Universidade Federal do Espírito Santo Programa de Pós-Graduação em Matemática

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❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❊s♣ír✐t♦ ❙❛♥t♦

Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛

  ❙♦❧✉çõ❡s ❞❡ ❱órt✐❝❡ ❞❛s ❊q✉❛çõ❡s ❞❡ ●✐♥③❜✉r❣✲▲❛♥❞❛✉ ❖❧❡s②❛ ●❛❧❦✐♥❛

  ❱✐tór✐❛

  

❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❊s♣ír✐t♦ ❙❛♥t♦

Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛

  ❖❧❡s②❛ ●❛❧❦✐♥❛

  

❙♦❧✉çõ❡s ❞❡ ❱órt✐❝❡ ❞❛s ❊q✉❛çõ❡s ❞❡ ●✐♥③❜✉r❣✲▲❛♥❞❛✉

  ❉✐ss❡rt❛çã♦ s✉❜♠❡t✐❞❛ ❝♦♠♦ ♣❛rt❡ ❞♦s r❡q✉✐s✐t♦s ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛ ♣❡❧♦ Pr♦✲ ❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥✐✈❡rs✐✲ ❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❊s♣ír✐t♦ ❙❛♥t♦✳ ❖r✐❡♥t❛❞♦r✿ ▼❛❣♥♦ ❇r❛♥❝♦ ❆❧✈❡s

  ❱ít♦r✐❛

  ❖❧❡s②❛ ●❛❧❦✐♥❛

  

❙♦❧✉çõ❡s ❞❡ ❱órt✐❝❡ ❞❛s ❊q✉❛çõ❡s ❞❡ ●✐♥③❜✉r❣✲▲❛♥❞❛✉

  ❉✐ss❡rt❛çã♦ s✉❜♠❡t✐❞❛ ❝♦♠♦ ♣❛rt❡ ❞♦s r❡q✉✐s✐t♦s ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛ ♣❡❧♦ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❊s♣ír✐t♦ ❙❛♥t♦✳

  ❇❆◆❈❆ ❊❳❆▼■◆❆❉❖❘❆ Pr♦❢✳ ❉r✳ ▼❛❣♥♦ ❇r❛♥❝♦ ❆❧✈❡s

  ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❊s♣ír✐t♦ ❙❛♥t♦ ❖r✐❡♥t❛❞♦r

  Pr♦❢✳ ❉r✳ ▲❡♦♥❛r❞♦ ▼❛❣❛❧❤ã❡s ▼❛❝❛r✐♥✐ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❘✐♦ ❞❡ ❏❛♥❡✐r♦

  Pr♦❢✳ ❉r✳ ▲❡♦♥❛r❞♦ ▼❡✐r❡❧❡s ❈â♠❛r❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❊s♣ír✐t♦ ❙❛♥t♦

  ❱ít♦r✐❛

  ❘❡s✉♠♦

  ◆❡st❛ ❞✐ss❡rt❛çã♦ ❡st✉❞❛♠♦s ✉♠ t❡♦r❡♠❛ ❞❡ ❈✳❍✳ ❚❛✉❜❡s s♦❜r❡ s♦❧✉çõ❡s ❞❡ ✈órt✐❝❡ ❞❛s ❡q✉❛çõ❡s ❞❡ ●✐♥③❜✉r❣✲▲❛♥❞❛✉✱ q✉❡ ❞❡s❝r❡✈❡♠ ❛ s✉♣❡r❝♦♥❞✉t✐✈✐✲ ❞❛❞❡✳ P❛r❛ ♣r♦✈❛r ♦ t❡♦r❡♠❛✱ ♣r❡❝✐s❛♠♦s ♠♦str❛r ❛ ❡①✐stê♥❝✐❛ ❞❛ s♦❧✉çã♦ ❞❡ ✉♠❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ♣❛r❝✐❛❧ ❡❧í♣t✐❝❛ ♥ã♦✲❧✐♥❡❛r ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠✳ P❛r❛ ♦❜t❡r ❛ ❡①✐stê♥❝✐❛ ❞❛ s♦❧✉çã♦✱ ❡st✉❞❛♠♦s ✉♠ ❢✉♥❝✐♦♥❛❧ ♥ã♦✲❧✐♥❡❛r ❞❡✜♥✐❞♦ ♥✉♠ ❝❡rt♦ ❡s♣❛ç♦ ❞❡ ❙♦❜♦❧❡✈✱ ❡ ❞❡t❛❧❤❛♠♦s ❛s ❝♦♥t❛s ❞♦ ❛rt✐❣♦ ❞❡ ❚❛✉❜❡s✳ ❚❛♠❜é♠ ✐♥❝❧✉í♠♦s ❞♦✐s ❝❛♣ít✉❧♦s ❛✉①✐❧✐❛r❡s s♦❜r❡ ✜❜r❛❞♦s ❡♠ r❡t❛s ❝♦♠♣❧❡✲ ①♦s ❡ ♣r❡❧✐♠✐♥❛r❡s ❛♥❛❧ít✐❝♦s✳

  P❛❧❛✈r❛s✲❝❤❛✈❡✿ s✉♣❡r❝♦♥❞✉t✐✈✐❞❛❞❡✱ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛s ❡❧í♣t✐❝❛s✱ ❡s♣❛✲ ç♦s ✜❜r❛❞♦s✱ ❡q✉❛çõ❡s ❞❡ ●✐♥③❜✉r❣✲▲❛♥❞❛✉✳

  ❆❜str❛❝t

  ■♥ t❤✐s ✇♦r❦ ✇❡ st✉❞② ❛ t❤❡♦r❡♠ ♦❢ ❈✳❍✳ ❚❛✉❜❡s ❝♦♥❝❡r♥✐♥❣ ✈♦rt❡① s♦❧✉t✐♦♥ t♦ t❤❡ ●✐♥③❜✉r❣✲▲❛♥❞❛✉ ❡q✉❛t✐♦♥s✱ ✇❤✐❝❤ ❞❡s❝r✐❜❡ s✉♣❡r❝♦♥❞✉❝t✐✈✐t②✳ ❚♦ ♣r♦✈❡ t❤❡ t❤❡♦r❡♠ ✇❡ ♥❡❡❞ t♦ s❤♦✇ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❛ s♦❧✉t✐♦♥ t♦ ❛ ♥♦♥✲❧✐♥❡❛r ❡❧✲ ❧✐♣t✐❝ ♣❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ♦❢ s❡❝♦♥❞ ♦r❞❡r✳ ❚♦ ♦❜t❛✐♥ t❤❡ ❡①✐st❡♥❝❡ ♦❢ s♦❧✉t✐♦♥ ✇❡ st✉❞② ❛ ♥♦♥✲❧✐♥❡❛r ❢✉♥❝t✐♦♥❛❧ ❞❡✜♥❡❞ ♦♥ ❛♥ ❛♣♣r♦♣r✐❛t❡ ❙♦❜♦❧❡✈ s♣❛❝❡✳ ❲❡ ❛❧s♦ ✐♥❝❧✉❞❡ t✇♦ ❛✉①✐❧✐❛r② ❝❤❛♣t❡rs ❝♦♥❝❡r♥✐♥❣ ❝♦♠♣❧❡① ❧✐♥❡ ❜✉♥✲ ❞❧❡s ❛♥❞ ❛♥❛❧②t✐❝❛❧ ♣r❡❧✐♠✐♥❛r✐❡s✳

  ❑❡②✲✇♦r❞s✿ s✉♣❡r❝♦♥❞✉❝t✐✈✐t②✱ ❡❧❧✐♣t✐❝ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✱ ❜✉♥❞❧❡ s♣❛❝❡s✱ ●✐♥③❜✉r❣✲▲❛♥❞❛✉ ❡q✉❛t✐♦♥s✳

  ❙✉♠ár✐♦

  ✶ ▲✐♥❡ ❇✉♥❞❧❡s ✼

  ✶✳✶ ▲✐♥❡ ❇✉♥❞❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✷ ❈♦♥❡①õ❡s ❡ ❈✉r✈❛t✉r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✶✳✸ ❈❧❛ss❡s ❞❡ ❈❤❡r♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸

  ✷ Pr❡❧✐♠✐♥❛r❡s ❆♥❛❧ít✐❝❛s ✷✻

  ✷✳✶ ▼❡❞✐❞❛ ❡ ■♥t❡❣r❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✷✳✷ ❊s♣❛ç♦s ❞❡ ❙♦❜♦❧❡✈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✷✳✸ ❆♥á❧✐s❡ ❋✉♥❝✐♦♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷

  ✸ ❚❡♦r❡♠❛ Pr✐♥❝✐♣❛❧ ✸✺

  ✸✳✶ ❱♦rt❡① ◆✉♠❜❡r ❡ ❋ór♠✉❧❛ ❞❡ ❇♦❣♦♠♦❧✬♥②✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✸✳✷ ❚❡♦r❡♠❛ Pr✐♥❝✐♣❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✸✳✸ Pr♦♣r✐❡❞❛❞❡s ❞♦ ❢✉♥❝✐♦♥❛❧ G ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✸✳✹ Pr♦✈❛ ❞♦ ❚❡♦r❡♠❛ Pr✐♥❝✐♣❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✸

  ❈❛♣ít✉❧♦ ✶ ▲✐♥❡ ❇✉♥❞❧❡s ✶✳✶ ▲✐♥❡ ❇✉♥❞❧❡s

  ❯♠ ❧✐♥❡ ❜✉♥❞❧❡ é ✉♠❛ tr✐♣❧❛ (L, M, π) ❢♦r♠❛❞❛ ♣♦r ✈❛r✐❡❞❛❞❡s L ❡ M ❡ ✉♠❛ ♣r♦❥❡çã♦ s✉❛✈❡ π : L → M t❛❧ q✉❡

  −1

  (m) ✶✳ ❈❛❞❛ ✜❜r❛ π é ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❝♦♠♣❧❡①♦ ❞❡ ❞✐♠❡♥sã♦ ❝♦♠✲

  ♣❧❡①❛ ✶❀

  

α

  ✷✳ ❊①✐st❡ ✉♠❛ ❝♦❜❡rt✉r❛ ❛❜❡rt❛ {U } ❞❡ M ❡ ❞✐❢❡♦♠♦r✜s♠♦s

  −1

  φ : π (U )

  α α α

  → U × C

  −1 α α (π (m))

  t❛✐s q✉❡ ♣❛r❛ t♦❞♦ ♣♦♥t♦ m ❡♠ U t❡♠♦s q✉❡ φ ⊂ {m} × C ❡ ❛ r❡str✐çã♦

  

−1

−1

  φ : π (m)

  α π (m)

  | → {m} × C é ✉♠ ✐s♦♠♦r✜s♠♦ C✲❧✐♥❡❛r✳

  α

  ❆ ❝♦❜❡rt✉r❛ {U } é ✉♠❛ ❝♦❜❡rt✉r❛ tr✐✈✐❛❧✐③❛❞♦r❛✳ ❊①❡♠♣❧♦ ✶✳✶✳✶ ✭▲✐♥❡ ❜✉♥❞❧❡ tr✐✈✐❛❧✮✳ ❈♦♥s✐❞❡r❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ M✳ ❖ ❧✐♥❡ ❜✉♥❞❧❡ tr✐✈✐❛❧ é ♦ ♣r♦❞✉t♦ L = M × C ❝♦♠ ❛ ♣r♦❥❡çã♦ π : L → M ❞❛❞❛ ♣♦r

  ❈❆P❮❚❯▲❖ ✶✳ ▲■◆❊ ❇❯◆❉▲❊❙ ✽

  ❱❛♠♦s ♠♦str❛r q✉❡ π : L → M é ✉♠ ❧✐♥❡ ❜✉♥❞❧❡✳

  

−1 −1

  (m) = (m) ✶✳ ❚❡♠♦s q✉❡ π {m}×C✳ P♦❞❡♠♦s ✈❡r q✉❡ π é ✉♠ ❡s♣❛ç♦

  ✈❡t♦r✐❛❧ ❝♦♠♣❧❡①♦ ❞❡ ❞✐♠❡♥sã♦ ❝♦♠♣❧❡①❛ ✶ ❝♦♠ ❛s ♦♣❡r❛çõ❡s (m, z) + (m, w) = (m, z + w) ,

  ❡ α

  (m, z) = (m, αz) , ♦♥❞❡ z✱ w ❡ α sã♦ ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✳

  ✷✳ ❚❡♠♦s q✉❡ {M} é ✉♠❛ ❝♦❜❡rt✉r❛ ❛❜❡rt❛ ❞❡ M ✳ ❆❧é♠ ❞✐ss♦✱ ❛ ❛♣❧✐❝❛çã♦ ✐❞❡♥t✐❞❛❞❡

  Id : L

  → M × C

  −1

  (m)) = s❛t✐s❢❛③ Id (π {m} × C✱ ❡ ♣♦rt❛♥t♦

  −1 −1

  Id : π (m)

  π (m)

  | → {m} × C é ✉♠ ✐s♦♠♦r✜s♠♦ C✲❧✐♥❡❛r✳

  ❈♦♥s✐❞❡r❡ ✉♠ ❧✐♥❡ ❜✉♥❞❧❡ π : L → M✳ ❯♠❛ ❛♣❧✐❝❛çã♦ s : U ⊂ M → L é ✉♠❛ s❡çã♦ ❧♦❝❛❧ s❡ ♣❛r❛ t♦❞♦ ♣♦♥t♦ m ❡♠ U✱ t❡♠♦s q✉❡ s (m) ❡stá ❡♠

  −1

  π (m) ✱ ♦✉ s❡❥❛✱ π .

  U

  ◦ s = Id ◗✉❛♥❞♦ U = M✱ ❞✐③❡♠♦s q✉❡ s é ✉♠❛ s❡çã♦ ❣❧♦❜❛❧✳ ❊①❡♠♣❧♦ ✶✳✶✳✷ ✭❙❡çõ❡s ❞♦ ❧✐♥❡ ❜✉♥❞❧❡ tr✐✈✐❛❧✮✳ ❈♦♥s✐❞❡r❡ ♦ ❧✐♥❡ ❜✉♥❞❧❡ tr✐✈✐❛❧ L = M × C✳ P❛r❛ t♦❞❛ ❢✉♥çã♦ f : M → C✱ t❡♠♦s q✉❡ ❛ ❛♣❧✐❝❛çã♦ s

  : M → L ❞❛❞❛ ♣♦r s (m) = (m, f (m)) é ✉♠❛ s❡çã♦ ❣❧♦❜❛❧✳ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱

  ♣❛r❛ t♦❞❛ s❡çã♦ ❣❧♦❜❛❧ s : M → L✱ t❡♠♦s q✉❡ s (m) ❡stá ❡♠ {m}×C✱ ♣♦rt❛♥t♦ ❡①✐st❡ ✉♠❛ ❢✉♥çã♦ f : M → C t❛❧ q✉❡ s (m) = (m, f (m))✳ Pr♦♣♦s✐çã♦ ✶✳✶✳✸✳ ❈♦♥s✐❞❡r❡ ✈❛r✐❡❞❛❞❡s L ❡ M ❡ ✉♠❛ ♣r♦❥❡çã♦ π : L → M

  ✱ s✉♣♦♥❤❛ q✉❡ ✈❛❧❡ ❛ ❝♦♥❞✐çã♦ ✶ ❞❛ ❞❡✜♥✐çã♦ ❞❡ ❧✐♥❡ ❜✉♥❞❧❡✳ ❚❡♠♦s q✉❡

  ❈❆P❮❚❯▲❖ ✶✳ ▲■◆❊ ❇❯◆❉▲❊❙ ✾

  ❉❡♠♦♥str❛çã♦✳ ❙✉♣♦♥❤❛ q✉❡ π : L → M é ✉♠ ❧✐♥❡ ❜✉♥❞❧❡✳ ❚♦♠❡ ✉♠❛

  α

  ❝♦❜❡rt✉r❛ ❛❜❡rt❛ {U } ❞❡ M ❡ ❞✐❢❡♦♠♦r✜s♠♦s

  −1

  φ : π (U )

  α α α

  → U × C

  −1

  (π (m))

  α α

  t❛✐s q✉❡ ♣❛r❛ t♦❞♦ ♣♦♥t♦ m ❡♠ U t❡♠♦s q✉❡ φ ⊂ {m} × C ❡ ❛ r❡str✐çã♦

  −1 −1

  φ : π (m)

  α π (m)

  | → {m} × C : U

  α α

  é ✉♠ ✐s♦♠♦r✜s♠♦ C✲❧✐♥❡❛r✳ P❛r❛ ❝❛❞❛ α✱ ❞❡✜♥✐♠♦s ❛ s❡çã♦ ❧♦❝❛❧ s → L ♣♦r

  −1

  s (m) = φ (m, 1) .

  α

α

  (m) ❚❡♠♦s q✉❡ s α 6= 0 ♣❛r❛ t♦❞♦ m ❡♠ U α ✳

  α

  ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s✉♣♦♥❤❛ q✉❡ ❡①✐st❡ ✉♠❛ ❝♦❜❡rt✉r❛ ❛❜❡rt❛ {U } ❞❡ M ❡

  α : U α

  s❡çõ❡s ❧♦❝❛✐s s → L q✉❡ ♥ã♦ s❡ ❛♥✉❧❛♠✳ ❉❡✜♥✐♠♦s ❛ ❛♣❧✐❝❛çã♦

  −1

  φ : π (U )

  α α α

  → U × C ♣♦r z

  −1

  φ m, (z) =

  α π (m) ✭✶✳✶✮

  | s (m)

  α −1 −1 −1

  (m) (U α ) (m) ♣❛r❛ ❝❛❞❛ π ❡♠ π ✳ ❆q✉✐ ✉s❛♠♦s q✉❡ π é ✉♠ ❡s♣❛ç♦

  α

  ✈❡t♦r✐❛❧ ❝♦♠♣❧❡①♦ ❞❡ ❞✐♠❡♥sã♦ ❝♦♠♣❧❡①❛ ✶ ❡ s ♥ã♦ s❡ ❛♥✉❧❛✳ ❚❡♠♦s q✉❡ ❛ ✐♥✈❡rs❛ é

  −1 −1

  φ : U (U )

  α α α × C → π

  ❞❛❞❛ ♣♦r

  −1 φ (m, λ) = λs (m) .

  α α −1 −1

  (U ) (m)

  α

  ❉❡ ❢❛t♦✱ ✭✶✮ ❉❛❞♦ ✉♠ z ❡♠ π ✱ t❡♠♦s q✉❡ z ❡stá π ♣❛r❛ ❛❧❣✉♠

  ❈❆P❮❚❯▲❖ ✶✳ ▲■◆❊ ❇❯◆❉▲❊❙ ✶✵ m

  α

  ❡♠ U ✱ ♣♦rt❛♥t♦

  −1 −1 −1

  φ (z) = φ (z)

  α α π (m) α ◦ φ α ◦ φ |

  z

  

−1

  = φ m,

  

α

  s (m)

  α

  z s = α (m) s

  (m)

  

α

  = z;

  −1

  (m) (m)

  α α

  ✭✷✮ ❉❛❞♦ ✉♠ (m, λ) ❡♠ U ×C✱ t❡♠♦s q✉❡ λs ❡stá ❡♠ π ✱ ♣♦rt❛♥t♦

  −1

  φ (m, λ) = φ (λs (m))

  α α α

  ◦ φ α

  −1

  = φ (λs (m))

  α π (m) α

  | λs (m)

  α

  m, = s

  (m)

  α = (m, λ) . −1

  (π (m))

  α

  ❆ ❊q✉❛çã♦ ✭✶✳✶✮ ♠♦str❛ q✉❡ φ ⊂ {m} × C ❡

  −1 −1 −1

  φ (z + w) = φ (z) + φ (w) ,

  α π (m) α π (m) α π (m)

  | | |

  −1 −1 −1

  φ (zw) = φ (z) φ (w) ,

  α | π (m) α | π (m) α | π (m)

  ♣♦rt❛♥t♦

  −1 −1

  φ

  α : π (m)

  | π (m) → {m} × C é ✉♠ ✐s♦♠♦r✜s♠♦ C✲❧✐♥❡❛r✳

  2

  ❊①❡♠♣❧♦ ✶✳✶✳✹ ✭❋✐❜r❛❞♦ t❛♥❣❡♥t❡ ❞❛ ❡s❢❡r❛✮✳ ▲❡♠❜r❡ q✉❡ ❛ ❡s❢❡r❛ S é

  3

  ❞❡✜♥✐❞❛ ❝♦♠♦ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♣♦♥t♦s (x, y, z) ❡♠ R t❛✐s q✉❡

  2

  2

  2 x + y + z = 1.

  2 P♦❞❡♠♦s ✈❡r q✉❡ ♦ ❡s♣❛ç♦ t❛♥❣❡♥t❡ ♥♦ ♣♦♥t♦ p ❡♠ S é ❞❛❞♦ ♣♦r

  ❈❆P❮❚❯▲❖ ✶✳ ▲■◆❊ ❇❯◆❉▲❊❙ ✶✶

  2

  ▲❡♠❜r❡ q✉❡ ♦ ✜❜r❛❞♦ t❛♥❣❡♥t❡ à ❡s❢❡r❛ S é ❞❡✜♥✐❞♦ ❝♦♠♦ [

  2

  2 S T S = . p 2 {p} × T p∈S

  2

  2

  ❈♦♥s✐❞❡r❡ ❛ ♣r♦❥❡çã♦ π : T S → S ❞❛❞❛ ♣♦r π (p, v) = p.

  2 2 −1

  (p) = ❱❛♠♦s ♠♦str❛r q✉❡ π : T S → S é ✉♠ ❧✐♥❡ ❜✉♥❞❧❡✳ ❚❡♠♦s q✉❡ π

2 S

  T

  p

  é ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❝♦♠♣❧❡①♦ ❞❡ ❞✐♠❡♥sã♦ ❝♦♠♣❧❡①❛ ✶ ❝♦♠ ❛s ♦♣❡r❛✲ çõ❡s

  (p, v) + (p, w) = (p, v + w) , ❡

  (α + iβ) (p, v) = (p, αv + βn

  p

  × v) ,

  p

  ♦♥❞❡ n é ♦ ✈❡t♦r ♥♦r♠❛❧ ✉♥✐tár✐♦ ♥♦ ♣♦♥t♦ p✳ ❖❜s❡r✈❡ q✉❡ iv é ✉♠❛ r♦t❛çã♦

  π

  ❞❡ ✉♠ â♥❣✉❧♦ ❞❛❞❛ ♣♦r

  2

  iv = n

  

p

× v.

  ❱❛♠♦s✱ ♣♦r ❡①❡♠♣❧♦✱ ♠♦str❛r q✉❡ [(α + iβ) (γ + iδ)] v = (α + iβ) [(γ + iδ) v] .

  ❚❡♠♦s q✉❡ [(α + iβ) (γ + iδ)] v = [αγ

  − βδ + i (αδ + βγ)] v = (αγ

  p − βδ) v + (αδ + βγ) n × v.

  2

  ◆❛ ❡s❢❡r❛ S ✱ t❡♠♦s q✉❡ i (iv) = i (n

  p

  × v) = n

  p p

  × (n × v) =

  −v,

  ❈❆P❮❚❯▲❖ ✶✳ ▲■◆❊ ❇❯◆❉▲❊❙ ✶✷

  ♣♦rt❛♥t♦ (α + iβ) [(γ + iδ) v] = (α + iβ) (γv + δn

  p

  × v) = α (γv + δn

  p p p

  × v) + βn × (γv + δn × v) = (aγ

  p

  − βδ) v + (aδ + βγ) n × v, ❧♦❣♦ [(α + iβ) (γ + iδ)] v = (α + iβ) [(γ + iδ) v] .

  ➱ ♣♦ssí✈❡❧ ✈❡r✐✜❝❛r q✉❡ ❛s ❞❡♠❛✐s ♣r♦♣r✐❡❞❛❞❡s ❞❡ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❝♦♠♣❧❡①♦

  −1

  (p) ✈❛❧❡♠ ❡♠ π ✳

  

2

  2

  ➱ ❢á❝✐❧ ✈❡r q✉❡ ❛s s❡çõ❡s ❞❡ T S → S sã♦ ♦s ❝❛♠♣♦s ❞❡ ✈❡t♦r❡s ♥❛

  2

  ❡s❢❡r❛ S ✳ P♦❞❡♠♦s ♦❜t❡r ❝❛♠♣♦s ❞❡ ✈❡t♦r❡s ♥❛ ❡s❢❡r❛ q✉❡ ❧♦❝❛❧♠❡♥t❡ ♥ã♦ s❡ ❛♥✉❧❛♠✳ ■st♦ ♣♦❞❡ ❢❡✐t♦✱ ♣♦r ❡①❡♠♣❧♦✱ ✉s❛♥❞♦ ❝♦♦r❞❡♥❛❞❛s ♣♦❧❛r❡s✳ P❡❧❛

2 Pr♦♣♦s✐çã♦ ✶✳✶✳✸✱ ❝♦♥❝❧✉í♠♦s q✉❡ T S é ✉♠ ❧✐♥❡ ❜✉♥❞❧❡✳

  1

  ❊①❡♠♣❧♦ ✶✳✶✳✺ ✭❋✐❜r❛❞♦ ❞❡ ❍♦♣❢✮✳ ❆ r❡t❛ ♣r♦❥❡t✐✈❛ ❝♦♠♣❧❡①❛ CP é ❞❡✜✲

  2

  ♥✐❞❛ ❝♦♠♦ ♦ ❡s♣❛ç♦ C \ {(0, 0)} ❝♦♠ ❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ ∼✱ ❞❡✜♥✐❞❛

  ∗

  , z ) , w )

  1

  2

  1

  2

  ♣♦r (z ∼ (w s❡ ❡①✐st❡ λ ❡♠ C t❛❧ q✉❡ w , = λz

  1

  1 w .

  = λz

  2

  2

  1

  , z , z ) ]

  ❆ ❝❧❛ss❡ ❞❡ (z

  1 2 é ❞❡♥♦t❛❞❛ ♣♦r [z

  

1

2 ✳ ❱❛♠♦s ♠♦str❛r q✉❡ CP é ✉♠❛

  ✈❛r✐❡❞❛❞❡ ❝♦♠♣❧❡①❛ ❞❡ ❞✐♠❡♥sã♦ ❝♦♠♣❧❡①❛ ✶✳ ❉❡✜♥✐♠♦s ♦s ❝♦♥❥✉♥t♦s ❛❜❡rt♦s

  1 U = [z , z ] : z ,

  1

  1

  2

  1

  ∈ CP 6= 0

  1 U = [z , z ] : z ,

  2

  1

  2

  2

  ∈ CP 6= 0 : U : U

  1

  

1

  2

  2

  ❡ ❞❡✜♥✐♠♦s ❞♦✐s ❞✐❢❡♦♠♦r✜s♠♦s ψ → C ❡ ψ → C ❞❛❞♦s ♣♦r z

  2

  ψ , z , ([z ]) =

  1

  1

  

2

  z

  1

  ❈❆P❮❚❯▲❖ ✶✳ ▲■◆❊ ❇❯◆❉▲❊❙ ✶✸

  ❘❡♣❛r❡ q✉❡

  ∗

  ψ ,

  1 (U

  1 2 ) = C

  ∩ U

  ∗

  ψ ,

  2 (U

  1 2 ) = C

  ∩ U ❡

  −1 ∗ ∗

  ψ : C

  2

  ◦ ψ

  1 → C

  1 z . 7→ z

  1

  ■st♦ ♠♦str❛ q✉❡ CP é ✈❛r✐❡❞❛❞❡ ❝♦♠♣❧❡①❛ ❞❡ ❞✐♠❡♥sã♦ ❝♦♠♣❧❡①❛ ✶✳ ❖ ✜❜r❛❞♦ ❞❡ ❍♦♣❢ H é ❞❡✜♥✐❞♦ ♣♦r

  2

  1

  2 H = : z .

  (z, [z]) ∈ C × CP ∈ C \ {(0, 0)} P♦❞❡♠♦s ✈❡r q✉❡ ♦ ✜❜r❛❞♦ ❞❡ ❍♦♣❢ H é ✉♠❛ ✈❛r✐❡❞❛❞❡ ❝♦♠♣❧❡①❛ ❞❡ ❞✐♠❡♥sã♦

  1

  ❝♦♠♣❧❡①❛ ✷✳ ❉❡✜♥✐♠♦s ❛ ♣r♦❥❡çã♦ π : H → CP ❞❛❞❛ ♣♦r π (z, [z]) = [z]✳

  1

  ❱❛♠♦s ♠♦str❛r q✉❡ π : H → CP é ✉♠ ❧✐♥❡ ❜✉♥❞❧❡✳ ❚❡♠♦s q✉❡ ❛ ✜❜r❛

  −1

  π ([z]) é ❞❛❞❛ ♣♦r

  

−1 ∗

  π ([z]) = {(λz, [z]) : λ ∈ C } .

  −1

  ([z]) P♦❞❡♠♦s ✈❡r q✉❡ π é ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❝♦♠♣❧❡①♦ ❞❡ ❞✐♠❡♥sã♦ ❝♦♠✲ ♣❧❡①❛ ✶ ❝♦♠ ❛s ♦♣❡r❛çõ❡s

  (λz, [z]) + (µz, [z]) = ((λ + µ) z, [z]) , λ (µz, [z]) = ((λµ) z, [z]) .

  : U : U

  1

  1

  2

  2

  ❉❡✜♥✐♠♦s ❛s s❡çõ❡s ❧♦❝❛✐s s → H ❡ s → H ❞❛❞❛s ♣♦r z

  2

  s , z , , z , ([z ]) = 1, [z ]

  1

  1

  2

  1

  2

  z

  1

  z

  1

  s ([z , z ]) = , 1 , [z , z ] .

  2

  1

  2

  1

  2

  ❈❆P❮❚❯▲❖ ✶✳ ▲■◆❊ ❇❯◆❉▲❊❙ ✶✹

  1

  2

  ❚❡♠♦s q✉❡ ❛s s❡çõ❡s ❧♦❝❛✐s s ❡ s ♥ã♦ s❡ ❛♥✉❧❛♠✳ P❡❧❛ Pr♦♣♦s✐çã♦ ✶✳✶✳✸✱

  1

  ❝♦♥❝❧✉í♠♦s q✉❡ H é ✉♠ ❧✐♥❡ ❜✉♥❞❧❡ s♦❜r❡ CP ✳ ❈♦♥s✐❞❡r❡ ✉♠ ❧✐♥❡ ❜✉♥❞❧❡ π : L → M ❝♦♠ ✉♠❛ ❝♦❜❡rt✉r❛ tr✐✈✐❛❧✐③❛❞♦r❛

  : U

  α α α

  {U } ❞❡ M✳ P❡❧❛ Pr♦♣♦s✐çã♦ ✶✳✶✳✸✱ ♣♦❞❡♠♦s t♦♠❛r s❡çõ❡s ❧♦❝❛✐s s → L

  α β

  q✉❡ ♥ã♦ s❡ ❛♥✉❧❛♠✳ ❊♠ U ∩ U ❡s❝r❡✈❛ s = g s .

  

α αβ β

  : U ❆s ❢✉♥çõ❡s g αβ α β sã♦ ❛s ❢✉♥çõ❡s ❞❡ tr❛♥s✐çã♦ ❞❡ s α ✳

  ∩ U → C ❊①❡♠♣❧♦ ✶✳✶✳✻ ✭❋✉♥çõ❡s ❞❡ tr❛♥s✐çã♦ ❞♦ ✜❜r❛❞♦ ❞❡ ❍♦♣❢✮✳ ▲❡♠❜r❡ q✉❡ ♦ ✜❜r❛❞♦ ❞❡ ❍♦♣❢ é ❞❛❞♦ ♣♦r

  2 H ,

  = (z, [z]) : z ∈ C \ {(0, 0)}

  1

  ❝♦♠ ❛ ♣r♦❥❡çã♦ π : H → CP ❞❛❞❛ ♣♦r π (z, [z]) = [z] ,

  ∗

  ♦♥❞❡ [z] = {λz : λ ∈ C }✳ ❈♦♥s✐❞❡r❡ ♦s ❝♦♥❥✉♥t♦s ❛❜❡rt♦s

  1 U , z

  = [z ] : z ,

  1

  1

  2

  1

  ∈ CP 6= 0 ❡

  1 U = [z , z ] : z ,

  2

  1

  2

  2

  ∈ CP 6= 0 : U : U

  1

  1

  2

  2

  ❡ ❛s s❡çõ❡s s → H ❡ s → H ❞❛❞❛s ♣♦r z

  2

  s , z , , z , ([z ]) = 1, [z ]

  1

  1

  2

  1

  2

  z

  1

  z

  1

  s ([z , z ]) = , 1 , [z , z ] .

  2

  1

  2

  1

  2

  z

  2

  ❚❡♠♦s q✉❡

  ❈❆P❮❚❯▲❖ ✶✳ ▲■◆❊ ❇❯◆❉▲❊❙ ✶✺ s

  β

  α

  ξ

  αβ

  = g

  β

  ✳ ❚❡♠♦s q✉❡ ξ

  β

  ∩ U

  α

  ❡♠ U

  s

  αβ

  β

  = ξ

  U β

  |

  , ξ

  α

  s

  α

  = ξ

  U α

  , ♦♥❞❡ g

  sã♦ ❛s ❢✉♥çõ❡s ❞❡ tr❛♥s✐çã♦ ❞❡ s

  ❈♦♥s✐❞❡r❡ ✉♠❛ s❡çã♦ ❣❧♦❜❛❧ ξ : M → L ❡ ❡s❝r❡✈❛ ξ

  g

  g

  

α

  = ξ

  β

  ✱ ♦❜t❡♠♦s ξ

  β

  ❉✐✈✐❞✐♥❞♦ ♦s ❞♦✐s ❧❛❞♦s ♣♦r s

  β .

  s

  αβ

  

α

  α

  = ξ

  α

  = ξ α s

  U α ∩U β

  |

  β = ξ

  s

  β

  ξ

  ✳ ❉❡♠♦♥str❛çã♦✳ ❚❡♠♦s q✉❡

  |

  α → L q✉❡ ♥ã♦ s❡ ❛♥✉❧❛♠✳

  1

  2

  1

  z

  1

  z

  2

  ] = z

  2

  , z

  1

  , 1 , [z

  z

  2

  1

  z

  

1

  z

  

2

  ]) = z

  2

  , z

  1

  ([z

  z

  , 1 , [z

  Pr♦♣♦s✐çã♦ ✶✳✶✳✼✳ ❈♦♥s✐❞❡r❡ ✉♠ ❧✐♥❡ ❜✉♥❞❧❡ π : L → M ❝♦♠ ✉♠❛ ❝♦❜❡rt✉r❛ tr✐✈✐❛❧✐③❛❞♦r❛ {U α } ❞❡ M ❡ s❡çõ❡s ❧♦❝❛✐s s α : U

  : U

  1 .

  z

  2

  z

  2 ]) =

  , z

  1

  12 ([z

  ❞❛❞❛ ♣♦r g

  1 ∩ U 2 → C ∗

  12

  1

  , z 2 ]) . ❖❜t❡♠♦s ❛ ❢✉♥çã♦ ❞❡ tr❛♥s✐çã♦ g

  1

  2 ([z

  s

  1

  z

  2

  ] = z

  2

  , z

  αβ .

  ❈❆P❮❚❯▲❖ ✶✳ ▲■◆❊ ❇❯◆❉▲❊❙ ✶✻

  Pr♦♣♦s✐çã♦ ✶✳✶✳✽✳ ❈♦♥s✐❞❡r❡ ✉♠ ❧✐♥❡ ❜✉♥❞❧❡ π : L → M ❝♦♠ ✉♠❛ ❝♦❜❡rt✉r❛

  α α : U α

  tr✐✈✐❛❧✐③❛❞♦r❛ {U } ❞❡ M ❡ s❡çõ❡s ❧♦❝❛✐s s → L q✉❡ ♥ã♦ s❡ ❛♥✉❧❛♠✳

  αβ α

  ❉❡♥♦t❡ ♣♦r g ❛s ❢✉♥çõ❡s ❞❡ tr❛♥s✐çã♦ ❞❡ s ✳ ❚❡♠♦s q✉❡ = 1

  αα α

  ✶✳ g ❡♠ U ❀ g = 1

  αβ βα α β

  ✷✳ g ❡♠ U ∩ U ✭s❡ ♥ã♦ ✈❛③✐♦✮❀ g g = 1

  αβ βγ γα α β γ

  ✸✳ g ❡♠ U ∩ U ∩ U ✭s❡ ♥ã♦ ✈❛③✐♦✮✳ s .

  = g =

  α αα α α

  ❉❡♠♦♥str❛çã♦✳ ✶✳ ❚❡♠♦s q✉❡ s P♦r ♦✉tr♦ ❧❛❞♦✱ t❡♠♦s q✉❡ s . 1 = 1.

  α αα

  · s P♦rt❛♥t♦ g ✷✳ ❚❡♠♦s q✉❡ s s

  α = g αβ β g s .

  = g αβ βα α = 1 . g = 1.

  α α αβ βα

  P♦r ♦✉tr♦ ❧❛❞♦✱ t❡♠♦s q✉❡ s · s P♦rt❛♥t♦ g ✸✳ ❚❡♠♦s q✉❡ s s

  = g

  α αβ β

  g s = g

  

αβ βγ γ

g g s .

  = g

  αβ βγ γα α

  . g g α = 1 α αβ βγ γα = 1. P♦r ♦✉tr♦ ❧❛❞♦✱ t❡♠♦s q✉❡s · s P♦rt❛♥t♦ g

  α

  ❖❜s❡r✈❛çã♦ ✶✳✶✳✾✳ ➱ ❝♦♥❤❡❝✐❞♦ q✉❡ ❞❛❞❛ ✉♠❛ ❝♦❜❡rt✉r❛ ❛❜❡rt❛ {U } ❞❡ ✉♠❛

  ∗

  : U

  

αβ α β

  ✈❛r✐❡❞❛❞❡ M✱ s❡ ❡①✐st❡♠ ❢✉♥çõ❡s g ∩U → C s❛t✐s❢❛③❡♥❞♦ ❛s ❝♦♥❞✐çõ❡s ✶✱ ✷ ❡ ✸ ❞❛ Pr♦♣♦s✐çã♦ ✶✳✶✳✽✱ ❡♥tã♦ ❡①✐st❡✱ ❛ ♠❡♥♦s ❞❡ ✐s♦♠♦r✜s♠♦✱ ✉♠ ❧✐♥❡

  : U

  α α

  ❜✉♥❞❧❡ L → M ❡ s❡çõ❡s ❧♦❝❛✐s s → L q✉❡ ♥ã♦ s❡ ❛♥✉❧❛♠ t❛✐s q✉❡ s = g s

  

α αβ β

  ❡♠ U α β ❬✽✱ ♣✳ ✺❪✳ ∩ U

  ❈❆P❮❚❯▲❖ ✶✳ ▲■◆❊ ❇❯◆❉▲❊❙ ✶✼

  ✶✳✷ ❈♦♥❡①õ❡s ❡ ❈✉r✈❛t✉r❛

  ❯♠❛ ❝♦♥❡①ã♦ ∇ ♥✉♠ ❧✐♥❡ ❜✉♥❞❧❡ L → M é ✉♠❛ ❛♣❧✐❝❛çã♦ ❧✐♥❡❛r ∇ : Γ (L) →

  ∗

  Γ (T M ⊗ L) t❛❧ q✉❡ ♣❛r❛ t♦❞❛ s❡çã♦ s : M → L ❡ ❢✉♥çã♦ f : M → C ✈❛❧❡ ❛ r❡❣r❛ ❞❡ ▲❡✐❜♥✐③

  ∇(fs) = df ⊗ s + f∇s. ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ ✉♠❛ ❝♦♥❡①ã♦ ∇ ♥✉♠ ❧✐♥❡ ❜✉♥❞❧❡ L → M é ✉♠❛

  ❛♣❧✐❝❛çã♦ q✉❡ ❛ss♦❝✐❛ ❝❛❞❛ s❡çã♦ s : M → L ❡ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s X ❡♠ M s : M

  X

  ❝♦♠ ✉♠❛ s❡çã♦ ∇ → L✱ ❡ t❛❧ q✉❡

  1

  2

  ✶✳ P❛r❛ ✉♠❛ s❡çã♦ s : M → L ❡ ❝❛♠♣♦s ❞❡ ✈❡t♦r❡s X ❡ X ❡♠ M✱ t❡♠♦s q✉❡ 1 2 = s s 1 2 ; s

  • X +X

  X X

  ∇ ∇ ∇ ✷✳ P❛r❛ ✉♠❛ s❡çã♦ s : M → L✱ ✉♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s X ❡♠ M ❡ ✉♠❛

  ❢✉♥çã♦ f : M → C✱ t❡♠♦s q✉❡ s = f s ;

  f X

  X

  ∇ ∇ , s : M

  1

  2

  ✸✳ P❛r❛ s❡çõ❡s s → L ❡ ✉♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s X ❡♠ M✱ t❡♠♦s q✉❡ s s

  (s + + s ) = ;

  X

  1

2 X

  1 X

  2

  ∇ ∇ ∇ ✹✳ P❛r❛ ✉♠❛ s❡çã♦ s : M → L✱ ✉♠❛ ❢✉♥çã♦ f : M → C ❡ ✉♠ ❝❛♠♣♦ ❞❡

  ✈❡t♦r❡s X ❡♠ M✱ t❡♠♦s q✉❡ s.

  X (f s) = df (X) s + f

  X

  ∇ ∇ ❊①❡♠♣❧♦ ✶✳✷✳✶ ✭❈♦♥❡①ã♦ ❞♦ ❧✐♥❡ ❜✉♥❞❧❡ tr✐✈✐❛❧✮✳ ❈♦♥s✐❞❡r❡ ♦ ❧✐♥❡ ❜✉♥❞❧❡ tr✐✈✐❛❧ L = M × C✳ ▲❡♠❜r❡ q✉❡ ❛s s❡çõ❡s ❡♠ L → M sã♦ ❞❛ ❢♦r♠❛ s

  (m) = (m, f (m)) ,

  ❈❆P❮❚❯▲❖ ✶✳ ▲■◆❊ ❇❯◆❉▲❊❙ ✶✽

  ♦♥❞❡ f : M → C✳ P♦❞❡♠♦s ❞❡✜♥✐r ❛ ❝♦♥❡①ã♦ ∇ ❞❛❞❛ ♣♦r ∇s = df.

  ▼❛✐s ♣r❡❝✐s❛♠❡♥t❡✱ ( s ) (m) = (m, df (X )) .

  

X m m

  ∇ ❱❛♠♦s ♠♦str❛r q✉❡ ∇ é ✉♠❛ ❝♦♥❡①ã♦ ❡♠ L → M✳ ❚♦♠❡ ❞✉❛s s❡çõ❡s s

  (m) = (m, f (m)) ,

  1

  ❡ s

  

2 (m) = (m, h (m)) ,

  ❡ ✉♠❛ ❢✉♥çã♦ φ : M → C✳ ❚❡♠♦s q✉❡ ( (s + s )) (m) = (m, d (f + h) (X ))

  X

  1

2 m

  ∇

  m

  = (m, df (X ) + dh (X ))

  m m m m

  = (m, df (X )) + (m, dh (X ))

  m m m m

  s s = ( ) (m) + ( ) (m) .

  X

  

1

X

  2

  ∇ ∇ ❆❧❡♠ ❞✐ss♦✱

  (

  X (φs 1 )) (m) = (m, d (φf ) (X m ))

  ∇

  m

  = (m, φ (m) df (X ) + f (m) dφ (X ))

  m m m m

  = φ (m) (m, df (X )) + dφ (X ) (m, f (m))

  m m m m = dφ (X ) s (m) + φ (m) ( s ) (m) . m m

1 X

  1

  ∇ P♦rt❛♥t♦✱

  • ,
  • s ) =

  1

  2

  1

  2

  ∇ (s ∇s ∇s .

  ) = dφ + φ

  1

  1

  1

  ∇ (φs ⊗ s ∇s Pr♦♣♦s✐çã♦ ✶✳✷✳✷✳ ❬✽✱ ♣✳ ✽❪ ❈♦♥s✐❞❡r❡ ✉♠ ❧✐♥❡ ❜✉♥❞❧❡ π : L → M ❝♦♠ ✉♠❛

  ❈❆P❮❚❯▲❖ ✶✳ ▲■◆❊ ❇❯◆❉▲❊❙ ✶✾ .

  U = U

  ∇˜s| ∇¯s| P♦rt❛♥t♦ ♣♦❞❡♠♦s ❞❡✜♥✐r .

  U

  ∇s = ∇˜s| ❈♦♥s✐❞❡r❡ ✉♠ ❧✐♥❡ ❜✉♥❞❧❡ L → M ❝♦♠ ✉♠❛ ❝♦♥❡①ã♦ ∇✱ ✉♠❛ ❝♦❜❡rt✉r❛

  : U

  α α α

  tr✐✈✐❛❧✐③❛❞♦r❛ {U } ❡ s❡çõ❡s ❧♦❝❛✐s s → L q✉❡ ♥ã♦ s❡ ❛♥✉❧❛♠✳ P❛r❛

  α α

  ❝❛❞❛ α✱ ❡①✐st❡ ✉♠❛ ú♥✐❝❛ ✶✲❢♦r♠❛ A ❡♠ U t❛❧ q✉❡ .

  = A

  α α α

  ∇s ⊗ s ▼❛✐s ♣r❡❝✐s❛♠❡♥t❡✱ s

  ( X α ) (m) = (A α ) (X m ) s α (m) .

  ∇

  m α α

  ❆s ✶✲❢♦r♠❛s A sã♦ ❝❤❛♠❛❞❛s ✶✲❢♦r♠❛s ❞❡ ❝♦♥❡①ã♦ ❞❡ s ✳ ❖❜s❡r✈❡ q✉❡ ❛ ❞❡✜♥✐çã♦ é ❥✉st✐✜❝❛❞❛ ♣❡❧❛ ♣r♦♣♦s✐çã♦ ❛❝✐♠❛✳

  Pr♦♣♦s✐çã♦ ✶✳✷✳✸✳ ❈♦♥s✐❞❡r❡ ✉♠ ❧✐♥❡ ❜✉♥❞❧❡ L → M ❝♦♠ ✉♠❛ ❝♦♥❡①ã♦ ∇✱ : U

  α α α

  ✉♠❛ ❝♦❜❡rt✉r❛ tr✐✈✐❛❧✐③❛❞♦r❛ {U } ❞❡ M ❡ s❡çõ❡s ❧♦❝❛✐s s → L q✉❡ ♥ã♦ s❡ ❛♥✉❧❛♠✳ ❚❡♠♦s q✉❡

  −1

  A dg , = A + g

  

α β αβ

αβ

  ♦♥❞❡ g αβ sã♦ ❛s ❢✉♥çõ❡s ❞❡ tr❛♥s✐çã♦ ❞❡ s α ❡ A α sã♦ ❛s ✶✲❢♦r♠❛s ❞❡ ❝♦♥❡①ã♦ ❞❡ s α ✳ ❉❡♠♦♥str❛çã♦✳ ❚❡♠♦s q✉❡

  = s )

  α αβ β

  ∇s ∇ (g = dg + g

  αβ β αβ β

  ⊗ s ∇s = dg + g A

  αβ β αβ β β

  ⊗ s ⊗ s = (dg + g A ) .

  αβ αβ β β

  ⊗ s

  ❈❆P❮❚❯▲❖ ✶✳ ▲■◆❊ ❇❯◆❉▲❊❙ ✷✵

  P♦r ♦✉tr♦ ❧❛❞♦ t❡♠♦s q✉❡ = A

  

α α α

  ∇s ⊗ s = g A .

  αβ α β

  ⊗ s P♦rt❛♥t♦ g A A .

  = dg + g

  

αβ α αβ αβ β

−1

  ▼✉❧t✐♣❧✐❝❛♥❞♦ ❛♠❜♦s ❧❛❞♦s ♣♦r g αβ ✱ ♦❜t❡♠♦s

  −1 A dg .

α = A β + g αβ

αβ

  Pr♦♣♦s✐çã♦ ✶✳✷✳✹✳ ❈♦♥s✐❞❡r❡ ✉♠ ❧✐♥❡ ❜✉♥❞❧❡ L → M ❝♦♠ ✉♠❛ ❝♦♥❡①ã♦ ∇✱ : U

  α α α

  ✉♠❛ ❝♦❜❡rt✉r❛ tr✐✈✐❛❧✐③❛❞♦r❛ {U } ❡ s❡çõ❡s ❧♦❝❛✐s s → L q✉❡ ♥ã♦ s❡

  ∇

  ❛♥✉❧❛♠✳ ❊①✐st❡ ✉♠❛ ✷✲❢♦r♠❛ F ❡♠ M t❛❧ q✉❡ F = dA ,

  

∇ U α α

  |

  α α

  ♦♥❞❡ A sã♦ ❛s ✶✲❢♦r♠❛s ❞❡ ❝♦♥❡①ã♦ ❞❡ s ✳

  ∇

  ❆ ✷✲❢♦r♠❛ F é ❛ ❝✉r✈❛t✉r❛ ❞❛ ❝♦♥❡①ã♦ ∇✳ = dA

  α β α β

  ❉❡♠♦♥str❛çã♦✳ ❇❛st❛ ♠♦str❛r q✉❡ dA ❡♠ U ∩ U ✳ ❚❡♠♦s q✉❡

  −1

  dA = d A + g dg

  α β αβ αβ −2 −1

  dg d = dA + g (dg )

  β αβ αβ αβ

  − g αβ ∧ dg αβ . = dA

  β

  2

  = 0 ❆❝✐♠❛ ✉s❛♠♦s ♦ ❢❛t♦ q✉❡ d ❡ ω ∧ ω = 0 ♣❛r❛ t♦❞❛ ✶✲❢♦r♠❛ ω✳

  ′

  Pr♦♣♦s✐çã♦ ✶✳✷✳✺✳ ❈♦♥s✐❞❡r❡ ✉♠ ❧✐♥❡ ❜✉♥❞❧❡ L → M ❝♦♠ ❝♦♥❡①õ❡s ∇ ❡ ∇ ✳ ❊①✐st❡ ✉♠❛ ✶✲❢♦r♠❛ η ❡♠ M t❛❧ q✉❡

  ❈❆P❮❚❯▲❖ ✶✳ ▲■◆❊ ❇❯◆❉▲❊❙ ✷✶

  ′

  F

∇ = F ∇ + dη.

  α α :

  ❉❡♠♦♥str❛çã♦✳ ❚♦♠❡ ✉♠❛ ❝♦❜❡rt✉r❛ tr✐✈✐❛❧✐③❛❞♦r❛ {U } ❞❡ M ❡ s❡çõ❡s s U

  α

  → L q✉❡ ♥ã♦ s❡ ❛♥✉❧❛♠✳ P♦❞❡♠♦s ❡s❝r❡✈❡r = A ,

  α α α

  ∇s ⊗ s

  ′ ′ s .

  = A

  α α

  ∇ α ⊗ s

  α

  ❉❡✜♥✐♠♦s ❛ ✶✲❢♦r♠❛ η ❡♠ U ♣♦r

  

η .

α = A α

  − A

  

α

  = η

  α β α β αβ

  ❱❛♠♦s ♠♦str❛r q✉❡ ❡♠ U ∩ U ✈❛❧❡ η ✳ ❉❡♥♦t❡ ♣♦r g ❛s ❢✉♥çõ❡s ❞❡

  α

  tr❛♥s✐çã♦ ❞❡ s ✳ ❚❡♠♦s q✉❡

  ′

  η = A

  α α α − A −1 −1

  ′

  = A + g dg A + g dg

  αβ β αβ β αβ − αβ ′

  = A β − A

  β = η . β

  ❚❡♠♦s q✉❡

  ′ ′

  s = A

  α

  ∇ α ⊗ s ,

  = (A + η )

  α α α

  ⊗ s ❧♦❣♦

  ′

  s = ∇ ∇s + η ⊗ s.

  ❚❛♠❜é♠ t❡♠♦s q✉❡

  ′ ′

  F = dA

  ∇ α

  = dA + dη

  α α

  ❈❆P❮❚❯▲❖ ✶✳ ▲■◆❊ ❇❯◆❉▲❊❙ ✷✷

  Pr♦♣♦s✐çã♦ ✶✳✷✳✻✳ ❈♦♥s✐❞❡r❡ ✉♠ ❧✐♥❡ ❜✉♥❞❧❡ L → M ❝♦♠ ✉♠❛ ❝♦♥❡①ã♦ ∇✳ ❙✉♣♦♥❤❛ q✉❡ ❡①✐st❡ ✉♠❛ s❡çã♦ ξ : M → L q✉❡ ♥ã♦ s❡ ❛♥✉❧❛ ❡ t❛❧ q✉❡ ∇ξ = 0.

  ❚❡♠♦s q✉❡ ❛ ❝✉r✈❛t✉r❛ ❞❛ ❝♦♥❡①ã♦ é ③❡r♦✱ ♦✉ s❡❥❛ F = 0.

  ∇ α

  ❉❡♠♦♥str❛çã♦✳ ❚♦♠❡ ✉♠❛ ❝♦❜❡rt✉r❛ tr✐✈✐❛❧✐③❛❞♦r❛ {U } ❞❡ M ❡ s❡çõ❡s ❧♦❝❛✐s s : U

  α α

  → L q✉❡ ♥ã♦ s❡ ❛♥✉❧❛♠✱ ❡ ❡s❝r❡✈❛ ξ = ξ s .

  

U α α α

  | ❚❡♠♦s q✉❡ s

  α = )

  ∇ξ| U ∇ (ξ α α = dξ α α + ξ α α

  ⊗ s ∇s A

  = dξ α α + ξ α α α ⊗ s ⊗ s = (dξ + ξ A ) .

  α α α α

  ⊗ s ❈♦♠♦ ∇ξ = 0✱ t❡♠♦s q✉❡

  ξ A = ,

  

α α α

  −dξ ❧♦❣♦ dξ

  α

  A

  α =

  − ξ

  α = d (ln ξ ) .

  α

  ❈❆P❮❚❯▲❖ ✶✳ ▲■◆❊ ❇❯◆❉▲❊❙ ✷✸

  P♦rt❛♥t♦ F = ddA

  

∇ α

= 0.

  ✶✳✸ ❈❧❛ss❡s ❞❡ ❈❤❡r♥

  ❚❡♦r❡♠❛ ✶✳✸✳✶✳ ❬✽✱ ♣✳ ✶✷❪ ❈♦♥s✐❞❡r❡ ✉♠ ❧✐♥❡ ❜✉♥❞❧❡ L → Σ s♦❜r❡ ✉♠❛ s✉♣❡r❢í❝✐❡ ❝♦♠♣❛❝t❛ s❡♠ ❜♦r❞♦ Σ ❝♦♠ ✉♠❛ ❝♦♥❡①ã♦ ∇✳ ❚❡♠♦s q✉❡ ˆ

1 F

  ∇

  2πi

  

Σ

  é ✉♠ ✐♥t❡✐r♦ q✉❡ ♥ã♦ ❞❡♣❡♥❞❡ ❞❡ ∇✳ ´

  1

  (L) = F

  1 ∇

  ❖ ♥ú♠❡r♦ c é ❛ ❝❧❛ss❡ ❞❡ ❈❤❡r♥✳

  2πi Σ

  ❊①❡♠♣❧♦ ✶✳✸✳✷✳ ❈♦♥s✐❞❡r❡ ♦ ❧✐♥❡ ❜✉♥❞❧❡ tr✐✈✐❛❧ L = Σ × C s♦❜r❡ ✉♠❛ s✉♣❡r❢í❝✐❡ ❝♦♠♣❛❝t❛ Σ✳ ❙❛❜❡♠♦s q✉❡ ∇ = d

  ∇

  é ✉♠❛ ❝♦♥❡①ã♦ ❡♠ L → Σ✳ ❙❛❜❡♠♦s q✉❡ ❛ ❝✉r✈❛t✉r❛ F é ♥✉❧❛✳ P♦rt❛♥t♦ ❛ ❝❧❛ss❡ ❞❡ ❈❤❡r♥ é ❞❛❞❛ ♣♦r

  ˆ

  1 c F (Σ

  

1 ∇

  × C) = 2πi

  Σ = 0.

  2

  2

  ❊①❡♠♣❧♦ ✶✳✸✳✸✳ ❈♦♥s✐❞❡r❡ ♦ ❧✐♥❡ ❜✉♥❞❧❡ T S → S ✳ ▲❡♠❜r❡ q✉❡ ❛s ❝♦♦r❞❡✲

  ❈❆P❮❚❯▲❖ ✶✳ ▲■◆❊ ❇❯◆❉▲❊❙ ✷✹

  ∇

  2

  ✳ ❈♦♥s✐❞❡ ❛ s❡çã♦ s : S

  2

  → T S

  2

  ❞❛❞❛ ♣♦r s = ( − sin θ, cos θ, 0) . ❚❡♠♦s q✉❡

  R 3

  2

  s = dθ ⊗ (− cos θ, − sin θ, 0) . ▲❡♠❜r❡ q✉❡ ❛ ❝♦♥❡①ã♦ ∇ ❞❡ T S

  2

  → S

  2

  é ❛ ♣r♦❥❡çã♦ ❞❡ ∇

  R 3

  sã♦ ❝❛♠♣♦s ❞❡ ✈❡t♦r❡s ❡♠ S

  → S

  ♥❛❞❛s ❡s❢ér✐❝❛s ❞❡ S

  ∂φ = (cos θ cos φ, sin θ cos φ, − sin φ) .

  2

  sã♦          x

  (θ, φ) = sin φ cos θ y (θ, φ) = sin φ sin θ z (θ, φ) = cos φ .

  ❆ ❜❛s❡ ❞♦ ♣❧❛♥♦ t❛♥❣❡♥t❡ é ❢♦r♠❛❞❛ ♣❡❧♦s ✈❡t♦r❡s ∂

  ∂θ = (

  − sin θ sin φ, cos θ sin φ, 0) , ∂

  ❖ ✈❡t♦r ♥♦r♠❛❧ ✉♥✐tár✐♦ n é ❞❛❞♦ ♣♦r n =

  2

  ∂ ∂φ

  ×

  ∂ ∂θ ∂ ∂φ

  ×

  ∂ ∂θ

  = (cos θ sin φ, sin θ sin φ, cos φ) . ▲❡♠❜r❡ q✉❡ ❛s s❡çõ❡s ❞♦ ❧✐♥❡ ❜✉♥❞❧❡ T S

  ♥♦ ♣❧❛♥♦ t❛♥❣❡♥t❡✱

  ❈❆P❮❚❯▲❖ ✶✳ ▲■◆❊ ❇❯◆❉▲❊❙ ✷✺

  ♣♦rt❛♥t♦ 3 R 3 s R s, n > n

  ∇s = ∇ − < ∇ 3 R = s + dθ

  ∇ ⊗ sin φn

  2

  2

  = dθ φ, φ, sin φ cos φ ⊗ − cos θ cos − sin θ cos

  = dθ ⊗ cos φ (n × s)

  = i cos φdθ ⊗ s. ❘❡s✉♠✐♥❞♦✱ t❡♠♦s q✉❡ ∇s = i cos φdθ ⊗ s.

  P♦rt❛♥t♦ ❛ ✶✲❢♦r♠❛ ❞❡ ❝♦♥❡①ã♦ ❞❛ s❡çã♦ s é ❞❛❞❛ ♣♦r A

  = i cos φdθ, ❡ ❝✉r✈❛t✉r❛ é ❞❛❞❛ ♣♦r

  F = d (i cos φdθ)

  ∇

  = −i sin φdφ ∧ dθ. P♦rt❛♥t♦ ❛ ❝❧❛ss❡ ❞❡ ❈❤❡r♥ é ❞❛❞❛ ♣♦r

  ˆ

  1

  2

  c T S = F

  1 ∇

2

  2πi S ˆ 2π ˆ π

  −1 = sin φdφ dθ

  2π = −2.

  ❈❛♣ít✉❧♦ ✷ Pr❡❧✐♠✐♥❛r❡s ❆♥❛❧ít✐❝❛s ✷✳✶ ▼❡❞✐❞❛ ❡ ■♥t❡❣r❛çã♦

  ❈♦♥s✐❞❡r❡ ✉♠ ❝♦♥❥✉♥t♦ X✳ ❯♠❛ ❢❛♠✐❧✐❛ A ❞❡ s✉❜❝♦♥❥✉♥t♦s ❞❡ X é ✉♠❛ σ

  ✲á❧❣❡❜r❛ s❡✿ ✶✳ ∅ ❡ X ❡stã♦ ❡♠ A✱ ✷✳ ❙❡ E ∈ A✱ ❡♥tã♦ ♦ ❝♦♠♣❧❡♠❡♥t❛r X \ E ∈ A❀

  S n , ..., E , ... E

  1 n i

  ✸✳ ❙❡ E ∈ A✱ ❡♥tã♦ ❛ ✉♥✐ã♦ i=1 ∈ A✳ ❖s ❡❧❡♠❡♥t♦s ❞❡ A sã♦ ❝♦♥❥✉♥t♦s ♠❡♥s✉rá✈❡✐s✳

  ❈♦♥s✐❞❡r❡ ✉♠ ❝♦♥❥✉♥t♦ X ❡ ✉♠❛ ❢❛♠í❧✐❛ S ❞❡ s✉❜❝♦♥❥✉♥t♦s ❞❡ X✳ ➱ ♣♦ssí✈❡❧ ♠♦str❛r q✉❡ ❛ ✐♥t❡rs❡çã♦ ❞❡ t♦❞❛s ❛s σ✲á❧❣❡❜r❛s ❝♦♥t❡♥❞♦ S é ✉♠❛ σ

  ✲á❧❣❡❜r❛✱ ❝❤❛♠❛❞❛ σ✲á❧❣❡❜r❛ ❣❡r❛❞❛ ♣♦r S✳ ❆ σ✲á❧❣❡❜r❛ ❣❡r❛❞❛ ♣❡❧❛ ❢❛♠í❧✐❛

  n

  ❞♦s ❝♦♥❥✉♥t♦s ❛❜❡rt♦s ❡♠ R é ❛ σ✲á❧❣❡❜r❛ ❞❡ ❇♦r❡❧ ❬✺✱ ✾❪✳ ❈♦♥s✐❞❡r❡ ✉♠ ❝♦♥❥✉♥t♦ X ❝♦♠ ✉♠❛ σ✲á❧❣❡❜r❛ A✳ ❯♠❛ ♠❡❞✐❞❛ é ✉♠❛

  ❢✉♥çã♦ µ :

  A → [0, ∞] , , E , ...

  1

  2

  t❛❧ q✉❡ ❞❛❞♦s ♦s ❝♦♥❥✉♥t♦s ❞✐s❥✉♥t♦s E ∈ A✱ t❡♠♦s q✉❡ !

  ∞ ∞

  [

  X

  ❈❆P❮❚❯▲❖ ✷✳ P❘❊▲■▼■◆❆❘❊❙ ❆◆❆▲❮❚■❈❆❙ ✷✼

  n

  ❚❡♦r❡♠❛ ✷✳✶✳✶✳ ❬✺✱ ✾❪ ❊①✐st❡ ú♥✐❝❛ ♠❡❞✐❞❛ µ ♥❛ σ✲á❧❣❡❜r❛ ❞❡ ❇♦r❡❧ ❞❡ R t❛❧ q✉❡ µ , b , b

  ([a

  1

1 ] n n ]) = (b n n ) ... (b

  1 1 ) .

  × ... × [a − a − a ❈♦♥s✐❞❡r❡ ✉♠ ❝♦♥❥✉♥t♦ X ❝♦♠ ✉♠❛ σ✲á❧❣❡❜r❛ A✳ ❯♠❛ ❢✉♥çã♦ f : X → R

  é ♠❡♥s✉rá✈❡❧ s❡ ♣❛r❛ t♦❞♦ a ❡♠ R t❡♠♦s q✉❡ {x ∈ X : f (x) > a} ∈ A.

  ❯♠ ❡s♣❛ç♦ ❞❡ ♠❡❞✐❞❛ é ✉♠❛ tr✐♣❧❛ (X, A, µ) ❢♦r♠❛❞❛ ♣♦r ✉♠ ❝♦♥❥✉♥t♦

  X ❝♦♠ ✉♠❛ σ✲á❧❣❡❜r❛ A ❡ ✉♠❛ ♠❡❞✐❞❛ µ✳

  Pr♦♣♦s✐çã♦ ✷✳✶✳✷✳ ❬✺✱ ✾❪ ❚❡♠♦s q✉❡ ✶✳ ❙❡ f ❡ g sã♦ ❛s ❢✉♥çõ❡s ♠❡♥s✉rá✈❡✐s✱ ❡♥tã♦ f + g ❡ fg sã♦ ❢✉♥çõ❡s

  ♠❡♥s✉rá✈❡✐s✳ ✷✳ ❙❡ f ❡ g sã♦ ❛s ❢✉♥çõ❡s ♠❡♥s✉rá✈❡✐s✱ ❡♥tã♦ max {f, g} ❡ min {f, g} sã♦

  ♠❡♥s✉rá✈❡✐s✳

  n

  ✸✳ ❙❡ ✉♠❛ s❡q✉❡♥❝✐❛ ❞❡ ❢✉♥çõ❡s ♠❡♥s✉rá✈❡✐s f ❝♦♥✈❡r❣❡ ♣♦♥t✉❛❧♠❡♥t❡ ♣❛r❛ ✉♠❛ ❢✉♥çã♦ f✱ ❡♥tã♦ f é ♠❡♥s✉rá✈❡❧✳

  n

  ❖❜s❡r✈❛çã♦ ✷✳✶✳✸✳ ❚♦❞❛ ❢✉♥çã♦ ❝♦♥t✐♥✉❛ f : R → R é ♠❡♥s✉rá✈❡❧ ✭❇♦r❡❧✮✳ ❆ r❡❝í♣r♦❝❛ é ❢❛❧s❛ ❬✺❪✳

  ❈♦♥s✐❞❡r❡ ✉♠ ❡s♣❛ç♦ ❞❡ ♠❡❞✐❞❛ (X, A, µ) . ❯♠❛ ❢✉♥çã♦ s : X → R é , ..., a s✐♠♣❧❡s s❡ ❡❧❛ ❛ss✉♠❡ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ✈❛❧♦r❡s {a

  1 n

  }✳ ❆ ✐♥t❡❣r❛❧ ❞❡ s é ❞❡✜♥✐❞❛ ♣♦r

  n

  ˆ

  X

  −1

  sdµ a µ s , = i (a i )

  X i=1 −1

  (a ) =

  i i ♦♥❞❡ s {x ∈ X : s (x) = a } .

  ❈♦♥s✐❞❡r❡ ✉♠ ❡s♣❛ç♦ ❞❡ ♠❡❞✐❞❛ (X, A, µ) ❡ ❢✉♥çã♦ ♠❡♥s✉rá✈❡❧ f : X → [0,

  ∞)✳ ❆ ✐♥t❡❣r❛❧ ❞❡ f é ❞❡✜♥✐❞❛ ♣♦r ˆ ˆ

  ❈❆P❮❚❯▲❖ ✷✳ P❘❊▲■▼■◆❆❘❊❙ ❆◆❆▲❮❚■❈❆❙ ✷✽ ❈♦♥s✐❞❡r❡ ✉♠ ❡s♣❛ç♦ ❞❡ ♠❡❞✐❞❛ (X, A, µ) ❡ ✉♠❛ ❢✉♥çã♦ ♠❡♥s✉rá✈❡❧ f :

  X → R✳ P♦❞❡♠♦s ❡s❝r❡✈❡r

  − +

  f , = f

  − f

  = max = max ♦♥❞❡ f {f, 0} ❡ f {−f, 0}✳ ❆ ❢✉♥çã♦ f é ✐♥t❡❣rá✈❡❧ s❡

  • ´ ´

  −

  f dµ < f dµ < ∞ ❡ ∞✳ ❆ ✐♥t❡❣r❛❧ ❞❡ f é ❞❡✜♥✐❞❛ ♣♦r

  X X

  ˆ ˆ ˆ

  − + f dµ f dµ f dµ.

  = −

  X X

  X

  • ´

  −

  f dµ < dµ < f ❖❜s❡r✈❡ q✉❡ ✈❛❧❡ ´ ∞ ❡ ∞ s❡✱ ❡ s♦♠❡♥t❡ s❡✱

  X X

  ´ |f| dµ < ∞✳

  X

  ❚❡♦r❡♠❛ ✷✳✶✳✹ ✭❚❡♦r❡♠❛ ❞❛ ❝♦♥✈❡r❣ê♥❝✐❛ ♠♦♥ót♦♥❛✮✳ ❬✺✱ ✾❪ ❈♦♥s✐❞❡r❡ ✉♠❛

  n

  s❡q✉❡♥❝✐❛ ❞❡ ❢✉♥çõ❡s ♠❡♥s✉rá✈❡✐s f ✳ ❙✉♣♦♥❤❛ q✉❡ (x) (x)

  1

  2 ≤ f ≤ f ≤ ... n

  ♣❛r❛ t♦❞♦ x✳ ❆ss✉♠❛ q✉❡ f ❝♦♥✈❡r❣❡ ♣♦♥t✉❛❧♠❡♥t❡ ♣❛r❛ ✉♠❛ ❢✉♥çã♦ f✳ ❊♥tã♦

  ˆ ˆ f = lim f .

  n

n

  ❚❡♦r❡♠❛ ✷✳✶✳✺ ✭❚❡♦r❡♠❛ ❞❛ ❝♦♥✈❡r❣ê♥❝✐❛ ❞♦♠✐♥❛❞❛✮✳ ❬✺✱ ✾❪ ❈♦♥s✐❞❡r❡ ✉♠❛

  n n

  s❡q✉❡♥❝✐❛ ❞❡ ❢✉♥çõ❡s ♠❡♥s✉rá✈❡✐s f ✳ ❙✉♣♦♥❤❛ q✉❡ f ❝♦♥✈❡r❣❡ ♣♦♥t✉❛❧♠❡♥t❡ ♣❛r❛ ✉♠❛ ❢✉♥çã♦ f✳ ❆ss✉♠❛ q✉❡ ❡①✐st❡ ✉♠❛ ❢✉♥çã♦ ✐♥t❡❣rá✈❡❧ g t❛❧ q✉❡

  n (x)

  |f | ≤ g (x) ♣❛r❛ t♦❞♦ n ❡ x✳ ❊♥tã♦

  ˆ ˆ f f .

  = lim

  n

n

✷✳✷ ❊s♣❛ç♦s ❞❡ ❙♦❜♦❧❡✈

  2 n n

  (R ) ❖ ❡s♣❛ç♦ L é ♦ ❝♦♥❥✉♥t♦ ❞❛s ❢✉♥çõ❡s ♠❡♥s✉rá✈❡✐s f ❡♠ R t❛✐s q✉❡

  ❈❆P❮❚❯▲❖ ✷✳ P❘❊▲■▼■◆❆❘❊❙ ❆◆❆▲❮❚■❈❆❙ ✷✾

  

2 n

  (R ) ❆ ♥♦r♠❛ ❞❡ ✉♠❛ ❢✉♥çã♦ f ❡♠ L é ❞❡✜♥✐❞❛ ♣♦r

  ˆ

  2 2 n

  2 f .

  L (R ) n

  = k f k

  R

  2

  2

  (R ) ❖ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❞❡ ❞✉❛s ❢✉♥çõ❡s f ❡ g ❡♠ L é ❞❡✜♥✐❞♦ ♣♦r 2 ˆ n f g.

  = hf, gi L (R )

  n R n

  ❯♠❛ ❢✉♥çã♦ ♠❡♥s✉rá✈❡❧ f ❡♠ R é ❧♦❝❛❧♠❡♥t❡ ✐♥t❡❣rá✈❡❧ s❡ ´ |f| < ∞

  K

n

  ♣❛r❛ t♦❞♦ ❝♦♥❥✉♥t♦ ❝♦♠♣❛❝t♦ K ❡♠ R ✳

  n

  ❈♦♥s✐❞❡r❡ ✉♠❛ ❢✉♥çã♦ ❧♦❝❛❧♠❡♥t❡ ✐♥t❡❣rá✈❡❧ f ❡♠ R ✳ ❯♠❛ ❢✉♥çã♦ ❧♦❝❛❧✲

  n

  f

  i

  ♠❡♥t❡ ✐♥t❡❣rá✈❡❧ ∂ ❡♠ R é ❛ ✐✲❡s✐♠❛ ❞❡r✐✈❛❞❛ ❢r❛❝❛ ❞❡ f s❡ ˆ ˆ f ∂ φ = φ∂ f

  i i

  −

  n n R R ∞ n

  (R ) ♣❛r❛ t♦❞❛ ❢✉♥çã♦ φ ❡♠ C c ✳ ❆ ❞❡✜♥✐çã♦ ❛❝✐♠❛ é ❥✉st✐✜❝❛❞❛ ♣❡❧❛s s❡✲ ❣✉✐♥t❡s ❛✜r♠❛çõ❡s✿

  ✶✳ ❙❡ ❡①✐st❡♠ ❢✉♥çõ❡s ❧♦❝❛❧♠❡♥t❡ ✐♥t❡❣rá✈❡✐s g ❡ h t❛✐s q✉❡ ˆ ˆ gφ = hφ

  n n R R ∞ n

  (R ) ♣❛r❛ t♦❞❛ ❢✉♥çã♦ φ ❡♠ C ✱ ❡♥tã♦ g = h✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ❛ ✐✲és✐♠❛

  c

  ❞❡r✐✈❛❞❛ ❢r❛❝❛✱ s❡ ❡①✐st✐r✱ é ú♥✐❝❛✳ ✷✳ ❙❡ ❛ ❢✉♥çã♦ f é s✉❛✈❡✱ ✐♥t❡❣r❛♥❞♦ ♣♦r ♣❛rt❡s✱ ♦❜t❡♠♦s

  ˆ ˆ f ∂ φ = φ (∂ f )

  i i

  −

  

n n

R R

∞ n

  (R ) ♣❛r❛ t♦❞❛ φ ❡♠ C ✳ P❡❧♦ ✐t❡♠ ✶✱ ❞❡❞✉③✐♠♦s q✉❡ ❛ ✐✲és✐♠❛ ❞❡r✐✈❛❞❛

  c

  ❢r❛❝❛ ❞❡ f é ✐❣✉❛❧ ❛ ✐✲és✐♠❛ ❞❡r✐✈❛❞❛ ♣❛r❝✐❛❧ ✭❝❧áss✐❝❛✮ ❞❡ f✳ ▼❛✐s ✐♥❢♦r♠❛çõ❡s s♦❜r❡ ❞❡r✐✈❛❞❛s ❢r❛❝❛s ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞❛s ❡♠ ❬✼✱ ❝❛♣

  ❈❆P❮❚❯▲❖ ✷✳ P❘❊▲■▼■◆❆❘❊❙ ❆◆❆▲❮❚■❈❆❙ ✸✵

  1 n

  (R ) ❖ ❡s♣❛ç♦ ❞❡ ❙♦❜♦❧❡✈ H é ♦ ❝♦♥❥✉♥t♦ ❞❛s ❢✉♥çõ❡s ❧♦❝❛❧♠❡♥t❡ ✐♥t❡✲

  n

  f, . . . , ∂ f

  1 n

  ❣rá✈❡✐s f ❡♠ R t❛✐s q✉❡ ❛s ❞❡r✐✈❛❞❛s ❢r❛❝❛s ∂ ❡①✐st❡♠✱ ❡ ❛❧❡♠ ❞✐ss♦

  ˆ

  

2

  f < ∞,

  n R

  ˆ

  2

  < |∇f| ∞,

  n R

  f, . . . , ∂ f )

  1 n

  ♦♥❞❡ ∇f = (∂ ✳

  

1 n

  (R ) ❆ ♥♦r♠❛ ❞❡ ✉♠❛ ❢✉♥çã♦ f ❡♠ H é ❞❡✜♥✐❞❛ ♣♦r

  ˆ ˆ

  2

  2

  2 1 + n = f .

  k f k H (R ) |∇f|

  n n R R 1 n

  (R ) ❖ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❞❡ ❢✉♥çõ❡s f ❡ g ❡♠ H é ❞❡✜♥✐❞♦ ♣♦r 1 ˆ ˆ

  n f g

  • H (R )

  = hf, gi h∇f, ∇gi .

  n n R R 1 n 1 n

  (R ) ❚❡♦r❡♠❛ ✷✳✷✳✶✳ ❬✶✱ ✹✱ ✼❪ ❖ ❡s♣❛ç♦ H ❝♦♠ ♣r♦❞✉t♦ ✐♥t❡r♥♦ hf, gi

  H (R )

  é ✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt✳

  1 n

  (R ) ▼❛✐s ✐♥❢♦r♠❛çõ❡s s♦❜r❡ H ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞❛s ❡♠ ❬✼✱ ❝❛♣ ✼❪ ❡

  ❬✹✱ ❝❛♣ ✺❪✳

  p n n

  (R ) ❖ ❡s♣❛ç♦ L é ♦ ❝♦♥❥✉♥t♦ ❞❛s ❢✉♥çõ❡s ♠❡♥s✉rá✈❡✐s f ❡♠ R t❛✐s q✉❡

  ˆ

  

p

  < |f| ∞.

  n R

p n

  (R ) ❆ ♥♦r♠❛ ❞❡ ✉♠❛ ❢✉♥çã♦ f ❡♠ L é ❞❡✜♥✐❞❛ ♣♦r

  ˆ

  p p p n = .

  k f k |f|

  L (R ) n

  R

  ❊①✐st❡♠ ✈ár✐♦s t❡♦r❡♠❛s s♦❜r❡ ✐♠❡rsõ❡s ❞❡ ❙♦❜♦❧❡✈✳ ❱❛♠♦s ✉s❛r ♦ s❡✲ ❣✉✐♥t❡ r❡s✉❧t❛❞♦ ♥♦ ❈❛♣ít✉❧♦ ✸✳ ❚❡♦r❡♠❛ ✷✳✷✳✷ ✭❉❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❙♦❜♦❧❡✈✮✳ ❬✼✱ ♣✳ ✷✵✻❪ P❛r❛ t♦❞❛ ❢✉♥çã♦ f

  ❈❆P❮❚❯▲❖ ✷✳ P❘❊▲■▼■◆❆❘❊❙ ❆◆❆▲❮❚■❈❆❙ ✸✶

  1

  2

  (R ) ❡♠ H ✱ t❡♠♦s q✉❡

  

k

k +2 k k

2

1 2 k 2

  L (R ) H (R )

  k k f k ≤ 2 k f k

  ♣❛r❛ t♦❞♦ 2 ≤ k < ∞✳

  k 2 +2 k

  k ❉❡♠♦♥str❛çã♦✳ ❚✐✈❡♠♦s ❞✐✜❝✉❧❞❛❞❡ ❞❡ ❡♥❝♦♥tr❛r ❛ ❝♦♥st❛♥t❡ ❡①♣❧í❝✐t❛ 2 ♥❛ r❡❢❡rê♥❝✐❛ ✉s❛❞❛ ♣❡❧♦ ❛rt✐❣♦ ❞♦ ❚❛✉❜❡s✳ ❊st❛ ❝♦♥st❛♥t❡ é ✐♠♣♦rt❛♥t❡ ♥♦ ❈❛♣ít✉❧♦ ✸✳ ❱❛♠♦s ♦❜t❡r ♦ r❡s✉❧t❛❞♦ ✉s❛♥❞♦ ♦✉tr❛ r❡❢❡rê♥❝✐❛✳ P❡❧♦ ❚❡♦r❡♠❛ ✽✳✺ ✐t❡♠ ✭✐✐✮ ❡♠ ❬✼✱ ♣✳ ✷✵✻❪✱ s❛❜❡♠♦s q✉❡ 2

  1 1 2

  k

  ,

  H (R )

  k f k L (R ) ≤ k f k pS

1

2,k 1 n 2 1 o −2

  

−1+

2

k−2 k 1− k k

  > k (k

  2,k

  ♦♥❞❡ S − 1) ✳ ❚❡♠♦s q✉❡

  8π 1 1 22 k

  1

  1

  1− k − 2 k

  < k 1

  1− k

  8π pS

  2,k

  (k − 1) 1 1 2 2 k − k

  1

  1− k 1

  ≤ k

  

1−

k k

  8π

  2 1 1 21 2

1

k

  1 2 − 1−

  k k

  2 ≤ k 1 2 3

  16 2 − −1+

  k k

  = k 1

  2 2 2 +

  ≤ k2 P♦rt❛♥t♦

  k .

  k

k +2 k k

2 1 2 k 2 k .

  k f k ≤ 2 k f k

  L (R ) H (R )

  ❊①✐st❡ ✉♠❛ t❡♦r✐❛ q✉❡ ❣❡♥❡r❛❧✐③❛ ❛ t❡♦r✐❛ ❝❧áss✐❝❛ ❞❛s ✐♠❡rsõ❡s ❞❡ ❙♦❜♦❧❡✈✳

  n n

  (R )

  A

  ❖ ❡s♣❛ç♦ L é ♦ ❝♦♥❥✉♥t♦ ❞❛s ❢✉♥çõ❡s ♠❡♥s✉rá✈❡✐s f ❡♠ R t❛✐s q✉❡ ˆ

  ❈❆P❮❚❯▲❖ ✷✳ P❘❊▲■▼■◆❆❘❊❙ ❆◆❆▲❮❚■❈❆❙ ✸✷ ♦♥❞❡ A é ✉♠❛ ❢✉♥çã♦ s❛t✐s❢❛③❡♥❞♦ ❝❡rt❛s ♣r♦♣r✐❡❞❛❞❡s ❬✶✱ ♣✳ ✷✻✷❪✳ ◆♦ ♥♦ss♦ 2

  t

  ❝❛s♦ A(t) = e − 1✳ ❱❛♠♦s ✉s❛r ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✳

  1

  2

  (R ) ❚❡♦r❡♠❛ ✷✳✷✳✸✳ ❬✶✱ ♣✳ ✷✼✼✲✷✽✵❪ P❛r❛ t♦❞❛ ❢✉♥çã♦ f ❡♠ H ✱ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ ♣♦s✐t✐✈❛ ρ ✭❞❡♣❡♥❞❡♥❞♦ ❞❡ f✮ t❛❧ q✉❡

  ˆ 2 2

  ρ f

  e < 2 − 1 ∞.

  R

  ❉❡♠♦♥str❛çã♦✳ ●♦st❛rí❛♠♦s ❞❡ ❛♣❧✐❝❛r ♦ t❡♦r❡♠❛ ✽✳✷✼ ❡♠ ❬✶✱ ♣✳ ✷✼✼❪ ❝♦♠ n = 2

  ✱ p = 2 ❡ m = 1✳ ▼❛s ❡❧❡ só ✈❛❧❡ ♣❛r❛ ❞♦♠í♥✐♦s ❧✐♠✐t❛❞♦s✳ ◆❛ ♣✳ ✷✽✵ ♦ ❛✉t♦r ❡①♣❧✐❝❛ ❛ ❛❞❛♣t❛çã♦ ♣❛r❛ ❞♦♠í♥✐♦s ♥ã♦ ❧✐♠✐t❛❞♦s✱ ❡ ♠♦str❛ ❝♦♠♦ ❡s❝♦❧❤❡r ❛ ❝♦♥st❛♥t❡ ρ✳

  ✷✳✸ ❆♥á❧✐s❡ ❋✉♥❝✐♦♥❛❧

  ▲❡♠❜r❡ q✉❡ ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ H ❝♦♠ ✉♠ ♣r♦❞✉t♦ ✐♥t❡r♥♦ h·, ·i é ✉♠

  H

  2

  = ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt s❡ H ❝♦♠ ❛ ♥♦r♠❛ kvk é ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦

  H hv, vi H

  ❝♦♠♣❧❡t♦✳ ▲❡♠❜r❡ q✉❡ ✉♠ ❢✉♥❝✐♦♥❛❧ G ♥✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt H é ✉♠❛ ❢✉♥çã♦ G : H → R✳

  ❈♦♥s✐❞❡r❡ ✉♠ ❢✉♥❝✐♦♥❛❧ G ❞❡✜♥✐❞♦ ♥✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt H✳ ❉❛❞♦s v ❡

  ′

  h (v, h) ❡♠ H✱ ❛ ❞❡r✐✈❛❞❛ ❞❡ ●ât❡❛✉① G é ❞❡✜♥✐❞❛ ♣♦r

  G (v + th)

  

′ − G (v)

  G (v, h) = lim ,

  t→0

  t s❡ ♦ ❧✐♠✐t❡ ❡①✐st✐r✳

  n

  ❯♠❛ s❡q✉ê♥❝✐❛ v ♥✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt H ❝♦♥✈❡r❣❡ ❢r❛❝❛♠❡♥t❡ ♣❛r❛ ✉♠ v ❡♠ H s❡

  , w

  n

  hv i H → hv, wi H ♣❛r❛ t♦❞♦ w ❡♠ H✳

  ❯♠ ❢✉♥❝✐♦♥❛❧ G ❞❡✜♥✐❞♦ ❡♠ H é s❡♠✐❝♦♥tí♥✉♦ ✐♥❢❡r✐♦r♠❡♥t❡ ♥♦ s❡♥t✐❞♦

  n

  ❢r❛❝♦ s❡ ♣❛r❛ t♦❞❛ s❡q✉ê♥❝✐❛ v ❡♠ H ❝♦♥✈❡r❣✐♥❞♦ ♣❛r❛ ✉♠ v ❡♠ H ♥♦ s❡♥t✐❞♦ ❢r❛❝♦✱ t❡♠♦s q✉❡

  ❈❆P❮❚❯▲❖ ✷✳ P❘❊▲■▼■◆❆❘❊❙ ❆◆❆▲❮❚■❈❆❙ ✸✸ ◆♦ ❝❛♣ít✉❧♦ ✸✱ ✈❛♠♦s ✉s❛r ❛ s❡❣✉✐♥t❡ ♣r♦♣♦s✐çã♦ ♣❛r❛ ♠♦str❛r q✉❡ ✉♠

  1

  2

  (R ) ❝❡rt♦ ❢✉♥❝✐♦♥❛❧ G ❞❡✜♥✐❞♦ ❡♠ H ♣♦ss✉✐ ✉♠ ♣♦♥t♦ ❞❡ ♠í♥✐♠♦ ❧♦❝❛❧✳ Pr♦♣♦s✐çã♦ ✷✳✸✳✶✳ ❬✶✶✱ ♣✳ ✶✵✵❪ ❈♦♥s✐❞❡r❡ ✉♠ ❢✉♥❝✐♦♥❛❧ G ❞❡✜♥✐❞♦ ♥✉♠

  ′

  (x, h) ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt H✳ ❙✉♣♦♥❤❛ q✉❡ ❛ ❞❡r✐✈❛❞❛ ❞❡ ●ât❡❛✉① G ❡①✐st❡ ♣❛r❛ t♦❞♦ x ❡ h ❡♠ H✳ ❆ss✉♠❛ q✉❡ G é s❡♠✐❝♦♥tí♥✉♦ ✐♥❢❡r✐♦r♠❡♥t❡ ♥♦ s❡♥t✐❞♦ ❢r❛❝♦✳ ❙✉♣♦♥❤❛ q✉❡ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ ♣♦s✐t✐✈❛ R t❛❧ q✉❡

  ′

  G (v, v) > 0 = R

  H

  ♣❛r❛ t♦❞♦ v ❡♠ H ❝♦♠ kvk ✳ ❊♥tã♦ ❡①✐st❡ ♣♦♥t♦ ❞❡ ♠í♥✐♠♦ ❧♦❝❛❧ ❞❡ G < R

  H

  ♥♦ ✐♥t❡r✐♦r ❞❛ ❜♦❧❛ |x| ≤ R✱ ♦✉ s❡❥❛✱ ❡①✐st❡ ✉♠ v ❡♠ H ❝♦♠ kv k t❛❧ q✉❡ G

  (v ) ≤ G(v)

  ♣❛r❛ t♦❞♦ v s✉✜❝✐❡♥t❡♠❡♥t❡ ♣ró①✐♠♦ ❞❡ v ✳ ❊♠ ♣❛rt✐❝✉❧❛r✱

  ′

  G (v ) = 0. ❯♠ ❢✉♥❝✐♦♥❛❧ G ❞❡✜♥✐❞♦ ♥✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt H é ❝♦♥✈❡①♦ s❡

  G (tv + (1 ) ) + (1 )

  1

  2

  1

  2

  − t) v ≤ tG (v − t) G (v ♣❛r❛ t♦❞♦ 0 ≤ t ≤ 1 ❡ v

  1 ❡ v 2 ❡♠ H✳

  ❯♠ ❢✉♥❝✐♦♥❛❧ G ❞❡✜♥✐❞♦ ♥✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt H é ❡str✐t❛♠❡♥t❡ ❝♦♥✈❡①♦ s❡ G

  (tv

  1 + (1

2 ) < tG (v

1 ) + (1 2 )

  − t) v − t) G (v

  1

  2

  ♣❛r❛ t♦❞♦ 0 < t < 1 ❡ v ❡ v ❡♠ H✳ ❱❛♠♦s ✉s❛r ❛ s❡❣✉✐♥t❡ ♣r♦♣♦s✐çã♦ ♣❛r❛ ♠♦str❛r ❛ ✉♥✐❝✐❞❛❞❡ ❞♦ ♣♦♥t♦ ❞❡

  ♠í♥✐♠♦ ❞♦ ❢✉♥❝✐♦♥❛❧ G ♠❡♥❝✐♦♥❛❞♦ ❛❝✐♠❛✳ Pr♦♣♦s✐çã♦ ✷✳✸✳✷✳ ❬✶✶✱ ♣✳ ✾✻❪ ❈♦♥s✐❞❡r❡ ✉♠ ❢✉♥❝✐♦♥❛❧ ❡str✐t❛♠❡♥t❡ ❝♦♥✈❡①♦ G

  ❞❡✜♥✐❞♦ ♥✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt H✳ ❆ss✉♠❛ q✉❡ G ♣♦ss✉✐ ✉♠ ♣♦♥t♦ ❞❡ ♠í♥✐♠♦✳ ❊♥tã♦ ❡st❡ ♣♦♥t♦ ❞❡ ♠í♥✐♠♦ é ú♥✐❝♦✳

  ❈❆P❮❚❯▲❖ ✷✳ P❘❊▲■▼■◆❆❘❊❙ ❆◆❆▲❮❚■❈❆❙ ✸✹ ♠✐❝♦♥tí♥✉♦ ✐♥❢❡r✐♦r♠❡♥t❡ ♥♦ s❡♥t✐❞♦ ❢r❛❝♦✳ ❱❛♠♦s ✉s❛r ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦ ♣❛r❛ ♠♦str❛r q✉❡ ♦ ❢✉♥❝✐♦♥❛❧ G é s❡♠✐❝♦♥tí♥✉♦ ✐♥❢❡r✐♦r♠❡♥t❡ ♥♦ s❡♥t✐❞♦ ❢r❛❝♦✳ Pr♦♣♦s✐çã♦ ✷✳✸✳✸✳ ❬✶✶✱ ♣✳ ✽✷❪ ❈♦♥s✐❞❡r❡ ✉♠ ❢✉♥❝✐♦♥❛❧ ❝♦♥✈❡①♦ G ♥✉♠

  ′

  (v, h) ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt H✳ ❙✉♣♦♥❤❛ q✉❡ ❛ ❞❡r✐✈❛❞❛ ❞❡ ●ât❡❛✉① G ❡①✐st❡

  ′

  (v, ♣❛r❛ t♦❞♦ v ❡ h ❡♠ H✳ ❆ss✉♠❛ q✉❡ G ·) é ❝♦♥tí♥✉♦ ♣❛r❛ t♦❞♦ v ❡♠ H✳ ❊♥tã♦ G é s❡♠✐❝♦♥tí♥✉♦ ✐♥❢❡r✐♦r♠❡♥t❡ ♥♦ s❡♥t✐❞♦ ❢r❛❝♦✳

  ′

  (v, ❋✐①❛❞♦ ✉♠ v ❡♠ H✱ ❧❡♠❜r❡ q✉❡ G ·) é ❝♦♥tí♥✉♦ ❡♠ h ∈ H s❡ ♣❛r❛

  n n H

  t♦❞❛ s❡q✉ê♥❝✐❛ h ❡♠ H t❛❧ q✉❡ kh − h k → 0✱ t❡♠♦s q✉❡

  ′ ′

  (v, h ) (v, h)

  n

  |G − G | → 0,

  ′ ′

  (v, (v, ❡ G ·) é ❝♦♥tí♥✉♦ s❡ G ·) é ❝♦♥tí♥✉♦ ❡♠ h ♣❛r❛ t♦❞♦ h ❡♠ H✳

  ❈❛♣ít✉❧♦ ✸ ❚❡♦r❡♠❛ Pr✐♥❝✐♣❛❧

✸✳✶ ❱♦rt❡① ◆✉♠❜❡r ❡ ❋ór♠✉❧❛ ❞❡ ❇♦❣♦♠♦❧✬♥②✐

  2

  2

  ❈♦♥s✐❞❡r❡ ♦ ❧✐♥❡ ❜✉♥❞❧❡ tr✐✈✐❛❧ R ×C → R ✳ P❡❧♦ ❊①❡♠♣❧♦ ✶✳✷✳✶ ❡ Pr♦♣♦s✐çã♦

  

2

  2

  ✶✳✷✳✺✱ s❛❜❡♠♦s q✉❡ t♦❞❛ ❝♦♥❡①ã♦ ❡♠ R × C → R é ❞❛ ❢♦r♠❛ d = d

  A

  − iA

  2

  ♣❛r❛ ❛❧❣✉♠❛ ✶✲❢♦r♠❛ A ❡♠ R ✳ ▲❡♠❜r❡ q✉❡ ❛ ❝✉r✈❛t✉r❛ ❞❛ ❝♦♥❡①ã♦ d A é ❞❛❞❛ ♣♦r

  F

A = dA.

  2

  2 P❡❧♦ ❊①❡♠♣❧♦ ✶✳✶✳✷✱ s❛❜❡♠♦s q✉❡ ❛s s❡çõ❡s ❞❡ R × C → R ♣♦❞❡♠ s❡r

  2

  ✐❞❡♥t✐✜❝❛❞❛s ❝♦♠ ❢✉♥çõ❡s ❝♦♠♣❧❡①❛s ❡♠ R ✳ ◆❡st❡ ❝❛♣ít✉❧♦ ✐❞❡♥t✐✜❝❛♠♦s s❡çõ❡s ❝♦♠ ❢✉♥çõ❡s ❝♦♠♣❧❡①❛s ❡ ❝♦♥❡①õ❡s ❝♦♠ ✶✲❢♦r♠❛s✳

  2

  2

  ❈♦♥s✐❞❡r❡ ✉♠❛ ❢✉♥çã♦ ❝♦♠♣❧❡①❛ φ ❡♠ R ❡ ✉♠❛ ✶✲❢♦r♠❛ A ❡♠ R ✳ ❙✉✲ ♣♦♥❤❛ q✉❡ φ ❡ A ♣♦ss✉❡♠ ❞❡r✐✈❛❞❛s ❢r❛❝❛s✳ ❆ ❡♥❡r❣✐❛ é ❞❡✜♥✐❞❛ ♣♦r

  2

  ˆ 1 λ

  1

  2

  2

  2 E φ ,

  (φ, A) = |d | |F

  

A A

2 | |φ| − 1 R

  2

  2

  8 ♦♥❞❡ λ é ✉♠❛ ❝♦♥st❛♥t❡✳

  ❖ ❡st✉❞♦ ❞❛ ❡♥❡r❣✐❛ é ✐♠♣♦rt❛♥t❡ ♥❛ t❡♦r✐❛ ❞❡ s✉♣❡r❝♦♥❞✉t✐✈✐❞❛❞❡✳ ◗✉❛♥❞♦

  ❈❆P❮❚❯▲❖ ✸✳ ❚❊❖❘❊▼❆ P❘■◆❈■P❆▲ ✸✻ ❣✐❛ ❞❡s❝r❡✈❡ s✉♣❡r❝♦♥❞✉t♦r❡s ❞♦ t✐♣♦ II ❬✸✱ ✻✱ ✶✷❪✳ ◆❡st❡ ❝❛♣ít✉❧♦ ❡st❛♠♦s ✐♥t❡r❡ss❛❞♦s ♥♦ ❝❛s♦ λ = 1 ✭✈❛❧♦r ❝rít✐❝♦✮✳ ◗✉❛♥❞♦ λ = 1✱ ❛ ❡♥❡r❣✐❛ é ❧✐♠✐t❛❞❛ ✐♥❢❡r✐♦r♠❡♥t❡ ♣♦r ✉♠ ♠ú❧t✐♣❧♦ ❞❡ ✈♦rt❡① ♥✉♠❜❡r ✭❚❡♦r❡♠❛ ✸✳✶✳✷✮✳ ◗✉❛❧q✉❡r ♣❛r (φ, A) q✉❡ ❛t✐♥❣❡ ♦ ♠í♥✐♠♦ ❞❡ ❡♥❡r❣✐❛ é ✉♠❛ s♦❧✉çã♦ ❞❛s ❡q✉❛çõ❡s ❞❡ ●✐♥③❜✉r❣✲▲❛♥❞❛✉ ✭❚❡♦r❡♠❛ ✸✳✷✳✶✮✳

  2

  ❚❡♦r❡♠❛ ✸✳✶✳✶✳ ❈♦♥s✐❞❡r❡ ✉♠❛ ❢✉♥çã♦ ❝♦♠♣❧❡①❛ φ ❡♠ R ❡ ✉♠❛ ✶✲❢♦r♠❛ A

  2

  ❡♠ R ✳ ❆ss✉♠❛ q✉❡ φ ❡ A ♣♦ss✉❡♠ ❞❡r✐✈❛❞❛s ❢r❛❝❛s✳ ❙✉♣♦♥❤❛ q✉❡ ❛ ❡♥❡r❣✐❛ E (φ, A) <

  ∞✳ ❊♥tã♦ ❛ ❡♥❡r❣✐❛ E é ✐♥✈❛r✐❛♥t❡ ♣♦r tr❛♥s❢♦r♠❛çõ❡s ❞❡ ❝❛❧✐❜r❡ φ

  → gφ

  −1

  A dg, → A − ig

  ♦♥❞❡ g é ✉♠❛ ❢✉♥çã♦ ❝♦♠♣❧❡①❛ ❝♦♠ |g| = 1✱ ♦✉ s❡❥❛✱

  −1 E gφ, A dg = E (φ, A) .

  − ig ❉❡♠♦♥str❛çã♦✳ ❚❡♠♦s q✉❡

  −1 −1

  d dg (gφ) = d (gφ) (gφ)

  A−ig dg

  − i A − ig = φdg + gdφ

  − iAgφ − φdg = g (dφ

  − iAφ) φ,

  = gd A ❡

  

−1

−1

  F = d A dg

  A−ig dg

  − ig

  −2 −1

  dg ddg = dA

  − i (−1) g ∧ dg − ig = dA

  , = F

  A

  ❈❆P❮❚❯▲❖ ✸✳ ❚❊❖❘❊▼❆ P❘■◆❈■P❆▲ ✸✼ ♣♦rt❛♥t♦

  ˆ λ

  2

  1

  1

  2

  2 −1

  2

  E gφ, A dg = (gφ) +

  • −1 −1

  A−ig dg A−ig dg

  − ig |d | 2 |F | |gφ| − 1

  R

  2

  2

  8 ˆ 1 λ

  2

  1

  2

  2

  2

  φ

  • = +

  

A A

2 |gd | − 1 |F | |gφ| R

  2

  2

  8 ˆ

  1 1 λ

  2

  2

  2

  2

  φ = A A + + 2 |d | |F | |φ| − 1

  R

  2

  2

  8 = E (φ, A) .

  ❆♥t❡s ❞❡ ❡♥✉♥❝✐❛r ♦ ♣ró①✐♠♦ t❡♦r❡♠❛✱ ✈❛♠♦s ✐♥tr♦❞✉③✐r ✉♠❛ ❢✉♥çã♦ ❝✉t✲

  R

  ♦✛ χ t❛❧ q✉❡   1 ,

  R

  ❡♠ B χ

  =

  R

  2

   ,

  ❡♠ R

  2R

  \ B

  

R

  ≤ χ ≤ 1, ❡

  C

  1

  ,

  R

  |dχ | ≤ R

  R

  ♦♥❞❡ B é ❛ ❜♦❧❛ ❞❡ ❝❡♥tr♦ ♥❛ ♦r✐❣❡♠ ❡ r❛✐♦ R✳ ❚♦♠❡ ✉♠❛ ❢✉♥çã♦ s✉❛✈❡ ˜χ ❡♠ [0, ∞) t❛❧ q✉❡

   

  1 ❡♠ [0, 1] ,

  χ ˜ = 

  ❡♠ [2, ∞), χ ≤ ˜ ≤ 1. ❚❡♠♦s q✉❡

  ′ ′

  sup χ χ |˜ | = sup |˜ |

  [1,2] .

  1

  ≤ C

  ❈❆P❮❚❯▲❖ ✸✳ ❚❊❖❘❊▼❆ P❘■◆❈■P❆▲ ✸✽ ❉❡✜♥✐♠♦s

  |x| χ χ .

  R (x) = ˜

  R ❚❡♠♦s q✉❡

   

  ,

1 R

  ❡♠ B χ

  =

  R

  2

   ,

  2R

  ❡♠ R \ B

  

R

≤ χ ≤ 1.

  ❆❧é♠ ❞✐ss♦✱

  1

  

′ |x|

  dχ χ d (x) = ˜

  R

  |x| , R R

  ❧♦❣♦ C

  1 (x) . R

  |dχ | ≤ R

  2

  ❚❡♦r❡♠❛ ✸✳✶✳✷✳ ❈♦♥s✐❞❡r❡ ✉♠❛ ❢✉♥çã♦ ❝♦♠♣❧❡①❛ φ ❡♠ R ❡ ✉♠❛ ✶✲❢♦r♠❛ A

  2

  ❡♠ R ✳ ❆ss✉♠❛ q✉❡ φ ❡ A ♣♦ss✉❡♠ ❞❡r✐✈❛❞❛s ❢r❛❝❛s✳ ❙✉♣♦♥❤❛ q✉❡ ❛ ❡♥❡r❣✐❛ E

  (φ, A) < ∞✳ ❖ ❧✐♠✐t❡

  ˆ

  1 χ F vort (φ, A) = lim 2 R A

R→∞

  2π R ❡①✐st❡✳ ❆❧é♠ ❞✐ss♦✱ vort (φ, A) é ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ ❡ ✐♥✈❛r✐❛♥t❡ ♣♦r tr❛♥s✲ ❢♦r♠❛çõ❡s ❞❡ ❝❛❧✐❜r❡✱ ❝❤❛♠❛❞♦ ✈♦rt❡① ♥✉♠❜❡r✳ ▼❛✐s ❛✐♥❞❛✱ s❡ ❛ ❢✉♥çã♦ φ é s✉❛✈❡ ❡ |φ| → 1 ♥♦ ✐♥✜♥✐t♦✱ ❡♥tã♦ vort (φ, A) é ✐❣✉❛❧ ❛♦ í♥❞✐❝❡ ❞❡ φ ♥♦ ✐♥✜♥✐t♦✳ ❉❡♠♦♥str❛çã♦✳ ❱❛♠♦s ❛ss✉♠✐r q✉❡ ❛ ❢✉♥çã♦ φ é s✉❛✈❡ ❡ |φ| → 1 q✉❛♥❞♦ |x| → ∞✳ ❖ ❝❛s♦ ❣❡r❛❧ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞♦ ❡♠ ❬✷❪✳

  ✶✳❛✳ ❱❛♠♦s ♠♦str❛r q✉❡

  1

  −2

  ¯ A φd φ φ = d ( .

  A A

  ❛r❣φ) − |φ| − φd ✭✸✳✶✮ 2i

  φ = dφ

  ▲❡♠❜r❛♥❞♦ q✉❡ d A − iAφ✱ t❡♠♦s q✉❡

  2

  ¯ φd φ φdφ ,

  A = ¯

  ❈❆P❮❚❯▲❖ ✸✳ ❚❊❖❘❊▼❆ P❘■◆❈■P❆▲ ✸✾ ❡

  2 φd φ . A = φdφ + iA

  |φ| ✭✸✳✸✮ ❙✉❜tr❛✐♥❞♦ ✭✸✳✷✮ ❞❡ ✭✸✳✸✮✱ ♦❜t❡♠♦s

  2 2i A = ¯ φdφ φ φ .

  A A

  |φ| − φdφ − ¯φd − φd ❈♦♠♦ |φ| → 1 q✉❛♥❞♦ x → ∞✱ ♣♦❞❡♠♦s ❞✐✈✐❞✐r ♣♦r |φ| ♣❛r❛ |x| s✉✜❝✐❡♥t❡✲ ♠❡♥t❡ ❣r❛♥❞❡✳ ❚❡♠♦s q✉❡

  1

  1

  −2 −2

  ¯ ¯ A φdφ φd φ φ

  = .

  A A

  |φ| − φdφ − |φ| − φd 2i 2i

  ❊s❝r❡✈❛

  

f

  φ = e 1 2

  f if e .

  = e ❚❡♠♦s q✉❡

  f

  dφ = e (df + idf ) ,

  1

  2

  ❡

  ¯ f

  dφ = e (df ) ,

  1

  2

  − idf ♣♦rt❛♥t♦

  1

  1 1 ¯

  −2 f f

  ¯ φdφ e dφ dφ

  = |φ| − φdφ − e 1

  2f

  2i 2i e

  1 1 ¯

  f +f f + ¯ f

  e = (df + idf ) (df )

  1

  2

  1

  2 1 − e − idf 2f

  2i e

  1 = (df + idf + idf )

  1

  2

  1

  2

  − df 2i

  = d ( ❛r❣φ) . ▲♦❣♦

  1

  −2

  ¯ A φd φ φ . = d ( A A

  ❛r❣φ) − |φ| − φd 2i

  ❈❆P❮❚❯▲❖ ✸✳ ❚❊❖❘❊▼❆ P❘■◆❈■P❆▲ ✹✵ ✶✳❜✳ ❯s❛♥❞♦ ✐♥t❡❣r❛çã♦ ♣♦r ♣❛rt❡s✱ ♦❜t❡♠♦s

  ˆ ˆ

  1

  1 χ F = χ dA 2 R A R 2

  2π R 2π R ˆ ˆ

  1

  1 = d (χ A ) dχ

  R R 2 − ∧ A. 2

  2π R 2π R ❈♦♠♦ χ R t❡♠ ♦ s✉♣♦rt❡ ❝♦♠♣❛❝t♦ ✭❛ ❢✉♥çã♦ s❡ ❛♥✉❧❛ ❢♦r❛ ❞❡ B

  2R ✮✱ ♣❡❧♦

  t❡♦r❡♠❛ ❞❡ ❙t♦❦❡s✱ t❡♠♦s q✉❡ ˆ

  1 d (χ A ) = 0, 2 R 2π R

  ♣♦rt❛♥t♦ ˆ ˆ

  1

  1 χ F dχ

  R A = R 2 − ∧ A. ✭✸✳✹✮ 2

  2π R 2π R ❙✉❜st✐t✉✐♥❞♦ ❛ ❊q✉❛çã♦ ✭✸✳✶✮ ♥❛ ❊q✉❛çã♦ ✭✸✳✹✮✱ ♦❜t❡♠♦s

  ˆ ˆ

  1

  1 χ F dχ

  R A = R 2 − ∧ d (❛r❣φ) ✭✸✳✺✮ 2

  2π R 2π R ˆ

  1

  1

  −2

  ¯ dχ φd φ φ .

  • R A A
  • 2 ∧ |φ| − φd 2π R 2i

      P❡❧♦ t❡♦r❡♠❛ ❞❡ ❙t♦❦❡s✱ t❡♠♦s q✉❡ ˆ ˆ

      1

      1 dχ d d (χ (

      R R ❛r❣φ))

      − ∧ d (❛r❣φ) = − 2 2π R 2π 2 B R \B R

      ˆ ˆ

      1

      1 χ d χ d

      = ( (

      R R

      − ❛r❣φ) + ❛r❣φ) 2π 2π 2

      ∂B R ∂B R

      ˆ

      1 = d (

      ❛r❣φ) 2π

      ∂B R = vort (φ, A) .

      P♦❞❡♠♦s ✈❡r q✉❡ vort (φ, A) é ♦ í♥❞✐❝❡ ❞❡ φ✳ ▲❡♠❜r❡ q✉❡ ♦ í♥❞✐❝❡ ❞❡ φ é ✉♠

      R

      ♠ú❧t✐♣❧♦ ✐♥t❡✐r♦ ❞♦ í♥❞✐❝❡ ❞❛ ❝✉r✈❛ ∂B ✱ ♣♦rt❛♥t♦ é ✉♠ ♠ú❧t✐♣❧♦ ✐♥t❡✐r♦ ❞❡ ✶✳ P♦❞❡♠♦s ❡s❝r❡✈❡r ❛ ❊q✉❛çã♦ ✭✸✳✺✮ ❝♦♠♦

      ˆ ˆ

    • φd

      B 2 R \B R

      A

      φ − φd

      A

      φ ≤

      C

      1

      πR ˆ

      |d

      −2

      A

      φ |

      2 1 2

      ˆ

      B 2 R \B R

      1 1 2

      C

      ¯ φd

      |φ|

      πR ˆ

      R

      C

      1

      πR ˆ

      B 2 R \B R

      |d

      A

      φ | . ❆❝✐♠❛ ✉s❛♠♦s ♦ ❢❛t♦ q✉❡ |dχ

      | ≤

      1 2i

      C 1 R

      ✳ P❡❧❛ ❞❡s✐❣✉❛❧✐❞❛❞❡ ❞❡ ❍♦❧❞❡r✱ t❡♠♦s q✉❡

      1 2π

      ˆ

      R 2

      dχ

      R

      ∧

      1

      B 2 R \B R

      φ |

      φ |

      |d

      A

      φ |

      2 1 2 .

      P♦❞❡♠♦s ❡s❝r❡✈❡r ˆ 2

      |d

      A

      2

      ˆ

      = ˆ 2

      |d

      A

      φ |

      2

      1 R 2

      \B R

      R 2 \B R

      √ π

      |d A φ

      πR √

      |

      2 1 2

      (areaB

      2R

      ) 1 2 =

      C

      1

      π

      1

      2R ˆ

      B 2 R \B R

      |d

      A

      φ |

      2 1 2

      ≤

      2C

      |φ| ≤

      A

      ❈❆P❮❚❯▲❖ ✸✳ ❚❊❖❘❊▼❆ P❘■◆❈■P❆▲ ✹✶ ❱❛♠♦s ♠♦str❛r q✉❡ q✉❛♥❞♦ R → ∞ ✈❛❧❡ q✉❡ lim

      R 2

      ¯ φd

      A

      φ − φd

      A

      φ ≤

      1 4π

      ˆ

      |φ|

      |φ|

      −2

      dχ

      R

      ∧ ¯ φd

      A

      φ − φd

      A

      −2

      1 2i

      1 4π

      

    A

    = vort (φ, A) .

      R→∞

      1 2π

      ˆ

      R 2

      χ

      R

      F

      P♦❞❡♠♦s ❛ss✉♠✐r q✉❡ |φ| ≥

      ∧

      1

      2

      ♣❛r❛ R s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡✳ ❚❡♠♦s q✉❡

      1 2π

      ˆ

      R 2

      dχ

      R

      φ ≤

      ˆ

      |d

      −2

      A

      φ ≤

      C

      1

      4πR ˆ

      B 2 R \B R

      |φ|

      ¯ φd

      A

      A

      φ

      A

      φ ≤

      C

      1

      2πR ˆ

      B 2 R \B R

      φ − φd

      φd

      R 2

      φ − φd

      |φ|

      −2

      |dχ

      R

      | ¯

      φd

      A

      A

      | ¯

      φ =

      1 4π

      ˆ

      B 2 R \B R

      |φ|

      −2

      |dχ

      R

      ,

      ❈❆P❮❚❯▲❖ ✸✳ ❚❊❖❘❊▼❆ P❘■◆❈■P❆▲ ✹✷ ♦♥❞❡

      

      2

       ,

      1 2 ❡♠ R \ B R

    1 R =

      \B R

      R

      ❡♠ B ❚❡♠♦s q✉❡

       .

      

    2

    2

      φ R lim 1 = 0.

      

    A \B R

      |d |

      R→∞

      ❆❧❡♠ ❞✐ss♦✱

      2 2

      2 φ R φ .

    1 A \B R A

      |d | ≤ |d | P❡❧♦ t❡♦r❡♠❛ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ❞♦♠✐♥❛❞❛ ✭❚❡♦r❡♠❛ ✷✳✶✳✺✮✱ t❡♠♦s q✉❡

      ˆ

      2 2

      φ R lim 1 = 0.

      A \B R 2 |d | R→∞ R

      \B R

      ❱♦❧t❛♥❞♦ à ❊q✉❛çã♦ ✭✸✳✻✮✱ t❡♠♦s q✉❡ ˆ

      1 χ F lim R A = vort (φ, A) . 2 R→∞

      2π R ✷✳ ❱❛♠♦s ♠♦str❛r q✉❡ vort (φ, A) é ✐♥✈❛r✐❛♥t❡ ♣♦r tr❛♥s❢♦r♠❛çõ❡s ❞❡

      ❝❛❧✐❜r❡ g → gφ

      −1

      A dg, → A − ig

      ♦♥❞❡ g é ✉♠❛ ❢✉♥çã♦ ❝♦♠♣❧❡①❛ ❝♦♠ |g| = 1✳ ❚❡♠♦s q✉❡ ˆ

      1

      −1 −1

      dg = lim χ F vort gφ, A R − ig A−ig dg 2 R→∞

      2π R ˆ

      1 χ F

      = lim R A 2 R→∞ 2π R = vort (φ, A) .

      ❚❡♦r❡♠❛ ✸✳✶✳✸ ✭❋ór♠✉❧❛ ❞❡ ❇♦❣♦♠♦❧✬♥②✐✮✳ ❈♦♥s✐❞❡r❡ ✉♠❛ ❢✉♥çã♦ ❝♦♠♣❧❡①❛

      ❈❆P❮❚❯▲❖ ✸✳ ❚❊❖❘❊▼❆ P❘■◆❈■P❆▲ ✹✸ ❢r❛❝❛s✳ ❙✉♣♦♥❤❛ q✉❡ ❛ ❡♥❡r❣✐❛ E(φ, A) < ∞✳ ❊♥tã♦

      E (φ, A)

      ≥ π vort (φ, A) , ❡ ✈❛❧❡ ❛ ✐❣✉❛❧❞❛❞❡ s❡ ❡ s♦♠❡♥t❡ s❡ ✈❛❧❡♠ ❛s ❡q✉❛çõ❡s ❞❡ ✈órt✐❝❡

      (∂ φ + A φ ) φ φ ) = 0,

      1

      1

      1

      2

      2

      2

      2

      1

      − (∂ − A ✭✸✳✼✮ φ φ φ φ

      (∂ + A ) + (∂ ) = 0,

      2

      1

      2

      2

      1

      2

      1

      1

      − A ✭✸✳✽✮ ❡

      1

      2

      2

    • F φ + φ = 0.

      12

      1 2 − 1 ✭✸✳✾✮

      2 ❉❡♠♦♥str❛çã♦✳ P❛r❛ ♣r♦✈❛r ♦ t❡♦r❡♠❛✱ ❜❛st❛ ♠♦str❛r q✉❡

      ˆ

      1

      2 E (φ, A) = φ + A φ ) φ φ )

      1

      1

      

    1

      2

      2

      2

      2

      1 2 {(∂ − (∂ − A }

    2 R

      ˆ

      1

      2

    • φ + A φ ) + (∂ φ φ )

      2

      1

      

    2

      2

      1

      2

      1

      1 2 {(∂ − A }

      2 R

      2

      ˆ ˆ

      1

      1

      1

      2

      2 + + + F φ + φ F .

      12

      12 2

      1 2 − 1 2

      2 R

      2

      2 R ❚❡♠♦s q✉❡

      2

      1

      1

      1

      2

      2

      2

      φ

    • |d A | |F A | |φ| − 1

      2

      2

      8

      2

      1

      1

      1

      1

      1

      2

      2

      

    2

      2

      2 = + + φ F φ + φ φ + φ .

    • F F

      A

      12

      12

      12

      |d |

      1 2 − 1 −

      1

      2

      2

      2

      2

      2

      2 ❚❡♠♦s q✉❡ d φ = ∂ φdx + ∂ φdx dx + A dx )

      A

      1

      1

      2

      

    2

      1

      1

      2

      2

      − iφ (A = (∂ φ φ ) dx + (∂ φ φ ) dx .

      1

      1

      

    1

      2

      2

      2

      − iA − iA

    • |∂
    • A
    • i (∂
    • A
    • i (∂
    • |∂
    • A
    • A
    • {(∂
    • 2 (∂
    • A
    • A

    • 1
    • 1
    • A
    • 1
    • A
    • 1
    • 1
    • φ
    • 1
    • (∂
    • A
    • A
    • φ

      φ

      2

      ) − (∂

      1

      φ

      2

      2

      − A

      φ

      2

      ) (∂

      2

      φ

      1

      1

      2

      φ

      2

      ) (∂

      1

      φ

      2

      − A

      1

      φ

      1

      ) −

      1

      12

      φ

      2

      1

      φ

      1

      φ

      2

      φ

      1

      ) }

      2

      2 {(∂

      2

      φ

      1

      2

      φ

      2

      ) + (∂

      1

      2

      12

      − A

      1

      φ

      1

      ) }

      2

      2 F

      12

      2 φ

      2

      1

      2 2 − 1

      2

      2 F

      1

      2 .

      2

      2 .

      1

      ∧ dx

      2

      −

      1

      12

      φ

      2

      1

      2

      2

      dx

      1

      ∧ dx

      ❉❡ ❢❛t♦✱ t❡♠♦s q✉❡ d i ¯ φd

      1

      A

      φ = id ¯ φ ∧ d

      A

      φ

      φdd

      A

      φ = id ¯ φ

      ∧ d

      A

      φ + i ¯ φd (dφ − iAφ)

      = id ¯ φ ∧ d

      A

      φ + ¯ φdφ ∧ A + |φ|

      2 dA.

      ) } dx

      φ

      ❆✜r♠❛♠♦s q✉❡ −

      2

      1

      2 d i ¯ φd

      A

      φ =

      {(∂

      1

      φ

      1

      1

      φ

      2

      ) (∂

      2

      φ

      − A

      1

      φ

      − A

      2

      φ

      1

      ) (∂

      2

      2

      2

      

    1

      φ

      2

      ) − (∂

      1

      φ

      − A

      2

      φ

      φ

      2

      φ

      2

      2

      φ

      2

      − A

      2

      φ

      1

      ) |

      2

      = {(∂

      1

      1

      φ

      1

      φ

      2

      ) − (∂

      2

      φ

      2

      − A

      2

      φ

      

    1

      ) }

      2

      2

      1

      2

      1

      2

      ❈❆P❮❚❯▲❖ ✸✳ ❚❊❖❘❊▼❆ P❘■◆❈■P❆▲ ✹✹ P♦rt❛♥t♦ |d

      A

      φ |

      2

      = |∂

      1

      φ − iA

      1

      φ |

      2

      2

      φ − iA

      2

      φ |

      = |∂

      2

      1

      φ

      1

      1

      φ

      2

      1

      φ

      2

      − A

      1

      φ

      1

      ) |

      2

      φ

      2

      |

      φ

      2

      − A

      1

      φ

      1 ) .

      ▲♦❣♦

      1

      2 |d

      A

      φ |

      2

      2 |F

      A

      2

      ) (∂

      8 |φ|

      2

      − 1

      2

      =

      1

      2 {(∂

      1

      φ

      1

      1

      φ

      2

      ) − (∂

      φ

      1

      2

      1

      2

      ) + (∂

      1

      φ

      2

      − A

      1

      φ

      1

      ) }

      2

      1

      φ

      φ

      1

      φ

      2

      2

      1

      φ

      2

      ) − 2 (∂

      1

      φ

      2

      − A

      2

      φ

      2

      ) (∂

    2 F

    • A
    • A
    • φ

    2 F

    • i ¯

      ❈❆P❮❚❯▲❖ ✸✳ ❚❊❖❘❊▼❆ P❘■◆❈■P❆▲ ✹✺ ▼❛s id ¯ φ φ

      A

      ∧ d ¯ ¯

      = i ∂ φdx + ∂ φdx φ φ ) dx + (∂ φ φ ) dx )

      1

      1

      2

      2

      1

      1

      1

      2

      2

      2

      ∧ ((∂ − iA − iA ¯ ¯

      = i ∂ φ (∂ φ φ ) φ (∂ φ φ ) dx

      1

      2

      2

      2

      1

      1

      1

      2

      − iA − ∂ − iA ∧ dx = (∂ φ + i∂ φ ) (∂ φ + A φ + i (∂ φ φ )) dx

      1

      2

      1

      1

      2

      1

      2

      2

      2

      2

      2

      1

      1

      2

      − A ∧ dx φ + i∂ φ ) (∂ φ + A φ + i (∂ φ φ )) dx ,

      2

      2

      2

      1

      1

      1

      1

      2

      1

      2

      1

      1

      1

      2

      − (∂ − A ∧ dx ❡

      2

      ¯ φdφ dA

      ∧ A + |φ|

      2

      2

      φ φdx φdx dx dx φ dx = ¯ (∂

      1 1 + ∂

      2 2 )

      1 1 + A

      2 2 ) + F 12 + φ

      1

      2

      ∧ (A

      1 2 ∧ dx

      2

      2

      = ¯ φ (A ∂ φ ∂ φ ) dx + F φ + φ dx

      2

      1

      1

      2

      1

      2

      

    12

      1

      2

      − A ∧ dx

      1 2 ∧ dx

      = (φ ) (A ∂ φ ∂ φ + i (A ∂ φ ∂ φ )) dx

      1

      2

      2

      1

      1

      1

      2

      1

      2

      1

      2

      1

      2

      2

      1

      2

      − iφ − A − A ∧ dx

      2

      2

    • F φ + φ dx ,

      12

      1

      2

      1 2 ∧ dx

      ♣♦rt❛♥t♦ d i ¯ φd φ

      A

      φ φ φ φ φ φ φ φ = + A ) (∂ ) + 2 (∂ + A ) (∂ )

      {−2 (∂

      1

      1

      1

      2

      2 2 − A

      2

      1

      2

      1

      2

      2

      1 2 − A

      1 1 } dx 1 ∧ dx

      2

      2

      2

      φ dx ,

    • F

      12 + φ

      1

      2

      ∧ dx

      1

      2

      ♦ q✉❡ ♣r♦✈❛ ❛ ❛✜r♠❛çã♦✳ P♦❞❡♠♦s ✈❡r q✉❡

      2

      1

      1

      1

      2

      2

      2

      φ dx

      A A

      1

      2

      |d | |F | |φ| − 1 ∧ dx

      2

      2

      8

      1

      2

      φ φ φ φ dx = + A ) )

      1

      1

      1

      2

      2

      2

      2

      1

      1

      2

      {(∂ − (∂ − A } ∧ dx

      2

      1

      2

      φ φ φ φ dx

    • 2
    • A ) + (∂ )

      1

      2

      2

      1

      2

      1

      1

      1

      2

      {(∂ − A } ∧ dx

      2

      

    2

      1

      1

      1

      2

      2

      φ F

      12 + φ

    • F dx dx

      1

      2

      12

      1

      2

      1 2 − 1 ∧ dx ∧ dx

      2

      2

      2

      1

      ❈❆P❮❚❯▲❖ ✸✳ ❚❊❖❘❊▼❆ P❘■◆❈■P❆▲ ✹✻

      

    R

      P♦❞❡♠♦s t♦♠❛r ✉♠❛ ❢✉♥çã♦ ❝✉t✲♦✛ χ ❝♦♠♦ ♥❛ ♣r♦✈❛ ❞♦ ❚❡♦r❡♠❛ ✸✳✶✳✷  

      ,

    1 R

      ❡♠ B χ =

      R

      2

       ,

      2R

      ❡♠ R \ B

      

    R

      ≤ χ ≤ 1, ❡

      C

      1

      ,

      R

      |dχ | ≤ R

      R

      ♦♥❞❡ B é ❛ ❜♦❧❛ ❞❡ ❝❡♥tr♦ ♥❛ ♦r✐❣❡♠ ❡ r❛✐♦ R✳ ▼✉❧t✐♣❧✐❝❛♥❞♦ ❛ ❊q✉❛çã♦

      R

      ✭✸✳✶✵✮ ♣♦r χ ❡ ✐♥t❡❣r❛♥❞♦✱ ♦❜t❡♠♦s ˆ

      2

      1

      1

      1

      2

      

    2

      2

    • χ φ +

      R A A 2 |d | |F | |φ| − 1 R

      2

      2

      8 ˆ

      1

      2

      χ φ φ φ φ = + A ) )

      R

      1

      1

      1

      

    2

      2

      2

      2

      1 2 {(∂ − (∂ − A }

    2 R

      ˆ

      1

      2

      χ φ φ φ φ

    • A ) + (∂ +

      )

      R

      2

      1

      2

      

    2

      1

      2

      1

      1 2 {(∂ − A }

    2 R

      2

      ˆ ˆ

      1

      1

      1

      2

      2

      χ

    • R
    • F φ χ F + φ +

      12 R

      12 2

      1 2 − 1 2

      2 R

      2

      2 R ˆ

      1 χ d i ¯ φd φ .

      R A

      − 2

      2 R P❡❧♦ t❡♦r❡♠❛ ❞❛ ❝♦♥✈❡r❣ê♥❝✐❛ ♠♦♥ót♦♥❛ ✭❚❡♦r❡♠❛ ✷✳✶✳✹✮✱ ♣❛r❛ ♣r♦✈❛r ♦ t❡♦✲ r❡♠❛✱ ❜❛st❛ ♠♦str❛r q✉❡

      ˆ χ d i ¯ φd φ = 0. lim R A 2 R→∞

      R

      ❖ ❛rt✐❣♦ ❬✷❪ ❛✜r♠❛ q✉❡ ❡st❡ ❧✐♠✐t❡ é ③❡r♦✳

      ✸✳✷ ❚❡♦r❡♠❛ Pr✐♥❝✐♣❛❧

      ❖ r❡s✉❧t❛❞♦ ♣r✐♥❝✐♣❛❧ ❞❡ ❬✶✵❪ ❡ ❞❡st❡ ❝❛♣ít✉❧♦ é ♦ s❡❣✉✐♥t❡✳

      2

      , . . . , a

      ❈❆P❮❚❯▲❖ ✸✳ ❚❊❖❘❊▼❆ P❘■◆❈■P❆▲ ✹✼

      2

      (φ, A) ❢♦r♠❛❞♦ ♣♦r ✉♠❛ ❢✉♥çã♦ ❝♦♠♣❧❡①❛ s✉❛✈❡ φ ❡♠ R ❡ ✉♠❛ ✶✲❢♦r♠❛ A

      2

      ❡♠ R t❛❧ q✉❡ E

      (φ, A) < ∞,

      E (φ, A) = π vort (φ, A) , ❡ ♣♦rt❛♥t♦ ♦ ♣❛r (φ, A) é s♦❧✉çã♦ ❞❛s ❡q✉❛çõ❡s ❞❡ ✈órt✐❝❡ ✭✸✳✼✮✱ ✭✸✳✽✮ ❡ ✭✸✳✾✮✳ ❆❧é♠ ❞✐ss♦✱

      , . . . , a

      1 n

      {③❡r♦s ❞❡ φ} = {a } ❝♦♠ ❛ ♦r❞❡♠ ❞❡ ❛♥✉❧❛♠❡♥t♦ ❞❡ φ ❡♠ a s❡♥❞♦ ❡①❛t❛♠❡♥t❡ ♦ ♥ú♠❡r♦ ❞❡ ✈❡③❡s

      , . . . , a

      1 n

      q✉❡ a ❛♣❛r❡❝❡ ♥♦ ❝♦♥❥✉♥t♦ {a }✳ P❛r❛ ♣r♦✈❛r ♦ ❚❡♦r❡♠❛ ✸✳✷✳✶ ❜❛st❛ tr❛♥s❢♦r♠❛r ♦ ♣r♦❜❧❡♠❛ ♥✉♠❛ ❡q✉❛çã♦

      ❞✐❢❡r❡♥❝✐❛❧ ♣❛r❝✐❛❧ ♥ã♦✲❧✐♥❡❛r ❡❧í♣t✐❝❛ ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠✳

      2

      , . . . , a ❚❡♦r❡♠❛ ✸✳✷✳✷✳ ❬✶✵❪ ❋✐①❡ ♦s ♣♦♥t♦s a

      1 n ❡♠ R ✭♥ã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡

      ❞✐st✐♥t♦s✮✳ ❈♦♥s✐❞❡r❡ ❛s ❢✉♥çõ❡s

      n

      X λ u (x) = ln 1 +

      − ✭✸✳✶✶✮

      2 k

      |x − a |

      k=1

      2

      , . . . , a ❞❡✜♥✐❞❛ ❡♠ R \ {a

      1 n }✱ ❡ n

      X λ g (x) = 4

      2

      2

      ( + λ) |x − a k |

      k=1

      2

      ❞❡✜♥✐❞❛ ❡♠ R ✱ ♦♥❞❡ λ > 4n. ❊♥tã♦ ❡①✐st❡ ✉♠❛ ú♥✐❝❛ ❢✉♥çã♦ r❡❛❧ ❛♥❛❧ít✐❝❛

      2

      v ❞❡✜♥✐❞❛ ❡♠ R s❛t✐s❢❛③❡♥❞♦

      u v

      e = 0 −∆v + g − 1 + e ✭✸✳✶✷✮

      ❡ lim v (x) = 0.

      |x|7→∞

      ❱❛♠♦s ♠♦str❛r q✉❡ ♦ ❚❡♦r❡♠❛ ✸✳✷✳✷ ✐♠♣❧✐❝❛ ♦ ❚❡♦r❡♠❛ ✸✳✷✳✶✳ ❈♦♥s✐❞❡r❡

      ❈❆P❮❚❯▲❖ ✸✳ ❚❊❖❘❊▼❆ P❘■◆❈■P❆▲ ✹✽

      1

      ♦♥❞❡ v ❡ u sã♦ ❛s ❢✉♥çõ❡s ❞♦ ❚❡♦r❡♠❛ ✸✳✷✳✷✳ P♦❞❡♠♦s ✈❡r q✉❡ f é s✉❛✈❡

      2

      , . . . , a

      1 n

      1

      ❡♠ R \ {a }✳ ➱ ♣♦ssí✈❡❧ ♠♦str❛r q✉❡ f s❛t✐s❢❛③

      1

      2f 1

    • e = 0

      1

      −∆f − 1 ✭✸✳✶✸✮

      2 ❡ n

      

    k

      2

      f , lim = ln (x )

      1 k

      − a

      |x|→a k

      2 , . . . , a

      k k 1 n

      ♦♥❞❡ n é ❛ ♦r❞❡♠ ❞❡ ❛♥✉❧❛♠❡♥t♦ ❞❡ φ ❡♠ a ✭❧❡♠❜r❡ q✉❡ ♦s ♣♦♥t♦s a ♥ã♦ sã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ❞✐st✐♥t♦s✮✳ P❛r❛ ❝❛❞❛ k✱ 1 ≤ k ≤ n✱ ❞❡✜♥❛ ♦ ❛♥❣✉❧♦

      (x k ) − a

      1 α .

      (x) = arctan

      k

      (x )

      k

      − a

      2 P♦❞❡♠♦s ✈❡r q✉❡

      α

    k (r, θ + 2π) = α (r, θ) + 2π.

    ❉❡✜♥❛ ❛ ❢✉♥çã♦

      

    n

      X f = α .

      

    2 k

    k=1

      2

      , . . . , a

      2

    1 n

      ❖❜s❡r✈❡ q✉❡ f é s✉❛✈❡ ❡♠ R \ {a } . ❚❡♠♦s q✉❡

      

    n

      X f (r, θ + 2π) = α (r, θ + 2π)

      2 k

    k=1

    n

      X = α (r, θ) + 2πn.

      k

    k=1

      ❉❡✜♥❛

      

    f

    1 +if 2

      φ = e , A = ∂ f + ∂ f ,

      1

      2

      

    1

      1

      2 A = f + ∂ f .

      2

      

    1

      1

      2

      2

      −∂

      ❈❆P❮❚❯▲❖ ✸✳ ❚❊❖❘❊▼❆ P❘■◆❈■P❆▲ ✹✾ (φ, A) s❛t✐s❢❛③ ❛s ❡q✉❛çõ❡s ❞❡ ✈órt✐❝❡ ✭✸✳✼✮✱ ✭✸✳✽✮ ❡ ✭✸✳✾✮✳ ❊s❝r❡✈❛

      ˆ A = A + iA ,

      

    1

      2

      ❡

      1 ¯ ∂ = (∂ + i∂ ) .

      

    1

      2

      2 ❚❡♠♦s q✉❡

      ˆ A f f f f

      = ∂ + ∂ + i ( + ∂ )

      2

      1

      1

      2

      1

      1

      2

      2

      −∂ = + i∂ ) (f + if )

      −i (∂

      1

      2

      1

      2

      ∂f, =

      −2i ¯

    • if

      1

      2

      ♦♥❞❡ f = f ✳ ▲♦❣♦ 2 ¯ ∂φ Aφ = 0.

      − i ˆ ❚❡♠♦s q✉❡ 2 ¯ ∂φ = (∂ + i∂ ) (φ + iφ )

      1

      2

      1

      2

      φ φ φ φ = ∂ + i (∂ + ∂ ) .

      1

      1

      2

      

    2

      1

      2

      2

      1

      − ∂ ❡

      Aφ =

      1 + iA 2 ) (φ 1 + iφ 2 )

      −i ˆ −i (A = A φ + A φ + i ( φ + A φ ) ,

      1

      2

      2

      

    1

      1

      1

      2

      2

      −A ♣♦rt❛♥t♦

      ∂φ Aφ 0 = 2 ¯ − i ˆ

      φ φ φ φ = (∂ + A ) )

      1

      1

      1

      2

      2

      2

      2

      1

      − (∂ − A φ φ φ φ + i ((∂ + A ) + (∂ )) .

      2

      1

      2

      

    2

      1

      2

      1

      1

      − A

      ❈❆P❮❚❯▲❖ ✸✳ ❚❊❖❘❊▼❆ P❘■◆❈■P❆▲ ✺✵ ❆❧é♠ ❞✐ss♦✱ t❡♠♦s q✉❡

      F = ∂ A A

      12

      1

      2

      2

      1

      − ∂ = ∂ ( f + ∂ f ) (∂ f + ∂ f )

      1

      1

      1

      

    2

      2

      2

      2

      1

      1

      2

      −∂ − ∂ = ∂ f + ∂ ∂ f ∂ f ∂ f

      1

      1

      1

      1

      

    2

      2

      1

      2

      2

      2

      2

      1

      −∂ − ∂ − ∂ = ∂ f ∂ f

      1

      1

      1

      2

      

    2

      1

      −∂ − ∂ = ,

      1

      −∆f ♣♦rt❛♥t♦

      1

      1

      2

      2

      2

    • F φ
    • φ +

      12

      1

      1 2 − 1 = −∆f |φ| − 1

      2

      2

      2

      1

      f +if

      = 1 2

    • e

      1

      −∆f − 1

      2

      1 1

      2f

      e =

    • 1 − 1

      −∆f

      2 = 0. ◆❛ ✉❧t✐♠❛ ✐❣✉❛❧❞❛❞❡ ✉s❛♠♦s ❛ ❊q✉❛çã♦ ✭✸✳✶✸✮✳ ■st♦ ♠♦str❛ q✉❡ (φ, A) é s♦❧✉çã♦ ❞❛s ❊q✉❛çõ❡s ✭✸✳✼✮✱ ✭✸✳✽✮ ❡ ✭✸✳✾✮✳

      ❖ ♦❜❥❡t✐✈♦ ❞♦ r❡st♦ ❞❡st❡ ❝❛♣ít✉❧♦ é ♣r♦✈❛r ♦ ❚❡♦r❡♠❛ ✸✳✷✳✷✳

      ✸✳✸ Pr♦♣r✐❡❞❛❞❡s ❞♦ ❢✉♥❝✐♦♥❛❧ G ∞

      2

      (R ) P❛r❛ t♦❞❛ ❢✉♥çã♦ v ❡♠ C c ✱ ❞❡✜♥✐♠♦s ♦ ❢✉♥❝✐♦♥❛❧

      ˆ

      1

      2 u v G (v) = ) + e (e . 2 |∇v| − v (1 − g − 1) ✭✸✳✶✹✮ R

      2 ◆❡st❛ s❡çã♦ ✈❛♠♦s ❡st✉❞❛r ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ G✳

      1

      2

      (R ) Pr♦♣♦s✐çã♦ ✸✳✸✳✶✳ ❖ ❢✉♥❝✐♦♥❛❧ G ♣♦❞❡ s❡r ❡st❡♥❞✐❞♦ ❛ H ✳ ❖✉ s❡❥❛✱

      1

      2

      (R ) ♣❛r❛ t♦❞❛ ❢✉♥çã♦ v ❡♠ H ✱ t❡♠♦s q✉❡ G (v) é ✜♥✐t♦✳

      ❈❆P❮❚❯▲❖ ✸✳ ❚❊❖❘❊▼❆ P❘■◆❈■P❆▲ ✺✶ ❉❡♠♦♥str❛çã♦✳ P♦❞❡♠♦s ❡s❝r❡✈❡r ❛ ❊q✉❛çã♦ ✭✸✳✶✹✮ ❝♦♠♦

      ˆ ˆ ˆ

      1

      2 u u v

      G v e (v) = (1 ) + (e 2 |∇v| − − g − e − 1 − v) 2 2 R

    2 R R

      = (I) − (II) + (III) .

      

    1

      2

      (R ) ❱❛♠♦s ♠♦str❛r q✉❡ s❡ v ❡stá ❡♠ H ✱ ❡♥tã♦ (I) < ∞✱ (II) < ∞ ❡ (III) <

      ∞✳ ✶✳ ❱❛♠♦s ♠♦str❛r q✉❡ (I) < ∞✳ P♦r ❤✐♣ós❡s❡✱ t❡♠♦s q✉❡

      ˆ

      

    2

      < 2 |∇v| ∞.

      R

      ✷✳ ❱❛♠♦s ♠♦str❛r q✉❡ (II) < ∞✳ ❚❡♠♦s q✉❡ ˆ ˆ

      u u

      v (1 ) 2 − g − e ≤ |v||1 − g − e |. 2 R R P❡❧❛ ❞❡s✐❣✉❛❧✐❞❛❞❡ ❞❡ ❍♦❧❞❡r✱ t❡♠♦s q✉❡ 1 1

      ˆ ˆ ˆ 2 2

      u 2 u

      2

      v .

      (1 ) 2 − g − e ≤ |v| |1 − g − e | 2 2 R R R P♦r ❤✐♣ót❡s❡✱ t❡♠♦s q✉❡

      ˆ

      

    2

      v < 2 ∞.

      R

      P❛r❛ ♠♦str❛r q✉❡ (II) < ∞✱ ❜❛st❛ ♠♦str❛r q✉❡ ˆ

      u

      2

      < 2 |1 − g − e | ∞.

      R

      = (0, 0)

      1 P❛r❛ ❡♥t❡♥❞❡r ♠❡❧❤♦r ❛ ♣r♦✈❛✱ ✈❛♠♦s s✉♣♦r n = 1 ❡ a ✳ ◆❡st❡ ❝❛s♦✱

      t❡♠♦s q✉❡ λ u ,

      (x) = 1 + − ln

      2

      |x| ❡

      4λ g , (x) =

      2

      2

      ❈❆P❮❚❯▲❖ ✸✳ ❚❊❖❘❊▼❆ P❘■◆❈■P❆▲ ✺✷ ♦♥❞❡ λ > 4✳ ❚❡♠♦s q✉❡

      −1

      λ

      u

      e = 1 +

      2

      |x|

      2

      |x| . =

      

    2

    • λ |x|

      P♦rt❛♥t♦ λ

      u 1 = .

      − e

      2

    • λ |x|

      P♦❞❡♠♦s ✈❡r q✉❡ 4λ λ

      u

    • 2

      |1 − g − e | =

      2

      2

      ( + λ) + λ |x| |x|

      C

      

    1

      ≤

      

    2

      |x| ♣❛r❛ |x| s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡✳ ❊❧❡✈❛♥❞♦ ❛♦ q✉❛❞r❛❞♦✱ ✈❡♠♦s q✉❡

      C

      2

    u

      2

      |1 − g − e | ≤

      4

      |x| ♣❛r❛ |x| s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡✳ P♦rt❛♥t♦

      ˆ ˆ ˆ

      u 2 u 2 u

      2 2 =

    • R |x|≤R |x|>R

      |1 − g − e | |1 − g − e | |1 − g − e |

      ˆ

      1 .

      3 + C

      2

      ≤ C

      4 |x|>R |x|

      ❯s❛♥❞♦ ❝♦♦r❞❡♥❛❞❛s ♣♦❧❛r❡s✱ t❡♠♦s q✉❡ ˆ ˆ ˆ

      2π ∞

      1

      1 rdrdθ = 2

      4

      4

      r

      R \R |x| R

      2π

      1 = ,

      2 R

      3 ❧♦❣♦

      ˆ

      u

      2

      ❈❆P❮❚❯▲❖ ✸✳ ❚❊❖❘❊▼❆ P❘■◆❈■P❆▲ ✺✸ ♣♦rt❛♥t♦ (II) < ∞✳

      ❱❛♠♦s ♠♦str❛r q✉❡ (III) < ∞✳ P❡❧❛ ❊q✉❛çã♦ ✭✸✳✶✶✮✱ ♣♦❞❡♠♦s ✈❡r q✉❡ ❛ ❢✉♥çã♦ u é ♥❡❣❛t✐✈❛✱ ♣♦rt❛♥t♦

      u

      e ≤ 1,

      ❧♦❣♦ ˆ ˆ

      u v u v

      e e (e 2 − 1 − v) ≤ |e − 1 − v| 2 R R

      ˆ

      v ≤ |e − 1 − v|. 2 R

      1

      2

      (R ) P❡❧♦ ❚❡♦r❡♠❛ ✷✳✷✳✸✱ ❝♦♠♦ v ❡stá ❡♠ H ✱ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ ♣♦s✐t✐✈❛ ρ

      ✭❞❡♣❡♥❞❡♥❞♦ ❞❡ v✮ t❛❧ q✉❡ ˆ 2 2

      ρ v

      e < 2 − 1 ∞. ✭✸✳✶✺✮

      R

      P♦❞❡♠♦s ❡s❝r❡✈❡r ˆ ˆ ˆ

      v v v 2 |e − 1 − v| = |e − 1 − v| + |e − 1 − v|. ✭✸✳✶✻✮ n o n o 1 1 R x:v(x)≥ 2 x:v(x)< 2

      ρ ρ x

      ❯s❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ e ≥ 1 + x✱ t❡♠♦s q✉❡ ˆ ˆ

      v v

      (e n o n o 1 |e − 1 − v| = − 1 − v) 1

      x:v(x)≥ x:v(x)≥ 2 2 ρ ρ

      ˆ

      v

      (e ≤ − 1) . n o 1

      x:v(x)≥ 2 ρ

      n 2 2

      2

      2 1 v ρ v 2

      v x : v(x)

      ❚❡♠♦s q✉❡ v ≤ ρ ♥♦ ❝♦♥❥✉♥t♦ ≥ o✱ ♣♦rt❛♥t♦ e ≤ e ♥♦

      ρ

      n

      1 2

      ρ

      x : v(x) o✱ ❧♦❣♦ ❝♦♥❥✉♥t♦ ≥

      ˆ ˆ 2 2

      v ρ v

      ❈❆P❮❚❯▲❖ ✸✳ ❚❊❖❘❊▼❆ P❘■◆❈■P❆▲ ✺✹ 2 2

      x ρ v

      ❯s❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ e ≥ 1+x✱ ✈❡♠♦s q✉❡ ❛ ❢✉♥çã♦ e −1 é ♥ã♦✲♥❡❣❛t✐✈❛✱ ♣♦rt❛♥t♦

      ˆ ˆ 2 2

      v ρ v e . n o 1 |e − 1 − v| ≤ − 1 2 R x:v(x)≥ 2

      ρ

      ❯s❛♥❞♦ ❛ ❊q✉❛çã♦ ✭✸✳✶✺✮✱ ♦❜t❡♠♦s ˆ

      

    v

    n o 1 |e − 1 − v| < ∞. ✭✸✳✶✼✮ x:v(x)≥ 2

      ρ

      P♦❞❡♠♦s ✈❡r q✉❡ ❛ ❢✉♥çã♦

      x

      e − 1 − x x

      →

      2

      x

      1 2

      ✱ ♣♦rt❛♥t♦ é ❧✐♠✐t❛❞❛ ♥♦ ✐♥t❡r✈❛❧♦ −∞,

      ρ v

      2

      v

      4

      |e − 1 − v| ≤ C n

      1

      x 2 : v(x) <

      ♥♦ ❝♦♥❥✉♥t♦ o✳ ▲♦❣♦

      ρ

      ˆ ˆ

      v

      2

      v

      4 n o n o 1 |e − 1 − v| ≤ C 1 x:v(x)< x:v(x)< 2 2

      ρ ρ

      ˆ

      2 v .

      4

      ≤ C 2 R P♦r ❤✐♣ót❡s❡✱ t❡♠♦s q✉❡

      ˆ

      2

      v < 2 ∞,

      R

      ♣♦rt❛♥t♦ ˆ

      

    v

    n o |e − 1 − v| < ∞. 1 ✭✸✳✶✽✮ x:v(x)< 2

      ρ

      P❡❧❛s ❊q✉❛çõ❡s ✭✸✳✶✻✮✱ ✭✸✳✶✼✮ ❡ ✭✸✳✶✽✮✱ t❡♠♦s q✉❡ ˆ

      v 2 |e − 1 − v| < ∞.

      R

      ❈❆P❮❚❯▲❖ ✸✳ ❚❊❖❘❊▼❆ P❘■◆❈■P❆▲ ✺✺

      1

      2

      (R ) Pr♦♣♦s✐çã♦ ✸✳✸✳✷✳ ❈♦♥s✐❞❡r❡ ❢✉♥çõ❡s v ❡ h ❡♠ H ✳ ❚❡♠♦s q✉❡ ❛ ❞❡✲

      ′

      (v, h) r✐✈❛❞❛ ❞❡ ●ât❡❛✉① G ❡①✐st❡ ❡ é ❞❛❞❛ ♣♦r ˆ

      ′ u u v

      G (v, h) = ) + he (e 2 {h∇v, ∇hi − h (1 − g − e − 1)} .

      R

      1 2 ′

      (R ) (v, ❆❧❡♠ ❞✐ss♦✱ ♣❛r❛ t♦❞❛ ❢✉♥çã♦ v ❡♠ H ✱ t❡♠♦s q✉❡ G ·) é ✉♠ ❢✉♥❝✐♦♥❛❧

      1

      2

      (R ) ❧✐♥❡❛r ❧✐♠✐t❛❞♦ ❡♠ H ✳ ❉❡♠♦♥str❛çã♦✳ ❚❡♠♦s q✉❡

      ˆ

      1

      2 u v

      G ,

      (v) = ) + e (e 2 |∇v| − v (1 − g − 1)

      R

      2 ❡

      ˆ

      1

      2 u v+th

      G e

      (v + th) = ) + e 2 |∇ (v + th) | − (v + th) (1 − g − 1

      R

      2 ˆ

      2

      1 t

      2

      2

      = + t 2 |∇v| h∇v, ∇hi + |∇h|

      R

      2

      2 ˆ

      u v+th

      e 2 ) + e , − (v + th) (1 − g

    • − 1

      R

      ♣♦rt❛♥t♦ ˆ

      th

      G t e (v + th)

      − G (v)

      2 u v − 1

      = ) + e e 2 h∇v, ∇hi + |∇h| − h (1 − g t R 2 t

      ˆ

      u +v

      = ) + he 2 h∇v, ∇hi − h (1 − g

      R

      ˆ

      th

      t e − 1 − th

      

    2 u v

    • e + e
    • 2 |∇h| t

        R

        2 ˆ

        u u v

        = ) + he (e 2 {h∇v, ∇hi − h (1 − g − e − 1)}

        R

        ˆ

        th

        t e − 1 − th

        

      2 u v

      • e ,
      • 2 + e |∇h| t

          R

          2

          ❈❆P❮❚❯▲❖ ✸✳ ❚❊❖❘❊▼❆ P❘■◆❈■P❆▲ ✺✻ ❧♦❣♦✱ ♣❡❧❛ ❞❡✜♥✐çã♦ ❞❡ ❞❡r✐✈❛❞❛ ❞❡ ●ât❡❛✉①✱ t❡♠♦s q✉❡

          G (v + th)

          ′ − G (v)

          G (v, h) = lim

          t→0

          t ˆ

          u u v

          = ) + he (e 2 {h∇v, ∇hi − h (1 − g − e − 1)}

          R

          ˆ

          th

          t e

          2 u v − 1 − th

        • lim + e e .

          |∇h|

          t→0 2 R

          2 t ✶✳ P♦r ❤✐♣ót❡s❡✱ t❡♠♦s q✉❡

          ˆ

          2

          < 2 |∇h| ∞,

          R

          ♣♦rt❛♥t♦ ˆ

          2

          t lim = 0. 2 |∇h|

          t→0 R

          ✷✳ ❱❛♠♦s ♠♦str❛r q✉❡ ˆ

          

        th

          e − 1 − th

          u v

          e e lim = 0. 2

          t→0 R t

          ❚❡♠♦s q✉❡ ˆ

          th

          e − 1 − th

          u v 2 e e t R

          ˆ th ˆ th e e − 1 − th − 1 − th

          

        u u v

          e e = 2 2 (e

        • t t

          − 1)

          R R

          ˆ th ˆ th e e − 1 − th − 1 − th

          u u v

          e e 2 2 (e ≤

        • t t

          − 1)

          R R = (I) + (II) .

          ✭✸✳✶✾✮ ✷✳❛✳ ❱❛♠♦s ♠♦str❛r q✉❡ lim (I) = 0.

          ✭✸✳✷✵✮

          t→0

          ❈❆P❮❚❯▲❖ ✸✳ ❚❊❖❘❊▼❆ P❘■◆❈■P❆▲ ✺✼ ❈♦♠♦ ❛ ❢✉♥çã♦ u é ♥❡❣❛t✐✈❛✱ t❡♠♦s q✉❡

          u

          e ≤ 1,

          ♣♦rt❛♥t♦ ˆ th ˆ e

          1 − 1 − th

          u th e e . 2 ≤ − 1 − th 2 t t R R

          ❯s❛♥❞♦ ❛ ❡①♣❛♥sã♦ ❡♠ sér✐❡ ❞❛s ♣♦t❡♥❝✐❛s ❞❛ ❢✉♥çã♦ ❡①♣♦♥❡♥❝✐❛❧✱ ♦❜t❡♠♦s

          ∞

          ˆ ˆ

          X

          1

          1

          

        th k−1 k

          e t h 2 − 1 − th ≤ 2 t k

          R R ! k=2

          !

          ∞

          ˆ

          X

          1

          k k−1

          t . ≤ |h| 2 R k !

          k=2

          P♦❞❡♠♦s ✈❡r q✉❡

          n

          X

          1

          k k−1

          n t 7→ |h| k !

          k=2

          é ✉♠❛ s❡q✉❡♥❝✐❛ ❝r❡s❝❡♥t❡ ❞❡ ❢✉♥çõ❡s ♥ã♦✲♥❡❣❛t✐✈❛s✳ P❡❧♦ t❡♦r❡♠❛ ❞❛ ❝♦♥✲ ✈❡r❣ê♥❝✐❛ ♠♦♥ót♦♥❛ ✭❚❡♦r❡♠❛ ✷✳✶✳✹✮✱ t❡♠♦s q✉❡

          !

          ∞ ∞

          ˆ ˆ

          X X

          1

          1

          k−1 k k−1 k t = t . 2 |h| |h| 2 k k R ! ! R k=2 k=2

          P♦rt❛♥t♦

          ∞

          X

          ˆ th ˆ e

          1 − 1 − th k

          u k−1 e t . 2 ≤ |h| 2 t k R ! R k=2

          ❆♣❧✐❝❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❙♦❜♦❧❡✈ ✭❚❡♦r❡♠❛ ✷✳✷✳✷✮✱ ♦❜t❡♠♦s ˆ

          k k/2+2 k k 1 2 k . 2 |h| ≤ 2 k h k H (R ) R

          P♦rt❛♥t♦

          

          ˆ th

          k k

          e

          u k−1 1 2 e t .

          X k √ − 1 − th

          2 ≤ 4 k h k H (R )

          ❈❆P❮❚❯▲❖ ✸✳ ❚❊❖❘❊▼❆ P❘■◆❈■P❆▲ ✺✽ ▲❡♠❜r❡ q✉❡ ❛ ❢ór♠✉❧❛ ❞❡ ❙t✐r❧✐♥❣ ❞❛ ❛♣r♦①✐♠❛çã♦ ❛ss✐♥tót✐❝❛ é ❞❛❞❛ ♣♦r

          k

          √ k k ,

          ! 2πk ∼ e

          ♦✉ s❡❥❛✱

          k

          k !e lim = 1.

          √

          k k→∞

          2πkk ❊♠ ♣❛rt✐❝✉❧❛r✱ ❡①✐st❡ ✉♠ ♥✉♠❡r♦ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ N t❛❧ q✉❡

          

        k k

          k ≤ k!e

          ♣❛r❛ t♦❞♦ k > N✳ ❚❡♠♦s q✉❡

          k

          k

          

        k−1 k−1 k

          t e ≤ t k !

          ♣❛r❛ t♦❞♦ k > N✱ ❡ ♣♦rt❛♥t♦

          

        N

          ˆ th

          k k

          e

          u k−1 1 2

          X k √ − 1 − th

          e t

          2 H (R ) 2 ≤ 4 k h k t k

          R !

        k=2

        k

          X √

          k−1 1 2 + 4 t 2e .

          H (R )

          k h k

          

        k=N +1

          ❖ ♣r✐♠❡✐r♦ t❡r♠♦ ❞♦ ❧❛❞♦ ❞✐r❡t♦ ❞❛ ❡q✉❛çã♦ ❛❝✐♠❛ é ✉♠❛ s♦♠❛ ✜♥✐t❛✱ ❡ ♦ √ −1 1 2

          2e

          H (R )

          s❡❣✉♥❞♦ t❡r♠♦ é ✉♠❛ sér✐❡ ❝♦♠ r❛✐♦ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ k h k ✳ P♦rt❛♥t♦ lim (I) = 0.

          t→0

          ✷✳❜✳ ❱❛♠♦s ♠♦str❛r q✉❡ lim (II) = 0.

          ✭✸✳✷✶✮

          t→0

          ❈❆P❮❚❯▲❖ ✸✳ ❚❊❖❘❊▼❆ P❘■◆❈■P❆▲ ✺✾ P❡❧❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❍♦❧❞❡r✱ t❡♠♦s q✉❡

          ˆ ˆ 1/2

          th

          e

          1 − 1 − th

          2 u v v

          e (e (e 2 − 1) ≤ − 1) 2 t t

          R R

          ˆ 1/2

          2 th

          e .

          ✭✸✳✷✷✮ · − 1 − th 2 R

          x

          ❯s❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ e − x − 1 ≥ 0✱ ♦❜t❡♠♦s

          2 v 2v v

          (e = e + 1 − 1) − 2e

          2v v

          = e − 2v − 1 − 2 (e − v − 1)

          2v e .

          ≤ − 2v − 1

          1

          2

          (R ) ❈♦♠♦ 2v ❡stá ❡♠ H ✱ ♣❡❧❛ ♣r♦✈❛ ❞❛ Pr♦♣♦s✐çã♦ ✸✳✸✳✶✱ s❛❜❡♠♦s q✉❡

          ˆ

          2v

          e < 2 − 2v − 1 ∞,

          R

          ♣♦rt❛♥t♦ ˆ ˆ

          2 v 2v

          (e e 2 − 1) ≤ − 2v − 1 2 R R <

          ∞. ✭✸✳✷✸✮ ❆❣♦r❛ tr❛❜❛❧❤❛♠♦s ❝♦♠ s❡❣✉♥❞♦ t❡r♠♦ ❞♦ ❧❛❞♦ ❞✐r❡t♦ ❞❛ ❊q✉❛çã♦ ✭✸✳✷✷✮✳ ❊s❝r❡✈❛ u = th✳ ❯s❛♥❞♦ ❛ ❡①♣❛♥sã♦ ❡♠ sér✐❡ ❞❛s ♣♦t❡♥❝✐❛s ❞❛ ❢✉♥çã♦ ❡①♣♦✲ ♥❡♥❝✐❛❧✱ ♦❜t❡♠♦s

          ∞

          ˆ ˆ

          X

          1

          2 u j+k

          . 2 |e − 1 − u| ≤ |u| 2 j

          R R !k! j,k=2

          P♦❞❡♠♦s ✈❡r q✉❡

          n

          X

          1

          j+k

          n 7→ |u| j

          !k!

          j,k=2

          ❈❆P❮❚❯▲❖ ✸✳ ❚❊❖❘❊▼❆ P❘■◆❈■P❆▲ ✻✵ ✈❡r❣ê♥❝✐❛ ♠♦♥ót♦♥❛ ✭❚❡♦r❡♠❛ ✷✳✶✳✹✮✱ t❡♠♦s q✉❡

          ∞ ∞

          ˆ ˆ

          X X

          1

          1

          j+k j+k

          , = 2 |u| |u| 2 R j !k! j !k! R

          j,k=2 j,k=2

          ♣♦rt❛♥t♦

          ∞

          ˆ ˆ

          X

          1

          2 u j+k 2 |e − 1 − u| ≤ |u| 2 j R !k! R j,k=2

          

        ∞ n

          ˆ

          X X

          1

          n = . 2 |u|

          (n R − k)!k!

          

        n=4 k=0

          ❆♣❧✐❝❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❙♦❜♦❧❡✈ ✭❚❡♦r❡♠❛ ✷✳✷✳✷✮✱ ♦❜t❡♠♦s ˆ

          

        n n/2+2 n n

        1 2

          n , 2 |u| ≤ 2 k u k H (R )

          R

          ♣♦rt❛♥t♦

          ∞ n

          ˆ

          n

          2

        u n/2 n

        1 2 2 . 2 |e − 1 − u| ≤ 4 k u k H (R ) R (n

          X X n

          − k)!k!

          n=4 k=0

          P❡❧❛ ❢ór♠✉❧❛ ❞❡ ❙t✐r❧✐♥❣✱ ❡①✐st❡ ✉♠ ♥✉♠❡r♦ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ N t❛❧ q✉❡

          

        n n

          n ≤ n!e

          ❈❆P❮❚❯▲❖ ✸✳ ❚❊❖❘❊▼❆ P❘■◆❈■P❆▲ ✻✶ ♣❛r❛ t♦❞♦ n > N✳ P♦rt❛♥t♦

          N n

          ˆ

          n

          u 2 n/2 n 1 2

          X X n

          2 2 |e − 1 − u| ≤ 4 k u k H (R )

          R (n

          − k)!k!

          n=4 k=0 ∞ n n

          X X n ! √ 1 2

        • 4 2e

          H (R )

          k u k (n

          − k)!k!

          n=N +1 k=0 N n n

          n/2 n 1 2

          X X n

          = 4

          2 k u k H (R ) (n

          − k)!k!

          n=4 k=0 ∞ n

          X √ 1 2 .

        • 4 2 2e k u k H (R )

          n=N +1

          ❆❝✐♠❛ ✉s❛♠♦s ♦ ❢❛t♦ q✉❡

          n

          X n !

          n .

          = 2 (n

          − k)!k!

          k=0

          ▲❡♠❜r❛♥❞♦ q✉❡ u = th✱ ♦❜t❡♠♦s

          N n

          ˆ

          n

          X X

          2 n

        th n n/2 n

        1 2

          e t

          2 − 1 − th ≤ 4 k h k 2 H (R ) R (n

          − k)!k!

          n=4

        k=0

        ∞ n

          X √

          n 1 2 t .

        • 4 2 2e k h k H (R ) ✭✸✳✷✹✮

          n=N +1

          ❖ ♣r✐♠❡✐r♦ t❡r♠♦ ❞♦ ❧❛❞♦ ❞✐r❡t♦ ❞❛ ❡q✉❛çã♦ ❛❝✐♠❛ é ✉♠❛ s♦♠❛ ✜♥✐t❛✱ ❡ ♦ √ −1 1 2

          2e s❡❣✉♥❞♦ t❡r♠♦ é ✉♠❛ sér✐❡ ❝♦♠ r❛✐♦ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ 2 k h k H (R ) ✳ P❡❧❛s ❊q✉❛çõ❡s ✭✸✳✷✷✮✱ ✭✸✳✷✸✮ ❡ ✭✸✳✷✹✮✱ t❡♠♦s q✉❡ lim (II) = 0.

          t→0

          ✷✳❝✳ P❡❧❛s ❊q✉❛çõ❡s ✭✸✳✶✾✮✱ ✭✸✳✷✵✮ ❡ ✭✸✳✷✶✮✱ ♦❜t❡♠♦s ˆ

          

        th

          e − 1 − th

          u v

          e e lim = 0. 2

          t→0 R t

          ❈❆P❮❚❯▲❖ ✸✳ ❚❊❖❘❊▼❆ P❘■◆❈■P❆▲ ✻✷ ✸✳ P❡❧♦s ✐t❡♥s ✶✳ ❡ ✷✳✱ ❝♦♥❝❧✉í♠♦s q✉❡

          ˆ

          ′ u u v

          G (v, h) = ) + he (e 2 {h∇v, ∇hi − h (1 − g − e − 1)} .

          R ′

          (v, P♦❞❡♠♦s ✈❡r q✉❡ G ·) é ❧✐♥❡❛r✳

          ✹✳ ❱❛♠♦s ♠♦str❛r q✉❡

          ′ 1 2 G (v, h) , H (R )

          ≤ C (v) k h k ♦♥❞❡ C (v) é ✉♠❛ ❝♦♥st❛♥t❡ ♣♦s✐t✐✈❛ ❞❡♣❡♥❞❡♥❞♦ ❞❡ v✳ ❚❡♠♦s q✉❡

          ˆ ˆ ˆ

          ′ u u v

          (v, h) e

          |G | ≤ |h∇v, ∇hi| + |h| |1 − g − e | + |h| |e − 1| 2 2 2 R R R ˆ ˆ ˆ

          u v

          ≤ |h∇v, ∇hi| + |h| |1 − g − e | + |h| |e − 1| , 2 2 2 R R R

          u

          ♣♦✐s e ≤ 1✳ P❡❧♦ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❍♦❧❞❡r t❡♠♦s 1 1 1 1 ˆ ˆ ˆ ˆ 2 2 2 2

          2

          2

          2 ′ 2 u

          h (v, h) 2 2 2 2

        • R R R R
        • 1

          1

            |G | ≤ |∇v| |∇h| |1 − g − e |

            ˆ 2 ˆ 2

            2 2 v

          • h
          • 2 2 |e − 1|

              R R 1

              ˆ 2

              2 u 1 2 1 2 1 2

            • H (R ) H (R ) H (R )

              ≤k h k k v k k h k |1 − g − e | 2

            1

            R ˆ 2

              2 v 1 2

            • .

              H (R )

              k h k |e − 1| 2 R P❡❧❛ ♣r♦✈❛ ❞❛ Pr♦♣♦s✐çã♦ ✸✳✸✳✶✱ s❛❜❡♠♦s q✉❡

              ˆ

              u

              2

              < 2 |1 − g − e | ∞,

              R

              ❡ ˆ

              2 v

              < 2 |e − 1| ∞,

              R

              ♣♦rt❛♥t♦

              ❈❆P❮❚❯▲❖ ✸✳ ❚❊❖❘❊▼❆ P❘■◆❈■P❆▲ ✻✸ ♦♥❞❡ 1 1

              ˆ ˆ 2 2

              2

              2 u v

              C . 1 2 + (v) =

            • R R ′

              H (R )

              k v k |1 − g − e | |e − 1| 2 2

              (v, ✺✳ P❡❧♦ ✐t❡♥s ✸✳ ❡ ✹✳ t❡♠♦s q✉❡ G ·) é ✉♠ ❢✉♥❝✐♦♥❛❧ ❧✐♥❡❛r ❧✐♠✐t❛❞♦ ❡♠

              1 2 ′

              H (R ) (v,

              ✳ ➱ ❝♦♥❤❡❝✐❞♦ q✉❡ ✐st♦ ✐♠♣❧✐❝❛ G ·) ❝♦♥tí♥✉♦✳

              Pr♦♣♦s✐çã♦ ✸✳✸✳✸✳ ❖ ❢✉♥❝✐♦♥❛❧ G é ❡str✐t❛♠❡♥t❡ ❝♦♥✈❡①♦✱ ✐st♦ é✱ ❞❛❞♦ ✉♠

              1

              2

              (R ) ♥ú♠❡r♦ 0 < t < 1 ❡ ❢✉♥çõ❡s v ❡ w ❡♠ H ✱ t❡♠♦s q✉❡

              G (tv + (1 − t) w) < tG (v) + (1 − t) G (w) . ❉❡♠♦♥str❛çã♦✳ ❊s❝r❡✈❛

              G (tv + (1

              − t) w) ˆ ˆ

              1

              2

              = (tv + (1 ) 2 |∇ (tv + (1 − t) w)| − − t) w) (1 − g 2

            2 R R

              ˆ

              u tv+(1−t)w

              e e 2 − 1

            • R

              = (I) − (II) + (III) . ❱❛♠♦s ♠♦str❛r q✉❡ (I) é ❝♦♥✈❡①♦✱ (II) é ❧✐♥❡❛r ❡ (III) é ❡str✐t❛♠❡♥t❡ ❝♦♥✲ ✈❡①♦✳

              ❚❡r♠♦ (I)✳ P❡❧❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❈❛✉❝❤②✲❙❝❤✇❛r③✱ t❡♠♦s q✉❡ ˆ

              1

              2

              2

              2

              2

              t (I) (tv + (1 + 2t (1

              − t) w) = |∇v| − t) h∇v, ∇wi + (1 − t) |∇w| 2

            2 R

              ˆ

              1

              2

              2

              2

              2

              t + 2t (1 . ≤ |∇v| − t) |∇v| |∇w| + (1 − t) |∇w| 2

            2 R

              ❈❆P❮❚❯▲❖ ✸✳ ❚❊❖❘❊▼❆ P❘■◆❈■P❆▲ ✻✹

              2

              2

            • b ❯s❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ 2ab ≤ a ✱ ♦❜t❡♠♦s

              ˆ

              1

              2

              2

              2

              2

              2

              2

              (I) (tv + (1 t + t (1 − t) w) ≤ |∇v| 2 − t) |∇v| |∇w| + (1 − t) |∇w|

            • 2 R ˆ

              1

              2

              2

              2

              2

              = t + t (1 + (1 + t (1 2 − t) |∇v| − t) − t) |∇w|

              2 R ˆ

              1

              2

              2

              = t + (1 2 |∇v| − t) |∇w|

              2 R = t (I) (v) + (1 − t) (I) (w) .

              ❚❡r♠♦ (II)✳ ❚❡♠♦s q✉❡ (II) (tv + (1 − t) w) = t (II) (v) + (1 − t) (II) (w) . ❚❡r♠♦ (III)✳ ❈♦♠♦ ❛ ❢✉♥çã♦ ❡①♣♦♥❡♥❝✐❛❧ é ❡str✐t❛♠❡♥t❡ ❝♦♥✈❡①❛✱ t❡♠♦s q✉❡

              tv+(1−t)w v w e < te .

            • (1

              − t) e

              u

              ❙✉❜tr❛✐♥❞♦ 1 ❡♠ ❛♠❜♦s ❧❛❞♦s ❡ ♠✉❧t✐♣❧✐❝❛♥❞♦ ❛♠❜♦s ❧❛❞♦s ♣♦r e ✱ ♦❜t❡♠♦s q✉❡

              u tv+(1−t)w u v w

              e e < e (te + (1 − 1 − t) e − 1) ,

              ❧♦❣♦ ˆ

              

            u tv+(1−t)w

              e e (III) (tv + (1

              − t) w) = − 1 2 R ˆ ˆ ˆ

              u v u w u

              < t e e e e e

            • (1
            • 2 − t) − 2 2 R R R ˆ ˆ

                u v u w

                e e = t (e (e 2 − 1) + (1 − t) − 1) 2 R R

                = t (III) (v) + (1 − t) (III) (w) . P♦rt❛♥t♦ G é ❡str✐t❛♠❡♥t❡ ❝♦♥✈❡①♦✳

                Pr♦♣♦s✐çã♦ ✸✳✸✳✹✳ ❖ ❢✉♥❝✐♦♥❛❧ G é s❡♠✐❝♦♥tí♥✉♦ ✐♥❢❡r✐♦r♠❡♥t❡ ♥♦ s❡♥t✐❞♦

                1

                2

                (R )

                n

                ❢r❛❝♦✳ ❖✉ s❡❥❛✱ s❡ v → v ♥♦ s❡♥t✐❞♦ ❢r❛❝♦ ❡♠ H ✱ ❡♥tã♦

                ❈❆P❮❚❯▲❖ ✸✳ ❚❊❖❘❊▼❆ P❘■◆❈■P❆▲ ✻✺

                ′

                (v, ❉❡♠♦♥str❛çã♦✳ P❡❧❛ Pr♦♣♦s✐çã♦ ✸✳✸✳✷✱ s❛❜❡♠♦s q✉❡ G ·) é ✉♠ ❢✉♥❝✐♦♥❛❧

                1

                2 ′

                (R ) (v,

                ❧✐♥❡❛r ❝♦♥tí♥✉♦ ❡♠ H ✳ P❡❧❛ Pr♦♣♦s✐çã♦ ✸✳✸✳✸✱ s❛❜❡♠♦s q✉❡ G ·) é ❡str✐t❛♠❡♥t❡ ❝♦♥✈❡①♦✳ ❯s❛♥❞♦ ❛ Pr♦♣♦s✐çã♦ ✷✳✸✳✸✱ ♦❜t❡♠♦s ♦ r❡s✉❧t❛❞♦✳ Pr♦♣♦s✐çã♦ ✸✳✸✳✺✳ ❊①✐st❡ ❝♦♥st❛♥t❡s α > 0✱ b ❡ k > 0 t❛✐s q✉❡

                2 1 2

                α k v k

                H (R ) ′

                G (v, v) ≥ − b 1 2 1 + k

                H (R )

                k v k

                1

                2

                (R ) ♣❛r❛ t♦❞❛ ❢✉♥çã♦ v ❡♠ H ✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ ♣♦s✐t✐✈❛ R t❛❧ q✉❡

                ′

                G (v, v) > 0

                1

                2 1 (R ) = R.

                ♣❛r❛ t♦❞❛ ❢✉♥çã♦ v ❡♠ H ❝♦♠ kvk H ❉❡♠♦♥str❛çã♦✳ ▲❡♠❜r❡ q✉❡

                n

                X λ u (x) = ln 1 + ,

                −

                2 k

                |x − a |

                k=1

                ❡

                n

                X λ g ,

                (x) = 4

                2

                2

                ( k + λ) |x − a |

                k=1

                ♦♥❞❡ λ > 4n✳ ✶✳ ❱❛♠♦s ♠♦str❛r q✉❡ 1 (x) ,

                1

                − g ≥ c ✭✸✳✷✺✮

                4n

                = 1

                1

                ♦♥❞❡ c − ✳ ❚❡♠♦s q✉❡

                λ

                λ

                1 ,

                ≤

                2

                2

                λ ( + λ)

                k

                |x − a |

                ❈❆P❮❚❯▲❖ ✸✳ ❚❊❖❘❊▼❆ P❘■◆❈■P❆▲ ✻✻ ♣♦rt❛♥t♦

                n

                X λ g (x) = 4

                2

                2

                ( + λ)

                k

                |x − a |

                k=1

                4n ≤

                λ = 1 ,

                1

                − c ❧♦❣♦ .

                1 (x)

                1

                − g ≥ c ✷✳ ❱❛♠♦s ♠♦str❛r q✉❡

                u (x)

                1 (x) − g − e ≥ 0. ✭✸✳✷✻✮

                ❚❡♠♦s q✉❡ !

                n

                2 X

              • λ

                k

                |x − a |

                u (x)

                e = exp ln −

                2 k

                |x − a |

                k=1 n

                2 Y k

                |x − a | = exp ln

                2 k + λ

                |x − a |

                k=1 n

                2 Y

              k

                |x − a | = ,

                

              2

              • λ

                k

                |x − a |

                k=1

                ♣♦rt❛♥t♦

                

              n n

                2 X Y

                λ

                k

              u (x) |x − a |

              • g (x) + e = 4 .

                ✭✸✳✷✼✮

                2

                2

                

              2

              • λ

                k

              • λ |x − a |

                

              k=1 k k=1

                |x − a | ❊s❝r❡✈❛

                4n γ =

                λ ❡

                λ z . =

                k

                2 k + λ

                |x − a |

                ❈❆P❮❚❯▲❖ ✸✳ ❚❊❖❘❊▼❆ P❘■◆❈■P❆▲ ✻✼ ◆♦t❡ q✉❡

                2 k

                |x − a | . 1 =

                k

                − z

                2 k + λ

                |x − a | P♦❞❡♠♦s r❡❡s❝r❡✈❡r ❛ ❊q✉❛çã♦ ✭✸✳✷✼✮ ❝♦♠♦

                n n

                X Y γ

                u (x)

                2

              • g z (x) + e = (1 ) .

                k k − z ✭✸✳✷✽✮

                n

                k=1 k=1

                ❖❜s❡r✈❡ q✉❡ λ z

                =

                k

                2 k + λ

                |x − a | < 1,

                ♣♦rt❛♥t♦ >

                1 0.

                

              k

                − z ▲❡♠❜r❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❛r✐t♠ét✐❝❛✲❣❡♦♠étr✐❝❛

                ! n

                n n

                Y

                X

                1 b b

                

              k k

                ≤ n

                k=1 k=1 i

                ♣❛r❛ b ♣♦s✐t✐✈♦✳ ❚❡♠♦s q✉❡ ! n

                n n

                Y

                X

                1 z (1 )

                1

                k k

                − z ≤ − n

                k=1 k=1 n

                X

                1 z . ≤ 1 − k ✭✸✳✷✾✮ n

                k=1

                ◆♦t❡ q✉❡ ♥❛ ✉❧t✐♠❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛ ✉s❛♠♦s ♦ ❢❛t♦ q✉❡

                

              n

                X

                1 z < 1 1.

                k

                − n

                

              k=1

                ❈❆P❮❚❯▲❖ ✸✳ ❚❊❖❘❊▼❆ P❘■◆❈■P❆▲ ✻✽ ❯s❛♥❞♦ ❛ ❉❡s✐❣✉❛❧❞❛❞❡ ✭✸✳✷✾✮ ♥❛ ❊q✉❛çã♦ ✭✸✳✷✽✮✱ ♦❜t❡♠♦s

                n n

                X X γ

                1

                u (x)

                2

                g z z (x) + e +

                k

                ≤ 1 − k n n

                k=1 k=1 n n

                X X γ

                1

              • z z

                k k

                ≤ 1 − n n

                k=1 k=1 ≤ 1.

                <

                1

                k

                ◆♦t❡ q✉❡ ♥❛ s❡❣✉♥❞❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✉s❛♠♦s z ❡ ♥❛ t❡r❝❡✐r❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✉s❛♠♦s γ < 1✳ Pr♦✈❛♠♦s q✉❡

                u (x)

                1 (x) − g − e ≥ 0. ✸✳ ❱❛♠♦s ♠♦str❛r q✉❡ s❡ v ≥ 0✱ ❡♥tã♦

                1

                2 u +v

                2

                v e (u + g )

                − 1 + g ≥ βv − ✭✸✳✸✵✮

                1 − β

                ♣❛r❛ t♦❞♦ 0 < β < 1✳ ❊s❝r❡✈❡♠♦s

                u +v 2 u +v v e = v + v (u + g ) + v e + v) .

                − 1 + g − 1 − (u

                x

                ➱ ♣♦ssí✈❡❧ ♠♦str❛r q✉❡ e − 1 − x ≥ 0 ♣❛r❛ t♦❞♦ x ❡♠ R✱ ❧♦❣♦

                u +v

                e

              • v) − 1 − (u ≥ 0,

                ♣♦rt❛♥t♦

                u +v

                2

                v e + v (u + g ) − 1 + g ≥ v

                2

                2

                = βv + (1 + v (u + g ) − β) v

                ❈❆P❮❚❯▲❖ ✸✳ ❚❊❖❘❊▼❆ P❘■◆❈■P❆▲ ✻✾ ♣❛r❛ t♦❞♦ 0 < β < 1✳ ❈♦♠♣❧❡t❛♥❞♦ q✉❛❞r❛❞♦s✱ ♦❜t❡♠♦s

                " #

              1

              2 !

                2

                u u

              • g + g

                u +v

                2

              2

                v e v 1 1

              • (1

                − 1 + g − β) − ≥ βv 2 2 2 (1 2 (1

                − β) − β)

                1

                2

                2

                (u + g ) ≥ βv − 4 (1

                − β)

                1

                2

                2

                (u + g ) , ≥ βv −

                1 − β

                ♣♦rt❛♥t♦

                1

                2 u +v 2 v e (u + g ) .

                − 1 + g ≥ βv −

                1 − β

                ✹✳ ❱❛♠♦s ♠♦str❛r q✉❡ s❡ v ≤ 0✱ ❡♥tã♦

                2

                c

                1

              u +v |v|

                v e .

                − 1 + g ≥ ✭✸✳✸✶✮ 1 + |v|

                ❚❡♠♦s q✉❡

                

              u +v u u −|v|

                v e ) + 1 .

                − 1 + g = |v| (1 − g − e |v| e − e ❯s❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ x

                −x

                1 − e ≥ 1 + x

                ♣❛r❛ x ≥ 0✱ ♦❜t❡♠♦s

                −|v| |v|

                1 , − e ≥ 1 +

                |v| ♣♦rt❛♥t♦

                2 u

                e

                u +v u |v|

                v e ) +

                − 1 + g ≥ |v| (1 − g − e 1 + |v|

                2

                2 u u

                ) + e |v| + |v| (1 − g − e |v| .

                = 1 + |v|

                ❯s❛♥❞♦ ❛s ❉❡s✐❣✉❛❧❞❛❞❡s ✭✸✳✷✺✮ ❡ ✭✸✳✷✻✮✱ ♦❜t❡♠♦s

                2

                c

                1

              u +v |v|

                ❈❆P❮❚❯▲❖ ✸✳ ❚❊❖❘❊▼❆ P❘■◆❈■P❆▲ ✼✵ ✺✳ ❱❛♠♦s ♠♦str❛r q✉❡

                ! ˆ

                2

                c β

                1 2 |v| ′

                G (v, v) 2

              • R 1 +

                ≥ |∇v| − b, ✭✸✳✸✷✮

                |v|

                1

                2 2 2

              2 L (R )

              • g ♦♥❞❡ β = ❡ b = 2 k u k ✳ P❡❧❛ Pr♦♣♦s✐çã♦ ✸✳✸✳✷✱ t❡♠♦s q✉❡

                ˆ

                2 ′ u u v

                G (v, v) = ) + ve (e 2 |∇v| − v (1 − g − e − 1)

                R

                ˆ

                2 u +v

                = + v e . 2 |∇v| − 1 + g

                R

                P♦❞❡♠♦s ❡s❝r❡✈❡r ˆ ˆ

                2 ′ u +v

                (v, v) = 2 |∇v| − 1 + g

              • G v e

                R {v≥0}

                ˆ

                

              u +v

              • {v≤0}

                v e .

                − 1 + g

                ❯s❛♥❞♦ ❛s ❊q✉❛çõ❡s ✭✸✳✸✵✮ ❡ ✭✸✳✸✶✮✱ ♦❜t❡♠♦s ˆ ˆ ˆ

                1

                2

                2 ′

                2 G βv

                (v, v) 2 (u + g ) ≥ |∇v| −

              • R

                1

                {v≥0} − β {v≥0}

                2

                ˆ c

                1

                |v| + . 1 +

                {v≤0} |v|

                ❖❜s❡r✈❡ q✉❡ ♥❛ s❡❣✉♥❞❛ ✐♥t❡❣r❛❧ ✈❛❧❡ q✉❡

                2

                β

                

              2 |v|

                β |v| ≥ 1 +

                |v|

                2

                c β

                1

                |v| ,

                ≥ 1 + |v|

                ❈❆P❮❚❯▲❖ ✸✳ ❚❊❖❘❊▼❆ P❘■◆❈■P❆▲ ✼✶ <

                1

                1 ♣♦✐s 0 < c ✳ ❯s❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛✱ ♦❜t❡♠♦s

                ˆ ˆ

                2 ˆ

                c β

                1

                2 1 |v|

                2 ′

                G (v, v)

                (u + + g ) ≥ |∇v| − 2 R 1 +

                1

                {v≥0} |v| − β {v≥0}

                ˆ

                2

                c

                1

                |v| .

              • 1 +

                {v≤0}

                |v| ❈♦♠♦ 0 < β < 1✱ t❡♠♦s q✉❡

                ˆ ˆ

                1

                1

                2

                2

                , (u + g ) (u + g )

                ≤ 2

                1

                1 R − β {v≥0} − β

                ❡ ˆ

                

              2 ˆ

                2

                c c β

                1

                1

                |v| |v| ,

                ≥ 1 + 1 +

                

              {v≤0} |v| {v≤0} |v|

                ❧♦❣♦

                2

                ˆ ˆ ˆ c β

                1

                2

                

              1

                2 ′ |v|

                G .

              • (v, v)

                (u + g ) ≥ |∇v| − 2 2 R 1 +

                1 R

                {v≥0} |v| − β

                ˆ

                2

                c β

                1

                |v|

              • 1 +

                {v≤0} |v|

                ! ˆ

                2 ˆ

                c β

                1

                1 2 |v| 2 .

                = (u + g ) 2 2 |∇v| −

              • R 1 +

                1 R |v| − β

                1

                2 2

              2

              2 L (R )

              • g ❊s❝r❡✈❛ β = ❡ b = 2 k u k ✳ ❖❜t❡♠♦s

                ! ˆ

                2

                c β

                1 2 |v| ′

                G (v, v) 2 ≥ |∇v| − b.

              • R 1 +

                |v| ✻✳ ❱❛♠♦s ♠♦str❛r q✉❡

                4

                ˆ

                2 2 2

                k v k

                L (R )

                |v| . 2 ≥ ✭✸✳✸✸✮

                

              2

                3 R 1 +

                |v| 2 2 3 2 k v k k v k

              • L (R ) L (R )

                ❚❡♠♦s q✉❡ ˆ ˆ 1

                2

                ❈❆P❮❚❯▲❖ ✸✳ ❚❊❖❘❊▼❆ P❘■◆❈■P❆▲ ✼✷ P❡❧❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❍♦❧❞❡r t❡♠♦s 1 1

                " # 2 ˆ ˆ

                

              2 ˆ

              2 2 |v|

                2

                3 . 2 2 2

              • R R 1 + R

                |v| ≤ |v| |v|

                |v|

                2 2 2

                3 3 2

              • L (R ) L (R )

                ✱ ❊❧❡✈❛♥❞♦ ❛♠❜♦s ♦s ❧❛❞♦s ❡♠ q✉❛❞r❛❞♦ ❡ ❞✐✈✐❞✐♥❞♦ ♣♦r k v k k v k

                ♦❜t❡♠♦s

                4

                2

                ˆ 2 2 k v k

                L (R )

                |v| .

                ≥

                2

                3 R 1 +

                |v| 2 2 3 2

              • 2 k v k k v k

                L (R ) L (R )

                ✼✳ ❱❛♠♦s t❡r♠✐♥❛r ❛ ♣r♦✈❛ ❞❛ ♣r♦♣♦s✐çã♦✳ ❯s❛♥❞♦ ❛ ❉❡s✐❣✉❛❧❞❛❞❡ ✭✸✳✸✸✮ ♥❛ ❉❡s✐❣✉❛❧❞❛❞❡ ✭✸✳✸✷✮✱ ♦❜t❡♠♦s

                4

                ˆ 2 2 c β

                1

                k v k

                L (R )

                2 ′

                G (v, v) ≥ |∇v| − b.

              • 2

                3 2 2 2 3

              • R
              • 2 k v k k v k L (R ) L (R )

                  ▲❡♠❜r❛♥❞♦ q✉❡ ˆ ˆ

                  2

                  2

                  2

                • ,
                • 1 2 = k v k |v| |∇v|

                    H (R ) 2 2 R R

                    t♦♠❡ 0 < σ < 1 t❛❧ q✉❡

                    2 2 2

                    2 1 2

                    = (1 , k v k L (R ) − σ) k v k H (R ) ❡

                    ˆ

                    2

                    2 1 2 .

                    = σ 2 |∇v| k v k H (R )

                    R

                    ❚❡♠♦s q✉❡

                    2

                    4 1 2

                    c β (1

                    1

                    − σ) k v k

                    H (R ) ′

                  2 G

                    (v, v) 1 2 ≥ σ k v k − b.

                  • H (R )

                    2

                    3

                    (1 1 2 3 2 − σ) k v k k v k

                  • H (R ) L (R )

                    P❡❧❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❙♦❜♦❧❡✈ ✭❚❡♦r❡♠❛ ✷✳✷✳✷✮✱ t❡♠♦s q✉❡

                    3 3 2

                    3 1 2

                    , k v k L (R ) ≤ k k v k H (R )

                    ❈❆P❮❚❯▲❖ ✸✳ ❚❊❖❘❊▼❆ P❘■◆❈■P❆▲ ✼✸ √

                    2 ♦♥❞❡ k = 8 · 27✳ ❊♥tã♦

                    2

                    4 1 2

                    c β (1

                    

                  1

                    − σ) k v k

                    H (R ) ′

                    2

                  • G (v, v)
                  • 1 2 ≥ σ k v k H (R ) − b

                      2 1 2

                      3 1 2

                      (1 +k − σ) k v k k v k

                      H (R ) H (R )

                      !

                      2

                      c β (1

                      1 2 − σ)

                      = σ 1 2 k v k H (R ) − b 1 2

                    • (1

                      − σ) + k k v k H (R ) !

                      2

                      (1

                      

                    2 − σ)

                    • β σ
                    • 1 2

                        1

                        ≥ c k v k H (R ) − b, 1 2 1 + k

                        H (R )

                        k v k β

                        1

                        ♣♦✐s 1 − σ < 1 ❡ 1 ≥ c ✳ P♦rt❛♥t♦

                        2 1 2

                        c β

                        1

                        k v k

                        H (R )

                        2 ′ 1 2 G (v, v) (1 + σ 1 + k H (R )

                        ≥ − σ) k v k − b 1 2 1 + k H (R ) k v k

                        2 1 2

                        c β

                        1

                        k v k

                        H (R )

                        2

                        (1 + σ ≥ − σ) − b 1 2 1 + k k v k H (R )

                        2 1 2

                        c β

                        1

                        3 k v k

                        H (R ) ≥ − b. 1 2 4 1 + k

                        k v k H (R )

                        3

                        c β

                        1

                        ❚♦♠❛♥❞♦ α = ✱ ♦❜t❡♠♦s

                        4

                        2 1 2

                        α k v k

                        H (R ) ′

                        G (v, v) ≥ − b. 1 2 1 + k

                        H (R )

                        k v k

                        ✸✳✹ Pr♦✈❛ ❞♦ ❚❡♦r❡♠❛ Pr✐♥❝✐♣❛❧

                        P❡❧❛s Pr♦♣♦s✐çõ❡s ✸✳✸✳✶✱ ✸✳✸✳✷ ❡ ✸✳✸✳✹✱ s❛❜❡♠♦s q✉❡ G é ✉♠ ♦♣❡r❛❞♦r ❞❡✜♥✐❞♦

                        1

                        2

                        (R ) ♥♦ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt H ✱ ❞✐❢❡r❡♥❝✐á✈❡❧ ♥♦ s❡♥t✐❞♦ ❞❡ ●ât❡❛✉① ❡ s❡♠✐✲ ❝♦♥tí♥✉♦ ✐♥❢❡r✐♦r♠❡♥t❡ ♥♦ s❡♥t✐❞♦ ❢r❛❝♦✳ P❡❧❛ Pr♦♣♦s✐çã♦ ✸✳✸✳✺✱ s❛❜❡♠♦s q✉❡ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ ♣♦s✐t✐✈❛ R t❛❧ q✉❡

                        ′

                        G (v, v) > 0

                        ❈❆P❮❚❯▲❖ ✸✳ ❚❊❖❘❊▼❆ P❘■◆❈■P❆▲ ✼✹

                        1

                        2 1 2

                        (R ) = R ♣❛r❛ t♦❞♦ v ❡♠ H ❝♦♠ kvk H (R ) ✳ P❡❧❛ Pr♦♣♦s✐çã♦ ✷✳✸✳✶✱ ❡①✐st❡ ✉♠❛

                        1 2 ′

                        (R ) (v ) = 0 ❢✉♥çã♦ v ❡♠ H ❞❡ ♠í♥✐♠♦ ❧♦❝❛❧ ❞❡ G ❝♦♠ G ✳ P❡❧❛ Pr♦♣♦s✐çã♦ ✸✳✸✳✸✱ s❛❜❡♠♦s q✉❡ G é ❡str✐t❛♠❡♥t❡ ❝♦♥✈❡①♦✳ P♦rt❛♥t♦✱ ♣❡❧❛ Pr♦♣♦s✐çã♦ ✷✳✸✳✷✱ ❛ ❢✉♥çã♦ v é ú♥✐❝❛✳ ▼♦str❛♠♦s q✉❡ ❡①✐st❡ ✉♠❛ ú♥✐❝❛ ❢✉♥çã♦ v ❡♠

                        1

                      2 H (R )

                        t❛❧ q✉❡

                        ′

                        G (v , w ) = 0

                        1

                        2

                        (R ) ♣❛r❛ t♦❞♦ w ❡♠ H ✳ ❖✉ s❡❥❛✱

                        ˆ

                        u u v

                        ) + we (e 2 {∇v · ∇w − w (1 − g − e − 1)} = 0

                        R

                        1

                        2

                        (R ) ♣❛r❛ t♦❞♦ w ❡♠ H ✳ P♦r ❞❡✜♥✐çã♦✱ s❡❣✉❡ q✉❡ v é ❛ s♦❧✉çã♦ ❢r❛❝❛ ❞❛ ❊q✉❛çã♦ ✭✸✳✶✷✮✳ ➱ ♣♦ssí✈❡❧ ♠♦str❛r q✉❡ ❛ ❢✉♥çã♦ v é ❛♥❛❧ít✐❝❛ r❡❛❧ ❬✶✵❪✳

                        ❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s

                        ❬✶❪ ❘✳ ❆✳ ❆❞❛♠s✱ ❏✳ ❏✳ ❋♦✉r♥✐❡r✳ ❙♦❜♦❧❡✈ ❙♣❛❝❡s✱ ✷ ❡❞✳ ❆❝❛❞❡♠✐❝ Pr❡ss ✭✷✵✵✸✮✳

                        ❬✷❪ ▼✳ ❆✐❣♥❡r✳ ❊①✐st❡♥❝❡ ♦❢ t❤❡ ●✐♥③❜✉r❣✲▲❛♥❞❛✉ ❱♦rt❡① ◆✉♠❜❡r✳ ❈♦♠✲ ♠✉♥✳ ▼❛t❤✳ P❤②s✳ ✷✶✻✱ ✶✼✲✷✷ ✭✷✵✵✶✮✳

                        ❬✸❪ ❍✳ ❏✳ ❉❡ ❱❡❣❛ ❡ ❋✳ ❆✳ ❙❝❤❛♣♦s♥✐❦✳ ❈❧❛ss✐❝❛❧ ✈♦rt❡① s♦❧✉t✐♦♥ ♦❢ t❤❡ ❆❜❡✲ ❧✐❛♥ ❍✐❣❣s ♠♦❞❡❧✳ P❤②s✐❝❛❧ ❘❡✈✐❡✇ ❉ ✶✹✳✹✱ ✶✶✵✵ ✭✶✾✼✻✮✳

                        ❬✹❪ ▲✳ ❈✳ ❊✈❛♥s✳ P❛rt✐❛❧ ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s✱ ✷ ❡❞✳ ❆▼❙ ✭✷✵✶✵✮✳ ❬✺❪ ●✳ ❇✳ ❋♦❧❧❛♥❞✳ ❘❡❛❧ ❆♥❛❧②s✐s✿ ▼♦❞❡r♥ ❚❡❝❤♥✐q✉❡s ❛♥❞ ❚❤❡✐r ❆♣♣❧✐❝❛t✐✲

                        ♦♥s✱ ✷ ❡❞✳ ❲✐❧❡②✲■♥t❡rs❝✐❡♥❝❡ ✭✶✾✾✾✮✳ ❬✻❪ ▲✳ ❏❛❝♦❜s ❡ ❈✳ ❘❡❜❜✐✳ ■♥t❡r❛❝t✐♦♥ ❡♥❡r❣② ♦❢ s✉♣❡r❝♦♥❞✉❝t✐♥❣ ✈♦rt✐❝❡s✳

                        P❤②s✐❝❛❧ ❘❡✈✐❡✇ ❇ ✶✾✳✾✱ ✹✹✽✻ ✭✶✾✼✾✮✳ ❬✼❪ ❊✳ ▲✐❡❜ ❡ ▼✳ ▲♦ss✳ ❆♥❛❧②s✐s✱ ✷ ❡❞✳ ❆▼❙ ✭✷✵✵✶✮✳ ❬✽❪ ▼✳ ▼✉rr❛②✳ ▲✐♥❡ ❇✉♥❞❧❡s✳ ❍♦♥♦✉rs ✶✾✾✾ ✭▲❡❝t✉r❡ ♥♦t❡s✮ ✭✷✵✵✷✮✳ ❬✾❪ ❲✳ ❘✉❞✐♥✳ ❘❡❛❧ ❛♥❞ ❝♦♠♣❧❡① ❛♥❛❧②s✐s✱ ✸ ❡❞✳ ▼❝●r❛✇✲❍✐❧❧ ✭✶✾✽✼✮✳ ❬✶✵❪ ❈✳ ❍✳ ❚❛✉❜❡s✳ ❆r❜✐tr❛r② ◆✲❱♦rt❡① ❙♦❧✉t✐♦♥s t♦ t❤❡ ❋✐rst ❖r❞❡r

                        ●✐♥③❜✉r❣✲▲❛♥❞❛✉ ❊q✉❛t✐♦♥s✳ ❈♦♠♠✉♥✳ ▼❛t❤✳ P❤②s✳ ✼✷✱ ✷✼✼✲✷✾✷ ✭✶✾✽✵✮✳ ❬✶✶❪ ▼✳ ▼✳ ❱❛✐♥❜❡r❣✳ ❱❛r✐❛t✐♦♥❛❧ ▼❡t❤♦❞ ❛♥❞ ▼❡t❤♦❞ ♦❢ ▼♦♥♦t♦♥❡ ❖♣❡r❛✲ t♦rs ✐♥ t❤❡ ❚❤❡♦r② ♦❢ ◆♦♥❧✐♥❡❛r ❊q✉❛t✐♦♥s✳ ❏♦❤♥ ❲✐❧❡② ✭✶✾✼✸✮✳

                        ❘❊❋❊❘✃◆❈■❆❙ ❇■❇▲■❖●❘➪❋■❈❆❙ ✼✻ ❬✶✷❪ ❊✳ ❲❡✐♥❜❡r❣✳ ▼✉❧t✐✈♦rt❡① s♦❧✉t✐♦♥s ♦❢ t❤❡ ●✐♥③❜✉r❣✲▲❛♥❞❛✉ ❡q✉❛t✐♦♥s✳

                        P❤②s✳ ❘❡✈✳ ❉ ✶✾✱ ✸✵✵✽ ✭✶✾✼✾✮✳

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0
0
165
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1
219
Arquiteta, Mestranda do Programa de Pós-Graduação em Engenharia Civil na Universidade Federal do Espírito Santo (UFES) marbisterra.com.br
0
0
10
Universidade Federal de Uberlândia Instituto de Química Programa de Pós-Graduação em Química
0
0
76
Programa de Pós-Graduação em Artes Instituto de Ciências da Arte Universidade Federal do Pará
0
3
140
Universidade Federal de Uberlândia Instituto de Química Programa de Pós-Graduação em Química
0
0
95
Serviço Público Federal Universidade Federal do Pará Centro Tecnológico Programa de Pós-Graduação em Engenharia Civil
0
0
151
Programa de Pós-Graduação em Artes Instituto de Ciências da Arte Universidade Federal do Pará
0
0
120
Universidade Federal do Pará Instituto de Letras e Comunicação Programa de Pós-Graduação em Letras
0
9
234
Universidade Federal do Pará Centro Tecnológico Programa de Pós-Graduação em Engenharia Civil
0
0
154
Universidade Federal de Santa Catarina Programa de Pós-Graduação em Engenharia e Gestão do Conhecimento
0
0
243
Universidade Federal de Uberlândia Faculdade de Matemática Licenciatura em Matemática
0
0
49
Universidade Federal de Uberlândia Instituto de Química Programa de Pós-Graduação em Química
0
1
160
Universidade Federal de Uberlândia Faculdade de Matemática Bacharelado em Matemática
0
0
51
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