✉❛çã♦ ❞♦ tr❛♥s♣♦rt❡ ❞❛ ✐♥t❡r❢❛❝❡✱ ❜❡♠ ❝♦♠♦ ✉♠ ♠ét♦❞♦ ❘✉♥❣❡✲❑✉tt❛ ❞❡ t❡r❝❡✐r❛ ♦r❞❡♠ ♣❛r❛ ❛ ✐♥t❡❣r❛çã♦ t❡♠♣♦r❛❧✳ ◆❡ss❡ ❈❛♣ít✉❧♦✱ ❡ ♥♦ ❈❛♣ít✉❧♦ s❡❣✉✐♥t❡✱ ❯t✐✲ ❧✐③❛r❡♠♦s ❛ ♥♦t❛çã♦ |x| ♣❛r❛ r❡♣r❡s❡♥t❛r ♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦ ✈❡t♦r x ✭♠❡❞✐❞♦ ❛tr❛✈és ❞❛
♥♦r♠❛ ❡✉❝❧✐❞✐❛♥❛✮✳
✷✳✶ ❘❡♣r❡s❡♥t❛çã♦ ❞❡ ■♥t❡r❢❛❝❡s
■♥t❡r❢❛❝❡s ♣♦❞❡♠ s❡r r❡♣r❡s❡♥t❛❞❛s ❞❡ ❞✉❛s ♠❛♥❡✐r❛s ❞✐st✐♥t❛s✿ ❞❡ ❢♦r♠❛ ❡①♣❧í❝✐t❛ ♦✉ ❞❡ ❢♦r♠❛ ✐♠♣❧í❝✐t❛✳ ◆❡st❛ s❡çã♦ é ❢❡✐t❛ ✉♠❛ ❜r❡✈❡ ✐♥tr♦❞✉çã♦ s♦❜r❡ ❝❛❞❛ ✉♠❛ ❞❛s ❞✉❛s ❛❜♦r❞❛❣❡♥s✱ ❛♣r❡s❡♥t❛♥❞♦ s✉❛s ✈❛♥t❛❣❡♥s ❡ ❞❡s✈❛♥t❛❣❡♥s✳ ❯♠❛ ❞❡s❝r✐çã♦ ♠❛✐s ❝♦♠♣❧❡t❛ s♦❜r❡ ♦ ❛ss✉♥t♦ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ❡♠[✷✻]✱ q✉❡ é ♦ t❡①t♦ ❜❛s❡ ✉t✐❧✐③❛❞♦ ♥❡st❛ s❡çã♦✳
✷✳✶✳✶ ❯♠❛ ❞✐♠❡♥sã♦
❈♦♥s✐❞❡r❡ ♣r✐♠❡✐r❛♠❡♥t❡ q✉❡ ♥♦ss♦ ❞♦♠í♥✐♦ ❞❡ ✐♥t❡r❡ss❡ s❡❥❛ ❛ r❡t❛ r❡❛❧✱ ❡ q✉❡ ❞❡s❡✲ ❥❛♠♦s ❞✐✈✐❞✐✲❧♦ ❡♠ ❞✉❛s r❡❣✐õ❡s ❞✐st✐♥t❛s✿
Ω1 = (−∞,−1)∪(1,∞)
Ω2 = (−1,1).
❉❡♥♦♠✐♥❡♠♦sΩ2 ❛ r❡❣✐ã♦ ✐♥t❡r♥❛ ❞♦ ❞♦♠í♥✐♦✱ ❡ Ω1 ❛ r❡❣✐ã♦ ❡①t❡r♥❛✳ ❆ ✐♥t❡r❢❛❝❡ ❡♥tr❡ ❛s
r❡❣✐õ❡s ✐♥t❡r♥❛ ❡ ❡①t❡r♥❛ ❝♦♥s✐st❡ ❞♦s ♣♦♥t♦s
∂Ω ={−1,1}.
❊ss❛ ❢♦r♠❛ ❞❡ r❡♣r❡s❡♥t❛r ❛ ✐♥t❡r❢❛❝❡✱ ❡①♣❧✐❝✐t❛♥❞♦ ♦s ♣♦♥t♦s q✉❡ ♣❡rt❡♥❝❡♠ ❛ ❡❧❛✱ é ❞❡♥♦♠✐♥❛❞❛ r❡♣r❡s❡♥t❛çã♦ ❡①♣❧í❝✐t❛✳ ❆♦ ✐♥✈és ❞✐ss♦✱ ❡ss❡s ♠❡s♠♦s ❞♦✐s ♣♦♥t♦s ♣❡rt❡♥✲ ❝❡♥t❡s ❛∂Ω♣♦❞❡♠ s❡r r❡♣r❡s❡♥t❛❞♦s ❛tr❛✈és ❞♦ ✐s♦❝♦♥t♦r♥♦ ♥✉❧♦ ❞❛ ❢✉♥çã♦φ(x) = x2−1✱
❝♦♠♦ ❛ s❡❣✉✐r✿
∂Ω ={x∈R;x2−1 = 0}.
✷✳ ❈♦♥❝❡✐t♦s ❇ás✐❝♦s ❡♠ ▼ét♦❞♦s ▲❡✈❡❧✲❙❡t ✷✷ ❆ r❡♣r❡s❡♥t❛çã♦ ❞❡ ✉♠❛ ✐♥t❡r❢❛❝❡ ❛tr❛✈és ❞❡ ❛❧❣✉♠ ✐s♦❝♦♥t♦r♥♦ ❞❡ ✉♠❛ ❢✉♥çã♦ é ❞❡♥♦♠✲ ✐♥❛❞❛ r❡♣r❡s❡♥t❛çã♦ ✐♠♣❧í❝✐t❛✳
❈♦♠ ❜❛s❡ ♥❡ss❡ ❡①❡♠♣❧♦✱ ♣♦❞❡♠♦s ❡♥✉♠❡r❛r ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ✐♠♣♦rt❛♥t❡s ❞❛s r❡♣r❡s❡♥t❛çõ❡s ❡①♣❧í❝✐t❛ ❡ ✐♠♣❧í❝✐t❛✳ Pr✐♠❡✐r❛♠❡♥t❡✱ s❡ ♦ ❞♦♠í♥✐♦ ❡st✉❞❛❞♦ ♣♦ss✉✐ ❞✐♠❡♥✲ sã♦n✱ ❛ ✐♥t❡r❢❛❝❡ ♣♦ss✉✐ ❞✐♠❡♥sã♦n−1✳ ■ss♦ ♣♦❞❡ s❡r ✈✐st♦ ❢❛❝✐❧♠❡♥t❡ ♥♦ ❡①❡♠♣❧♦ ❛❝✐♠❛✱
❡♠ q✉❡ ♦ ❞♦♠í♥✐♦ é ❛ r❡t❛ r❡❛❧✱ q✉❡ ♣♦ss✉✐ ❞✐♠❡♥sã♦1✱ ❡ ❛ ✐♥t❡r❢❛❝❡ é ❢♦r♠❛❞❛ ♣♦r ❛♣❡♥❛s ❞♦✐s ♣♦♥t♦s✱ ❝❛❞❛ ✉♠ ❞❡❧❡s ❝♦♠ ❞✐♠❡♥sã♦ 0✳ ❖✉tr❛ ♣r♦♣r✐❡❞❛❞❡ ✐♠♣♦rt❛♥t❡ é q✉❡✱ s❡ ♦ ❞♦♠í♥✐♦ ❞❡ ✐♥t❡r❡ss❡ ♣♦ss✉✐ ❞✐♠❡♥sã♦n✱ q✉❛❧q✉❡r ❢✉♥çã♦ ✐♠♣❧í❝✐t❛ q✉❡ s❡❥❛ ✉t✐❧✐③❛❞❛ ♣❛r❛
r❡♣r❡s❡♥t❛r ✉♠❛ ✐♥t❡r❢❛❝❡ ♥❡ss❡ ❞♦♠í♥✐♦ ❞❡✈❡ ❡st❛r ❞❡✜♥✐❞❛ ♥✉♠ ❡s♣❛ç♦ ❞❡ ❞✐♠❡♥sã♦ n✱
❡♥q✉❛♥t♦ q✉❡ ❛ r❡♣r❡s❡♥t❛çã♦ ❡①♣❧í❝✐t❛ ❧✐❞❛ ❝♦♠ ♣r♦❜❧❡♠❛s ❞❡ ❞✐♠❡♥sã♦ n−1✳
P♦r ✜♠✱ ✈❛❧❡ ❞✐③❡r q✉❡ ❛ ✐♥t❡r❢❛❝❡ ♣♦❞❡ s❡r ❞❡t❡r♠✐♥❛❞❛ ✐♠♣❧✐❝✐t❛♠❡♥t❡ ♣♦r q✉❛❧q✉❡r ✐s♦❝♦♥t♦r♥♦ ❞❡s❡❥❛❞♦✱ ❡ ♥ã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ♣❡❧♦ ✐s♦❝♦♥t♦r♥♦ ♥✉❧♦✳ P♦❞❡♠♦s r❡♣r❡s❡♥t❛r ❛ ♠❡s♠❛ ✐♥t❡r❢❛❝❡ ∂Ω ❛❝✐♠❛ ♣❡❧♦ ✐s♦❝♦♥t♦r♥♦ ✶ ❞❛ ❢✉♥çã♦ x2✳ ❈♦♥t✉❞♦✱ é ♠❛✐s ✉s✉❛❧
✉t✐❧✐③❛r ♦ ✐s♦❝♦♥t♦r♥♦ ♥✉❧♦✱ ❡ ❡❧❡ s❡rá ✉t✐❧✐③❛❞♦ ❛♦ ❧♦♥❣♦ ❞❡ss❡ ❞❡st❡ tr❛❜❛❧❤♦✳
✷✳✶✳✷ ❉✉❛s ❉✐♠❡♥sõ❡s
❈♦♥s✐❞❡r❡ ❛❣♦r❛ ♦ ❝❛s♦ ❜✐❞✐♠❡♥s✐♦♥❛❧✱ ❡♠ q✉❡ ♥♦ss♦ ❞♦♠í♥✐♦ ❞❡ ✐♥t❡r❡ss❡ é ♦R2✳ ◆❡ss❡
❝❛s♦✱ ❛ ✐♥t❡r❢❛❝❡ s❡rá ✉♠❛ ✭♦✉ ♠❛✐s✮ ❝✉r✈❛✭s✮ ❢❡❝❤❛❞❛✭s✮ q✉❡ s❡♣❛r❛rá✭ã♦✮R2 ❡♠ ❞✉❛s ♦✉
♠❛✐s r❡❣✐õ❡s ❞❡ ár❡❛s ♥ã♦ ♥✉❧❛s✳ ❈♦♥s✐❞❡r❡ ❛✐♥❞❛ q✉❡ ❛s r❡❣✐õ❡s ✐♥t❡r♥❛s ❡ ❡①t❡r♥❛s ❞♦ ❞♦♠í♥✐♦ s❡❥❛♠ ❞❛❞❛s ♣♦r
Ω1 ={x∈R2;|x|>1}
Ω2 ={x∈R2;|x|<1}.
❆ss✐♠✱ ❛ ✐♥t❡r❢❛❝❡ é ♦ ❝ír❝✉❧♦ ❞❡ r❛✐♦ 1 ❡ ❝❡♥tr♦ ♥❛ ♦r✐❣❡♠✱ q✉❡ ♣♦❞❡ s❡r ❞❡✜♥✐❞♦ ❞❡ ♠❛♥❡✐r❛ ❡①♣❧í❝✐t❛ ♣♦r
∂Ω ={x∈R2;|x|= 1}. ✭✷✳✶✮
◆♦t❡ q✉❡✱ ♥✉♠ ❝❛s♦ ♠❛✐s ❣❡r❛❧ ❡♠ q✉❡ ❛ ✐♥t❡r❢❛❝❡ s❡❥❛ ✉♠❛ ❝✉r✈❛ ♠❛✐s ❝♦♠♣❧❡①❛✱ ♥ã♦ é ♣♦ssí✈❡❧ ❡s❝r❡✈❡r ∂Ω ♥✉♠❛ ❢♦r♠❛ ❛♥❛❧ít✐❝❛✱ ❡ ♣r❡❝✐s❛♠♦s ❛r♠❛③❡♥❛r t♦❞♦s ♦s ♣♦♥t♦s ❞❛ ✐♥t❡r❢❛❝❡ ❞✐r❡t❛♠❡♥t❡✳ ❊♥tr❡t❛♥t♦✱ ❝♦♠♦ ❡❧❛ é ✉♠❛ ❝✉r✈❛ q✉❡ ♣♦ss✉✐ ✐♥✜♥✐t♦s ♣♦♥t♦s✱ ❡❧❡s ♥ã♦ ♣♦❞❡♠ s❡r t♦❞♦s ❛r♠❛③❡♥❛❞♦s ♥✉♠ ❝♦♠♣✉t❛❞♦r✱ ❞❡ ♠♦❞♦ q✉❡ é ♣r❡❝✐s♦ ✉t✐❧✐③❛r ✉♠❛ ❞✐s❝r❡t✐③❛çã♦✳
Pr✐♠❡✐r❛♠❡♥t❡✱ ❞❡✈❡♠♦s ♣❛r❛♠❡tr✐③❛r ❛ ❝✉r✈❛ ❝♦♠♦ ✉♠❛ ❢✉♥çã♦ ✈❡t♦r✐❛❧ φ(s)✱ ♥✉♠ ✐♥t❡r✈❛❧♦ ❢❡❝❤❛❞♦ I = [s0, sf]✱ ♦♥❞❡ φ(si) ✭0≤ i≤ f✮ ❝♦rr❡s♣♦♥❞❡♠ ❛♦s ♣♦♥t♦s ❞❛ ❝✉r✈❛
q✉❡ r❡♣r❡s❡♥t❛ ❛ ✐♥t❡r❢❛❝❡✳ ❖❜s❡r✈❡ q✉❡✱ ❝♦♠♦ ❛ ❝✉r✈❛ é ❢❡❝❤❛❞❛✱ φ(s0) = φ(sf)✳ ❆
s❡❣✉✐r✱ ❞✐s❝r❡t✐③❛♠♦sI ❡♠ ✉♠ ❝♦♥❥✉♥t♦ ✜♥✐t♦ ❞❡ ♣♦♥t♦ss0 < s1 < . . . < sf✱ ❝♦♠ ♦s s✉❜✐♥✲
t❡r✈❛❧♦s [si, si+1] ♥ã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ❞❡ ♠❡s♠♦ ❝♦♠♣r✐♠❡♥t♦✳ ❊♥tã♦✱ ♣❛r❛ ❝❛❞❛ ♣♦♥t♦
si✱ ❛r♠❛③❡♥❛♠♦s ♦ ♣♦♥t♦ ❞❛ ❝✉r✈❛ r❡♣r❡s❡♥t❛❞♦ ♣♦r φ(si)✱ ♣❛r❛ ♦❜t❡r ✉♠❛ r❡♣r❡s❡♥t❛çã♦
❡①♣❧í❝✐t❛ ❞✐s❝r❡t✐③❛❞❛ ❞❛ ✐♥t❡r❢❛❝❡✳
❆ r❡♣r❡s❡♥t❛çã♦ ✐♠♣❧í❝✐t❛ ❞❛ ✐♥t❡r❢❛❝❡ ❞❡s❝r✐t❛ ❡♠ ✭2.1✮ ♣♦❞❡ s❡r ❢❡✐t❛ ❛tr❛✈és ❞♦
✐s♦❝♦♥t♦r♥♦ ♥✉❧♦ ❞❛ ❢✉♥çã♦φ(x) = |x| −1✳ ◆♦t❡ q✉❡✱ s❡φ(x) = 0✱x♣❡rt❡♥❝❡ à ✐♥t❡r❢❛❝❡
∂Ω❞❡s❝r✐t❛ ❞❡ ❢♦r♠❛ ❡①♣❧í❝✐t❛ ❡♠ ✭2.1✮✳ ❈♦♥t✉❞♦✱ ❛ss✐♠ ❝♦♠♦ ♥♦ ❝❛s♦ ❡①♣❧í❝✐t♦✱ ♣♦❞❡ s❡r ❜❛st❛♥t❡ ❞✐❢í❝✐❧ ♦❜t❡r ✉♠❛ ❢✉♥çã♦ ❛♥❛❧ít✐❝❛ φ q✉❡ ❞❡✜♥❛ ✐♠♣❧✐❝✐t❛♠❡♥t❡ ✉♠❛ ❝✉r✈❛ ♠❛✐s
❝♦♠♣❧❡①❛✳
P❛r❛ ✐ss♦✱ ❛ s♦❧✉çã♦ é s❡♠❡❧❤❛♥t❡ à ❞♦ ❝❛s♦ ❡①♣❧í❝✐t♦✿ ❞❡✈❡♠♦s ❞✐s❝r❡t✐③❛r ♦ ❞♦♠í♥✐♦ ❞❡
φ✱ q✉❡ é✱ ♥♦ ❝❛s♦✱ ♦ ❝♦♥❥✉♥t♦ R2✳ ❈♦♥t✉❞♦✱ ❝♦♠♦ R2 é ✐❧✐♠✐t❛❞♦✱ é ♥❡❝❡ssár✐♦ r❡str✐♥❣✐r ❛
❞✐s❝r❡t✐③❛çã♦ ❛ ✉♠ s✉❜❞♦♠í♥✐♦ D⊂R2✳ ❆ s❡❣✉✐r✱ ❡s❝♦❧❤❡♠♦s ♣♦♥t♦sxi = (xi, yi)∈D✱ ❡
❛r♠❛③❡♥❛♠♦s ♦s ✈❛❧♦r❡s ❞❡φ(xi)✳ ◆♦t❡ q✉❡✱ ❛♣❡s❛r ❞❡φ♥ã♦ s❡r ❝♦♥❤❡❝✐❞❛ ❛♥❛❧✐t✐❝❛♠❡♥t❡✱
✷✳ ❈♦♥❝❡✐t♦s ❇ás✐❝♦s ❡♠ ▼ét♦❞♦s ▲❡✈❡❧✲❙❡t ✷✸
x xc y
∂Ω
❋✐❣✉r❛ ✷✳✶✿ ❖ ❝♦♥t♦r♥♦ ❞❛ ❡str❡❧❛ ✈❡r♠❡❧❤❛ r❡♣r❡s❡♥t❛ ❛ ✐♥t❡r❢❛❝❡∂Ω✱ ♦ ❝ír❝✉❧♦ ❛❧❛r❛♥❥❛❞♦
♣♦ss✉✐ r❛✐♦|x−xc|❡ ♦ ❝ír❝✉❧♦ ❡♠ ❛♠❛r❡❧♦ ♣♦ss✉✐ r❛✐♦|y−xc|✳ ❙❡xc é ♦ ♣♦♥t♦ ❞❛ ✐♥t❡r❢❛❝❡
♠❛✐s ♣ró①✐♠♦ ❞❡ x✱ ❡❧❡ t❛♠❜é♠ é ♦ ♣♦♥t♦ ❞❛ ✐♥t❡r❢❛❝❡ ♠❛✐s ♣ró①✐♠♦ ❞❡ q✉❛❧q✉❡r ♣♦♥t♦
q✉❡ ♣❡rt❡♥ç❛ ❛♦ s❡❣♠❡♥t♦ ❞❡ r❡t❛ q✉❡ ✈❛✐ ❞❡x ❛ xc✳
◆❡♠ ❛ ❛❜♦r❞❛❣❡♠ ❡①♣❧í❝✐t❛ ♥❡♠ ❛ ✐♠♣❧í❝✐t❛ ♥♦s ❞✐③❡♠ ❡①❛t❛♠❡♥t❡ ♦♥❞❡ ❛ ✐♥t❡r❢❛❝❡ ❡stá ❧♦❝❛❧✐③❛❞❛❀ ❛♦ ✐♥✈és ❞✐ss♦✱ ❡❧❛s ❝♦♥tê♠ ❛♣❡♥❛s ❛s ✐♥❢♦r♠❛çõ❡s ❞❡ ❛❧❣✉♥s ♣♦♥t♦s ❞❡❧❛✳ P❛r❛ ❝♦♥str✉✐r ❛ ✐♥t❡r❢❛❝❡ ❝♦♠♣❧❡t❛✱ é ♥❡❝❡ssár✐♦ ❢❛③❡r ❛♣r♦①✐♠❛çõ❡s ♥✉♠ér✐❝❛s✱ ❝♦♠♦ ✐♥t❡r♣♦❧❛çõ❡s✳ ❊♠ [✷✻]✱ é ❞✐t♦ q✉❡ ♦ ♣r♦❝❡❞✐♠❡♥t♦ ♣❛❞rã♦ ♣❛r❛ ❡ss❛ r❡❝♦♥str✉çã♦ ❝♦♠ ❛ ✐♥t❡r❢❛❝❡ r❡♣r❡s❡♥t❛❞❛ ❞❡ ❢♦r♠❛ ❡①♣❧í❝✐t❛ é ♦ ✉s♦ ❞❡ s♣❧✐♥❡s✳ ❏á ♥❛ r❡♣r❡s❡♥t❛çã♦ ✐♠♣❧í❝✐t❛✱ ❛ s✐t✉❛çã♦ é ❛✐♥❞❛ ♠❛✐s ❝♦♠♣❧✐❝❛❞❛✱ ♣♦rq✉❡✱ ❛ ♠❡♥♦s q✉❡ ❛ ❞✐s❝r❡t✐③❛çã♦ s❡❥❛ ❡s❝♦❧❤✐❞❛ ❞❡ ♠❛♥❡✐r❛ ❡①tr❡♠❛♠❡♥t❡ ♣r❡❝✐s❛✱ ♣♦❞❡ ♥ã♦ ❤❛✈❡r ♥❡♥❤✉♠ ♣♦♥t♦x∈Dt❛❧ q✉❡ φ(x) = 0 ♣❛r❛ q✉❡ ❛ ✐♥t❡r♣♦❧❛çã♦ ♣♦ss❛ s❡r ❢❡✐t❛✳ ◆❡ss❡ ❝❛s♦✱ ❛ ✐♥t❡r♣♦❧❛çã♦ ❞❡✈❡ s❡r ❛❞❛♣t❛❞❛ ♣❛r❛ q✉❡ ♣♦ss❛ s❡r ❢❡✐t❛ ❛♣❡♥❛s ❝♦♠ ♦s ✈❛❧♦r❡s ❝♦♥❤❡❝✐❞♦s ❞❡φ✳
✷✳✷ ❋✉♥çã♦ ❉✐stâ♥❝✐❛
❉❛❞♦ ✉♠ ❞♦♠í♥✐♦ ❞❡ ✐♥t❡r❡ss❡Ωs❡♣❛r❛❞♦ ❡♠ ❞✉❛s r❡❣✐õ❡s ♣♦r ✉♠❛ ✐♥t❡r❢❛❝❡ ∂Ω✱ ✉♠❛
❢✉♥çã♦ ❞✐stâ♥❝✐❛ d: Ω→R é ❞❡✜♥✐❞❛ ❝♦♠♦
d(x) = min xc∈∂Ω{|
x−xc|}. ✭✷✳✷✮
❖❜s❡r✈❡ q✉❡d s❡ ❛♥✉❧❛ ♣❛r❛ t♦❞♦s ♦s ♣♦♥t♦s ❞❛ ✐♥t❡r❢❛❝❡✳ ❆❧é♠ ❞✐ss♦✱ ♥♦t❡ q✉❡✱ s❡xc
é ♦ ♣♦♥t♦ ❞❛ ✐♥t❡r❢❛❝❡ ♠❛✐s ♣ró①✐♠♦ ❞❡ ✉♠ ❝❡rt♦ ♣♦♥t♦ x✱ ❡♥tã♦ xc t❛♠❜é♠ é ♦ ♣♦♥t♦
❞❛ ✐♥t❡r❢❛❝❡ ♠❛✐s ♣ró①✐♠♦ ❞❡ t♦❞♦s ♦s ♣♦♥t♦s ❞♦ s❡❣♠❡♥t♦ ❞❡ r❡t❛ q✉❡ ❝♦♥❡❝t❛ x ❛ xc✳
■ss♦ ♣♦❞❡ s❡r ✈✐st♦ ❛tr❛✈és ❞❛ ❋✐❣✉r❛ 2.1✳ ❈♦♠♦ xc é ♦ ♣♦♥t♦ ❞❛ ✐♥t❡r❢❛❝❡ ♠❛✐s ♣ró①✐♠♦
❛x✱ ♣♦❞❡♠♦s ❞❡s❡♥❤❛r ✉♠ ❝ír❝✉❧♦ ❝❡♥tr❛❞♦ ❡♠ x ❝✉❥❛ ú♥✐❝❛ ✐♥t❡rs❡❝çã♦ ❝♦♠ ❛ ✐♥t❡r❢❛❝❡
s❡❥❛ ❡①❛t❛♠❡♥t❡ ❡♠ xc✳ ❙❡❥❛ ❛❣♦r❛ y ✉♠ ♣♦♥t♦ ❞♦ s❡❣♠❡♥t♦ ❞❡ r❡t❛ q✉❡ ♣❛ss❡ ♣♦r x ❡
xc✱ ❡ ❞❡s❡♥❤❡♠♦s ✉♠ ♥♦✈♦ ❝ír❝✉❧♦ ❝❡♥tr❛❞♦ ❡♠y q✉❡ ✐♥t❡rs❡❝t❡ ❛ ✐♥t❡r❢❛❝❡ ❡♠ xc✳ ❊ss❡
❝ír❝✉❧♦ ♠❡♥♦r é ✐♥t❡r♥♦ ❛♦ ❝ír❝✉❧♦ ♠❛✐♦r✱ ❞❡ ♠♦❞♦ q✉❡✱ s❡ ❤♦✉✈❡r ✉♠ s❡❣✉♥❞♦ ♣♦♥t♦ ❞❛ ✐♥t❡r❢❛❝❡ q✉❡ ♣❡rt❡♥ç❛ ❛♦ ❝ír❝✉❧♦ ♠❡♥♦r✱ ❡❧❡ t❛♠❜é♠ ♣❡rt❡♥❝❡rá ❛♦ ❝ír❝✉❧♦ ♠❛✐♦r✱ ♦ q✉❡ ♥ã♦ é ♣♦ssí✈❡❧ s❡xc ❢♦r ♦ ♣♦♥t♦ ❞❛ ✐♥t❡r❢❛❝❡ ♠❛✐s ♣ró①✐♠♦ ❞❡x✳ P♦rt❛♥t♦✱ ♥ã♦ ♣♦❞❡ ❤❛✈❡r
♣♦♥t♦ ❞❛ ✐♥t❡r❢❛❝❡ ♠❛✐s ♣ró①✐♠♦ ❞❡ y ❞♦ q✉❡ xc✳
❯♠❛ ♣r♦♣r✐❡❞❛❞❡ út✐❧ ❡ s✐♠♣❧❡s ❞❡ s❡r ✈❡r✐✜❝❛❞❛ é q✉❡✱ ♣❛r❛ t♦❞❛ ❢✉♥çã♦ ❞✐stâ♥❝✐❛ d✱
✈❛❧❡
