Curvaturas médias anisotrópicas : estabilidade e resultados para hipersuperfícies nãoconvexas

❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❖ ❈❊❆❘➪

❈❊◆❚❘❖ ❉❊ ❈■✃◆❈■❆❙

PÓ❙✲●❘❆❉❯❆➬➹❖ ❊▼ ▼❆❚❊▼➪❚■❈❆

❏♦♥❛t❛♥ ❋❧♦r✐❛♥♦ ❞❛ ❙✐❧✈❛

❈✉r✈❛t✉r❛s ▼é❞✐❛s ❆♥✐s♦tró♣✐❝❛s✿ ❊st❛❜✐❧✐❞❛❞❡ ❡ ❘❡s✉❧t❛❞♦s

♣❛r❛ ❍✐♣❡rs✉♣❡r❢í❝✐❡s ◆ã♦✲❈♦♥✈❡①❛s

❏♦♥❛t❛♥ ❋❧♦r✐❛♥♦ ❞❛ ❙✐❧✈❛

❈✉r✈❛t✉r❛s ▼é❞✐❛s ❆♥✐s♦tró♣✐❝❛s✿ ❊st❛❜✐❧✐❞❛❞❡ ❡ ❘❡s✉❧t❛❞♦s

♣❛r❛ ❍✐♣❡rs✉♣❡r❢í❝✐❡s ◆ã♦✲❈♦♥✈❡①❛s

❚❡s❡ s✉❜♠❡t✐❞❛ à ❈♦♦r❞❡♥❛çã♦ ❞♦ ❈✉rs♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛✱ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❈❡❛rá✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ❉♦✉t♦r ❡♠ ▼❛t❡♠át✐❝❛✳

➪r❡❛ ❞❡ ❝♦♥❝❡♥tr❛çã♦✿ ▼❛t❡♠át✐❝❛✳

❖r✐❡♥t❛❞♦r✿

Pr♦❢✳ ❉r✳ ❆♥t♦♥✐♦ ●❡r✈ás✐♦ ❈♦❧❛r❡s✳

❙✐❧✈❛✱ ❏♦♥❛t❛♥ ❋❧♦r✐❛♥♦ ❞❛

❙✺✽❝ ❈✉r✈❛t✉r❛s ♠é❞✐❛s ❛♥✐s♦tró♣✐❝❛s✿ ❡st❛❜✐❧✐❞❛❞❡ ❡ r❡s✉❧t❛❞♦s ♣❛r❛ ❤✐♣❡rs✉♣❡r❢í❝✐❡s ♥ã♦✲❝♦♥✈❡①❛s ✴ ❏♦♥❛t❛♥ ❋❧♦r✐❛♥♦ ❞❛ ❙✐❧✈❛✳ ✕ ❋♦rt❛❧❡③❛ ✿ 2011✳

75❢✳

❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❆♥t♦♥✐♦ ●❡r✈ás✐♦ ❈♦❧❛r❡s ➪r❡❛ ❞❡ ❝♦♥❝❡♥tr❛çã♦✿ ▼❛t❡♠át✐❝❛

❚❡s❡ ✭❉♦✉t♦r❛❞♦✮ ✲ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❈❡❛rá✱ ❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s✱ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛✱ ❋♦rt❛❧❡③❛✱2011✳

✶✳●❡♦♠❡tr✐❛ ❞✐❢❡r❡♥❝✐❛❧✳ ■✳ ❈♦❧❛r❡s✱ ❆♥t♦♥✐♦ ●❡r✈ás✐♦ ✭❖r✐❡♥t✳✮

❆●❘❆❉❊❈■▼❊◆❚❖❙

❊♠ ♣r✐♠❡✐r♦ ❧✉❣❛r ❛ ❉❡✉s✱ ♣♦r s✉❛ ✐♥✜♥✐t❛ ❣r❛ç❛ ❡ ♠✐s❡r✐❝ór❞✐❛❀

❆♦s ♠❡✉s ♣❛✐s ❆❢♦♥s♦ ❳❛✈✐❡r ❡ ▼❛r✐❛ ■♥ê③✱ ♣♦r t♦❞❛ ❛ss✐stê♥❝✐❛ ❡ ♦r❛çõ❡s ✭s❡♠ ❡❧❡s ♥ã♦ ❡st❛r✐❛ ❛q✉✐✮❀

➚ ♠✐♥❤❛ q✉❡r✐❞❛ ❡ ❛♠❛❞❛ ❡s♣♦s❛ ❙t❡❢❛♥✐❡ ❈❛✈❛❧❝❛♥t✐✱ ♣❡❧♦ ❛♠♦r✱ ❝❛r✐♥❤♦✱ ❝♦♠♣❛♥❤✐❛ ❡ ❝♦♠♣r❡❡♥sã♦❀ ❡ ❛ s✉❛ ❢❛♠í❧✐❛ ♣❡❧❛ ❝♦♥s✐❞❡r❛çã♦ ❞❡♠♦♥str❛❞❛ ❡ ❛♦s ♥♦ss♦s ❛♠✐❣♦s❀

❆♦ ♣r♦❢❡ss♦r ●❡r✈ás✐♦ ❈♦❧❛r❡s✱ ♣♦r t♦❞♦ ❛♣♦✐♦✱ ✐♥❝❡♥t✐✈♦ ❡ ♦r✐❡♥t❛çã♦ ❝♦♠♣❡t❡♥t❡✱ ❛❧é♠ ❞❛ ❡s❝♦❧❤❛ ❞❡st❡ ❜❡❧♦ t❡♠❛ ♦ q✉❛❧ t✐✈❡ ♦ ♣r✐✈✐❧é❣✐♦ ❞❡ ❡st✉❞❛r❀

❆❣r❛❞❡ç♦ ❛♦s ♣r♦❢❡ss♦r❡s ❏♦r❣❡ ❍❡r❜❡rt✱ ❆❜❞ê♥❛❣♦ ❆❧✈❡s✱ P❛♦❧♦ P✐❝❝✐♦♥❡ ❡ ❖s❝❛r P❛❧♠❛s ♣♦r t❡r❡♠ ❛❝❡✐t❛❞♦ ♦ ❝♦♥✈✐t❡ ❞❡ ♣❛rt✐❝✐♣❛r ❞❛ ❜❛♥❝❛ ❡①❛♠✐♥❛❞♦r❛ ❡ ♣❡❧❛s ❝♦♥✲ tr✐❜✉✐çõ❡s ❞❛❞❛s ❛ ❡st❡ tr❛❜❛❧❤♦ ❛tr❛✈és ❞❡ s✉❣❡stõ❡s ❡ ❝♦rr❡çõ❡s✳

❆♦s ♠❡✉s ❝♦❧❡❣❛s ❡ ❡①✲❝♦❧❡❣❛s ❞❡ ♣ós✲❣r❛❞✉❛çã♦ ❋❧á✈✐♦ ❋r❛♥ç❛✱ ❈í❝❡r♦ ❞❡ ❆q✉✐♥♦✱ ❯❧✐ss❡s P❛r❡♥t❡✱ ▼❛r❝♦ ❱❡❧ásq✉❡③✱ ❏♦❜s♦♥ ❞❡ ◗✉❡✐r♦③✱ ❚✐❛❣♦ ❈❛ú❧❛✱ ❉❛♠✐ã♦ ❏ú♥✐♦✱ ▼✐❝❤❡❧ P✐♥❤♦✱ ▲✉✐③ ❋❡r♥❛♥❞♦✱ ❱❛❧❜❡r ▼❛r❝✐♦ ❡ ❛ t♦❞♦s q✉❡ ❝♦♥tr✐❜✉✐r❛♠ ❞✐r❡t❛ ❡ ✐♥❞✐r❡t❛♠❡♥t❡ ❛té ❛q✉✐❀

❆♦s s❡❝r❡tár✐♦s ❞❛ ♣ós✲❣r❛❞✉❛çã♦ ❆♥❞ré❛ ❉❛♥t❛s✱ ❆❞r✐❛♥♦ ◆❡✈❡s ❡ ❈❛t❛r✐♥❛ ●♦♠❡s✱ ♣❡❧❛ s✐♠♣❛t✐❛ ❡ ❛❥✉❞❛ ❡♠ r❡s♦❧✈❡r ❛ss✉♥t♦s ❞❡ ♥❛t✉r❡③❛ ❛❞♠✐♥✐str❛t✐✈❛ ❡ ❛♦s ❜✐❜❧✐♦t❡❝ár✐♦s ❘♦❝✐❧❞❛ ❈❛✈❛❧❝❛♥t❡✱ ❋❡r♥❛♥❞❛ ❋r❡✐t❛s ❡ ❊r✐✈❛♥ ❈❛r♥❡✐r♦✱ ♣❡❧♦ ❡✜❝✐❡♥t❡ ❞❡s❡♠♣❡♥❤♦ ❞❡ s✉❛s ❛t✐✈✐❞❛❞❡s❀

❉♦ ❙❊◆❍❖❘ é ❛ t❡rr❛ ❡ ❛ s✉❛ ♣❧❡♥✐t✉❞❡✱ ♦ ♠✉♥❞♦ ❡ ❛q✉❡❧❡s q✉❡ ♥❡❧❡ ❤❛❜✐t❛♠✳

P♦rq✉❡ ❡❧❡ ❛ ❢✉♥❞♦✉ s♦❜r❡ ♦s ♠❛r❡s ❡ ❛ ✜r♠♦✉ s♦❜r❡ ♦s r✐♦s✳

◗✉❡♠ s✉❜✐rá ❛♦ ♠♦♥t❡ ❞♦ ❙❊◆❍❖❘ ♦✉ q✉❡♠ ❡st❛rá ♥♦ s❡✉ ❧✉❣❛r s❛♥t♦❄

❆q✉❡❧❡ q✉❡ é ❧✐♠♣♦ ❞❡ ♠ã♦s ❡ ♣✉r♦ ❞❡ ❝♦r❛çã♦✱ q✉❡ ♥ã♦ ❡♥tr❡❣❛ ❛ s✉❛ ❛❧♠❛ à ✈❛✐❞❛❞❡✱ ♥❡♠ ❥✉r❛ ❡♥❣❛♥♦s❛♠❡♥t❡✳

❊st❡ r❡❝❡❜❡rá ❛ ❜ê♥çã♦ ❞♦ ❙❊◆❍❖❘ ❡ ❛ ❥✉st✐ç❛ ❞♦ ❉❡✉s ❞❛ s✉❛ s❛❧✈❛çã♦✳

❊st❛ é ❛ ❣❡r❛çã♦ ❞❛q✉❡❧❡s q✉❡ ❜✉s❝❛♠✱ ❞❛q✉❡❧❡s q✉❡ ❜✉s❝❛♠ ❛ t✉❛ ❢❛❝❡✱ ó ❉❡✉s ❞❡ ❏❛❝ó✳

▲❡✈❛♥t❛✐✱ ó ♣♦rt❛s✱ ❛s ✈♦ss❛s ❝❛❜❡ç❛s❀ ❧❡✈❛♥t❛✐✲✈♦s✱ ó ❡♥tr❛❞❛s ❡t❡r♥❛s✱ ❡ ❡♥tr❛rá ♦ ❘❡✐ ❞❛ ●❧ór✐❛✳

◗✉❡♠ é ❡st❡ ❘❡✐ ❞❛ ●❧ór✐❛❄ ❖ ❙❊◆❍❖❘ ❢♦rt❡ ❡ ♣♦❞❡r♦s♦✱ ♦ ❙❊◆❍❖❘ ♣♦❞❡r♦s♦ ♥❛ ❣✉❡rr❛✳

▲❡✈❛♥t❛✐✱ ó ♣♦rt❛s✱ ❛s ✈♦ss❛s ❝❛❜❡ç❛s❀ ❧❡✈❛♥t❛✐✲✈♦s✱ ó ❡♥tr❛❞❛s ❡t❡r♥❛s✱ ❡ ❡♥tr❛rá ♦ ❘❡✐ ❞❛ ●❧ór✐❛✳

◗✉❡♠ é ❡st❡ ❘❡✐ ❞❛ ●❧ór✐❛❄ ❖ ❙❊◆❍❖❘ ❞♦s ❊①ér❝✐t♦s❀ ❡❧❡ é ♦ ❘❡✐ ❞❛ ●❧ór✐❛✳

❘❊❙❯▼❖

❊st❡ tr❛❜❛❧❤♦ ❝♦♥s✐st❡ ❡♠ ❞✉❛s ♣❛rt❡s✳

◆❛ ♣r✐♠❡✐r❛ ♣❛rt❡✱ ❡st✉❞❛r❡♠♦s ❤✐♣❡rs✉♣❡r❢í❝✐❡s ❝♦♠♣❛❝t❛s s❡♠ ❜♦r❞♦ ✐♠❡rs❛s ♥♦ ❡s✲ ♣❛ç♦ ❊✉❝❧✐❞✐❛♥♦ ❝♦♠ ♦ q✉♦❝✐❡♥t❡ ❞❛s ❝✉r✈❛t✉r❛s ♠é❞✐❛s ❛♥✐s♦tró♣✐❝❛s (r+1)Cnr+1HrF+1

a(k+1)Cnk+1H_kF₊₁−b ❂ ❝♦♥st❛♥t❡✳ Pr♦✈❛r❡♠♦s q✉❡ t❛✐s ❤✐♣❡rs✉♣❡r❢í❝✐❡s sã♦ ♣♦♥t♦s ❝rít✐❝♦s ♣❛r❛ ✉♠ ♣r♦❜❧❡♠❛ ✈❛r✐❛❝✐♦♥❛❧ ❞❡ ♣r❡s❡r✈❛r ✉♠❛ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞❛(k, F)✲ár❡❛ ❡ ❞♦(n+ 1)✲✈♦❧✉♠❡ ❞❡t❡r✲

♠✐♥❛❞♦ ♣♦r M✳ ❉❡♠♦str❛r❡♠♦s q✉❡ ❛ ❤✐♣❡rs✉♣❡r❢í❝✐❡ é (r, k, a, b)✲❡stá✈❡❧ s❡✱ ❡ s♦♠❡♥t❡

s❡✱ ❛ ♠❡♥♦s ❞❡ tr❛♥s❧❛çã♦ ❡ ❤♦♠♦t❡t✐❛✱ ❡❧❛ é ❛ ❲✉❧✛ s❤❛♣❡ ❞❡ F ✭✈❡❥❛ ❙❡çã♦ ✷✳✶✮✱ s♦❜

❛❧❣✉♠❛s ❝♦♥❞✐çõ❡s ❛❝❡r❝❛ ❞❡ a, b∈R✳

◆❛ s❡❣✉♥❞❛ ♣❛rt❡ ❞❡ss❡ tr❛❜❛❧❤♦✱ ♦❜t❡♠♦s ♦✉tr❛s ❝❛r❛❝t❡r✐③❛çõ❡s ♣❛r❛ ❛ ❲✉❧✛ s❤❛♣❡ ❡♥✈♦❧✈❡♥❞♦ ❛s ❝✉r✈❛t✉r❛s ♠é❞✐❛s ❛♥✐s♦tró♣✐❝❛s ❞❡ ♦r❞❡♠ s✉♣❡r✐♦r ❞❡ ✉♠❛ ❤✐♣❡rs✉♣❡r❢í✲ ❝✐❡ M ❡♠ Rn+1 _{❡ ♦ ❝♦♥❥✉♥t♦ W = R}n+1 ₋S

❆❇❙❚❘❆❈❚

❚❤✐s ✇♦r❦ ❝♦♥s✐sts ♦❢ t✇♦ ♣❛rts✳

■♥ t❤❡ ✜rst ♣❛rt ✇❡ ❞❡❛❧ ✇✐t❤ ❛ ❝♦♠♣❛❝t ❤②♣❡rs✉r❢❛❝❡ ✇✐t❤♦✉t ❜♦✉♥❞❛r② ✐♠♠❡rs❡❞ ✐♥ t♦ t❤❡ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡ ✇✐t❤ t❤❡ q✉♦t✐❡♥t ♦❢ ❛♥✐s♦tr♦♣✐❝ ♠❡❛♥ ❝✉r✈❛t✉r❡s (r+1)Cnr+1HrF+1

a(k+1)Cnk+1H_kF₊₁−b =

constant✳ ❙✉❝❤ ❛ ❤②♣❡rs✉r❢❛❝❡ ✐s ❛ ❝r✐t✐❝❛❧ ♣♦✐♥t ❢♦r t❤❡ ✈❛r✐❛t✐♦♥❛❧ ♣r♦❜❧❡♠ ♣r❡s❡r✈✐♥❣ ❛

❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡(k, F)✲❛r❡❛ ❛♥❞ (n+ 1)✲✈♦❧✉♠❡ ❡♥❝❧♦s❡❞ ❜②M✳ ❲❡ s❤♦✇ t❤❛t

✐t ✐s (r, k, a, b)✲st❛❜❧❡ ✐❢✱ ❛♥❞ ♦♥❧② ✐❢✱ ✉♣ t♦ tr❛♥s❧❛t✐♦♥s ❛♥❞ ❤♦♠♦t❤❡t✐❡s✱ ✐t ✐s t❤❡ ❲✉❧✛

s❤❛♣❡✱ ✉♥❞❡r s♦♠❡ ❛ss✉♠♣t✐♦♥s ♦♥ a, b∈R✳

■♥ t❤❡ s❡❝♦♥❞ ♣❛rt ✇❡ ♦❜t❛✐♥ ❢✉rt❤❡r ❝❤❛r❛❝t❡r✐③❛t✐♦♥s ❢♦r t❤❡ ❲✉❧✛ s❤❛♣❡ ✐♥✈♦❧✈✐♥❣ t❤❡ ❛♥✐s♦tr♦♣✐❝ ♠❡❛♥ ❝✉r✈❛t✉r❡s ♦❢ ❤✐❣❤❡r ♦r❞❡r ♦❢ ❛ ❤②♣❡rs✉r❢❛❝❡ M ✐♥ Rn+1 _{❛♥❞ t❤❡}

s❡t W = Rn+1 ₋S

❙✉♠ár✐♦

✶ ■♥tr♦❞✉çã♦ ✽

✷ Pr❡❧✐♠✐♥❛r❡s ✶✻

✷✳✶ ❆ ❲✉❧✛ ❙❤❛♣❡ ❞❡ F ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻

✷✳✷ ❖s ❖♣❡r❛❞♦r❡s Pr✱ Tr ❡Lr ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶

✸ ❊st❛❜✐❧✐❞❛❞❡ ❞❡ ❍✐♣❡rs✉♣❡r❢í❝✐❡s ❝♦♠ ◗✉♦❝✐❡♥t❡ ❈♦♥st❛♥t❡ ❞❛s ❈✉r✈❛✲ t✉r❛s ▼é❞✐❛s ❆♥✐s♦tró♣✐❝❛s Pr❡s❡r✈❛♥❞♦ ✉♠❛ ❈♦♠❜✐♥❛çã♦ ❞❛ ➪r❡❛ ❡

❞♦ ❱♦❧✉♠❡ ✷✻

✸✳✶ ❖ Pr♦❜❧❡♠❛ ❱❛r✐❛❝✐♦♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✸✳✷ (r, k, a, b)✲❊st❛❜✐❧✐❞❛❞❡ ❞❡ ❍✐♣❡rs✉♣❡r❢í❝✐❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶

✹ ❙♦❜r❡ Pr♦♣r✐❡❞❛❞❡s ❞❛s ❈✉r✈❛t✉r❛s ▼é❞✐❛s ❆♥✐s♦tró♣✐❝❛s ❞❡ ❍✐♣❡rs✉✲

♣❡r❢í❝✐❡s ❈♦♠♣❛❝t❛s ◆ã♦✲❈♦♥✈❡①❛s ✹✽

✹✳✶ ❯♠❛ ❋ór♠✉❧❛ ■♥t❡❣r❛❧ ❞♦ ❚✐♣♦ ▼✐♥❦♦✇s❦✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✹✳✷ ❈✉r✈❛t✉r❛s ▼é❞✐❛s ❆♥✐s♦tró♣✐❝❛s ▲✐♥❡❛r♠❡♥t❡ ❘❡❧❛❝✐♦♥❛❞❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✹✳✸ ❆♣❧✐❝❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺

❈❛♣ít✉❧♦ ✶

■♥tr♦❞✉çã♦

❊st❡ tr❛❜❛❧❤♦ ❝♦♥s✐st❡ ❡♠ ❞✉❛s ♣❛rt❡s✳

◆❛ ♣r✐♠❡✐r❛ ♣❛rt❡✱ ❡st✉❞❛r❡♠♦s ❤✐♣❡rs✉♣❡r❢í❝✐❡s ❝♦♠♣❛❝t❛s s❡♠ ❜♦r❞♦ ✐♠❡rs❛s ♥♦ ❡s✲ ♣❛ç♦ ❊✉❝❧✐❞✐❛♥♦ ❝♦♠ ♦ q✉♦❝✐❡♥t❡ ❞❛s ❝✉r✈❛t✉r❛s ♠é❞✐❛s ❛♥✐s♦tró♣✐❝❛s (r+1)Cnr+1HrF+1

a(k+1)Cnk+1HkF+1−b ❂

❝♦♥st❛♥t❡✳ Pr♦✈❛r❡♠♦s q✉❡ t❛✐s ❤✐♣❡rs✉♣❡r❢í❝✐❡s sã♦ ♣♦♥t♦s ❝rít✐❝♦s ♣❛r❛ ✉♠ ♣r♦❜❧❡♠❛ ✈❛r✐❛❝✐♦♥❛❧ ❞❡ ♣r❡s❡r✈❛r ✉♠❛ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞❛(k, F)✲ár❡❛ ❡ ❞♦(n+ 1)✲✈♦❧✉♠❡ ❞❡t❡r✲

♠✐♥❛❞♦ ♣♦r M✳ ❉❡♠♦str❛r❡♠♦s q✉❡ ❛ ❤✐♣❡rs✉♣❡r❢í❝✐❡ é (r, k, a, b)✲❡stá✈❡❧ s❡✱ ❡ s♦♠❡♥t❡

s❡✱ ❛ ♠❡♥♦s ❞❡ tr❛♥s❧❛çã♦ ❡ ❤♦♠♦t❡t✐❛✱ ❡❧❛ é ❛ ❲✉❧✛ s❤❛♣❡ ❞❡ F✱ ♦♥❞❡ F : Sn _{−→ R}+ _é

✉♠❛ ❢✉♥çã♦ ♣♦s✐t✐✈❛ ✭✈❡❥❛ ❙❡çã♦ ✷✳✶✮✱ s♦❜ ❛❧❣✉♠❛s ❝♦♥❞✐çõ❡s ❛❝❡r❝❛ ❞❡ a, b∈R✳

❉❛❞♦ ✉♠❛ ✐♠❡rsã♦ x : M −→ Rn+1 _{❞❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ M ❝♦♠♣❛❝t❛✱ s❡♠ ❜♦r❞♦ ❡}

♦r✐❡♥t❛❞❛✱ ♣❛r❛ ❝❛❞❛ r✱ 0 ≤ r ≤ n✱ ❞❡✜♥✐♠♦s ❛ (r, F)✲❢✉♥çã♦ ár❡❛ A_r,F : (−ǫ, ǫ) −→ R

❛ss♦❝✐❛❞❛ ❛ ✈❛r✐❛çã♦ X :M ×(−ǫ, ǫ)−→Rn+1 _{✭✈❡❥❛ ❙❡çã♦ ✸✳✶✮ ❞❡x ♣♦r}

A_r,F(t) = Z

M

F(Nt)SrdMt.

❖(n+ 1)✲✈♦❧✉♠❡ ❛❧❣é❜r✐❝♦ ❞❡t❡r♠✐♥❛❞♦ ♣♦rM é ❞❛❞♦ ♣♦r

V(t) = 1

n+ 1 Z

M

hXt, NtidMt. ✭✶✳✶✮

❖ ♣r♦❜❧❡♠❛ ✈❛r✐❛❝✐♦♥❛❧ ❛♥✐s♦tró♣✐❝♦ ♣r❡s❡r✈❛♥❞♦ ♦ ✈♦❧✉♠❡ t❡♠ s✐❞♦ ❡st✉❞❛❞♦ ♣♦r ✈ár✐♦s ❛✉t♦r❡s ✭✈❡❥❛ ❬✷✼❪✱ ❬✷✾❪✱ ❬✶✾❪✮✳ P❛r❛ ✉♠ t❛❧ ♣r♦❜❧❡♠❛ ✈❛r✐❛❝✐♦♥❛❧✱ ❡❧❡s ❝❤❛♠❛r❛♠ ✉♠❛ ✐♠❡rsã♦ ❝rít✐❝❛ x ❞♦ ❢✉♥❝✐♦♥❛❧ Ar,F ✭✐st♦ é✱ ✉♠❛ ❤✐♣❡rs✉♣❡r❢í❝✐❡ ❝♦♠ HrF+1 = ❝♦♥st❛♥t❡✮

❡stá✈❡❧ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❛ s❡❣✉♥❞❛ ✈❛r✐❛çã♦ ❞❡Ar,F é ♥ã♦ ♥❡❣❛t✐✈❛ ♣❛r❛ t♦❞❛ ✈❛r✐❛çã♦ ❞❡

x♣r❡s❡r✈❛♥❞♦ ♦ (n+ 1)✲✈♦❧✉♠❡ V✳

❍✐st♦r✐❝❛♠❡♥t❡✱ ❛ r❡❢❡rê♥❝✐❛ ❝❧áss✐❝❛ é ♦ ❛rt✐❣♦ ❞❡ ●✳ ❲✉❧✛ ❬✸✵❪ ♠♦t✐✈❛❞♦ ♣♦r ❛♣❧✐❝❛çõ❡s ❡♠ ❈r✐st❛❧♦❣r❛✜❛✳ Pr♦♣r✐❡❞❛❞❡s ❢ís✐❝❛s ❞❡ ✉♠ s✐♠♣❧❡s ❝r✐st❛❧ ❞✐❢❡r❡♠ ❝♦♠ ❛ ❞✐r❡çã♦✱ ✐st♦ é✱ ❡❧❡s sã♦ ❛♥✐s♦tró♣✐❝♦s✳ ▼❛✐s r❡❝❡♥t❡♠❡♥t❡✱ ❇✳ ❆♥❞r❡✇s ❬✸❪ tr❛t♦✉ ❞❛ ❡✈♦❧✉çã♦ ♣❡❧❛ ❝✉r✈❛t✉r❛ ♠é❞✐❛ ❛♥✐s♦tró♣✐❝❛✱ ✉♠❛ ❢♦r♠✉❧❛çã♦ ❣❡r❛❧ ❞♦s ✢✉①♦s ❝r✐st❛❧✐♥♦s ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ♠♦❞❡❧♦s ❞❡ ❝r❡s❝✐♠❡♥t♦ ❞❡ ❝r✐st❛✐s ❬✻❪✳

❊♠1998✱ P❛❧♠❡r ❡♠ ❬✷✼❪ ✭✈❡❥❛ t❛♠❜é♠ ❲✐♥❦❧♠❛♥♥ ❡♠ ❬✷✾❪✮ ❝♦♥s✐❞❡r♦✉ ♦ ❝❛s♦r= 0

❡ ♣r♦✈♦✉✿

❚❡♦r❡♠❛ ❆ ✭ ❬✷✼❪ ✮ ❙❡❥❛ x:M −→Rn+1 _{✉♠❛ ❤✐♣❡rs✉♣❡r❢í❝✐❡ ❢❡❝❤❛❞❛ ❝♦♠}

HF

1 =❝♦♥st❛♥t❡✳ ❊♥tã♦ x é ❡stá✈❡❧ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❛ ♠❡♥♦s ❞❡ tr❛♥s❧❛çã♦ ❡

❤♦♠♦t❡t✐❛✱ x(M) é ❛ ❲✉❧✛ s❤❛♣❡ ❞❡ F✳

❊♠2008✱ ❍❡ ❡ ▲✐ ❡♠ ❬✶✾❪ ❡st✉❞❛r❛♠ ♦ ❝❛s♦ ❞❡ ✐♠❡rsõ❡s ❞❡ ❤✐♣❡rs✉♣❡r❢í❝✐❡s ❡♠ Rn+1

❝♦♠ ❛(r+ 1)✲és✐♠❛ ❝✉r✈❛t✉r❛ ♠é❞✐❛ ❛♥✐s♦tró♣✐❝❛ ❝♦♥st❛♥t❡ ❡ ♣r♦✈❛r❛♠✿

❚❡♦r❡♠❛ ❇ ✭ ❬✶✾❪ ✮ ❙✉♣♦♥❤❛ 0 ≤ r ≤ n − 1✳ ❙❡❥❛ x : M −→ Rn+1

✉♠❛ ❤✐♣❡rs✉♣❡r❢í❝✐❡ ❢❡❝❤❛❞❛ ❝♦♠ HF

r+1 = ❝♦♥st❛♥t❡✳ ❊♥tã♦✱ x é ❡stá✈❡❧ s❡✱ ❡

s♦♠❡♥t❡ s❡✱ ❛ ♠❡♥♦s ❞❡ tr❛♥s❧❛çã♦ ❡ ❤♦♠♦t❡t✐❛✱ x(M)é ❛ ❲✉❧✛ s❤❛♣❡ ❞❡ F✳

❊♠2008✱ ❍❡ ❡ ▲✐ ❡♠ ❬✶✻❪ ❝♦♥s✐❞❡r❛r❛♠ ♦ ❝❛s♦ ❞❡ ✐♠❡rsõ❡s ❞❡ ❤✐♣❡rs✉♣❡r❢í❝✐❡s ❡♠Rn+1

❝♦♠ ♦ q✉♦❝✐❡♥t❡ ❝♦♥st❛♥t❡ ❡♥tr❡ ❛(r+ 1)✲és✐♠❛ ❝✉r✈❛t✉r❛ ♠é❞✐❛ ❛♥✐s♦tró♣✐❝❛ ❡ ❛r✲és✐♠❛

❝✉r✈❛t✉r❛ ♠é❞✐❛ ❛♥✐s♦tró♣✐❝❛✱ q✉❡ sã♦ ♣♦♥t♦s ❝rít✐❝♦s ♣❛r❛ ✉♠ ♣r♦❜❧❡♠❛ ✈❛r✐❛❝✐♦♥❛❧ ❞❡ ♠✐♥✐♠✐③❛r ♦ ❢✉♥❝✐♦♥❛❧Ar,F✱ ♣r❡s❡r✈❛♥❞♦ ❛(r−1)✲ár❡❛✱A(r−1),F✱ ❞❛ ❤✐♣❡rs✉♣❡r❢í❝✐❡ ❡♠ ✈❡③ ❞♦(n+ 1)✲✈♦❧✉♠❡✳ P❛r❛ t❛❧ ♣r♦❜❧❡♠❛ ✈❛r✐❛❝✐♦♥❛❧✱ ❡❧❡s ❝❤❛♠❛r❛♠ ✉♠❛ ✐♠❡rsã♦ ❝rít✐❝❛ x

❞♦ ❢✉♥❝✐♦♥❛❧Ar,F ✭✐st♦ é✱ ✉♠❛ ❤✐♣❡rs✉♣❡r❢í❝✐❡ ❝♦♠ HF

r+1

HF

r =❝♦♥st❛♥t❡✮ ♣♦♥t♦ ❝rít✐❝♦ ❡stá✈❡❧ ❞❡ Ar,F s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❛ s❡❣✉♥❞❛ ✈❛r✐❛çã♦ ❞❡ Ar,F é ♥ã♦ ♥❡❣❛t✐✈❛ ♣❛r❛ t♦❞❛ ✈❛r✐❛çã♦ ❞❡x ♣r❡s❡r✈❛♥❞♦ ❛ (r−1)✲ár❡❛ ❞❛ ❤✐♣❡rs✉♣❡r❢í❝✐❡✳ ❊❧❡s ♣r♦✈❛r❛♠✿

❚❡♦r❡♠❛ ❈ ✭ ❬✶✻❪ ✮ ❙❡❥❛ M ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ n✲❞✐♠❡♥s✐♦♥❛❧✱

♦r✐❡♥tá✈❡❧✱ ❝♦♥❡①❛ ❡ ❝♦♠♣❛❝t❛ s❡♠ ❜♦r❞♦✳ ❯♠❛ ✐♠❡rsã♦ ✐s♦♠étr✐❝❛x:M −→ Rn+1 _{é ♣♦♥t♦ ❝rít✐❝♦ ❡stá✈❡❧ ❞❡A}

r,F s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❛ ♠❡♥♦s ❞❡ tr❛♥s❧❛çã♦ ❡ ❤♦♠♦t❡t✐❛✱ x(M) é ❛ ❲✉❧✛ s❤❛♣❡ ❞❡ F✱1≤r≤n−1.

❘❡s✉❧t❛❞♦s s✐♠✐❧❛r❡s ❢♦r❛♠ ♣r♦✈❛❞♦s ❡♠ ❬✶✽❪ ♣❡❧♦s ♠❡s♠♦s ❛✉t♦r❡s ♣❛r❛ ❢♦r♠❛s ❡s♣❛✲ ❝✐❛✐s q✉❛♥❞♦ F = 1✳

❈♦♥s✐❞❡r❛♠♦s ♥❡ss❛ ♣r✐♠❡✐r❛ ♣❛rt❡ ❤✐♣❡rs✉♣❡r❢í❝✐❡s ❡♠ Rn+1 _{q✉❡ sã♦ ♣♦♥t♦s ❝rít✐❝♦s}

♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ✈❛r✐❛❝✐♦♥❛❧ ❞❡ ♠✐♥✐♠✐③❛r ♦ ❢✉♥❝✐♦♥❛❧Ar,F ♣r❡s❡r✈❛♥❞♦ ✉♠❛ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞❛ (k, F)✲ár❡❛✱A_k,F✱ ❡ ❞♦ ✈♦❧✉♠❡ V ❞❡M✱ ♦♥❞❡ 0≤k < r ≤n−1✱ ❞❡♥♦t❛❞❛ ♣♦r

Ca,b;k✳ P♦r ✉♠ ❛r❣✉♠❡♥t♦ ♣❛❞rã♦ ❡♥✈♦❧✈❡♥❞♦ ♠✉❧t✐♣❧✐❝❛❞♦r❡s ❞❡ ▲❛❣r❛♥❣❡✱ ✐st♦ s✐❣♥✐✜❝❛

q✉❡ ❡st❛♠♦s ❝♦♥s✐❞❡r❛♥❞♦ ♦s ♣♦♥t♦s ❝rít✐❝♦s ❞♦ ❢✉♥❝✐♦♥❛❧

Jr,k,a,b(t) =Ar,F(t) +λCa,b;k(t),

♦♥❞❡ Ca,b;k(t) = aAk,F(t) + bV(t)✱ λ é ✉♠❛ ❝♦♥st❛♥t❡ ❛ s❡r ❞❡t❡r♠✐♥❛❞❛ ❡ a, b ∈ R sã♦ ❛♠❜♦s ♥ã♦ ♥✉❧♦s✳ ▼♦str❛r❡♠♦s q✉❡ ❛ ❡q✉❛çã♦ ❞❡ ❊✉❧❡r✲▲❛❣r❛♥❣❡ ❞❡ J_r,k,a,b é ✭✈❡❥❛

Pr♦♣♦s✐çã♦ ✸✳✹✮✿

(r+ 1)Sr+1+λ[a(k+ 1)Sk+1−b] = 0.

❆ss✐♠ ♦s ♣♦♥t♦s ❝rít✐❝♦s sã♦ ❡①❛t❛♠❡♥t❡ ❛s ❤✐♣❡rs✉♣❡r❢í❝✐❡s ❝♦♠

(r+ 1)Cr+1

n HrF+1

a(k+ 1)Ck+1

n HkF+1−b

= (r+ 1)Sr+1

a(k+ 1)Sk+1−b

=constante.

■♥tr♦❞✉③✐♠♦s ♦ ❝♦♥❝❡✐t♦ ❞❡ (r, k, a, b)✲❡st❛❜✐❧✐❞❛❞❡ q✉❡ ♣❛r❛ ♦ ❝❛s♦ k =b = 0✱ a = 1

❡ F ≡ 1 ❝♦✐♥❝✐❞❡ ❝♦♠ ♦ ❞❛❞♦ ❡♠ ❬✶✽❪✱ ♣❛r❛ ♦ ❝❛s♦ ❞♦ ❡s♣❛ç♦ ❊✉❝❧✐❞✐❛♥♦✳ P❛r❛ ✉♠

t❛❧ ♣r♦❜❧❡♠❛ ✈❛r✐❛❝✐♦♥❛❧✱ ❝❤❛♠❛♠♦s ✉♠❛ ✐♠❡rsã♦ ❝rít✐❝❛ x ❞♦ ❢✉♥❝✐♦♥❛❧ A_r,F ✭✐st♦ é✱

✉♠❛ ❤✐♣❡rs✉♣❡r❢í❝✐❡ ❝♦♠ (r+1)Cnr+1HrF+1

a(k+1)Ck+1

n HkF+1−b

=❝♦♥st❛♥t❡✮(r, k, a, b)✲❡stá✈❡❧ s❡✱ ❡ s♦♠❡♥t❡ s❡✱

❛ s❡❣✉♥❞❛ ✈❛r✐❛çã♦ ❞❡ Ar,F é ♥ã♦ ♥❡❣❛t✐✈❛ ♣❛r❛ t♦❞❛ ✈❛r✐❛çã♦ ❞❡ x ♣r❡s❡r✈❛♥❞♦ Ca,b;k✳

❚❛♠❜é♠✱ ♣❛r❛a= 0❡b = 1♥♦ss♦ ❝♦♥❝❡✐t♦ ❝♦✐♥❝✐❞❡ ❝♦♠ ♦ ❞❛❞♦ ❡♠ ❬✶✾❪ ♣❛r❛ ❡st❛❜✐❧✐❞❛❞❡✳

❊♠ ♣❛rt✐❝✉❧❛r✱ ♣❛r❛ a = 0, b = 1 ❡ F = 1✱ ♦❜t❡♠♦s ❛ r✲❡st❛❜✐❧✐❞❛❞❡ ❞❡ ❤✐♣❡rs✉♣❡r❢í❝✐❡s

❡♠ Rn+1 _{❝♦♠♦ ❞❡✜♥✐❞♦ ❡♠ ❬✹❪✳}

Pr♦✈❛♠♦s ❡♥tã♦ ♦ s❡❣✉✐♥t❡✳

❚❡♦r❡♠❛ ✸✳✽ ❙❡❥❛ x : M −→ Rn+1 _{✉♠❛ ✐♠❡rsã♦ ❞❡ ✉♠❛ ❤✐♣❡rs✉♣❡r❢í❝✐❡}

s✉❛✈❡✱ ❢❡❝❤❛❞❛ ❡ ♦r✐❡♥t❛❞❛✳ ❙❡ x(M) é✱ ❛ ♠❡♥♦s ❞❡ tr❛♥s❧❛çã♦ ❡ ❤♦♠♦t❡t✐❛✱ ❛

❲✉❧✛ s❤❛♣❡ ❞❡ F✱ ❡♥tã♦ x é (r, k, a, b)✲❡stá✈❡❧ ♣❛r❛ 

 

 

a= 0 ❡ b6= 0;

a6= 0 ❡ b= 0;

a, b6= 0 ❡ a b ∈/

h

k!(n−k−1)!

n! ,

r+1

r−k.

k!(n−k−1)!

n!

.

❖ ❚❡♦r❡♠❛ 3.8❣❡♥❡r❛❧✐③❛ r❡s✉❧t❛❞♦s ❡♠ ❬✷✼❪✱ ❬✶✾❪✱ ❬✶✽❪ ❡ ❬✶❪✳

❖ ♣ró①✐♠♦ r❡s✉❧t❛❞♦ é ✉♠❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞❛ ❲✉❧✛ s❤❛♣❡ ❝♦♠♦ ♣♦♥t♦ ❝rít✐❝♦ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ✈❛r✐❛❝✐♦♥❛❧ ♥♦ ❝❛s♦ ♦♥❞❡b = 0✳

❚❤❡♦r❡♠ ✸✳✶✵ ❙✉♣♦♥❤❛ 0 ≤ k < r ≤ n−1✳ ❙❡❥❛ x : M −→ Rn+1 _✉♠❛

❤✐♣❡rs✉♣❡r❢í❝✐❡ s✉❛✈❡✱ ❢❡❝❤❛❞❛ ❡ ♦r✐❡♥t❛❞❛✳ ❊♥tã♦✱ x é ♣♦♥t♦ ❝rít✐❝♦ ❞♦ ❢✉♥✲

❝✐♦♥❛❧ Fr,k,a,0 s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❛ ♠❡♥♦s ❞❡ tr❛♥s❧❛çã♦ ❞❡ ❤♦♠♦t❡t✐❛✱x(M)é ❛

❲✉❧✛ s❤❛♣❡ ❞❡ F✳

❖ ♣ró①✐♠♦ r❡s✉❧t❛❞♦✱ s♦❜ ❝❡rt❛s ❝♦♥❞✐çõ❡s ❛❝❡r❝❛ ❞❡a, b∈R✱ ❛♣r❡s❡♥t❛ ✉♠❛ r❡❝í♣r♦❝❛

❞♦ ❚❡♦r❡♠❛ ✸✳✽✳

❚❡♦r❡♠❛ ✸✳✶✸ ❙✉♣♦♥❤❛ 0 ≤ k < r ≤ n −1✳ ❙❡❥❛ x : M −→ Rn+1 _✉♠❛

❤✐♣❡rs✉♣❡r❢í❝✐❡ s✉❛✈❡✱ ❢❡❝❤❛❞❛ ❡ ♦r✐❡♥t❛❞❛✳ ❙❡xé(r, k, a, b)✲❡stá✈❡❧ ♣❛r❛a, b∈ R ♥ã♦ ♥✉❧♦s✱ t❛❧ q✉❡λa,b≤0✱ ❡♥tã♦ ❛ ♠❡♥♦s ❞❡ tr❛♥s❧❛çã♦ ❡ ❤♦♠♦t❡t✐❛✱x(M) é ❛ ❲✉❧✛ s❤❛♣❡ ❞❡ F✱ ♦♥❞❡λa,b é ❞❛❞♦ ♣♦r ✭✸✳✶✷✮✳

❋✐♥❛❧♠❡♥t❡✱ ❛♣r❡s❡♥t❛♠♦s ✉♠❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞❛ ❲✉❧✛ s❤❛♣❡ ❞❡F ❝♦♠♦ ♣♦♥t♦ ❝rít✐❝♦

♣❛r❛ ✉♠ ♣r♦❜❧❡♠❛ ✈❛r✐❛❝✐♦♥❛❧ ❞❡ ♠✐♥✐♠✐③❛r ♦ ❢✉♥❝✐♦♥❛❧Ar,F ♣r❡s❡r✈❛♥❞♦ ✉♠❛ ❝❡rt❛ ❝♦♠✲ ❜✐♥❛çã♦ ❧✐♥❡❛r ❞❛ (k, F)✲ár❡❛✱ A_k,F✱ ❡ ❞♦ ✈♦❧✉♠❡ ❞❡ M✱ V✱ ♦♥❞❡ 0 ≤ k < r ≤ n−1✳

◆❡st❡ r❡s✉❧t❛❞♦ ♦❜t❡♠♦s ✉♠❛ ♥♦✈❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞❛ ❲✉❧✛ s❤❛♣❡✱ ❛❧é♠ ❞❡ r❡❝✉♣❡r❛r ♦ ❚❡♦r❡♠❛ A✱ ♦ ❚❡♦r❡♠❛ B ❡ ❣❡♥❡r❛❧✐③❛r ♦ ❚❡♦r❡♠❛ C ♣❛r❛ ♦ q✉♦❝✐❡♥t❡ ❞❛s ❝✉r✈❛t✉r❛s

♠é❞✐❛s ❛♥✐s♦tró♣✐❝❛s✳

❚❡♦r❡♠❛ ✸✳✶✺ ❙✉♣♦♥❤❛ 0 ≤ k < r ≤ n −1✳ ❙❡❥❛ x : M −→ Rn+1 _✉♠❛

❤✐♣❡rs✉♣❡r❢í❝✐❡ s✉❛✈❡✱ ❢❡❝❤❛❞❛ ❡ ♦r✐❡♥t❛❞❛ ♣❛r❛ q✉❛❧ (r+1)Sr+1

a(k+1)Sk+1−b é ✉♠❛ ❝♦♥s✲

t❛♥t❡✱ ♦♥❞❡ a= 0 ♦✉b = 0 ♦✉ a, b∈R ♥ã♦ ♥✉❧♦s t❛❧ q✉❡ λa,b <0✳ ❊♥tã♦✱ x é

(r, k, a, b)✲❡stá✈❡❧ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❛ ♠❡♥♦s ❞❡ tr❛♥s❧❛çã♦ ❡ ❤♦♠♦t❡t✐❛✱ x(M)

é ❛ ❲✉❧✛ s❤❛♣❡ ❞❡ F✳

❖❜s❡r✈❛♠♦s q✉❡ ♣❛r❛k =r =a= 0 ❡b = 1✱ ♦ ❚❡♦r❡♠❛ ✸✳✶✺ ❝♦rr❡s♣♦♥❞❡ ❛♦ ❚❡♦r❡♠❛ 1 ❡♠ ❬✷✼❪✳ P❛r❛ a = 0 ❡ b = 1✱ ♦❜t❡♠♦s ♦ ❚❡♦r❡♠❛ 1.3 ❡♠ ❬✶✾❪✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ s❡

❛❧é♠ ❞❡ a = 0 ❡ b = 1✱ ✜③❡r♠♦s F = 1✱ ♦❜t❡r❡♠♦s ♦ t❡♦r❡♠❛ ❞❡ r✲❡st❛❜✐❧✐❞❛❞❡ ♣❛r❛

❤✐♣❡rs✉♣❡r❢í❝✐❡s ❝♦♠♣❛❝t❛s ❡♠Rn+1 _{♣r♦✈❛❞♦ ♣♦r ❆❧❡♥❝❛r✱ ❉♦ ❈❛r♠♦ ❡ ❘♦s❡♥❜❡r❣ ❡♠ ❬✶❪✳}

❈♦♥❝❧✉í♠♦s ❡♥tã♦ q✉❡ ♥♦ss♦s r❡s✉❧t❛❞♦s ❡ ♠ét♦❞♦s✱ ❛♣r❡s❡♥t❛❞♦s ♥❡ss❛ ♣r✐♠❡✐r❛ ♣❛rt❡✱ ❣❡♥❡r❛❧✐③❛♠ ❡ ✉♥✐✜❝❛♠ ❛❧❣✉♥s t❡♦r❡♠❛s ❡ tr❛t❛♠❡♥t♦s ❞❡ ♣r♦✈❛s s♦❜r❡ ❛ ❡st❛❜✐❧✐❞❛❞❡ ❞❡ ❤✐♣❡rs✉♣❡r❢í❝✐❡s ❝♦♠♣❛❝t❛s ❡♠Rn+1 _{♥♦s ❝❛s♦s ❛♥✐s♦tró♣✐❝♦s ❡ ❡✉❝❧✐❞✐❛♥♦s✳}

◆❛ s❡❣✉♥❞❛ ♣❛rt❡ ❞❡ss❡ tr❛❜❛❧❤♦✱ ♦❜t❡♠♦s ♦✉tr❛s ❝❛r❛❝t❡r✐③❛çõ❡s ♣❛r❛ ❛ ❲✉❧✛ s❤❛♣❡ ❡♥✈♦❧✈❡♥❞♦ ❛s ❝✉r✈❛t✉r❛s ♠é❞✐❛s ❛♥✐s♦tró♣✐❝❛s ❞❡ ♦r❞❡♠ s✉♣❡r✐♦r ❞❡ ✉♠❛ ❤✐♣❡rs✉♣❡r❢í✲ ❝✐❡ M ❡♠ Rn+1 _{❡ ♦ ❝♦♥❥✉♥t♦ W = R}n+1 ₋S

p∈MTp✳ ❖s r❡s✉❧t❛❞♦s sã♦ ♦❜t✐❞♦s ♣❛r❛ ❤✐♣❡rs✉♣❡r❢í❝✐❡s ❝♦♠♣❛❝t❛s ♥ã♦ ❝♦♥✈❡①❛s s❛t✐s❢❛③❡♥❞♦ W6=∅✳

■♥✐❝✐❛❧♠❡♥t❡ ❞❡♠♦str❛♠♦s ❛ s❡❣✉✐♥t❡ ❢ór♠✉❧❛ ✐♥t❡❣r❛❧ ❞♦ t✐♣♦ ▼✐♥❦♦✇s❦✐ ♣❛r❛ ❤✐♣❡rs✉✲ ♣❡r❢í❝✐❡s ❝♦♠♣❛❝t❛s ❡♠ Rn+1_{✱ ♣❛r❛ ♦ ❝❛s♦ ❛♥✐s♦tró♣✐❝♦✱ q✉❡ ❣❡♥❡r❛❧✐③❛ ❛ ❢ór♠✉❧❛ ✐♥t❡❣r❛❧}

♣r♦✈❛❞❛ ♣♦r ❍❡ ❡ ▲✐ ❡♠ ❬✶✼❪✱ ❝♦♠♦ ❛♣r❡s❡♥t❛❞❛ ❛❜❛✐①♦ ✭✈❡❥❛ ❚❡♦r❡♠❛ ✹✳✷ ♥❛ ❙❡ç❛♦ ✹✳✶✮✿

❙❡❥❛ x:M −→Rn+1 _{✉♠❛ ✐♠❡rsã♦ ✐s♦♠étr✐❝❛ ❞❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛}

❢❡❝❤❛❞❛ ❡ ♦r✐❡♥tá✈❡❧ Mn_{✱ F : S}n _{−→ R}+ _{s❛t✐s❢❛③❡♥❞♦ ❛ ❝♦♥❞✐çã♦ ✭✷✳✶✮ ❡} 1 ≤ p≤ n,1≤ q ≤ n ✐♥t❡✐r♦s✳ ❊♥tã♦✱ ♣❛r❛ 0≤ r ≤n−1✱ t❡♠♦s ❛ s❡❣✉✐♥t❡

❢ór♠✉❧❛ ✐♥t❡❣r❛❧ ❞♦ t✐♣♦ ▼✐♥❦♦✇s❦✐✿

Z

M

{

"

q

|x|2 2

q−1

−phx, Nip−1

#

APr(∇SnF), xT

+ ✭✶✳✷✮

+ Fhx, Nip−1−F q

|x|2 2

q−1!

div(PrxT)−(n−r)

n r

H_rF

−

− F q(q−1)

|x|2 2

q−2

PrxT, xT

−

− (n−r)

n r

F q

|x|2 2

q−1

H_rF +H_rF₊₁hx, Nip

!

}dM = 0.

❉❡ ❢❛t♦✱ s❡ p=q = 1 ♥♦ ❚❡♦r❡♠❛ ✹✳✷✱ ♦❜t❡♠♦s q✉❡ Z

Mn

F(N)H_rF +H_rF₊₁hx, Ni

dM = 0,

r❡s✉❧t❛❞♦ ♣r♦✈❛❞♦ ❡♠ ❬✶✼❪✳

◆❛ ❙❡çã♦ ✹✳✷✱ ♦❜t❡♠♦s r❡s✉❧t❛❞♦s ❡♥✈♦❧✈❡♥❞♦ ✉♠❛ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞❛s ❝✉r✈❛t✉r❛s ♠é❞✐❛s ❛♥✐s♦tró♣✐❝❛s ❞❡ ♦r❞❡♠ s✉♣❡r✐♦r✱ ❝♦♠♦ ❡♠ ❬✷❪✱ r❡s✉❧t❛♥❞♦ ♥✉♠❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞❛ ❲✉❧✛ s❤❛♣❡✱ ❝♦♠♦ ❛♣r❡s❡♥t❛❞♦ ❛❜❛✐①♦✳ ❆ ❡①♣r❡ssã♦ ❞❡ ✉♠ HF

s ❝♦♠♦ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞♦s ♦✉tr♦s HF

r ✬s ✭❝♦♠ ❝♦❡✜❝✐❡♥t❡s ❝♦♥st❛♥t❡s✮ ❢♦r❛♠ ❝♦♥s✐❞❡r❛❞❛s ♣♦r ♠✉✐t♦s ❛✉✲ t♦r❡s ✭✈❡❥❛✱ ♣♦r ❡①❡♠♣❧♦✱ ❬✾❪✱ ❬✽❪✮✳ ◆♦ ♣r✐♠❡✐r♦ r❡s✉❧t❛❞♦ ❞❡ss❛ s❡çã♦✱ ♦s ❝♦❡✜❝✐❡♥t❡s ❞❛ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r sã♦ ❢✉♥çõ❡s✱ ❝♦♠♦ s❡❣✉❡ ✭✈❡❥❛ ❚❡♦r❡♠❛ ✹✳✺ ♥❛ ❙❡çã♦ ✹✳✷✮✿

❙❡❥❛ x:M −→Rn+1 _{✉♠❛ ✐♠❡rsã♦ ✐s♦♠étr✐❝❛ ❞❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛}

❢❡❝❤❛❞❛ ❡ ♦r✐❡♥tá✈❡❧ Mn _{❡ F : S}n _{−→ R}+ _{s❛t✐s❢❛③❡♥❞♦ ❛ ❝♦♥❞✐çã♦ ✭✷✳✶✮✳}

❆ss✉♠❛ q✉❡ ♣❛r❛ ✐♥t❡✐r♦s r ❡ s✱ ❝♦♠ 0 < r < s−1 ≤ n−1✱ ❛s ❝✉r✈❛t✉r❛s

♠é❞✐❛s ❛♥✐s♦tró♣✐❝❛s ❞❡ ♦r❞❡♠ s✉♣❡r✐♦r sã♦ ❧✐♥❡❛r♠❡♥t❡ r❡❧❛❝✐♦♥❛❞❛s ♣♦r

H_sF =arHrF +...+as−1HsF−1, ✭✶✳✸✮

♣❛r❛ ❢✉♥çõ❡s ❝♦♥tí♥✉❛s ♥ã♦ ♥❡❣❛t✐✈❛s ar, ..., as−1 :M −→R s❛t✐s❢❛③❡♥❞♦ βi ≤

αi+2✱∀r ≤ i ≤ s −3, βs−1 ≤ η, βs−2 ≤ η, αr ≥ ξ, αr+1 ≥ ξ✱ ♦♥❞❡ αi :=

infM{ai}✱ βi := supM{ai} ❡ η, ξ sã♦ ❝♦♥st❛♥t❡s ♣♦s✐t✐✈❛s t❛✐s q✉❡ HsF−1 ≤

ξ ηH

F

r−1 ❡ HsF−1 ≤ HsF+1✳ ❊♥tã♦✱ ❛ ♠❡♥♦s ❞❡ tr❛♥s❧❛çã♦ ❞❡ ❤♦♠♦t❡t✐❛✱ x(M) é

❛ ❲✉❧✛ s❤❛♣❡ ❞❡ F✳

◆♦ ❚❡♦r❡♠❛ ✹✳✺✱ ♦❜s❡r✈❡ q✉❡ s❡ al >0 ♣❛r❛ ❛❧❣✉♠ l ∈ {r, ..., s−1} ❛s ❝♦♥st❛♥t❡s ξ ❡η s❡♠♣r❡ ❡①✐st❡♠✳ P♦r ❡①❡♠♣❧♦✱ ❜❛st❛ t♦♠❛r η_ξ =infM

n_HF r−1

HF s−1

o

>0✭✈❡❥❛ ▲❡♠❛ ✹✳✹ ♥❛

❙❡çã♦ ✹✳✷✮✳

❉❡✜♥✐♠♦s ♦ ❝♦♥❥✉♥t♦W=Rn+1₋S

p∈MTp✱ ♦♥❞❡ Tp é ♦ ❤✐♣❡r♣❧❛♥♦ ❞❡Rn+1 ❡♠ x(p)✱ ❞❛❞♦ ♣❡❧♦ ❡s♣❛ç♦ t❛♥❣❡♥t❡ ❛x(M)❡♠ x(p) ✭✈❡❥❛ ❉❡✜♥✐çã♦ ✹✳✼ ♥❛ ❙❡çã♦ ✹✳✷✮✳

P♦❞❡♠♦s r❡♠♦✈❡r ❛ ❤✐♣ót❡s❡HF

s−1 ≤HsF+1 ♥♦ ❚❡♦r❡♠❛ ✹✳✺ ❡ ❛✐♥❞❛ ♦❜t❡r♠♦s ♦ r❡s✉❧✲

t❛❞♦✳ P❛r❛ ✐st♦ ♥❡❝❡ss✐t❛♠♦s q✉❡ ❛s ❢✉ñçõ❡sai✬s s❡❥❛♠ ❝♦♥st❛♥t❡s✳ ◆♦ss❛ ♦✉tr❛ ❛❧t❡r♥❛t✐✈❛

♣❛r❛ s❡ ❧✐✈r❛r ❞❛ ❤✐♣ót❡s❡HF

s−1 ≤HsF+1 é s✉♣♦r q✉❡ ♦ ❝♦♥❥✉♥t♦ W⊂Rn+1 é ♥ã♦ ✈❛③✐♦ ✭❡

♠❡❧❤♦r❛r ❛ ✈❛r✐❛çã♦ ❞❡r ❡s✮ ❝♦♠♦ s❡❣✉❡ ✭✈❡❥❛ ❚❡♦r❡♠❛ ✹✳✾ ♥❛ ❙❡çã♦ ✹✳✷✮✿

❙❡❥❛ x:M −→Rn+1 _{✉♠❛ ✐♠❡rsã♦ ✐s♦♠étr✐❝❛ ❞❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛}

❢❡❝❤❛❞❛ ❡ ♦r✐❡♥tá✈❡❧ Mn _{❡ F : S}n _{−→ R}+ _{s❛t✐s❢❛③❡♥❞♦ ❛ ❝♦♥❞✐çã♦ ✭✷✳✶✮✳}

❙✉♣♦♥❤❛ q✉❡ ✉♠❛ ❞❛s ♦♣çõ❡s ❛❜❛✐①♦ ❛❝♦rr❡✳

✶✳ ❊①✐st❡♠ ✐♥t❡✐r♦s r❡ s✱ ❝♦♠ 0< r < s < n✱ t❛✐s q✉❡ ❛s ❝✉r✈❛t✉r❛s ♠é❞✐❛s

❛♥✐s♦tró♣✐❝❛s ❞❡ ♦r❞❡♠ s✉♣❡r✐♦r sã♦ ❧✐♥❡❛r♠❡♥t❡ r❡❧❛❝✐♦♥❛❞❛s ♣♦r

H_sF =arHrF +...+as−1HsF−1 ✭✶✳✹✮

♣❛r❛ ♥ú♠❡r♦s ♥ã♦ ♥❡❣❛t✐✈♦s ar, ..., as−1❀

✷✳ W 6= ∅ ❡ ❡①✐st❡♠ ✐♥t❡✐r♦s r ❡ s✱ ❝♦♠ 0 ≤ r < s < n ♦✉ 0 < r < s ≤ n✱ t❛✐s q✉❡ ❛s ❝✉r✈❛t✉r❛s ♠é❞✐❛s ❛♥✐s♦tró♣✐❝❛s ❞❡ ♦r❞❡♠ s✉♣❡r✐♦r sã♦

❧✐♥❡❛r♠❡♥t❡ r❡❧❛❝✐♦♥❛❞❛s ♣♦r

H_sF =arHrF +...+as−1HsF−1 ✭✶✳✺✮

♣❛r❛ ♥ú♠❡r♦s ♥ã♦ ♥❡❣❛t✐✈♦s ar, ..., as−1✳

❊♥tã♦✱ ❛ ♠❡♥♦s ❞❡ tr❛♥s❧❛çã♦ ❡ ❤♦♠♦t❡t✐❛✱ x(M) é ❛ ❲✉❧✛ s❤❛♣❡ ❞❡ F✳

P❛r❛ ♣r♦✈❛r ♦ r❡s✉❧t❛❞♦ ❛❝✐♠❛ ✉s❛♠♦s ❛ ❢ór♠✉❧❛ ✐♥t❡❣r❛❧ t✐♣♦ ▼✐♥❦♦✇s❦✐✳

◆♦t❡ q✉❡W ♥ã♦ ✈❛③✐♦ é ✉♠❛ ❤✐♣ót❡s❡ ♠❛✐s ❢r❛❝❛ q✉❡ ❛ ❤✐♣ót❡s❡ ❞❡ ❝♦♥✈❡①✐❞❛❞❡ ♣❛r❛

✉♠❛ ❤✐♣❡rs✉♣❡r❢í❝✐❡x(M)✳ ❉❡ ❢❛t♦✱ ♣❛r❛ M ❝♦♥❡①♦ ❡ ❢❡❝❤❛❞♦✱ ❛ ❤✐♣ót❡s❡ ✏W ♥ã♦ ✈❛③✐♦✑

é ❡q✉✐✈❛❧❡♥t❡ ❛ ❤✐♣ót❡s❡ ✏x(M) s❡r ♦ ❜♦r❞♦ ❞❡ ✉♠ ✭ú♥✐❝♦✮ ❞♦♠í♥✐♦ ❡str❡❧❛❞♦ ❛❜❡rt♦ V

❡♠Rn+1_{✑ ✭❬✶✹❪✮✳ ❆❧é♠ ❞✐ss♦✱ ♥❡st❡ ❝❛s♦Wé ♦ ✐♥t❡r✐♦r ❞♦ s✉❜❝♦♥❥✉♥t♦ ❞❛❞♦ ♣❡❧♦s ♣♦♥t♦s}

❞❡V q✉❡ ♣♦❞❡♠ s❡r ❝♦♥❡❝t❛❞♦s ❛ q✉❛❧q✉❡r ♦✉tr♦ ♣♦♥t♦ ❞❡ V ♣♦r ✉♠ s❡❣♠❡♥t♦ ❞❡ r❡t❛

❝♦♥t✐❞♦ ❡♠ V✳ P♦rt❛♥t♦✱ q✉❛♥❞♦ x(M) é ❝♦♥✈❡①♦✱ ❡♥tã♦ t❛♠❜é♠ V é ❝♦♥✈❡①♦ ❡ W

❝♦✐♥❝✐❞❡ ❝♦♠ ♦ ✐♥t❡r✐♦r ❞❡ V ✭✈❡❥❛ ❖❜s❡r✈❛çã♦ ✸✳✶✹ ♥❛ ❙❡❝ã♦ ✹✳✷✮✳

❈♦♠♦ ✉♠❛ ❛♣❧✐❝❛çã♦ ❞♦ ❚❡♦r❡♠❛ ✹✳✾ ♣r♦✈❛♠♦s ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦ ✭✈❡❥❛ ❈♦r♦❧ár✐♦ ✹✳✶✵ ♥❛ ❙❡çã♦ ✹✳✸✮✿

❙❡❥❛ x:M −→Rn+1 _{✉♠❛ ✐♠❡rsã♦ ✐s♦♠étr✐❝❛ ❞❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛}

❢❡❝❤❛❞❛ ❡ ♦r✐❡♥tá✈❡❧ Mn_{✱ F : S}n _{−→ R}+ _{s❛t✐s❢❛③❡♥❞♦ ❛ ❝♦♥❞✐çã♦ ✭✷✳✶✮ ❡}

W 6=∅✳ ❙❡ HsF HF

r =constante ♣❛r❛ 0≤ r < s < n ♦✉ 0 < r < s≤ n✱ ❡♥tã♦✱ ❛ ♠❡♥♦s ❞❡ tr❛♥s❧❛çã♦ ❡ ❤♦♠♦t❡t✐❛✱ x(M) é ❛ ❲✉❧✛ s❤❛♣❡ ❞❡ F✳

Pr♦✈❛♠♦s ♠❛✐s ❞✉❛s ❛♣❧✐❝❛çõ❡s ❞♦ ❚❡♦r❡♠❛ ✹✳✾✳ ❆❜❛✐①♦ ❛ ♣r✐♠❡✐r❛ ❛♣❧✐❝❛çã♦ ✭✈❡❥❛ ❈♦r♦❧ár✐♦ ✹✳✶✶ ♥❛ ❙❡çã♦ ✹✳✸✮✿

❙❡❥❛ x:M −→Rn+1 _{✉♠❛ ✐♠❡rsã♦ ✐s♦♠étr✐❝❛ ❞❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛}

❢❡❝❤❛❞❛ ❡ ♦r✐❡♥tá✈❡❧ Mn _{❡ F :S}n_−→R+ _{s❛t✐s❢❛③❡♥❞♦ ❛ ❝♦♥❞✐çã♦ ✭✷✳✶✮✱ ❝♦♠} (r+ 1)✲és✐♠❛ ❝✉r✈❛t✉r❛ ♠é❞✐❛ ❛♥✐s♦tró♣✐❝❛ HF

r+1 ❝♦♥st❛♥t❡✱ 0 ≤ r ≤ n−1✳

❊♥tã♦✱ ♦ ❝♦♥❥✉♥t♦ ❞❡ ♣♦♥t♦s W é ♥ã♦ ✈❛③✐♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❛ ♠❡♥♦s ❞❡

tr❛♥s❧❛çã♦ ❡ ❤♦♠♦t❡t✐❛✱ x(M) é ❛ ❲✉❧✛ s❤❛♣❡ ❞❡ F✳

◆♦ ❝♦r♦❧ár✐♦ ❛❝✐♠❛✱ ♦❜t❡♠♦s ♦ ❈♦r♦❧ár✐♦1.1❡♠ ❬✶✼❪✱ ❝♦♠ ❛ ❝♦♥❞✐ç❛♦W♥ã♦ ✈❛③✐♦ ❡♠

✈❡③ ❞❛ ❝♦♥✈❡①✐❞❛❞❡ ❞❛ ❤✐♣❡rs✉♣❡r❢í❝✐❡✳

❆ ♦✉tr❛ ❛♣❧✐❝❛çã♦ s❡❣✉❡ ❛❜❛✐①♦ ✭✈❡❥❛ ❚❡♦r❡♠❛ ✹✳✶✷ ♥❛ ❙❡çã♦ ✹✳✸✮✿

❙❡❥❛ x:M −→Rn+1 _{✉♠❛ ✐♠❡rsã♦ ✐s♦♠étr✐❝❛ ❞❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛}

❢❡❝❤❛❞❛ ❡ ♦r✐❡♥tá✈❡❧ Mn _{❡ F : S}n _{−→ R}+ _{s❛t✐s❢❛③❡♥❞♦ ❛ ❝♦♥❞✐çã♦ ✭✷✳✶✮✳}

❙❡ HF s HF

r = constante ♣❛r❛ 1 ≤ r < s ≤ n✱ ❡♥tã♦ ❛ ♠❡♥♦s ❞❡ tr❛♥s❧❛çã♦ ❡ ❤♦♠♦t❡t✐❛✱ x(M) é ❛ ❲✉❧✛ s❤❛♣❡ ❞❡ F✳

❖ ❚❡♦r❡♠❛ ✹✳✶✷✱ ❣❡♥❡r❛❧✐③❛ ♣❛r❛ ❛ ♦ ❝❛s♦ ❛♥✐s♦tró♣✐❝♦ ♦ t❡♦r❡♠❛ ♦❜t✐❞♦ ♣♦r ❬✷✷❪ ♣❛r❛ ♦ ❝❛s♦ ❊✉❝❧✐❞✐❛♥♦✳

❚❛♠❜é♠✱ ♥♦ ❚❡♦r❡♠❛ ✹✳✶✷✱ r❡♦❜t❡♠♦s ♦ ❚❡♦r❡♠❛ 1.3✱ ✭♣❛r❛ k ≥ 1✮ ♦ ❚❡♦r❡♠❛ 1.5 ❡

✭♣❛r❛k ≥1✮ ♦ ❚❡♦r❡♠❛ 1.4✱ ❡♠ ❬✶✼❪✱ s❡♠ ❛ ❤✐♣ót❡s❡ ❞❡ ❝♦♥✈❡①✐❞❛❞❡✳

❆✐♥❞❛ ♥❛ ❙❡çã♦ ✹✳✸✱ ♣r♦✈❛♠♦s ✭✈❡❥❛ ❚❡♦r❡♠❛ ✹✳✶✸✮✿

❙❡❥❛ x:M −→Rn+1 _{✉♠❛ ✐♠❡rsã♦ ✐s♦♠étr✐❝❛ ❞❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛}

❢❡❝❤❛❞❛ ❡ ♦r✐❡♥tá✈❡❧ Mn _{❡ F :S}n_−→R+ _{s❛t✐s❢❛③❡♥❞♦ ❛ ❝♦♥❞✐çã♦ ✭✷✳✶✮✱ ❝♦♠}

❛r✲és✐♠❛ ❝✉r✈❛t✉r❛ ♠é❞✐❛ ❛♥✐s♦tró♣✐❝❛HF

r+1 ❝♦♥st❛♥t❡✱0≤r≤n−1✳ ❊♥tã♦✱

♦ ❝♦♥❥✉♥t♦ ❞❡ ♣♦♥t♦s W é ♥ã♦ ✈❛③✐♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❛ ♠❡♥♦s ❞❡ tr❛♥s❧❛çã♦

❡ ❤♦♠♦t❡t✐❛✱ x(M) é r✲❡stá✈❡❧✳

❆q✉✐✱ r✲❡stá✈❡❧ s✐❣♥✐✜❝❛ ♦ s❡❣✉✐♥t❡✿ Mn _{é ♣♦♥t♦ ❝rít✐❝♦ ❞♦ ❢✉♥❝✐♦♥❛❧}

F_r,F;Λ = Z

M

F(N)HF

♦♥❞❡ Λ é ✉♠❛ ❝♦♥st❛♥t❡ ❡ V = _n+11 R

Mhx, NidM é ♦ (n+ 1)✲✈♦❧✉♠❡ ❞❡t❡r♠✐♥❛❞♦ ♣♦r

M✱ ❡ ❛ s❡❣✉♥❞❛ ✈❛r✐❛çã♦ ❞❡ Fr,F;Λ✱

F′′_r,F;Λ =−(r+ 1) Z

M

ψ{Lrψ+ψhTr◦dN, dNi}dM, é ♥ã♦ ♥❡❣❛t✐✈❛ ♣❛r❛ t♦❞♦ ψ ❝♦♠ R

MψdM = 0 ✭✈❡❥❛ ❬✶✾❪✮✳

❖ r❡s✉❧t❛❞♦ ❛❝✐♠❛ ♠♦str❛ q✉❡ ❛ss✉♠✐♥❞♦ q✉❡ ❛ ❤✐♣❡rs✉♣❡r❢í❝✐❡ é ❝♦♠♣❛❝t❛ ❝♦♠

HF

r+1 =constante✱ ❛ ❤✐♣ót❡s❡ W ♥ã♦ ✈❛③✐♦ é ❡q✉✐✈❛❧❡♥t❡ ❛ r✲st❛❜✐❧✐❞❛❞❡✳

❚❛♠❜é♠ ♣r♦✈❛♠♦s ♥❛ ❙❡çã♦ ✹✳✸ q✉❡ s❡ ♦ ✐♥t❡❣r❛♥❞♦ ❞❛ ❢ór♠✉❧❛ t✐♣♦ ▼✐♥❦♦✇s❦✐ ❞❛❞♦ ♥♦ ❈♦r♦❧ár✐♦ ✷✳✸ ♥ã♦ ♠✉❞❛ ❞❡ s✐♥❛❧✱ ❡♥tã♦ x(M) é ❛ ❲✉❧✛ s❤❛♣❡ ❞❡ F ✭✈❡❥❛ ❚❡♦r❡♠❛

✹✳✶✺✮✿

❙❡❥❛ x:M −→Rn+1 _{✉♠❛ ✐♠❡rsã♦ ✐s♦♠étr✐❝❛ ❞❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛}

❢❡❝❤❛❞❛ ❡ ♦r✐❡♥tá✈❡❧ Mn_{✱ F : S}n _{−→ R}+ _{s❛t✐s❢❛③❡♥❞♦ ❛ ❝♦♥❞✐çã♦ ✭✷✳✶✮ ❡ t❛❧}

q✉❡

F H_rF +H_rF₊₁hx, Ni

♥ã♦ ♠✉❞❛ ❞❡ s✐♥❛❧ ♣❛r❛ ❛❧❣✉♠ 0 ≤r ≤ n−1✳ ❊♥tã♦✱ x(M) é ❛ ❲✉❧✛ s❤❛♣❡

❞❡ F✳

❋✐♥❛❧♠❡♥t❡✱ ♥♦ss♦ ú❧t✐♠♦ r❡s✉❧t❛❞♦ ♠♦str❛ q✉❡ ❡♠ ❤✐♣❡rs✉♣❡r❢í❝✐❡s ❝♦♠♣❛❝t❛s ❛ ❤✐✲ ♣ót❡s❡W ♥ã♦ ✈❛③✐♦ ❝❛r❛❝t❡r✐③❛ ❛ ❲✉❧✛ s❤❛♣❡ ❛ ♠❡♥♦s ❞❡ ❞✐❢❡♦♠♦r✜s♠♦s✳ ❉❡ ❢❛t♦✱ s❡W

é ♥ã♦ ✈❛③✐♦✱ ♦❜t❡♠♦s q✉❡M é ❞✐❢❡♦♠♦r❢❛ ❛ ❲✉❧✛ s❤❛♣❡ ❞❡F ❡x:M −→Rn+1 _{é ❞❡ ❢❛t♦}

✉♠ ♠❡r❣✉❧❤♦ ✭✈❡❥❛ ❚❡♦r❡♠❛ ✹✳✶✻ ♥❛ ❙❡çã♦ ✹✳✸✮✿

❙❡❥❛ x:M −→Rn+1 _{✉♠❛ ✐♠❡rsã♦ ✐s♦♠étr✐❝❛ ❞❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛}

❢❡❝❤❛❞❛ ❡ ♦r✐❡♥tá✈❡❧ Mn _{❡ F : S}n _{−→ R}+ _{s❛t✐s❢❛③❡♥❞♦ ❛ ❝♦♥❞✐çã♦ ✭✷✳✶✮✳ ❙❡}

W6=∅✱ ❡♥tã♦M é ❞✐❢❡♦♠♦r❢❛ ❛ ❲✉❧✛ s❤❛♣❡ ❞❡F ❡ xé ❞❡ ❢❛t♦ ✉♠ ♠❡r❣✉❧❤♦✳

◆❛ ♣r♦✈❛ ❞♦ ❚❡♦r❡♠❛ ✹✳✶✻ ❡①✐❜✐♠♦s ♦ ❞✐❢❡♦♠♦r✜s♠♦✳

❈❛♣ít✉❧♦ ✷

Pr❡❧✐♠✐♥❛r❡s

❊st❡ ❝❛♣ít✉❧♦ ♦❜❥❡t✐✈❛ ❡st❛❜❡❧❡❝❡r ❛s ♥♦t❛çõ❡s q✉❡ s❡rã♦ ✉t✐❧✐③❛❞❛s ♥♦s ❞❡♠❛✐s ❝❛♣ít✉❧♦s ❞❡st❡ tr❛❜❛❧❤♦✱ ❜❡♠ ❝♦♠♦ ♦s ❢❛t♦s ❜ás✐❝♦s ❞❛ t❡♦r✐❛ q✉❡ ❢❛r❡♠♦s ✉s♦ ♣♦st❡r✐♦r♠❡♥t❡✳

✷✳✶ ❆ ❲✉❧✛ ❙❤❛♣❡ ❞❡

F

❙❡❥❛ F : Sn _{−→ R}+ _{✉♠❛ ❢✉♥çã♦ s✉❛✈❡ ♣♦s✐t✐✈❛ q✉❡ s❛t✐s❢❛③ ❛ s❡❣✉✐♥t❡ ❝♦♥❞✐çã♦ ❞❡}

❝♦♥✈❡①✐❞❛❞❡✿

(D2F +F I)x >0, ∀x∈Sn, ✭✷✳✶✮

♦♥❞❡D2_{F ❞❡♥♦t❛ ♦ ❍❡ss✐❛♥♦ ✐♥trí♥s❡❝♦ ❞❡F ♥❛n✲❡s❢❡r❛S}n_⊂Rn+1_{✱ ■ ❞❡♥♦t❛ ❛ ✐❞❡♥t✐❞❛❞❡}

❡♠ TxSn ❡ >0 s✐❣♥✐✜❝❛ q✉❡ ❛ ♠❛tr✐③ é ♣♦s✐t✐✈❛ ❞❡✜♥✐❞❛✳ ❈♦♥s✐❞❡r❡♠♦s ❛ ❛♣❧✐❝❛çã♦

φ :Sn −→Rn+1,

x−→F(x)x+ (gradSnF)_x, ✭✷✳✷✮

❝✉❥❛ ✐♠❛❣❡♠ WF = φ(Sn) é ✉♠❛ ❤✐♣❡rs✉♣❡r❢í❝✐❡ s✉❛✈❡✱ ❝♦♥✈❡①❛ ❡♠ Rn+1 ❝❤❛♠❛❞❛ ❛ ❲✉❧✛ s❤❛♣❡ ❞❡ F✭✈❡❥❛ ❬✶✵❪✱ ❬✶✼❪✱ ❬✶✻❪✱ ❬✷✸❪✱ ❬✷✼❪✱ ❬✷✽❪✱ ❬✷✾❪✮✳ ◆♦ ❝❛s♦ ❡♠ q✉❡ F = 1 t❡♠♦s

q✉❡ ❛ ❲✉❧✛ s❤❛♣❡ ❞❡F é ❛ n✲❡s❢❡r❛ Sn _⊂Rn+1_✳

❆❣♦r❛ s❡❥❛

x:Mn −→Rn+1

✉♠❛ ✐♠❡rsã♦ s✉❛✈❡ ❞❡ ✉♠❛ ❤✐♣❡rs✉♣❡r❢í❝✐❡ ❢❡❝❤❛❞❛ ❡ ♦r✐❡♥t❛❞❛ ❡

N :Mn −→Sn

✷✳✶ ❆ ❲✉❧✛ ❙❤❛♣❡ ❞❡ F

s✉❛ ❛♣❧✐❝❛çã♦ ❞❡ ●❛✉ss✱ ✐st♦ é✱N é ♦ ✈❡t♦r ♥♦r♠❛❧ ✉♥✐tár✐♦ ❞❡M✳

❉❡✜♥✐♠♦s

AF :=D2F +F I ✭✷✳✸✮

❡

NF :M −→WF, NF :=φ◦N ❛ ❛♣❧✐❝❛çã♦ ❞❡ ●❛✉ss ❣❡♥❡r❛❧✐③❛❞❛ ❞❛ ❲✉❧✛ s❤❛♣❡✳ ❊♥tã♦

SF :=−dNF =−AF ◦dN

é ❝❤❛♠❛❞♦ ♦ ♦♣❡r❛❞♦rF✲❲❡✐♥❣❛rt❡♥✱ ❡ ♦s ❛✉t♦✈❛❧♦r❡s ❞❡ SF sã♦ ❝❤❛♠❛❞♦s ❞❡ ❝✉r✈❛t✉r❛s ♣r✐♥❝✐♣❛✐s ❛♥✐s♦tró♣✐❝❛s✳ ❱❡r❡♠♦s ♥♦ ▲❡♠❛ ✷✳✷ ❛ s❡❣✉✐r q✉❡✱ ❛s r❛✐③❡s ❞♦ ♣♦❧✐♥ô♠✐♦ ❝❛r❛❝t❡ríst✐❝♦ ❞❡SF sã♦ t♦❞❛s r❡❛✐s✳

❱❡❥❛ q✉❡✱ ❡♠ ❝❛❞❛ ♣♦♥t♦ p∈Mn_{✱ S}

F(p) :TpM −→TpM é ✉♠ ♦♣❡r❛❞♦r ❧✐♥❡❛r✳ P❛r❛ ❝❛❞❛1≤r≤n✱ s❡❥❛Sr(p)❛r✲és✐♠❛ ❢✉♥çã♦ s✐♠étr✐❝❛ ❡❧❡♠❡♥t❛r ♥♦s ❛✉t♦✈❛❧♦r❡s ❞❡SF(p)✱ ❛ss✐♠ ♦❜t❡♠♦sn ❢✉♥çõ❡s s✉❛✈❡s Sr :Mn −→R✱ t❛❧ q✉❡

det(tI −SF) = n

X

k=0

(−1)kSktn−k, ✭✷✳✹✮

♦♥❞❡S0 = 1 ♣♦r ❞❡✜♥✐çã♦✳ ❙❡p∈Mn❡{λk}sã♦ ♦s ❛✉t♦✈❛❧♦r❡s ❝♦♠ r❡s♣❡✐t♦ ❛♦ ♦♣❡r❛❞♦r

SF(p)✱ t❡♠♦s q✉❡

Sr =σr(λ1, ..., λn) :=

X

1≤i1<...<ir≤n

λi1...λir,

♦♥❞❡σr ∈R[X1, ..., Xn]é ♦r✲és✐♠♦ ♣♦❧✐♥ô♠✐♦ s✐♠étr✐❝♦ ❡❧❡♠❡♥t❛r ♥❛s ✐❞❡♥t✐❞❛❞❡sX1, ..., Xn✳

P❛r❛ 1≤r ≤n✱ ❞❡✜♥✐♠♦s ❛r✲és✐♠❛ ❝✉r✈❛t✉r❛ ♠é❞✐❛ ❛♥✐s♦tró♣✐❝❛HF

r ❞❡x ♣♦r

Cr

nHrF =σr(λ1, ..., λn), ✭✷✳✺✮

♦♥❞❡Cr n=

n r

✳

❯♠❛ ✈❛r✐❛çã♦ ❞❡ ✉♠❛ ✐♠❡rsã♦ s✉❛✈❡x:Mn_−→Rn+1 _{é ✉♠❛ ❛♣❧✐❝❛çã♦ s✉❛✈❡}

X :Mn×(−ǫ, ǫ)−→Rn+1

s❛t✐s❢❛③❡♥❞♦ ❛ s❡❣✉✐♥t❡ ❝♦♥❞✐çã♦✿

✷✳✶ ❆ ❲✉❧✛ ❙❤❛♣❡ ❞❡ F

✶✳ P❛r❛ t ∈ (−ǫ, ǫ)✱ ❛ ❛♣❧✐❝❛çã♦ Xt : Mn −→ Rn+1 ❞❛❞❛ ♣♦r Xt(p) = X(t, p) é ✉♠❛ ✐♠❡rsã♦ s✉❛✈❡ t❛❧ q✉❡ X0 =x✱

✷✳ Xt|∂M =x|∂M✱ ♣❛r❛ t♦❞♦ t ∈(−ǫ, ǫ✮✳

❙❡❥❛ x : M −→ Rn+1 _{✉♠❛ ❤✐♣❡rs✉♣❡r❢í❝✐❡ s✉❛✈❡✱ ❢❡❝❤❛❞❛ ❡ ♦r✐❡♥t❛❞❛ ❝♦♠ ❛♣❧✐❝❛çã♦}

❞❡ ●❛✉ss N : M −→ Sn_{✱ ✐st♦ é✱ N é ✉♠ ❝❛♠♣♦ ✈❡t♦r✐❛❧ ♥♦r♠❛❧ ✉♥✐tár✐♦✳ ❙❡❥❛ X} t ✉♠❛ ✈❛r✐❛çã♦ ❞❡ x✱ ❡Nt:M −→Sn ❛ ❛♣❧✐❝❛çã♦ ❞❡ ●❛✉ss ❞❡ Xt✳

❖ ❝❛♠♣♦ ✈❛r✐❛❝✐♦♥❛❧ ❛ss♦❝✐❛❞♦ ❛ ✈❛r✐❛çã♦ X é ♦ ❝❛♠♣♦ ✈❡t♦r✐❛❧ ∂X_∂t ✳ ❉❡♥♦t❛♥❞♦ ♣♦r f =_∂X

∂t , Nt

✱ ♦❜t❡♠♦s

∂X

∂t =f Nt+

∂X ∂t

⊤

, ✭✷✳✻✮

♦♥❞❡⊤ r❡♣r❡s❡♥t❛ ❛ ❝♦♠♣♦♥❡♥t❡ t❛♥❣❡♥t❡✳ ❉❡✜♥✐♠♦s

g =

∂X ∂t

⊤

. ✭✷✳✼✮

❆ ♣r✐♠❡✐r❛ ✈❛r✐❛çã♦ ❝♦rr❡s♣♦♥❞❡♥t❡ ❛♦ ✈❡t♦r ♥♦r♠❛❧ ✉♥✐tár✐♦ é ❞❛❞❛ ♣♦r ✭✈❡❥❛ ❬✷✸❪✱ ❬✷✼❪✱ ❬✷✾❪✮

∂tNt=−gradf +dNt(g), ✭✷✳✽✮

❛ ♣r✐♠❡✐r❛ ✈❛r✐❛çã♦ ❞♦ ❡❧❡♠❡♥t♦ ❞❡ ✈♦❧✉♠❡ é

∂tdMt = (divg−nHf)dMt ✭✷✳✾✮

❡ ❛ ♣r✐♠❡✐r❛ ✈❛r✐❛çã♦ ❞♦ ✈♦❧✉♠❡ V é

V′(t) = Z

M

f dMt, ✭✷✳✶✵✮

♦♥❞❡ grad✱ div ❡ H r❡♣r❡s❡♥t❛♠ ♦ ❣r❛❞✐❡♥t❡✱ ♦ ❞✐✈❡r❣❡♥t❡ ❡ ❛ ❝✉r✈❛t✉r❛ ♠é❞✐❛ ❝♦♠

r❡s♣❡✐t♦ ❛Xt✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳

❙❡❥❛♠ {E1,· · · , En} ✉♠ r❡❢❡r❡♥❝✐❛❧ ♦rt♦❣♦♥❛❧ ❧♦❝❛❧ ❡♠ Sn✱ ei =ei(t) =Ei◦Nt✱ ♦♥❞❡

i= 1,· · · , n✳ ❚❡♠♦s q✉❡{e1,· · · , en}é ✉♠ r❡❢❡r❡♥❝✐❛❧ ♦rt♦❣♦♥❛❧ ❧♦❝❛❧ ❞❡x:M −→Rn+1✳ ❆s ❡q✉❛çõ❡s ❡str✉t✉r❛✐s ❞❛ ✐♠❡rsã♦ ❝❛♥ô♥✐❝❛ y:Sn_−→Rn+1 _sã♦✿

   

  

dy =P

iθiEi;

dEi =PjθijEj−θiy;

dθi =Pjθij ∧θj;

dθij −P_kθik∧θkj =−1₂P_klR¯ijklθk∧θl =−θi∧θj.

✭✷✳✶✶✮

✷✳✶ ❆ ❲✉❧✛ ❙❤❛♣❡ ❞❡ F

♦♥❞❡θij +θji = 0 ❡ R¯ijkl =δikδjl−δilδjk.

❆s ❡q✉❛çõ❡s ❡str✉t✉r❛✐s ❞❡Xt sã♦ ✭✈❡❥❛ ❬✷✺❪✱ ❬✷✹❪✮

     

    

dXt=Piωiei;

dNt =−

P

ijhijωjei;

dei =P_jωijej +P_jhijωjNt;

dωi =Pjωij ∧ωj;

dωij −Pkωik∧ωkj =−1₂PklRijklθk∧θl,

✭✷✳✶✷✮

♦♥❞❡ ωij +ωji = 0✱ Rijkl +Rijlk = 0✱ ❡ Rijkl sã♦ ♦s ❝♦♠♣♦♥❡♥t❡s ❞♦ t❡♥s♦r ❝✉r✈❛t✉r❛ ❘✐❡♠❛♥♥✐❛♥♦ ❞❡ M ❝♦♠ r❡s♣❡✐t♦ ❛ ♠étr✐❝❛ ✐♥❞✉③✐❞❛ ♣♦rXt✳ ❆q✉✐ ♦♠✐t✐♠♦s ❛ ✈❛r✐á✈❡❧ t

❞❡ ❛❧❣✉♠❛s q✉❛♥t✐❞❛❞❡s ❣❡♦♠étr✐❝❛s✳

❉❡♥♦t❡♠♦s ♣♦rsij ♦s ❝♦❡✜❝✐❡♥t❡s ❞❡SF ❝♦♠ r❡s♣❡✐t♦ ❛♦ r❡❢❡r❡♥❝✐❛❧{e1, ..., en}✱ ✐st♦ é✱

SF :=−dNF =−d(φ◦N) =−AF ◦dN =

X

i,j

sijωjej, ✭✷✳✶✸✮

♦♥❞❡φ é ❞❡✜♥✐❞♦ ❡♠ ✭✷✳✷✮✳

P❡❧❛ ❞❡✜♥✐çã♦ ❛❝✐♠❛✱ ♥♦t❡ q✉❡ ❡♠❜♦r❛ AF ❡ dN s❡❥❛♠ s✐♠étr✐❝♦s✱ SF é s✐♠étr✐❝♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱AF ❡ dN ❝♦♠✉t❛♠✱ ♦ q✉❡ ♥ã♦ ♦❝♦rr❡ ❡♠ ❣❡r❛❧

❆❣♦r❛ ✐r❡♠♦s ❞❡♠♦♥str❛r q✉❡✱ ❡♠ q✉❛❧q✉❡r ❝❛s♦✱ t♦❞❛s ❛s r❛✐③❡s ❞♦ ♣♦❧✐♥ô♠✐♦ ❝❛r❛❝✲ t❡ríst✐❝♦ ❞❡SF sã♦ r❡❛✐s✳ ❆♥t❡s✱ ✈❡❥❛♠♦s ♦ s❡❣✉✐♥t❡ ❧❡♠❛✳

▲❡♠❛ ✷✳✶✳ ❙❡❥❛ V ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❞❡ ❞✐♠❡♥sã♦ ✜♥✐t❛ ❝♦♠ ♣r♦❞✉t♦ ✐♥t❡r♥♦ h,i ❡ T

✉♠ ♦♣❡r❛❞♦r ❧✐♥❡❛r ❡♠V✳ ❊♥tã♦T é ♣♦s✐t✐✈♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❡①✐st❡ ✉♠ ♦♣❡r❛❞♦r ❧✐♥❡❛r

✐♥✈❡rtí✈❡❧L ❡♠ V t❛❧ q✉❡ T ❂ L∗_{L✱ ♦♥❞❡ L}∗ _{é ♦ ♦♣❡r❛❞♦r ❧✐♥❡❛r ❛❞❥✉♥t♦ ❞❡ L✳}

❉❡♠♦♥str❛çã♦✳ ❙✉♣♦♥❤❛T =L∗_{L✱ ♦♥❞❡ L é ✉♠ ♦♣❡r❛❞♦r ❧✐♥❡❛r ✐♥✈❡rtí✈❡❧✳ ❊♥tã♦ T}∗ ₌

(L∗_L)∗ _=L∗_{L=T✱ ❞♦♥❞❡T é s✐♠étr✐❝♦✳ ❆❧é♠ ❞✐ss♦✱hT v, vi=hL}∗_{Lv, vi=hLv, Lvi ≥0✳}

❱✐st♦ q✉❡ Lé ✐♥✈❡rtí✈❡❧✱ ♣❛r❛ v 6= 0✱ t❡♠♦s Lv 6= 0 ❡ hT v, vi>0✳ ❆ss✐♠✱ T é ♣♦s✐t✐✈♦✳

P♦r ♦✉tr♦ ❧❛❞♦✱ s❡ T é ♣♦s✐t✐✈♦✱ ❡♥tã♦ (v, w) = hT v, wi é ✉♠ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❡♠ V✳ ❙❡❥❛♠ {α1,· · · , αn} ❡ {β1, ..., βn} ❜❛s❡s ♦rt♦♥♦r♠❛✐s ❞❡ V ❝♦♠ r❡s♣❡✐t♦ ❛ h,i ❡ (,)✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❊♥tã♦✱

(βi, βj) = δij =hαi, αji.

❉❡✜♥✐♠♦sL ❝♦♠♦ ♦ ú♥✐❝♦ ♦♣❡r❛❞♦r ❡♠ V t❛❧ q✉❡ Lβi =αi✱ ♦♥❞❡ i ∈ {1,· · · , n}✳ ❈♦♠♦

L❧❡✈❛ ❜❛s❡ ❡♠ ❜❛s❡✱ t❡♠♦s q✉❡Lé ✐♥✈❡rtí✈❡❧✳ ◆♦t❡ q✉❡✱ ❞❛❞♦sv =P

aiβi ❡w=Pbjβj

✷✳✶ ❆ ❲✉❧✛ ❙❤❛♣❡ ❞❡ F

✈❡t♦r❡s q✉❛✐sq✉❡r ❡♠V t❡♠♦s✱

(v, w) = X

i

aiβi,

X

j

bjβj

!

=X

i,j

aibj(βi, βj) =

X

i,j

aibjhαi, αji=

= X

i,j

aibjhLβi, Lβji=

DX

aiLβi,

X

bjLβj

E

=hLv, Lwi.

❆ss✐♠✱

hT v, wi=: (v, w) =hLv, Lwi=hL∗Lv, wi,

♣❛r❛ t♦❞♦v, w∈V✳ P♦rt❛♥t♦✱ T =L∗_L✳

▲❡♠❛ ✷✳✷✳ ❚♦❞❛s ❛s r❛✐③❡s ❞♦ ♣♦❧✐♥ô♠✐♦ ❝❛r❛❝t❡ríst✐❝♦ ❞❡ SF sã♦ r❡❛✐s✳ ❆❧é♠ ❞✐ss♦✱ s❡ ❛s ❝✉r✈❛t✉r❛s ♣r✐♥❝✐♣❛✐s ❞❡x sã♦ ♣♦s✐t✐✈❛s✱ ♦ ♠❡s♠♦ ♦❝♦rr❡ ❝♦♠ ❛s ❝✉r✈❛t✉r❛s ♣r✐♥❝✐♣❛✐s

❛♥✐s♦tró♣✐❝❛s✳

❉❡♠♦♥str❛çã♦✳ ❉❡ ✭✷✳✶✸✮ t❡♠♦s q✉❡ SF = −AF ◦dN✳ ❉❡♥♦t❡♠♦s ❛s ♠❛tr✐③❡s ❞♦s ❝♦✲ ❡✜❝✐❡♥t❡s ❞❡ AF ❡ −dN ♣♦r A = (Aij) ❡ B = (hij)✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ P❡❧♦ ❢❛t♦ ❞❡

A s❡r ♣♦s✐t✐✈♦ ❡ ♣❡❧♦ ▲❡♠❛ ✷✳✶ ✭❡♠ s✉❛ ❢♦r♠❛ ♠❛tr✐❝✐❛❧✮✱ ❡①✐st❡ ✉♠❛ ♠❛tr✐③ ✐♥✈❡rtí✈❡❧ C t❛❧ q✉❡ A = Ct_{C✳ ❆ss✐♠✱ s❡❣✉❡ ❞❡ |λI − S}

F| = |λI − AB| = |λI − CtCB| =

|Ct_{(λI −CBC}t_)(Ct₎−1_{| = |λI −CBC}t_{|✱ q✉❡ S}

F t❡♠ ♦s ♠❡s♠♦s ❛✉t♦✈❛❧♦r❡s ❞❛ ♠❛tr✐③ s✐♠étr✐❝❛ CBCt_{✱ ❝✉❥♦s ❛✉t♦✈❛❧♦r❡s ✭❝♦♠♦ s❛❜❡♠♦s✮ sã♦ t♦❞♦s r❡❛✐s✳ ❆❧é♠ ❞✐ss♦✱ s❡ ❛s} ❝✉r✈❛t✉r❛s ♣r✐♥❝✐♣❛✐s sã♦ ♣♦s✐t✐✈❛s✱ t❡♠♦s q✉❡B é ♣♦s✐t✐✈♦ ❞❡✜♥✐❞♦✳ ❚♦♠❛♥❞♦ ✉♠ ❛✉t♦✲

✈❛❧♦rλ❞❡SF✱ ❞♦♥❞❡ t❛♠❜é♠ ❛✉t♦✈❛❧♦r ❞❡ CBCt✱ ❡v ✉♠ ❛✉t♦✈❡t♦r ❛ss♦❝✐❛❞♦ ❛ λ✱ t❡♠♦s q✉❡

λhv, vi=

CBCtv, v =

BCtv, Ctv

>0,

✈✐st♦ q✉❡ Ct_{v 6= 0 ❡ B é ♣♦s✐t✐✈♦ ❞❡✜♥✐❞♦✳ ❆ss✐♠✱ λ >0✳}

❖s ❛✉t♦✈❛❧♦r❡s ❞❡SF sã♦ ❝❤❛♠❛❞♦s ❞❡ ❝✉r✈❛t✉r❛s ♣r✐♥❝✐♣❛✐s ❛♥✐s♦tró♣✐❝❛s ❡ ❞❡♥♦t❛❞♦s ♣♦rλ1, ..., λn✳

❚❡♠♦s q✉❡ SF(ej) =P_isijei✳ SF é ❝❤❛♠❛❞♦ ♦ ♦♣❡r❛❞♦r F✲❲❡✐♥❣❛rt❡♥✳ ❖❜s❡r✈❡ q✉❡✱ ❞❡ ✭✷✳✹✮✱ t❡♠♦s

Sk=

1

k!

X

i1,···,ik;j1,···,jk

δj1,···,jk

i1,···,iksi1j1.· · ·.sikjk, ✭✷✳✶✹✮

♦♥❞❡ δj1,···,jk

i1,···,ik é ♦ sí♠❜♦❧♦ ❞❡ ❑r♦♥❡❝❦❡r ❣❡♥❡r❛❧✐③❛❞♦ ✉s✉❛❧✱ ✐st♦ é✱ δ

j1,···,jk

i1,···,ik é ✐❣✉❛❧ ❛ +1

✷✳✷ ❖s ❖♣❡r❛❞♦r❡s Pr✱ Tr ❡ Lr

❞❡(i1,· · · , ik) ❡ ♥♦s ❞❡♠❛✐s ❝❛s♦s é ✐❣✉❛❧ ❛ ③❡r♦✳

❈✐t❛r❡♠♦s três ❧❡♠❛s q✉❡ s❡rã♦ ♥❡❝❡ssár✐♦s ♣❛r❛ ❛s ♣r♦✈❛s ❞♦s ♣ró①✐♠♦s t❡♦r❡♠❛s✳ ▲❡♠❛ ✷✳✸ ✭❬✶✼❪✱ ❬✶✻❪✮✳ ❙❡❥❛ x: M −→ Rn+1 _{✉♠❛ ✐♠❡rsã♦ ✐s♦♠étr✐❝❛ ❞❡ ✉♠❛ ✈❛r✐❡❞❛❞❡}

❘✐❡♠❛♥♥✐❛♥❛ ❢❡❝❤❛❞❛ ❡ ♦r✐❡♥tá✈❡❧ Mn ❡ F : Sn −→ R+ s❛t✐s❢❛③❡♥❞♦ ❛ ❝♦♥❞✐çã♦ ✭✷✳✶✮✳

P❛r❛ ❝❛❞❛r = 0,1,· · · , n−1✱ ❛ s❡❣✉✐♥t❡ ❢ór♠✉❧❛ ❞♦ t✐♣♦ ▼✐♥❦♦✇s❦✐ ♦❝♦rr❡✿ Z

M

HF

r F(N) +HrF+1hx, NidM = 0. ✭✷✳✶✺✮

▲❡♠❛ ✷✳✹ ✭❬✶✼❪✱ ❬✶✻❪✱ ❬✷✼❪✮✳ ❙❡❥❛ x : M −→ Rn+1 _{✉♠❛ ✐♠❡rsã♦ ✐s♦♠étr✐❝❛ ❞❡ ✉♠❛}

✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ ❢❡❝❤❛❞❛ ❡ ♦r✐❡♥tá✈❡❧Mn_{❡F :S}n _−→R+ _{s❛t✐s❢❛③❡♥❞♦ ❛ ❝♦♥❞✐çã♦}

✭✷✳✶✮✳ ❙❡λ1 =λ2 =· · ·=λn =constante6= 0✱ ❡♥tã♦ ❛ ♠❡♥♦s ❞❡ tr❛♥s❧❛çã♦ ❡ ❤♦♠♦t❡t✐❛✱

x(M) é ❛ ❲✉❧✛ s❤❛♣❡ ❞❡ F✳

▲❡♠❛ ✷✳✺ ✭❬✶✶❪✮✳ ❙❡❥❛ x : M −→ Rn+1 _{✉♠❛ ✐♠❡rsã♦ ✐s♦♠étr✐❝❛ ❞❡ ✉♠❛ ✈❛r✐❡❞❛❞❡}

❘✐❡♠❛♥♥✐❛♥❛ ❢❡❝❤❛❞❛ ❡ ♦r✐❡♥tá✈❡❧ Mn ❡ F : Sn −→ R+ s❛t✐s❢❛③❡♥❞♦ ❛ ❝♦♥❞✐çã♦ ✭✷✳✶✮✳

❆ss✉♠❛ q✉❡ ♣❛r❛ ❛❧❣✉♠ 0≤ r ≤n−1✱ HF

r+1 > 0 ❡♠ ❝❛❞❛ ♣♦♥t♦ ❞❡ M✳ ❊♥tã♦ HkF >0 ❡♠ ❝❛❞❛ ♣♦♥t♦ ❞❡ M✱ ♣❛r❛ ❝❛❞❛k = 1,· · · , r✳

✷✳✷ ❖s ❖♣❡r❛❞♦r❡s

P

✱

T

❡

L

❙❡❥❛♠x:Mn_−→Rn+1_{✉♠❛ ✐♠❡rsã♦ s✉❛✈❡ ❞❡ ✉♠❛ ❤✐♣❡rs✉♣❡r❢í❝✐❡ ❢❡❝❤❛❞❛ ❡ ♦r✐❡♥tá✈❡❧}

❡SF ♦ ♦♣❡r❛❞♦r F✲❲❡✐♥❣❛rt❡♥ ❝♦♠♦ ♥❛ ❙❡çã♦ 2.1✳ ■r❡♠♦s ✐♥tr♦❞✉③✐r ❛s tr❛♥s❢♦r♠❛çõ❡s

Pk :X(M)−→X(M), 0≤k≤n,

❛ss♦❝✐❛❞❛s ❛♦ ♦♣❡r❛❞♦r F✲❲❡✐♥❣❛rt❡♥SF ✭✈❡❥❛ ❬✶✼❪✱ ❬✶✻❪✮✳ ❙❡❣✉♥❞♦ ❛ ♥♦ss❛ ❞❡✜♥✐çã♦ ❞❛

r✲és✐♠❛ ❝✉r✈❛t✉r❛ ♠é❞✐❛ ❛♥✐s♦tró♣✐❝❛✱ ❡st❛s tr❛♥s❢♦r♠❛çõ❡s sã♦ ❞❡✜♥✐❞❛s ✐♥❞✉t✐✈❛♠❡♥t❡

❛ ♣❛rt✐r ❞❡SF ♣♦r

P0 =I e Pk =

n k

H_kFI−Pk−1◦SF, ✭✷✳✶✻✮

♣❛r❛ ❝❛❞❛k = 1, ..., n✱ ♦♥❞❡ I ❞❡♥♦t❛ ❛ ✐❞❡♥t✐❞❛❞❡ ❡♠ X(M)✳ ❊q✉✐✈❛❧❡♥t❡♠❡♥t❡✱

Pk = k

X

j=0 (−1)j

n k−j

H_kF_−jS_Fj. ✭✷✳✶✼✮

✷✳✷ ❖s ❖♣❡r❛❞♦r❡s Pr✱ Tr ❡ Lr

◆♦t❡ q✉❡ ❝❛❞❛ Pk(p) é t❛♠❜é♠ ✉♠ ♦♣❡r❛❞♦r ❧✐♥❡❛r ❡♠ ❝❛❞❛ ❡s♣❛ç♦ t❛♥❣❡♥t❡ TpM q✉❡ ❝♦♠✉t❛ ❝♦♠ SF(p)✳ P❛r❛ ❝❛s♦s ♦♥❞❡ SF(p)✱ ❡ ♣♦rt❛♥t♦ Pk(p)✱ é ❞✐❛❣♦♥❛❧✐③á✈❡❧ ✭♣♦r ❡①❡♠♣❧♦ q✉❛♥❞♦ F é ❝♦♥st❛♥t❡✮✱ ❡❧❡s ♣♦❞❡♠ s❡r s✐♠✉❧t❛♥❡❛♠❡♥t❡ ❞✐❛❣♦♥❛❧✐③á✈❡✐s✳ ❆❧é♠

❞✐ss♦✱ ♥❡ss❡s ❝❛s♦s✱ s❡ {e1, ..., en} é ✉♠ r❡❢❡r❡♥❝✐❛❧ ♦rt♦♥♦r♠❛❧ ❡♠ TpM q✉❡ ❞✐❛❣♦♥❛❧✐③❛

SF✱ SF(p)ei =λiei✱ ❡♥tã♦

(Pr)p(ei) =λi,r(p)ei, ✭✷✳✶✽✮

♦♥❞❡

λi,r=

X

i1<...<ir, ij6=i

λi1...λir, ✭✷✳✶✾✮

♣❛r❛ ❝❛❞❛1 ≤r ≤ n ❡ λi,0 := 1✳ ❉❡ ❢❛t♦ ✱ s❡♥❞♦ j ∈ {1, ..., n}✱ ♣♦r ✐♥❞✉çã♦ ❡♠ r t❡♠♦s

q✉❡

P0ei =Iei =ei =λi,0ei. ❆❣♦r❛ s✉♣♦♥❤❛

Pr−1ei =λi,r−1(p)ei, ♦♥❞❡

λi,r−1 =

X

i1<...<ir−1, ij6=i

λi1...λir−1.

❆ss✐♠✱ ❞❡ ✭✷✳✶✻✮

Prei =

n r

H_rFei−(SF ◦Pr−1)ei =

n r

H_rFei −SF(λi,r−1ei)

= X

i1<···<ir

λi1· · ·λir !

ei−λi,r−1SFei

= 

 X

i1<···<ir

λi1· · ·λir −

X

i1<···<ir−1 ij6=i

λi1· · ·λir−1λi



ei

= 



X

i1<···<ir ij6=i

λi1· · ·λir 

ei =λi,rei.

✷✳✷ ❖s ❖♣❡r❛❞♦r❡s Pr✱ Tr ❡ Lr

❖ ♦♣❡r❛❞♦r Tk :X(M)−→X(M), 0≤k ≤n✱ é ❞❡✜♥✐❞♦ ♣♦r

Tk =Pk◦AF. ✭✷✳✷✵✮

◆♦t❡ q✉❡ ♦s ♦♣❡r❛❞♦r❡sTk sã♦ t♦❞♦s ♦♣❡r❛❞♦r❡s ❛✉t♦❛❞❥✉♥t♦s✳ ❉❡ ❢❛t♦✱ ❞❛❞♦sX ❡Y ❡♠

T M✱ ♣♦r ✭✷✳✶✼✮✱ t❡♠♦s

hTkX, Yi = h(Pk◦AF)X, Yi

=

* _k X

j=0 (−1)j

n k−j

H_kF_−jS_Fj ◦AF

!

X, Y

+

=

k

X

j=0 (−1)j

n k−j

H_kF_−j

S_Fj ◦AF

X, Y

= −

k

X

j=0 (−1)j

n k−j

H_kF_−jh(AF ◦dN ◦AF)X, Yi

= −

k

X

j=0 (−1)j

n k−j

HF

k−jhX,(AF ◦dN ◦AF)Yi

= hX,(Pk◦AF)Yi=hX, TkYi,

♦♥❞❡ ❛ q✉✐♥t❛ ✐❣✉❛❧❞❛❞❡ ♦❝♦rr❡ ❞❡✈✐❞♦ AF ❡ dN s❡r❡♠ s✐♠étr✐❝♦s✳ ❉❛ ♠❡s♠❛ ❢♦r♠❛

SF ◦AF✱dN ◦SF ❡dN ◦Pk sã♦ s✐♠étr✐❝♦s✳ ❆❧é♠ ❞✐ss♦✱ ❝♦♠♦

Tk−1◦dN =Pk−1◦AF ◦dN =−Pk−1◦SF, t❡♠♦s q✉❡

Pk =

n k

H_kFI−Pk−1◦SF =

n k

H_kFI+Tk−1◦dN . ✭✷✳✷✶✮

❆ s❡❣✉✐r✱ ❛♣r❡s❡♥t❛♠♦s ✉♠ ❧❡♠❛ q✉❡ s❡rá ✉t✐❧✐③❛❞♦ ♣♦st❡r✐♦r♠❡♥t❡ ✭✈❡❥❛ ❬✶✻❪✮✳ ▲❡♠❛ ✷✳✻✳ ❆ ♠❛tr✐③ ❞❡ Pk é ❞❛❞❛ ♣♦r✿

(Pk)ij =

1

k!

X

i1,···,ik;j1,···,jk

δj1,···,jki

i1,···,ikjsi1j1.· · · .sikjk. ✭✷✳✷✷✮

✷✳✷ ❖s ❖♣❡r❛❞♦r❡s Pr✱ Tr ❡ Lr

❉❡♠♦♥str❛çã♦✳ Pr♦✈❛r❡♠♦s ♣♦r ✐♥❞✉çã♦ ❡♠ k✳ P❛r❛ k = 0✱ é ❢á❝✐❧ ✈❡r✐✜❝❛r q✉❡ ✭✷✳✷✷✮

é ✈❡r❞❛❞❡✐r❛✳ ❙✉♣♦♥❤❛ q✉❡ ♦ r❡s✉❧t❛❞♦ é ✈❡r❞❛❞❡✐r♦ ♣❛r❛ k = r✳ ❱❛♠♦s ♠♦str❛r q✉❡ é

✈❡r❞❛❞❡✐r♦ ♣❛r❛k =r+ 1✳ ❉❡ ✭✷✳✷✶✮✱ ✭✷✳✺✮ ❡ ✭✷✳✶✹✮✱ t❡♠♦s q✉❡ (Pr+1)ij = Sr+1δij−(PrSF)ij =Sr+1δji −

X

l

(Pr)ilslj

= 1

(r+ 1)!

X

i1,···,ir+1;j1,···,jr+1

δj1,···,jr+1

i1,···,ir+lsi1j1.· · · .sir+1jr+1δ

i j

− 1

r!

X

i1,···,ir;j1,···,jr;l

δj1,···,jri

i1,···,irlsi1j1.· · · .sirjrslj

= 1

(r+ 1)! X

(δj1,···,jr+1

i1,···,ir+lδ i j −

X

l

δj1,···,jl−1,i,jl+1,···,jr+1

i1,···,il−l,il,il+1,···,ir+1)si1j1.· · · .sir+1jr+1

= 1

(r+ 1)! X det       

δj1

i1 δ

j2

i1 · · · δ

jr+1

i1 δ

i i1

δj1

i2 δ

j2

i2 · · · δ

jr+1

i2 δ

i i2

· · · · δj1

ir+1 δ

j2

ir+1 · · · δ

jr+1

ir+1 δ

i ir+1

δj1

j δ

j2

j · · · δ jr+1

j δji

      

si1j1.· · · .sir+1jr+1

= 1

(r+ 1)!

X

i1,···,ir+1;j1,···,jr+1

δj1,···,jr+1i

i1,···,ir+ljsi1j1.· · · .sir+1jr+1,

❞❡✈✐❞♦

δj1,···,jl

i1,···,il =det       

δj1

i1 δ

j2

i1 · · · δ

jl−1

i1 δ

jl i1

δj1

i2 δ

j2

i2 · · · δ

jl−1

i2 δ

jl i2

· · · · δj1

il−1 δ

j2

il−1 · · · δ

jl₋1

il−1 δ

jl il−1

δj1

il δ j2

il · · · δ jl−1

il δ jl il        .

❆❣♦r❛ ❞❡✜♥✐♠♦s ♦ ♦♣❡r❛❞♦rLk :C∞(M)−→C∞(M)♣♦r

Lk(f) =div(Tk(gradf)). ✭✷✳✷✸✮

❊q✉✐✈❛❧❡♥t❡♠❡♥t❡✱

Lk(f) =

X

i,j

h

(Tk)ijfj

i

i, ✭✷✳✷✹✮

✷✳✷ ❖s ❖♣❡r❛❞♦r❡s Pr✱ Tr ❡ Lr

♦♥❞❡ ❞❡♥♦t❛♠♦s ♦s ❝♦❡✜❝✐❡♥t❡s ❞❛ ❞✐❢❡r❡♥❝✐❛❧ ❝♦✈❛r✐❛♥t❡ ❞❡f❡Tk❝♦♠ r❡s♣❡✐t♦ ❛{ei}i=1,···,n ♣♦rfi ❡(Tk)ij✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳