❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❊ ❙➹❖ ❈❆❘▲❖❙
❈❊◆❚❘❖ ❉❊ ❈■✃◆❈■❆❙ ❊❳❆❚❆❙ ❊ ❉❊ ❚❊❈◆❖▲❖●■❆
P❘❖●❘❆▼❆ ❉❊ PÓ❙✲●❘❆❉❯❆➬➹❖ ❊▼ ▼❆❚❊▼➪❚■❈❆
❈♦♥✜♥❛♠❡♥t♦ ❞❡ ♣❛rtí❝✉❧❛s q✉â♥t✐❝❛s ❛ ❝✉r✈❛s ❞♦
❡s♣❛ç♦
❆❧❡ss❛♥❞r❛ ❆♣❛r❡❝✐❞❛ ❱❡rr✐
✳
❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❊ ❙➹❖ ❈❆❘▲❖❙
❈❊◆❚❘❖ ❉❊ ❈■✃◆❈■❆❙ ❊❳❆❚❆❙ ❊ ❉❊ ❚❊❈◆❖▲❖●■❆
P❘❖●❘❆▼❆ ❉❊ PÓ❙✲●❘❆❉❯❆➬➹❖ ❊▼ ▼❆❚❊▼➪❚■❈❆
❈♦♥✜♥❛♠❡♥t♦ ❞❡ ♣❛rtí❝✉❧❛s q✉â♥t✐❝❛s ❛ ❝✉r✈❛s ❞♦
❡s♣❛ç♦
❆❧❡ss❛♥❞r❛ ❆♣❛r❡❝✐❞❛ ❱❡rr✐
❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❈és❛r ❘♦❣ér✐♦ ❞❡ ❖❧✐✈❡✐r❛✳
❚❡s❡ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥✐✲ ✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙ã♦ ❈❛r❧♦s✱ ❝♦♠♦ ♣❛rt❡ ❞♦s r❡q✉✐s✐t♦s ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❚ít✉❧♦ ❞❡ ❉♦✉t♦r ❡♠ ▼❛t❡♠át✐❝❛✳
Ficha catalográfica elaborada pelo DePT da Biblioteca Comunitária/UFSCar
V554cp
Verri, Alessandra Aparecida.
Confinamento de partículas quânticas a curvas do espaço / Alessandra Aparecida Verri. -- São Carlos : UFSCar, 2010. 97 f.
Tese (Doutorado) -- Universidade Federal de São Carlos, 2010.
1. Teoria espectral (Matemática). 2. Átomo de hidrogênio. 3. Coulomb, Potencial de. 4. Redução de dimensão. I. Título.
Banca Examinadora:
rfilLt/,
Prof. Dr. Cesar Rogério de Oliveira
. DM
,-UFSCar
7:L,~
~
Prof. Dr. Túlio de Oliveira Carvalho
DMA- UEL
~êf~~Jl~
t?'prof. Dr. João Carlos Alves Barata
DFMA- USP
ç~
Prof. Dr. Paulo Afonso Faria da Veiga
ICMC - USP
✳
❆❣r❛❞❡❝✐♠❡♥t♦s
❆ ❉❡✉s ♣♦r ❡s❝♦❧❤❡r ♣❛r❛ ♠✐♠ ❡st❡ ❝❛♠✐♥❤♦ ❡ t♦r♥❛✲❧♦ ✐♥❝r✐✈❡❧♠❡♥t❡ ❛❧❡❣r❡✳ ❆♦ ♠❡✉ ♦r✐❡♥t❛❞♦r ♣❡❧❛ s✉❛ ♣❛❝✐ê♥❝✐❛ ✐♥✜♥✐t❛✱ s✉❛ ❝♦♠♣r❡❡♥sã♦✱ s✉❛s sá❜✐❛s ♦r✐❡♥t❛çõ❡s✱ s✉❛ ❞❡❞✐❝❛çã♦✱ ♣❡❧❛ ❝♦♥✜❛♥ç❛ ❡ ♠♦t✐✈❛çã♦ ❛ ❡st❡ tr❛❜❛❧❤♦✳
❆♦s ♠❡✉s ♣❛✐s✱ ♠♦❞❡❧♦s ❞❡ ❧✉t❛ ❡ ♠♦t✐✈♦ ❞❡ ♦r❣✉❧❤♦ s❡♠ ✜♠✱ s❡♠ ♦s q✉❛✐s ♥ã♦ s❡r✐❛ ♣♦ssí✈❡❧ r❡❛❧✐③❛r ❡st❡ s♦♥❤♦✳
❆♦s ❛♠✐❣♦s ❞♦ ❞❡♣❛rt❛♠❡♥t♦✱ ♣❡❧❛ ❛♠✐③❛❞❡✱ ♣❡❧♦ ❝❛r✐♥❤♦✱ ♣❡❧❛s r✐s❛❞❛s ❡ ♣♦r ❡st❛r❡♠ s❡♠♣r❡ ♣r♦♥t♦s ❛ ❛❥✉❞❛r✳
❆ t♦❞♦s q✉❡ ❞❡ ✉♠❛ ♠❛♥❡✐r❛ ♦✉ ❞❡ ♦✉tr❛ ❡st✐✈❡r❛♠ ❝♦♠✐❣♦✱ ♦❝❛s✐♦♥❛❧♠❡♥t❡ ♦✉ ❝♦♥st❛♥t❡♠❡♥t❡✱ ♥❡ss❡s ú❧t✐♠♦s ❛♥♦s✳
❘❡s✉♠♦
◆❡st❡ tr❛❜❛❧❤♦ ❡st✉❞❛♠♦s r❡❞✉çõ❡s ❞❡ ❞✐♠❡♥sõ❡s ❡♠ ❛❧❣✉♥s s✐st❡♠❛s q✉â♥✲ t✐❝♦s❀ t❛✐s r❡❞✉çõ❡s ♦❝♦rr❡♠ ❞❡✈✐❞♦ ❛♦ ❝♦♥✜♥❛♠❡♥t♦ ❞♦ ♠♦✈✐♠❡♥t♦ ❞❡ ♣❛rtí❝✉❧❛s✱ ✐♥✐❝✐❛❧♠❡♥t❡ ❡♠ t✉❜♦s ♥♦ ❡s♣❛ç♦✱ ❛ ❝✉r✈❛s✳ ◆♦ss♦ ♣r✐♥❝✐♣❛❧ ♦❜❥❡t✐✈♦ é ❡♥❝♦♥tr❛r ♦ ♦♣❡r❛❞♦r ❡❢❡t✐✈♦ q✉❡ ❞❡s❝r❡✈❡ ♦ ♠♦✈✐♠❡♥t♦ ❞❛ ♣❛rtí❝✉❧❛ ❛♣ós ♦ ❝♦♥✜♥❛♠❡♥t♦✳ ✭✶✮ ◆❛ ♣r✐♠❡✐r❛ s✐t✉❛çã♦ ❡st✉❞❛♠♦s ✉♠ t✉❜♦ ✐♥✜♥✐t♦ ❣❡r❛❞♦ ♣♦r ✉♠❛ ❝✉r✈❛ ❝♦♠ t♦rçã♦ ❡ ❝✉r✈❛t✉r❛s ♥ã♦✲tr✐✈✐❛✐s✳ ❆q✉✐ ❛s s❡çõ❡s tr❛♥s✈❡rs❛✐s ♣♦ss✉❡♠ s❡♠♣r❡ ♦ ♠❡s♠♦ ❞✐â♠❡tr♦✳ ✭✷✮ ❊st✉❞❛♠♦s t❛♠❜é♠ t✉❜♦s ♥♦ ❡s♣❛ç♦ ❞❡❢♦r♠❛❞♦s ❞❡ ✉♠❛ ❢♦r♠❛ ❡s✲ ♣❡❝í✜❝❛✱ ♦✉ s❡❥❛✱ ♦ ❞✐â♠❡tr♦ ❞❛s s❡çõ❡s tr❛♥s✈❡rs❛✐s ♣♦ss✉✐ ✉♠ ú♥✐❝♦ ♠á①✐♠♦ ❣❧♦❜❛❧✳ ❚❛✐s t✉❜♦s t❛♠❜é♠ ❛♣r❡s❡♥t❛♠ ❝✉r✈❛t✉r❛ ❡ t♦rçã♦ ♥ã♦✲tr✐✈✐❛✐s✳ ✭✸✮ ❋✐♥❛❧♠❡♥t❡ ❛♥❛✲ ❧✐s❛♠♦s ❛ q✉❡stã♦ ❞❡ q✉❛❧ ❡①t❡♥sã♦ ❛✉t♦✲❛❞❥✉♥t❛ ❞♦ át♦♠♦ ❞❡ ❤✐❞r♦❣ê♥✐♦ ✉♥✐❞✐♠❡♥✲ s✐♦♥❛❧ s❡r✐❛ ✜s✐❝❛♠❡♥t❡ r❡❧❡✈❛♥t❡✳ ❈♦♥s✐❞❡r❛♠♦s t❛❧ át♦♠♦ ♥✉♠ t✉❜♦ tr✐❞✐♠❡♥s✐♦♥❛❧ ❡ ❡st✉❞❛♠♦s ♦ ❧✐♠✐t❡ ❞❡ q✉❛♥❞♦ ♦ t✉❜♦ ❝♦♥✈❡r❣❡ ❛♦ ❡✐①♦✲x✱ ❡ ✐ss♦ ♠♦str♦✉ q✉❡ ❛
❆❜str❛❝t
■♥ t❤✐s ✇♦r❦ ✇❡ st✉❞② ❞✐♠❡♥s✐♦♥❛❧ r❡❞✉❝t✐♦♥s ✐♥ s♦♠❡ q✉❛♥t✉♠ s②st❡♠s❀ s✉❝❤ r❡❞✉❝t✐♦♥s ♦❝❝✉r ❞✉❡ t♦ ❝♦♥✜♥❡♠❡♥t ♦❢ t❤❡ ♣❛rt✐❝❧❡ ❢r♦♠ ❛ t✉❜❡ ✐♥ s♣❛❝❡ t♦ ❛ ❝✉r✈❡✳ ❖✉r ♠❛✐♥ ❣♦❛❧ ✐s t♦ ✜♥❞ t❤❡ ❡✛❡❝t✐✈❡ ❤❛♠✐❧t♦♥✐❛♥ ♦♣❡r❛t♦r t❤❛t ❞❡s❝r✐❜❡s t❤❡ ♠♦t✐♦♥ ♦❢ t❤❡ ♣❛rt✐❝❧❡ ❛❢t❡r ❝♦♥✜♥❡♠❡♥t✳ ❲❡ ❝♦♥s✐❞❡r t❤r❡❡ ♣❛rt✐❝✉❧❛r s✐t✉❛t✐♦♥s✳ ✭✶✮ ■♥ t❤❡ ✜rst s✐t✉❛t✐♦♥✱ ✇❡ st✉❞② ❛♥ ✐♥✜♥✐t❡❧② ❧♦♥❣ t✉❜❡ ❣❡♥❡r❛t❡❞ ❜② ❛ ❝✉r✈❡ ✇✐t❤ ♥♦♥✲tr✐✈✐❛❧ t♦rs✐♦♥ ❛♥❞ ❝✉r✈❛t✉r❡✳ ❍❡r❡ t❤❡ t✉❜❡ ❝r♦ss s❡❝t✐♦♥s ❛❧✇❛②s ❤❛✈❡ t❤❡ s❛♠❡ ❞✐❛♠❡t❡r✳ ✭✷✮ ❲❡ ❛❧s♦ st✉❞② t✉❜❡s ✐♥ s♣❛❝❡ ❞❡❢♦r♠❡❞ ✐♥ ❛ s♣❡❝✐✜❝ ✇❛②✱ ✐✳❡✳✱ t❤❡ ❞✐❛♠❡t❡r ♦❢ t❤❡ ❝r♦ss s❡❝t✐♦♥s ❤❛✈❡ ❛ ✉♥✐q✉❡ ❣❧♦❜❛❧ ♠❛①✐♠✉♠✳ ❙✉❝❤ t✉❜❡s ❛❧s♦ ❤❛✈❡ ♥♦♥✲tr✐✈✐❛❧ t♦rs✐♦♥ ❛♥❞ ❝✉r✈❛t✉r❡✳ ✭✸✮ ❋✐♥❛❧❧②✱ ✇❡ ❛♥❛❧②③❡ t❤❡ q✉❡st✐♦♥ ♦❢ ✇❤✐❝❤ s❡❧❢✲❛❞❥♦✐♥t ❡①t❡♥s✐♦♥ ♦❢ t❤❡ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ ❤②❞r♦❣❡♥ ❛t♦♠ ✇♦✉❧❞ ❜❡ ♣❤②s✐❝❛❧❧② r❡❧❡✈❛♥t✳ ❲❡ ❝♦♥s✐❞❡r s✉❝❤ ❛t♦♠ ✐♥ ❛ t❤r❡❡✲❞✐♠❡♥s✐♦♥❛❧ t✉❜❡ ❛♥❞ t❛❦❡ t❤❡ ❧✐♠✐t ❛s t❤❡ t✉❜❡ ❝♦♥✈❡r❣❡s t♦ t❤❡ x ❛①✐s✱ ❛♥❞ ✐t ✐s s❤♦✇♥ t❤❛t t❤❡ ❉✐r✐❝❤❧❡t ✭❛t t❤❡ ♦r✐❣✐♥✮ ❡①t❡♥s✐♦♥
❙✉♠ár✐♦
✶ ■♥tr♦❞✉çã♦ ✶✵
✷ ❘❡✈✐sã♦ ❞❡ Γ✲❈♦♥✈❡r❣ê♥❝✐❛ ✶✼
✷✳✶ ❉❡✜♥✐çõ❡s ❡ ♣r♦♣r✐❡❞❛❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✷✳✷ Γ✲❝♦♥✈❡r❣ê♥❝✐❛ ❡ ❝♦♥✈❡r❣ê♥❝✐❛ ❞♦s r❡s♦❧✈❡♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾
✸ Γ✲❈♦♥✈❡r❣ê♥❝✐❛ ❡ ❛ ♣❛rtí❝✉❧❛ ❧✐✈r❡ ✷✶
✸✳✶ ●❡♦♠❡tr✐❛ ❞♦ ❞♦♠í♥✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✸✳✷ P❛rtí❝✉❧❛ ❧✐✈r❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✸✳✸ ▼✉❞❛♥ç❛ ❞❡ ✈❛r✐á✈❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✸✳✹ ❆❧❣✉♥s r❡s✉❧t❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼
✹ Γ✲❈♦♥✈❡r❣ê♥❝✐❛ ❡ ♦ ♣♦t❡♥❝✐❛❧ ❞❡ ❈♦✉❧♦♠❜ ✸✵
✹✳✶ ❖ ❝❛s♦ r❡♣✉❧s✐✈♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✹✳✷ ❖ ❝❛s♦ ❛tr❛t✐✈♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾
✺❈♦♥✈❡r❣ê♥❝✐❛ ✉♥✐❢♦r♠❡ ✲ P❛rtí❝✉❧❛ ❧✐✈r❡ ✹✺
✺✳✶ ❉❡✜♥✐çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✺✳✷ ❈♦♥✈❡r❣ê♥❝✐❛ ❯♥✐❢♦r♠❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽
✻ ❈♦♥✈❡r❣ê♥❝✐❛ ✉♥✐❢♦r♠❡ ✲ ❈♦✉❧♦♠❜ ❛tr❛t✐✈♦ ✺✺
✻✳✶ ❘❡❞✉çã♦ ❞❡ ❞✐♠❡♥sã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻
✻✳✷ ❈♦♥✈❡r❣ê♥❝✐❛ ✉♥✐❢♦r♠❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵
✼ ❖ ▲❛♣❧❛❝✐❛♥♦ ❡♠ t✉❜♦s ❞❡❢♦r♠❛❞♦s ✻✸
✼✳✶ ●❡♦♠❡tr✐❛ ❞♦ ❞♦♠í♥✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹ ✼✳✷ ❋♦r♠❛ q✉❛❞rát✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺ ✼✳✸ ❘❡❞✉çã♦ ❞❡ ❞✐♠❡♥sã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✽ ✼✳✹ ❈❛s♦ ❉✐r✐❝❤❧❡t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✻ ✼✳✺ ❉❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ✼✳✻ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✽ ✼✳✻ ❈❛s♦ ◆❡✉♠❛♥♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✶ ✼✳✼ ❈❛s♦ I =R ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✸
✼✳✽ ❖ ❡s♣❡❝tr♦ ❞✐s❝r❡t♦ ♥♦ ❝❛s♦ I =R ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✻
❆ ✽✾
❆✳✶ ❆ ❡①t❡♥sã♦ ❞❡ ❉✐r✐❝❤❧❡t ❡ s✉❛ ❢♦r♠❛ q✉❛❞rát✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✾ ❆✳✷ ❯♠ r❡s✉❧t❛❞♦ s♦❜r❡ ❝♦♥✈❡r❣ê♥❝✐❛ ✉♥✐❢♦r♠❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✶ ❆✳✸ ❆ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❍❛r❞② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✸ ❆✳✹ ❖s ▲❛♣❧❛❝✐❛♥♦s ❞❡ ❉✐r✐❝❤❧❡t ❡ ◆❡✉♠❛♥♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✹ ❆✳✺ ❚❡♦r❡♠❛ ❞❛ ❈♦♥✈❡r❣ê♥❝✐❛ ❉♦♠✐♥❛❞❛ ♣❛r❛ ♦♣❡r❛❞♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✺
❈❛♣ít✉❧♦ ✶
■♥tr♦❞✉çã♦
❉❡ ❢♦r♠❛ ❣❡r❛❧✱ ♥❡st❛ t❡s❡ ❡st✉❞❛♠♦s ❛❧❣✉♠❛s r❡❞✉çõ❡s ❞❡ ❞✐♠❡♥sã♦ ❡♠ ❝❡rt♦s s✐st❡♠❛s q✉â♥t✐❝♦s❀ t❛❧ r❡❞✉çã♦ ♦❝♦rr❡ ❞❡✈✐❞♦ ❛♦ ❝♦♥✜♥❛♠❡♥t♦ ❞♦ ♠♦✈✐♠❡♥t♦ ❞❛ ♣❛rtí❝✉❧❛✱ ✐♥✐❝✐❛❧♠❡♥t❡ r❡str✐t♦ ❛ ✉♠ t✉❜♦ ❡♠ R3✱ ❛ ✉♠❛ ❝✉r✈❛✳ ◆♦ss♦ ♣r✐♥❝✐♣❛❧
♦❜❥❡t✐✈♦ é❡♥❝♦♥tr❛r ♦ ♦♣❡r❛❞♦r ❡❢❡t✐✈♦ q✉❡ ❞❡s❝r❡✈❡ ♦ ♠♦✈✐♠❡♥t♦ ❞❛ ♣❛rtí❝✉❧❛ ❛♣ós ♦ ❝♦♥✜♥❛♠❡♥t♦✳
❈❧❛r❛♠❡♥t❡ t❛❧ ❝♦♥✜♥❛♠❡♥t♦ é✉♠ ♣r♦❝❡ss♦ s✐♥❣✉❧❛r ❡✱ ❛ss✐♠✱ ❞❡✈❡ s❡r ❝♦♥s✐✲ ❞❡r❛❞♦ ❝♦♠ ♠✉✐t♦ ❝✉✐❞❛❞♦❀ ♣♦r ❡①❡♠♣❧♦✱ ❞❡✈✐❞♦ ❛♦ ♣r✐♥❝í♣✐♦ ❞❡ ✐♥❝❡rt❡③❛ ❛ r❡❞✉çã♦ ❞❡ ❛❧❣✉♠❛ ❞✐♠❡♥sã♦ ✐♠♣❧✐❝❛ ❡♠ q✉❡ ♦s ♠♦♠❡♥t♦s ✭♥❛s ❞✐r❡çõ❡s q✉❡ ✏s♦♠❡♠✑✮ ❞✐✈✐r✲ ❥❛♠ ❡ ✐ss♦ ❞❡✈❡ s❡r ❝♦♠♣❡♥s❛❞♦ ✭✏r❡♥♦r♠❛❧✐③❛❞♦✑✮✳ ❉❡✈❡r❡♠♦s tr❛t❛r t❛✐s ❧✐♠✐t❡s ❞❡ ❢♦r♠❛ ❛♣r♦♣r✐❛❞❛✱ t❛♥t♦ ♥♦ s❡♥t✐❞♦ ❞♦s r❡s♦❧✈❡♥t❡s ❝♦♠♦ ♥♦ ✉s♦ ❞❛ Γ✲❝♦♥✈❡r❣ê♥❝✐❛❀
❝♦♠♦ ❡st❡ ú❧t✐♠♦ ❝♦♥❝❡✐t♦ ♥ã♦ é❞❡ ✉s♦ ❝♦♠✉♠ ❡♥tr❡ ♦s ❡st✉❞✐♦s♦s ❞❛ ♠❛t❡♠át✐❝❛ ❞❛ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛✱ ❞❡✈♦t❛r❡♠♦s ❛❧❣✉♠ ❡s♣❛ç♦ ♥♦ ✐♥í❝✐♦ ❞❛ t❡s❡ ❛ ❡❧❡✳
❆♥á❧✐s❡s ❞❡ t❛✐s r❡❞✉çõ❡s t❡♠ s✐❞♦ ❝♦♠✉♥s ♥♦ ❡st✉❞♦ ❞❡ ❣r❛❢♦s q✉â♥t✐❝♦s ❬✽❪✱ ♠❛s ❛q✉✐ ❝♦♥s✐❞❡r❛♠♦s s✐t✉❛çõ❡s ❞✐❢❡r❡♥t❡s✳ ❈♦♥s✐❞❡r❛r❡♠♦s três s✐t✉❛çõ❡s✱ ❝♦♠ r❡s✉❧t❛❞♦s ♦r✐❣✐♥❛✐s ❡♠ ❝❛❞❛ ✉♠❛ ❞❡❧❛s✳
✶✳ ◆❛ ♣r✐♠❡✐r❛ s✐t✉❛çã♦ t❡♠♦s ✉♠ t✉❜♦ ✐♥✜♥✐t♦ ❣❡r❛❞♦ ♣♦r ✉♠❛ ❝✉r✈❛ ❝♦♠ t♦rçã♦ ❡ ❝✉r✈❛t✉r❛s ♥ã♦✲tr✐✈✐❛✐s✳ ◆♦ss♦ ♣♦♥t♦ ❞❡ ♣❛rt✐❞❛ ♥❡st❡ tó♣✐❝♦ ❢♦✐ ❛ r❡❢❡rê♥❝✐❛ ❬✶❪✱ ❛ q✉❛❧ ❡st✉❞♦✉ ♦ ❝❛s♦ ❞❡ t✉❜♦s ❝♦♠♣❛❝t♦s✳ P❛r❛ t✉❜♦s ✐♥✜♥✐t♦s ❡①✐st❡♠ r❡✲
s✉❧t❛❞♦s ✉s❛♥❞♦Γ✲❝♦♥✈❡r❣ê♥❝✐❛ ❡ ❝♦♥✈❡r❣ê♥❝✐❛ ❢♦rt❡ ♥♦ s❡♥t✐❞♦ ❞♦s r❡s♦❧✈❡♥t❡s
❬✺❪✳ ◆♦ ❈❛♣ít✉❧♦ ✺ ❝♦♥s❡❣✉✐♠♦s ❛ ❝♦♥✈❡r❣ê♥❝✐❛ ✉♥✐❢♦r♠❡ ❞♦s r❡s♦❧✈❡♥t❡s✳ ❆q✉✐ ❛s s❡çõ❡s tr❛♥s✈❡rs❛✐s ♣♦ss✉❡♠ s❡♠♣r❡ ♦ ♠❡s♠♦ ❞✐â♠❡tr♦✳
✷✳ ❊st✉❞❛♠♦s t❛♠❜é♠ t✉❜♦s ❞❡❢♦r♠❛❞♦s ❞❡ ✉♠❛ ❢♦r♠❛ ❡s♣❡❝í✜❝❛✱ ♦✉ s❡❥❛✱ ♦ ❞✐â♠❡tr♦ ❞❛s s❡çõ❡s tr❛♥s✈❡rs❛✐s ♣♦ss✉✐ ✉♠ ú♥✐❝♦ ♠á①✐♠♦ ❣❧♦❜❛❧ ✭❝♦♠ ❝❡rt❛s ❝❛r❛❝t❡ríst✐❝❛s✮✳ ◆♦ss♦ ♣♦♥t♦ ❞❡ ♣❛rt✐❞❛ ♥❡st❡ tó♣✐❝♦ ❢♦r❛♠ ♦s tr❛❜❛❧❤♦s ❬✾❪ ❡ ❬✶✵❪✱ ♥♦s q✉❛✐s t✉❜♦s ♣❧❛♥❛r❡s ❡ s❡♠ ❝✉r✈❛t✉r❛ ❢♦r❛♠ ❝♦♥s✐❞❡r❛❞♦s✳ ◆♦ ❈❛♣ít✉❧♦ ✼ ❝♦♥s✐❞❡r❛♠♦s t✉❜♦s s✐♠✐❧❛r❡s ♠❛s ♥♦ ❡s♣❛ç♦ R3 ❡ ❝♦♠ ❝✉r✈❛t✉r❛ ❡
t♦rçã♦ ♥ã♦✲tr✐✈✐❛✐s✳
✸✳ ❋✐♥❛❧♠❡♥t❡ ❛♥❛❧✐s❛♠♦s ❛ q✉❡stã♦ ❞❡ q✉❛❧ ❡①t❡♥sã♦ ❛✉t♦✲❛❞❥✉♥t❛ ❞♦ át♦♠♦ ❞❡ ❤✐❞r♦❣ê♥✐♦ ✉♥✐❞✐♠❡♥s✐♦♥❛❧ s❡r✐❛ ✜s✐❝❛♠❡♥t❡ r❡❧❡✈❛♥t❡✳ P❛r❛ ✐ss♦ ❝♦♥s✐❞❡r❛♠♦s ❡ss❡ át♦♠♦ ♥✉♠ t✉❜♦ tr✐❞✐♠❡♥s✐♦♥❛❧ ❡ ❡st✉❞❛♠♦s ♦ ❧✐♠✐t❡ ❞❡ q✉❛♥❞♦ ♦ t✉❜♦ ❝♦♥✈❡r❣❡ ❛♦ ❡✐①♦✲x✱ ♣r♦❝✉r❛♥❞♦ s❡❧❡❝✐♦♥❛r ❛❧❣✉♠❛ ❡①t❡♥sã♦ ❞♦ ❝❛s♦ ❡♠ ✉♠❛
❞✐♠❡♥sã♦✳ ◆♦ss♦ ♣♦♥t♦ ❞❡ ♣❛rt✐❞❛ ❢♦r❛♠ ❛s ❝❛r❛❝t❡r✐③❛çõ❡s ❞❡ss❛s ❡①t❡♥sõ❡s ❞✐s❝✉t✐❞❛s ❡♠ ❬✻❪✱ ❡ ❛ ❡①t❡♥sã♦ ❞❡ ❉✐r✐❝❤❧❡t ❢♦✐ s❡♠♣r❡ ♦❜t✐❞❛ ❛♣ós ♦ ❝♦♥✜♥❛✲ ♠❡♥t♦✳ ❊ss❡s r❡s✉❧t❛❞♦s ❛♣❛r❡❝❡♠ ♥♦s ❈❛♣ít✉❧♦s ✹ ❡ ✻ ❞❛ t❡s❡✳
◆♦ q✉❡ s❡❣✉❡ ♣r♦❝✉r❛♠♦s ❛♣r❡s❡♥t❛r ✉♠ ♣♦✉❝♦ ♠❛✐s ❞❡ ❞❡t❛❧❤❡s ❞♦ q✉❡ é ❞✐s❝✉t✐❞♦ ♥❡st❛ t❡s❡✳
❙❡❥❛I ♦ ♦♣❡r❛❞♦r ✐❞❡♥t✐❞❛❞❡ ❡ 0≤Tj ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ♦♣❡r❛❞♦r❡s ♣♦s✐t✐✈♦s ❡
❛✉t♦✲❛❞❥✉♥t♦s ❛t✉❛♥❞♦ ❡♠ ✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt H✳ ❆ ♠♦♥♦t♦♥✐❝✐❞❛❞❡ ❞❛ s❡q✉ê♥❝✐❛
❞♦s ♦♣❡r❛❞♦r❡s r❡s♦❧✈❡♥t❡sR−λ(Tj) := (Tj+λI)−1 ✭λ >0✮ ✐♠♣❧✐❝❛ ❛ ♠♦♥♦t♦♥✐❝✐❞❛❞❡
❞❛ ❝♦rr❡s♣♦♥❞❡♥t❡ s❡q✉ê♥❝✐❛ ❞❡ ❢♦r♠❛s s❡sq✉✐❧✐♥❡❛r❡s ❡ ✈✐❝❡✲✈❡rs❛✳ ❉❡✈✐❞♦ à ♠♦♥♦✲ t♦♥✐❝✐❞❛❞❡✱ ❡♠ ❛❧❣✉♠❛s s✐t✉❛çõ❡s t❡♠♦s ❛ ❡①✐stê♥❝✐❛ ❞❡ ❧✐♠✐t❡s✱ ❝♦♠♦ ❝♦♥✈❡r❣ê♥❝✐❛ ❢♦rt❡ ❞♦s r❡s♦❧✈❡♥t❡s ✭✈❡❥❛ ❙❡çã♦ ✶✵✳✹ ❡♠ ❬✹❪✮✳ ❙✉♣♦♥❞♦ s✐t✉❛çõ❡s ♠❛✐s ❣❡r❛✐s✱ ❛ ♣r✐♥❝í♣✐♦ ♥ã♦ é ❝❧❛r❛ ❛ r❡❧❛çã♦ ❡♥tr❡ ❝♦♥✈❡r❣ê♥❝✐❛ ❢♦rt❡ ❞♦s r❡s♦❧✈❡♥t❡s ❡ ❝♦♥✈❡r❣ê♥✲ ❝✐❛ ❞❛s r❡s♣❡❝t✐✈❛s ❢♦r♠❛s q✉❛❞rát✐❝❛s✱ ♦✉ s❡❥❛✱ ♦ q✉❡ ❛❝♦♥t❡❝❡ q✉❛♥❞♦ s✉♣♦♠♦s q✉❡ ❡ss❛s ❢♦r♠❛s sã♦ ❛♣❡♥❛s✱ ♣♦r ❡①❡♠♣❧♦✱ ❧✐♠✐t❛❞❛s ✐♥❢❡r✐♦r♠❡♥t❡❄ ❊st❛ q✉❡stã♦ ❡stá ❞✐r❡t❛♠❡♥t❡ r❡❧❛❝✐♦♥❛❞❛ ❝♦♠ ♦ ❝♦♥❝❡✐t♦ ❞❡ Γ✲❝♦♥✈❡r❣ê♥❝✐❛ q✉❡ é ♦ ❛ss✉♥t♦ ❞♦ ❈❛✲
♣ít✉❧♦ ✷✳ ◆❡❧❡ ❛♣r❡s❡♥t❛♠♦s ❛ ❞❡✜♥✐çã♦ ❞❡ Γ✲❝♦♥✈❡r❣ê♥❝✐❛ ❡ ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s
❜ás✐❝❛s✳ ❉❡♥tr❡ ❡❧❡s ❛ r❡❧❛çã♦ ❝♦♠ r❡s♣❡✐t♦ à ❝♦♥✈❡r❣ê♥❝✐❛ ❢♦rt❡ ❞♦s r❡s♦❧✈❡♥t❡s✳ ❈♦♠♦ ✉♠❛ ❛♣❧✐❝❛çã♦ ❞❛Γ✲❝♦♥✈❡r❣ê♥❝✐❛ ❞❡ ❢♦r♠❛s✱❝♦♥s✐❞❡r❡♠♦s ♦ ❝❛s♦ ❞❡ ✉♠❛
♣❛rtí❝✉❧❛ s❡ ♠♦✈❡♥❞♦ ❡♠ ✉♠ t✉❜♦ ❡♠ R3 s❡♠ ❛ ✐♥✢✉ê♥❝✐❛ ❞❡ q✉❛❧q✉❡r ♣♦t❡♥❝✐❛❧✳
❆ ❡st❛ s✐t✉❛çã♦ ✈❛♠♦s ❝❤❛♠❛r s✐♠♣❧❡s♠❡♥t❡ ❞❡ ♣❛rtí❝✉❧❛ ❧✐✈r❡ ❡♠ ✉♠ t✉❜♦ ❡♠ R3✳
❋❛③❡♥❞♦ ❡ss❡ t✉❜♦ ✏❡♥❝♦❧❤❡r✑ ❛ ✉♠❛ ❝✉r✈❛ s✉❛✈❡✱❛ q✉❡stã♦ é ❡♥❝♦♥tr❛r ♦ ♦♣❡r❛❞♦r ❧✐♠✐t❡ s♦❜r❡ ❡ss❛ ❝✉r✈❛✳ ▼❛✐s ♣r❡❝✐s❛♠❡♥t❡✱♦ ♦♣❡r❛❞♦r ✐♥✐❝✐❛❧ ❛ss♦❝✐❛❞♦ ❛♦ ♣r♦❜❧❡♠❛ é ♦ ▲❛♣❧❛❝✐❛♥♦ ❞❡ ❉✐r✐❝❤❧❡t
ψ → −∆εψ, ψ ∈H2(Ωε)∩H01(Ωε),
❡♠ q✉❡Ωε ⊂R3é ✉♠ ❝♦♥❥✉♥t♦ ❣❡r❛❞♦ ♣♦r ✉♠❛ s❡çã♦ tr❛♥s✈❡rs❛❧ Sε =εS✭S ⊂R2✮ ♦
q✉❛❧ r♦t❛❝✐♦♥❛ ❛♦ ❧♦♥❣♦ ❞❡ ✉♠❛ ❝✉r✈❛ r(s) :I ⊆R→R3✳ ◆♦ ❡♥t❛♥t♦✱q✉❛♥❞♦ε →0
❛ r❡❣✐ã♦ Ωε t♦r♥❛✲s❡ ❝❛❞❛ ✈❡③ ♠❛✐s ❡str❡✐t❛ ❡ ❛s ♦s❝✐❧❛çõ❡s tr❛♥s✈❡rs❛✐s ❞❛ ♣❛rtí❝✉❧❛
✜❝❛♠ ♠✉✐t♦ rá♣✐❞❛s✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱❛ ❡♥❡r❣✐❛ t♦t❛❧ ❞♦ s✐st❡♠❛ ❞✐✈❡r❣❡ ❡ ✈❛✐ ❛ ✐♥✜♥✐t♦ q✉❛♥❞♦ ε → 0✳ P❛r❛ ❡♥①❡r❣❛r♠♦s ♠❡❧❤♦r ❡st❛ s✐t✉❛çã♦ ❝♦♥s✐❞❡r❡♠♦s
♦ s❡❣✉✐♥t❡ ❡①❡♠♣❧♦ ❜✐❞✐♠❡♥s✐♦♥❛❧✳ ❯♠❛ ♣❛rtí❝✉❧❛ s❡ ♠♦✈❡ ❧✐✈r❡♠❡♥t❡ ❡♠ Σε :=
[0, π]×[0, επ]✳ ❆s ❛✉t♦❢✉♥çõ❡s ❡ ♦s r❡s♣❡❝t✐✈♦s ❛✉t♦✈❛❧♦r❡s ❞♦ ▲❛♣❧❛❝✐❛♥♦ ❞❡ ❉✐r✐❝❤❧❡t
❛ss♦❝✐❛❞♦s ❛♦ ✐♥t❡r✈❛❧♦ [0, επ] sã♦
uεn(y) = senny
ε
, λεn= n
2
ε2, n = 1,2,3,· · ·.
❋✐❝❛ ❝❧❛r♦ q✉❡ q✉❛♥❞♦ ❛ r❡❣✐ã♦ Σε s❡ r❡❞✉③ ❛♦ ✐♥t❡r✈❛❧♦ [0, π] ♣❛r❛ ε → 0✱
❛s ❡♥❡r❣✐❛s ❞♦ ❡st❛❞♦ tr❛♥s✈❡rs❛❧ t❡♥❞❡♠ ❛ ✐♥✜♥✐t♦✳ ❆ss✐♠✱✈♦❧t❛♥❞♦ ❛♦ ❝❛s♦ tr✐❞✐✲ ♠❡♥s✐♦♥❛❧✱s❡❥❛ λ0 ♦ ♣r✐♠❡✐r♦ ❛✉t♦✈❛❧♦r ❞♦ ▲❛♣❧❛❝✐❛♥♦ ❞❡ ❉✐r✐❝❤❧❡t r❡str✐t♦ à S ❡u0 ❛ ❛✉t♦❢✉♥çã♦ ❝♦rr❡s♣♦♥❞❡♥t❡✳ ❖ ❡st✉❞♦ é ❡♥tã♦ ❢❡✐t♦ ❝♦♠ ❢♦r♠❛s q✉❛❞rát✐❝❛s✱♥ã♦ s♦♠❡♥t❡ ❞♦ ▲❛♣❧❛❝✐❛♥♦ ❞❡ ❉✐r✐❝❤❧❡t ❡♠ Ωε✱♠❛s s✐♠ ❞❛ ❢❛♠í❧✐❛
Ωα ε
|∇xφ|2−
λ0
ε2|φ| 2
dx, φ∈H01(Ωε). ✭✶✳✶✮
❉❡ ❢❛t♦✱✈✐♠♦s q✉❡ ❛s ❡♥❡r❣✐❛s ❞♦ ❡st❛❞♦ tr❛♥✈❡rs❛❧ t❡♥❞❡♠ ❛ ✐♥✜♥✐t♦ ♥♦ ❧✐♠✐t❡
♦❝♦rr❛ ❡ ❛ss✐♠ s♦♠❡♥t❡ ❛ ❞✐♥â♠✐❝❛ s♦❜r❡ ❛ ❝✉r✈❛ ♣❡r♠❛♥❡ç❛ ♥❡st❡ ❧✐♠✐t❡✳ ❖ ❞♦♠í♥✐♦ ❞❛ ❢♦r♠❛ q✉❛❞rát✐❝❛ ❧✐♠✐t❡ t❛♠❜é♠ ❡stá ❞✐r❡t❛♠❡♥t❡ r❡❧❛❝✐♦♥❛❞♦ ❝♦♠ ❡st❡ ❢❛t♦✱ ✈❡r❡♠♦s q✉❡ t❛❧ ❝♦♥❥✉♥t♦ é {wu0 :w∈H1(R)}✳
◆❛ ❞❡✜♥✐çã♦ ❞❡ Ωε✱ ♦ ❝❛s♦ ❡♠ q✉❡ ■ é ✉♠ ✐♥t❡r✈❛❧♦ ❧✐♠✐t❛❞♦ ❢♦✐ ❛♥❛❧✐s❛❞♦ ❡♠
❬✶❪✳ ❋♦✐ ♠♦str❛❞♦ q✉❡ ♦s ❛✉t♦✈❛❧♦r❡s {λε
i :i∈N} ❞❡ −∆ε ❡♠ Ωε ❝♦♠ ❛ ❝♦♥❞✐çã♦ ❞❡
❉✐r✐❝❤❧❡t ♥❛ ❢r♦♥t❡✐r❛✱ t❡♠ ♦ s❡❣✉✐♥t❡ ❝♦♠♣♦rt❛♠❡♥t♦✿
λεi =
λ0
ε2 +μ
ε
i, μεi →μi,
❡♠ q✉❡ μi sã♦ ❛✉t♦✈❛❧♦r❡s ❞❡ ✉♠ ♣r♦❜❧❡♠❛ ✉♥✐❞✐♠❡♥s✐♦♥❛❧
−w′′(s) +q(s)w(s) =μw(s), w∈H01(I).
❖ ♣♦t❡♥❝✐❛❧ q(s) ❞❡♣❡♥❞❡ ❞♦s ❡❢❡✐t♦s ❣❡♦♠étr✐❝♦s ❞♦ t✉❜♦✳ ▼❛✐s t❛r❞❡ ❛❧❣✉♥s r❡✲
s✉❧t❛❞♦s ❢♦r❛♠ ♦❜t✐❞♦s ♣❛r❛ t✉❜♦s ✐❧✐♠✐t❛❞♦s ❬✺❪✳ ◆♦ ❈❛♣ít✉❧♦ ✸ ❛♣r❡s❡♥t❛♠♦s ✉♠ r❡s✉♠♦ ❞♦s ♣r✐♥❝✐♣❛✐s r❡s✉❧t❛❞♦s ❞❡ss❡s ❞♦✐s tr❛❜❛❧❤♦s✱ ♦✉ s❡❥❛✱ ❬✶❪ ❡ ❬✺❪✳
❆❣♦r❛✱ ♣❛r❛ ❛♣r❡s❡♥t❛r♠♦s ❛❧❣✉♥s ❞♦s ♦❜❥❡t✐✈♦s ❞♦ tr❛❜❛❧❤♦✱ ❝♦♥s✐❞❡r❡♠♦s ♦ ♦♣❡r❛❞♦r ❞❡ ❙❝❤rö❞✐♥❣❡r ❝♦♠ ♣♦t❡♥❝✐❛❧ ❞❡ ❈♦✉❧♦♠❜ V(x) = − κ
|x|✱ 0 = κ ∈ R✳
■♥❞❡♣❡♥❞❡♥t❡♠❡♥t❡ ❞❛ ❞✐♠❡♥sã♦ ❝♦♥s✐❞❡r❛❞❛✱ ❝❤❛♠❛♠♦s ❡st❡ ♠♦❞❡❧♦ ❞❡ át♦♠♦ ❞❡ ❤✐❞r♦❣ê♥✐♦✳ ❆✐♥❞❛ ♠❛✐s✱ ❝❤❛♠❛♠♦s ❞❡ ❝❛s♦ ❛tr❛t✐✈♦ s❡ κ >0❡ ❝❛s♦ r❡♣✉❧s✐✈♦ ❝❛s♦
❝♦♥trár✐♦✳ P❛r❛ ♦ ❝❛s♦ tr✐❞✐♠❡♥s✐♦♥❛❧✱ ❝♦♥s✐❞❡r❡♠♦s
( ˙Hψ)(x) = −(∆ψ)(x)− κ
|x|ψ(x), dom ˙H=C
∞
0 (R3).
❖ ❚❡♦r❡♠❛ ❞❡ ❑❛t♦ ❘❡❧❧✐❝❤ ❬✹❪ ♥♦s ❞✐③ q✉❡ ❡st❡ ♦♣❡r❛❞♦r é ❡ss❡♥❝✐❛❧♠❡♥t❡ ❛✉t♦✲ ❛❞❥✉♥t♦✱ ♦✉ s❡❥❛✱ ♣♦ss✉✐ ✉♠❛ ú♥✐❝❛ ❡①t❡♥sã♦ ❛✉t♦✲❛❞❥✉♥t❛✳ ◆♦ ❝❛s♦ ✉♥✐❞✐♠❡♥s✐♦♥❛❧ ♦ ♦♣❡r❛❞♦r é
( ˙Hw)(s) =−w′′(s)− κ
|s|w(s), dom ˙H =C
∞
0 (R\{0}). ✭✶✳✷✮ ➱ ❝♦♥❤❡❝✐❞♦ q✉❡ H˙ tê♠ í♥❞✐❝❡s ❞❡ ❞❡✜❝✐ê♥❝✐❛s ✐❣✉❛✐s ❛ 2❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱
♣♦ss✉✐ ✐♥✜♥✐t❛s ❡①t❡♥sõ❡s ❛✉t♦✲❛❞❥✉♥t❛s ❬✻❪✳ ◆❛ ✐♥tr♦❞✉çã♦ ❞♦ ❈❛♣ít✉❧♦ ✹ ❤á ✉♠❛ ❞❡s❝r✐çã♦ ❞❡ t♦❞❛s ❡ss❛s ❡①t❡♥sõ❡s✳ ❉✐❢❡r❡♥t❡♠❡♥t❡ ❞❛ ✈❡rsã♦ ✸❉✱ ❛ s✐♥❣✉❧❛r✐❞❛❞❡ ❞♦
♣♦t❡♥❝✐❛❧ ❞❡ ❈♦✉❧♦♠❜ ✶❉ ❡①✐❣❡ ❝♦♥❞✐çõ❡s ❞❡ ❝♦♥t♦r♥♦ ♥❛ ♦r✐❣❡♠ ♣❛r❛ ❛s ❡①t❡♥sõ❡s✳ ❯♠ ❡①❡♠♣❧♦ ❞❡ ✉♠❛ ❡①t❡♥sã♦ ❛✉t♦✲❛❞❥✉♥t❛ ❞❡ H˙ é
(HDw)(s) =−w′′(s)−
κ
|s|w(s) ✭✶✳✸✮
❡♠ q✉❡
domHD ={w∈H2(R\{0}) :w(0−) = 0 =w(0+)}.
❈❤❛♠❛♠♦s HD ❞❡ ❡①t❡♥sã♦ ❞❡ ❉✐r✐❝❤❧❡t ❞♦ ♦♣❡r❛❞♦r ✭✶✳✷✮✳
❉✐s❝✉ssõ❡s s♦❜r❡ q✉❛❧ ❡①t❡♥sã♦ ❛✉t♦✲❛❞❥✉♥t❛ ♠❡❧❤♦r r❡♣r❡s❡♥t❛ ♦ át♦♠♦ ❞❡ ❤✐❞r♦❣ê♥✐♦ ✉♥✐❞✐♠❡♥s✐♦♥❛❧ t❡♠ ❛♣❛r❡❝✐❞♦ ❜❛st❛♥t❡ ♥❛ ❧✐t❡r❛t✉r❛ ❬✶✶✱ ✶✸❪✳ ❯♠ ❛r❣✉✲ ♠❡♥t♦ ❛♣r❡s❡♥t❛❞♦ ❡♠ ❬✶✸❪✱ ♣❛r❛ ♦ ❝❛s♦ κ >0✱ ♣❛r❛ ❛❞♦t❛r ❛ ❝♦♥❞✐çã♦ ❞❡ ❝♦♥t♦r♥♦
❞❡ ❉✐r✐❝❤❧❡t ♥❛ ♦r✐❣❡♠ é ♦ s❡❣✉✐♥t❡✿ ❝♦♥s✐❞❡r❡ ♦s ♦♣❡r❛❞♦r❡s ✉♥✐❞✐♠❡♥s✐♦♥❛✐s
(Haw)(s) = −w′′(s)−κ
w(s)
|s|+a, a >0.
❋♦✐ ♠♦str❛❞♦ q✉❡ ❛ s❡q✉ê♥❝✐❛ Ha ❝♦♥✈❡r❣❡ ♣❛r❛ HD ♥♦ s❡♥t✐❞♦ ✉♥✐❢♦r♠❡ ❞♦s r❡s♦❧✲
✈❡♥t❡s q✉❛♥❞♦ a→0✳
❏✉♥t❛♥❞♦ ♦ ♣r♦❜❧❡♠❛ ❞❡ s❡❧❡❝✐♦♥❛r ❡①t❡♥sõ❡s ❛✉t♦✲❛❞❥✉♥t❛s ❞❡ ✭✶✳✷✮ ❡ ❛s ✐❞é✐❛s ❞❡ ❬✶❪ ❡ ❬✺❪ ❛♣r❡s❡♥t❛❞❛s ❛❝✐♠❛✱ ♥♦ss❛ ♣r♦♣♦st❛ ♥❡st❛ t❡s❡ é ❛ s❡❣✉✐♥t❡✿ ❝♦♥s✐❞❡r❛r ♦ át♦♠♦ ❞❡ ❤✐❞r♦❣ê♥✐♦ s❡ ♠♦✈❡♥❞♦ ❡♠ ✉♠ t✉❜♦ ❡♠ R3 ❡ s❛❜❡r q✉❛❧ ❞❛s ❡①t❡♥sõ❡s é s❡✲
❧❡❝✐♦♥❛❞❛ ♥♦ ♣r♦❝❡ss♦ ❞❡ ❝♦♥✜♥❛♠❡♥t♦✳ ◆♦ ❡♥t❛♥t♦✱ ♣❛r❛ q✉❡ r❡❝✉♣❡r❡♠♦s ♦ át♦♠♦ ❞❡ ❤✐❞r♦❣ê♥✐♦ ✉♥✐❞✐♠❡♥s✐♦♥❛❧ ♥♦ ❧✐♠✐t❡ε →0❛ r❡❣✐ã♦ Ωε ❞❡✈❡ s❡r ❣❡r❛❞❛ ♣❡❧❛ ❝✉r✈❛
r(s) = (s,0,0)✳ P❛r❛ ♦ ❝❛s♦ r❡♣✉❧s✐✈♦ ❛♣❧✐❝❛♠♦s ❛ t❡♦r✐❛ ❞❡ Γ✲❝♦♥✈❡r❣ê♥❝✐❛❀ ❥á ♣❛r❛
♦ ❝❛s♦ ❛tr❛t✐✈♦ ❛♥❛❧✐s❛♠♦s ❛ ♠♦♥♦t♦♥✐❝✐❞❛❞❡ ❞❛s ❢♦r♠❛s q✉❛❞rát✐❝❛s✳ ❈♦♠♦ ♣r✐♥❝✐✲ ♣❛❧ r❡s✉❧t❛❞♦ ❞❡ ♥♦ss♦s ❡st✉❞♦s✱ ❛ ❡①t❡♥sã♦ ❞❡ ❉✐r✐❝❤❧❡t é s❡❧❡❝✐♦♥❛❞❛ ♥♦ ♣r♦❝❡ss♦ ❞❡ ❝♦♥✜♥❛♠❡♥t♦ ❡♠ ❛♠❜♦s ♦s ❝❛s♦s ❛❝✐♠❛✳ ❊st❡s r❡s✉❧t❛❞♦s sã♦ ❛♣r❡s❡♥t❛❞♦s ♥♦ ❈❛♣ít✉❧♦ ✹✳
❉❡s❞❡ q✉❡ ❛ t❡♦r✐❛ ❞❡ Γ✲❝♦♥✈❡r❣ê♥❝✐❛ ♥♦s ♣❡r♠✐t❡ t✐r❛r ❝♦♥❝❧✉sõ❡s s♦❜r❡ ❝♦♥✲
✈❡r❣ê♥❝✐❛ ❢♦rt❡ ❞♦s r❡s♦❧✈❡♥t❡s✱ ♥♦ ❈❛♣ít✉❧♦ ✻ ❝♦♥s✐❞❡r❛♠♦s ✉♠❛ s✐t✉❛çã♦ s✐♠✐❧❛r ❛ ❛❝✐♠❛ ❡ ❝♦♥s❡❣✉✐♠♦s ✉♠❛ ❝♦♥✈❡r❣ê♥❝✐❛ ✉♥✐❢♦r♠❡ ❞♦s r❡s♦❧✈❡♥t❡s ♥♦ ♣r♦❝❡ss♦ ❞❡ ❝♦♥✜♥❛♠❡♥t♦✳ ❈♦♠♦ ❡r❛ ❞❡ s❡ ❡s♣❡r❛r ❛ ❡①t❡♥sã♦ ❞❡ ❉✐r✐❝❤❧❡t ❢♦✐ ❛ s❡❧❡❝✐♦♥❛❞❛✳ ❆
❞✐❢❡r❡♥ç❛ é q✉❡ ♥❡ss❡ ❝❛s♦ ❡♠ ❝❛❞❛ r❡❣✐ã♦ Ωε ♥ós ❝♦♥s✐❞❡r❛♠♦s ♦ ♣♦t❡♥❝✐❛❧ ♣❡rt✉r✲
❜❛❞♦
Vε(x) :=−
κ
|x|+εα, x∈Ωε, 0< α <1.
❱♦❧t❛♥❞♦ ❛♦ ❝❛s♦ ❞❛ ♣❛rtí❝✉❧❛ ❧✐✈r❡✱ ✈❡r❡♠♦s ♥❛ ❙❡çã♦ ✸✳✹ ❞♦ ❈❛♣ít✉❧♦ ✸ q✉❡ ♦s r❡s✉❧t❛❞♦s ❞❡ ❬✺❪ ❣❛r❛♥t❡♠ ❛ ❝♦♥✈❡r❣ê❝✐❛ ❢♦rt❡ ♥♦ s❡♥t✐❞♦ ❞♦s r❡s♦❧✈❡♥t❡s ❞♦s ♦♣❡r❛❞♦r❡s ❛ss♦❝✐❛❞♦s ❛s ❢♦r♠❛s ✭✶✳✶✮ ♥♦ ❧✐♠✐t❡ ε → 0✳ ❆q✉✐✱ ♥♦ss❛ ♣r✐♥❝✐♣❛❧ ❝♦♥✲
tr✐❜✉✐çã♦ é ❛ ❝♦♥✈❡r❣ê♥❝✐❛ ✉♥✐❢♦r♠❡ ❞♦s r❡s♦❧✈❡♥t❡s✳ ❊st❡ r❡s✉❧t❛❞♦ é ❛♣r❡s❡♥t❛❞♦ ♥♦ ❈❛♣ít✉❧♦ ✸✳
❖✉tr❛ s✐t✉❛çã♦ ❞❡ ❝♦♥✜♥❛♠❡♥t♦ q✉❡ ❝♦♥s✐❞❡r❛♠♦s é ❛ s❡❣✉✐♥t❡✳ ❙❡❥❛ h(s)✉♠❛
❢✉♥çã♦ ❝♦♥t✐♥✉❛ ❞❡✜♥✐❞❛ ❡♠ I = [−a, b]✱0≤a, b≤ ∞✱ ❡ s✉♣♦♥❤❛♠♦s q✉❡
✭✐✮ s= 0 é ♦ ú♥✐❝♦ ♣♦♥t♦ ❞❡ ♠á①✐♠♦ ❣❧♦❜❛❧ ❞❡ h ❡♠ I❀
✭✐✐✮ h∈C1 ❡♠ I\{0}❡ ♥✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ s= 0 ❛❞♠✐t❡ ❛ ❡①♣❛♥sã♦
h(s) =
⎧ ⎨ ⎩
M −c+sm+O(sm+1) s❡ s >0;
M −c−|s|m+O(|s|m+1) s❡ s <0
❡♠ q✉❡ M✱ m✱ c± sã♦ ♥ú♠❡r♦s r❡❛✐s ❡ M, c± > 0✱ m ≥ 1✳ ❈♦♥s✐❞❡r❡♠♦s ✉♠❛
♣❛rtí❝✉❧❛ s❡ ♠♦✈❡♥❞♦ ❧✐✈r❡♠❡♥t❡ ♥❛ s❡❣✉✐♥t❡ r❡❣✐ã♦ ❞❡ R2
Ωε={(s, y)∈R2 :s∈I,0≤y≤εh(s)}.
❖ ❝❛s♦ ❡♠ q✉❡ I é ✉♠ ✐♥t❡r✈❛❧♦ ❧✐♠✐t❛❞♦ ❢♦✐ ❛♥❛❧✐s❛❞♦ ❡♠ ❬✾❪ ❡ ♦ ❝❛s♦ I =R
❡♠ ❬✶✵❪✳ ❘❡s✉♠✐❞❛♠❡♥t❡✱ ❡♠ ❛♠❜♦s ♦s ❝❛s♦s✱ ❢♦✐ ♠♦str❛❞♦ q✉❡ ♥♦ ❧✐♠✐t❡ ε → 0 ♦s
❛✉t♦✈❛❧♦r❡s lj(ε) ❞♦ ▲❛♣❧❛❝✐❛♥♦ ❞❡ ❉✐r✐❝❤❧❡t ❡♠ Ωε t❡♠ ♦ s❡❣✉✐♥t❡ ❝♦♠♣♦rt❛♠❡♥t♦
❛ss✐♥tót✐❝♦✿
μj = lim ε→0ε
2β
lj(ε)−
π2
ε2M2
, β= 2
m+ 2,
❡♠ q✉❡ μj sã♦ ♦s ❛✉t♦✈❛❧♦r❡s ❞♦ ♦♣❡r❛❞♦r ❡♠ L2(R) ✭✐ss♦ ♠❡s♠♦✱ ❡♠ L2(R)✱ ✐♥❞❡✲
♣❡♥❞❡♥t❡ s❡ I ❢♦r ❧✐♠✐t❛❞♦ ♦✉ ♥ã♦✮ ❞❛❞♦ ♣♦r
(Hu)(s) =−u′′(s) +q(s)u(s), q(s) =
⎧ ⎨ ⎩
2π2M−3c
+sm s❡ s >0
2π2M−3c
−|s|m s❡ s <0
.
◆♦ ❈❛♣ít✉❧♦ ✼ ♥ós ❣❡♥❡r❛❧✐③❛♠♦s ❡ss❡s r❡s✉❧t❛❞♦s ♣❛r❛✿ ❛ ♣❛rtí❝✉❧❛ ❧✐✈r❡ s❡ ♠♦✈❡♥❞♦ ❛❣♦r❛ ❡♠ ✉♠ t✉❜♦ ❡♠ R3 ❡ ❛ s❡çã♦ tr❛♥s✈❡rs❛❧ ❞❡ss❡ t✉❜♦ é ♠✉❧t✐♣❧✐❝❛❞❛
♣❡❧❛ ♠❡s♠❛ ❢✉♥çã♦ h(s)❛❝✐♠❛✳ ❯♠ ❢❛t♦ ❛❞✐❝✐♦♥❛❧ ♥♦ ♥♦ss♦ ❡st✉❞♦✱ ❛❧é♠ ❞❛ ❞✐♠❡♥✲
sã♦✱ é q✉❡ ❛ r❡❣✐ã♦ ♣♦❞❡ ❛♣r❡s❡♥t❛r ❝✉r✈❛t✉r❛ ❡ t♦rçã♦ ♥ã♦✲♥✉❧❛s✳ ◆♦ss❛s ❝♦♥❝❧✉sõ❡s ✜♥❛✐s ♦❜t✐❞❛s sã♦ s✐♠✐❧❛r❡s às ♦❜t✐❞❛s ❡♠ ❬✾✱ ✶✵❪✱ t❛♥t♦ ♣❛r❛ t✉❜♦s ❧✐♠✐t❛❞♦s q✉❛♥t♦ ♣❛r❛ t✉❜♦s ✐❧✐♠✐t❛❞♦s✳ ❖ ♠❛✐s s✉r♣r❡❡♥❞❡♥t❡ é q✉❡ ✈❡r❡♠♦s q✉❡ ♦s ❡❢❡✐t♦s ❝♦♠♦ ❝✉r✈❛t✉r❛ ❡ t♦rçã♦ ♥ã♦ ✐♥✢✉❡♥❝✐❛♠ ♦ ♦♣❡r❛❞♦r ❧✐♠✐t❡ ❡❢❡t✐✈♦ s♦❜r❡ ❛ ❝✉r✈❛✳
❚❡r♠✐♥❛♠♦s ❛ss✐♠ ✉♠❛ ✈✐sã♦ ❣❡r❛❧ ❞❡st❛ t❡s❡❀ ♦s ❞❡t❛❧❤❡s sã♦ ❛♣r❡s❡♥t❛❞♦s ♥♦s ❝❛♣ít✉❧♦s s❡❣✉✐♥t❡s✳
❆❣♦r❛ ❛❧❣✉♠❛s ♥♦t❛çõ❡s ❡ ♦❜s❡r✈❛çõ❡s q✉❡ s❡rã♦ ✉s❛❞❛s ❛♦ ❧♦♥❣♦ ❞❡ t♦❞♦ ♦ tr❛❜❛❧❤♦✳ P❛r❛ y = (y1, y2) ∈ R2 ❡s❝r❡✈❡♠♦s y2 := y12 +y22✳ ❙❡❥❛♠ z, z′ ∈ R2✱ ♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦z, z′às ✈❡③❡s é ❞❡♥♦t❛❞♦ ♣♦r z·z′✳ ❊♠ t♦❞♦ ♦ tr❛❜❛❧❤♦ ♦s ❡s♣❛ç♦s
❞❡ ❍✐❧❜❡rt sã♦ r❡❛✐s ❡♠❜♦r❛ ♠✉✐t♦s ❞♦s r❡s✉❧t❛❞♦s ♣♦ss❛♠ s❡r ❣❡♥❡r❛❧✐③❛❞♦s ♣❛r❛ ❡s♣❛ç♦s ❞❡ ❍✐❧❜❡rt ❝♦♠♣❧❡①♦s✳ ❙❡♠♣r❡ q✉❡ ♥♦s r❡❢❡r✐♠♦s ❛♦ ♦♣❡r❛❞♦r ▲❛♣❧❛❝✐❛♥♦ ❡st❛♠♦s ❛ss✉♠✐♥❞♦ q✉❡ é ♦ ▲❛♣❧❛❝✐❛♥♦ ♥❡❣❛t✐✈♦✱ ♦✉ s❡❥❛✱ −∆✳ ❙❡ Ω é ✉♠ s✉❜❝♦♥✲
❥✉♥t♦ ❛❜❡rt♦ ❞❡ Rn✱ ♦s ❡s♣❛ç♦s Hm(Ω) ✭m = 1,2✮ ❞❡♥♦t❛♠ ♦s ❡s♣❛ç♦s ❞❡ ❙♦❜♦❧❡✈✳
▲❡♠❜r❡♠♦s q✉❡ Hm
0 (Ω) ❞❡♥♦t❛ ♦ ❝♦♥❥✉♥t♦ ❞❛s ❢✉♥çõ❡s ❡♠ Hm(Ω) ❝♦♠ ❛ ❤✐♣ót❡s❡ ❛❞✐❝✐♦♥❛❧ ❞❡ q✉❡ s❡ ❛♥✉❧❛♠ ♥❛ ❢r♦♥t❡✐r❛ ∂Ω✳
❈❛♣ít✉❧♦ ✷
❘❡✈✐sã♦ ❞❡
Γ
✲❈♦♥✈❡r❣ê♥❝✐❛
◆❡st❡ ❝❛♣ít✉❧♦ ❛♣r❡s❡♥t❛♠♦s ❛♣❡♥❛s ✉♠❛ ♥♦çã♦ ❞❛ t❡♦r✐❛ ❞❡ Γ✲❝♦♥✈❡r❣ê♥❝✐❛❀
♣❛r❛ ✉♠ tr❛t❛♠❡♥t♦ ♠❛✐s ❞❡t❛❧❤❛❞♦ ✐♥❞✐❝❛♠♦s ❬✸❪✳ ❆s ❞❡✜♥✐çõ❡s ❡ r❡s✉❧t❛❞♦s ❛♣r❡✲ s❡♥t❛❞♦s ❛q✉✐ s❡rã♦ ♠✉✐t♦ út❡✐s ♣❛r❛ ♦s ♣ró①✐♠♦s ❝❛♣ít✉❧♦s✳
✷✳✶ ❉❡✜♥✐çõ❡s ❡ ♣r♦♣r✐❡❞❛❞❡s
❙❡❥❛(Tε)✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ♦♣❡r❛❞♦r❡s ❛✉t♦✲❛❞❥✉♥t♦s ❝♦♠ ❞♦♠í♥✐♦ domTε ❡♠
✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt H✳ ❉❡♥♦t❛♠♦s ♣♦r bε ❛s r❡s♣❡❝t✐✈❛s ❢♦r♠❛s s❡sq✉✐❧✐♥❡❛r❡s
❛ss♦❝✐❛❞❛s ❛ ❡st❡s ♦♣❡r❛❞♦r❡s✳ ❱❛♠♦s ♣❡♥s❛r ❡♠ ε →0 ❡ ❛♥❛❧✐s❛r ♦ ❧✐♠✐t❡ T ✭r❡s♣✳ b✮ ❞❡ (Tε)✭r❡s♣✳ (bε)✮✳ ❖ ❞♦♠í♥✐♦ ❞❡ T ♥ã♦ s❡rá s✉♣♦st♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ❞❡♥s♦ ❡♠
H ❡ s❡✉ ❢❡❝❤♦ s❡rá ❞❡♥♦t❛❞♦ ♣♦r H0 = domT ✭❝♦♠ imgT ⊂ H0✮✳
❆ ❢✉♥çã♦ζ →b(ζ, ζ)s❡rá s✐♠♣❧❡s♠❡♥t❡ ❞❡♥♦t❛❞❛ ♣♦rb(ζ)❡ ❝❤❛♠❛❞❛ ❞❡ ❢♦r♠❛
q✉❛❞rát✐❝❛ ❛ss♦❝✐❛❞❛✳ ❱❛♠♦s ❛ss✉♠✐r t❛♠❜é♠ q✉❡ b é ♣♦s✐t✐✈❛ ✭♦✉ s❡❥❛✱ b(ζ) ≥ 0✱
∀ζ ∈ domb✮ ❡ b(ζ) =∞ s❡ ζ ∈ H\domb✳ ❆ss✐♠✱ ❣❛r❛♥t✐♠♦s q✉❡ b é s❡♠✐✲❝♦♥tí♥✉❛
✐♥❢❡r✐♦r♠❡♥t❡✱ ♦ q✉❛❧ é ❡q✉✐✈❛❧❡♥t❡ ❛ ❞✐③❡r q✉❡ ❛ ❢♦r♠❛ q✉❛❞rát✐❝❛ b é ❣❡r❛❞❛ ♣♦r
✉♠ ♦♣❡r❛❞♦r ❛✉t♦✲❛❞❥✉♥t♦ ❡ ♣♦s✐t✐✈♦ T✱ ♦✉ s❡❥❛✱
b(ζ, η) = T1/2ζ, T1/2η, ζ, η∈domb= domT1/2;
✈❡❥❛ ❚❡♦r❡♠❛ ✾✳✸✳✶✶ ❡♠ ❬✹❪✳ ❙❡ λ ∈ R✱ ❡♥tã♦ b +λ ✐♥❞✐❝❛ ❛ ❢♦r♠❛ s❡sq✉✐❧✐♥❡❛r
(b+λ)(ζ, η) :=b(ζ, η)+λζ, η❝✉❥❛ ❝♦rr❡s♣♦♥❞❡♥t❡ ❢♦r♠❛ q✉❛❞rát✐❝❛ é b(ζ)+λ||ζ||2✳ ➱ ❝♦♥❤❡❝✐❞♦ q✉❡ ✭▲❡♠❛ ✶✵✳✹✳✹ ❡♠ ❬✹❪✮✱ ♣❛r❛ λ > 0✱ t❡♠✲s❡ bε1 ≤ bε2 s❡✱ ❡ s♦✲
♠❡♥t❡ s❡✱Rλ(Tε2)≤Rλ(Tε1)✱ ♦✉ s❡❥❛✱ ❛ s❡q✉ê♥❝✐❛ ❞❡ ❢♦r♠❛s q✉❛❞rát✐❝❛s é ♠♦♥ót♦♥❛
s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❛ ❝♦rr❡s♣♦♥❞❡♥t❡ s❡q✉ê♥❝✐❛ ❞❡ ♦♣❡r❛❞♦r❡s r❡s♦❧✈❡♥t❡s é ♠♦♥ót♦♥❛✳ ❊st❛ s✐t✉❛çã♦ t❡♠ s✐❞♦ ♠✉✐t♦ ❡①♣❧♦r❛❞❛ ♥❛ ❧✐t❡r❛t✉r❛✱ ✉♠❛ ✈❡③ q✉❡ é ♣♦ssí✈❡❧ ♦❜t❡r ❝♦♥✈❡r❣ê♥❝✐❛ ❢♦rt❡ ❞♦s r❡s♦❧✈❡♥t❡s ❛tr❛✈és ❞♦ ❡st✉❞♦ ❞❡ss❛s ❢♦r♠❛s q✉❛❞rát✐❝❛s ✭♣♦r ❡①❡♠♣❧♦✱ ✈❡❥❛ ❙❡çã♦ ✶✵✳✹ ❡♠ ❬✹❪✮✳ P❛r❛ s❡q✉ê♥❝✐❛s ♠❛✐s ❣❡r❛✐s ❞❡ ♦♣❡r❛❞♦r❡s ❛ ❝♦♥✲ ✈❡r❣ê♥❝✐❛ ❢♦rt❡ ❞♦s r❡s♦❧✈❡♥t❡s ♥ã♦ é ❛ss✐♠ tã♦ ❞✐r❡t❛ ❡ r❡❧❛❝✐♦♥❛✲s❡ ❝♦♠ ✉♠ ❝♦♥❝❡✐t♦ ❝❤❛♠❛❞♦ ❞❡Γ✲❝♦♥✈❡r❣ê♥❝✐❛ ❬✸❪✳ ❖ ❝♦♥❝❡✐t♦ ❞❡ Γ✲❝♦♥✈❡r❣ê♥❝✐❛ ♥ã♦ é r❡str✐t♦ ❛♣❡♥❛s
à ❢♦r♠❛s q✉❛❞rát✐❝❛s ❡ ♣♦❞❡ s❡r ❛♣❧✐❝❛❞♦ ❛ ❡s♣❛ç♦s t♦♣♦❧ó❣✐❝♦s ♠❛✐s ❣❡r❛✐s✳ ◆♦ ❡♥t❛♥t♦✱ ❛q✉✐ ♥❡st❡ ❝❛♣ít✉❧♦✱ ✈❛♠♦s r❡str✐♥❣✐r ❛ ❞✐s❝✉ssã♦ ❡♠ ❡♣❛ç♦s ❞❡ ❍✐❧❜❡rt ❡ ❢♦r♠❛s s❡sq✉✐❧✐♥❡❛r❡s s❡♠✐✲❝♦♥tí♥✉❛s ✐♥❢❡r✐♦r♠❡♥t❡✳ ❆ t❡♦r✐❛ ❣❡r❛❧ ♣♦❞❡ s❡r ❡♥❝♦♥✲ tr❛❞❛ ❡♠ ❬✸❪✳ ❆q✉✐✱ H s❡♠♣r❡ ❞❡♥♦t❛ ✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt r❡❛❧ ❡ B(ζ;δ) ❛ ❜♦❧❛
❛❜❡rt❛ ❞❡ ❝❡♥tr♦ ζ ∈ H ❡ r❛✐♦ δ >0✳ ❉❡✜♥✐♠♦s t❛♠❜é♠ R:=R∪ {∞}✳
❉❡✜♥✐çã♦ ✷✳✶ ❖ Γ✲❧✐♠✐t❡ ✐♥❢❡r✐♦r ❞❡ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❢✉♥çõ❡s s❡♠✐✲❝♦♥tí♥✉❛s ✐♥✲
❢❡r✐♦r♠❡♥t❡ fε :H → Ré ❛ ❢✉♥çã♦ f− :H →R ❞❛❞❛ ♣♦r
f−(ζ) = lim
δ→0lim infε→0 inf{fε(η) :η ∈B(ζ;δ)}, ζ ∈ H. ❖ Γ✲❧✐♠✐t❡ s✉♣❡r✐♦r f+(ζ) ❞❡ f
ε é ❞❡✜♥✐❞♦ s✉❜st✐t✉✐♥❞♦ lim inf ❛❝✐♠❛ ♣♦r
lim sup✳ ❙❡ f− = f+ := f✱ ❞✐③❡♠♦s q✉❡ t❛❧ ❢✉♥çã♦ é ♦ Γ✲❧✐♠✐t❡ ❞❡ f
ε ❡ s❡rá ❞❡✲
♥♦t❛❞♦ ♣♦r
f = Γ−lim
ε→0fε.
❋♦✐ ❛ss✉♠✐❞♦ ♥❛ ❞❡✜♥✐çã♦ ❛❝✐♠❛ q✉❡ ❛ t♦♣♦❧♦❣✐❛ ❞❡ H é ❛ t♦♣♦❧♦❣✐❛ ❞❛ ♥♦r♠❛
✉s✉❛❧ ❡ ♥❡st❡ ❝❛s♦ ❢❛❧❛♠♦s ❞❡ Γ✲❝♦♥✈❡r❣ê♥❝✐❛ ❢♦rt❡✳ ❙❡ ❛ t♦♣♦❧♦❣✐❛ ❢r❛❝❛ é ❝♦♥s✐❞❡✲
r❛❞❛✱ ❛s ❜♦❧❛s ❞❡✈❡♠ s❡r s✉❜st✐t✉✐❞❛s ♣❡❧♦s ❝♦♥❥✉♥t♦s ❛❜❡rt♦s ❞❛ t♦♣♦❧♦❣✐❛ ❢r❛❝❛✱ ❡ ❛ss✐♠ ❢❛❧❛♠♦s ❞❡ Γ✲❝♦♥✈❡r❣ê♥❝✐❛ ❢r❛❝❛✳
❊①❡♠♣❧♦ ✷✳✶ ❆ s❡q✉ê♥❝✐❛ fε : R → R✱ fε(x) = sen (x/ε)✱ Γ✲❝♦♥✈❡r❣❡ à ❢✉♥çã♦
❝♦♥st❛♥t❡ −1 q✉❛♥❞♦ ε →0✳ ❊st❡ s✐♠♣❧❡s ❡①❡♠♣❧♦ ✐❧✉str❛ ❜❡♠ ❛ ✏❝♦♥✈❡r❣ê♥❝✐❛ ❞♦
♠í♥✐♠♦✑ q✉❡ ❢♦✐ ✉♠❛ ♠♦t✐✈❛çã♦ ♣❛r❛ ✐♥tr♦❞✉çã♦ ❞❛ Γ✲❝♦♥✈❡r❣ê♥❝✐❛✳
❖❜s❡r✈❛çã♦ ✷✳✶ ❆❧❣✉♥s ❢❛t♦s s♦❜r❡ Γ✲❝♦♥✈❡r❣ê♥❝✐❛✿
✭✐✮ ❡♠ ❣❡r❛❧ Γ−lim
ε→0fε =−Γ−εlim→0(−fε)❀ ✭✐✐✮ ❛ss✉♠❛ q✉❡ f = Γ−lim
ε→0fε ❡g = Γ−limε→0gε✳ P♦❞❡ ❛❝♦♥t❡❝❡r ❞❡ fε+gε ♥ã♦ s❡r
Γ✲❝♦♥✈❡r❣❡♥t❡❀
✭✐✐✐✮ ♥ã♦ é ♥❡❝❡ssár✐♦ r❡str✐♥❣✐r ❛ ❞❡✜♥✐çã♦ ❞❡Γ✲❝♦♥✈❡r❣ê♥❝✐❛ ❛ ❢✉♥çõ❡s s❡♠✐✲❝♦♥tí♥✉❛s
✐♥❢❡r✐♦r♠❡♥t❡✳ P♦r ❡①❡♠♣❧♦✱ s❡ fε =f✱ ♣❛r❛ t♦❞♦ ε✱ ❡ f ♥ã♦ é s❡♠✐✲❝♦♥tí♥✉❛
✐♥❢❡r✐♦r♠❡♥t❡✱ ❡♥tã♦✱ ♦Γ−lim
ε→0f é ❛ ♠❛✐♦r ❢✉♥çã♦ s❡♠✐✲❝♦♥tí♥✉❛ ♠❛❥♦r❛❞❛ ♣♦r
f✱ ❡ ❛ss✐♠✱ ❞✐❢❡r❡♥t❡ ❞❡ f✳
❊♠ ❬✸❪ ❛♣❛r❡❝❡ ❛ ❞❡♠♦♥str❛çã♦ ❞❛ s❡❣✉✐♥t❡ ♣r♦♣♦s✐çã♦✳
Pr♦♣♦s✐çã♦ ✷✳✷ ❙❡❥❛♠ fε :H →R ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❢✉♥çõ❡s s❡♠✐✲❝♦♥tí♥✉❛s ✐♥❢❡✲
r✐♦r♠❡♥t❡ ❡ f :H → R✳ ❆ s❡q✉ê♥❝✐❛ fε Γ✲❝♦♥✈❡r❣❡ ❢♦rt❡♠❡♥t❡ ❛ f s❡✱ ❡ s♦♠❡♥t❡ s❡✱
❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s sã♦ s❛t✐s❢❡✐t❛s✿
✭✐✮ ♣❛r❛ ❝❛❞❛ ζ ∈ H ❡ t♦❞❛ s❡q✉ê♥❝✐❛ ζε →ζ ❡♠ H t❡♠✲s❡
lim inf
ε→0 fε(ζε)≥f(ζ).
✭✐✐✮ ♣❛r❛ ❝❛❞❛ ζ ∈ H ❡①✐st❡ ✉♠❛ s❡q✉ê♥❝✐❛ ζε →ζ ❡♠ H ❞❡ ❢♦r♠❛ q✉❡
f(ζ) = lim
ε→0fε(ζε).
❖❜s❡r✈❛çã♦ ✷✳✷ ❙❡ ❡♠ ✈❡③ ❞❛ ❝♦♥✈❡r❣ê♥❝✐❛ ❢♦rt❡ ζε →ζ ❢♦r ❝♦♥s✐❞❡r❛❞❛ ❛ ❝♦♥✈❡r✲
❣ê♥❝✐❛ ❢r❛❝❛ ζε ⇀ ζ ♥❛ ♣r♦♣♦s✐çã♦ ❛❝✐♠❛✱ ❡♥tã♦ ❞✐③❡♠♦s q✉❡ fε Γ✲❝♦♥✈❡r❣❡ ❢r❛❝❛✲
♠❡♥t❡ ❛ f✳
✷✳✷
Γ
✲❝♦♥✈❡r❣ê♥❝✐❛ ❡ ❝♦♥✈❡r❣ê♥❝✐❛ ❞♦s r❡s♦❧✈❡♥t❡s
❱❛♠♦s ❡♥✉♥❝✐❛r ♦ ♣r✐♥❝✐♣❛❧ r❡s✉❧t❛❞♦ s♦❜r❡ Γ✲❝♦♥✈❡r❣ê♥❝✐❛ ❡ ❝♦♥✈❡r❣ê♥❝✐❛ ❢♦rt❡
❚❡♦r❡♠❛ ✷✳✸ ❙❡❥❛♠ bε✱ b ❢♦r♠❛s s❡sq✉✐❧✐♥❡❛r❡s ❢❡❝❤❛❞❛s ❡ ♣♦s✐t✐✈❛s ❡♠ ✉♠ ❡s♣❛ç♦
❞❡ ❍✐❧❜❡rt H✳ ❙❡❥❛♠ Tε✱ T ♦s r❡s♣❡❝t✐✈♦s ♦♣❡r❛❞♦r❡s ❛✉t♦✲❛❞❥✉♥t♦s ❡ ♣♦s✐t✐✈♦s
❛ss♦❝✐❛❞♦s✳ ❙ã♦ ❡q✉✐✈❛❧❡♥t❡s✿
✭✐✮ bε Γ✲❝♦♥✈❡r❣❡ ❢♦rt❡♠❡♥t❡ ❛ b ❡ ♣❛r❛ ❝❛❞❛ ζ ∈ H t❡♠✲s❡ lim inf
ε→0 bε(ζε) ≥ b(ζ)✱ ♣❛r❛ t♦❞❛ s❡q✉ê♥❝✐❛ ζε ⇀ ζ✳
✭✐✐✮ bε Γ✲❝♦♥✈❡r❣❡ ❢♦rt❡♠❡♥t❡ ❡ ❢r❛❝❛♠❡♥t❡ ❛ b✳
✭✐✐✐✮ bε+λ Γ✲❝♦♥✈❡r❣❡ ❢♦rt❡♠❡♥t❡ ❡ ❢r❛❝❛♠❡♥t❡ ❛ b+λ✱ ♣❛r❛ ❛❧❣✉♠λ >0✭❡ ❛ss✐♠✱
♣❛r❛ t♦❞♦ λ≥0✮✳
✭✐✈✮ P❛r❛ ❝❛❞❛ η ∈ H ❡λ >0✱ ❛ s❡q✉ê♥❝✐❛
min
ζ∈H
bε(ζ) +λ||ζ||2+ζ, η
❝♦♥✈❡r❣❡ ❛
min
ζ∈H
b(ζ) +λ||ζ||2+ζ, η
.
✭✈✮ Tε ❝♦♥✈❡r❣❡ ❛T ♥♦ s❡♥t✐❞♦ ❢♦rt❡ ❞♦s r❡s♦❧✈❡♥t❡s ❡♠ H0 = domT ⊂ H✱ ♦✉ s❡❥❛✱
lim
ε→0R−λ(Tε)ζ =R−λ(T)P0ζ, ∀ζ ∈ H,∀λ >0, ❡♠ q✉❡ P0 é ♦ ♣r♦❥❡t♦r ♦rt♦❣♦♥❛❧ ❡♠ H0.
❈❛♣ít✉❧♦ ✸
Γ
✲❈♦♥✈❡r❣ê♥❝✐❛ ❡ ❛ ♣❛rtí❝✉❧❛ ❧✐✈r❡
◆❡st❡ ❝❛♣ít✉❧♦ ✈❛♠♦s ❞✐s❝✉t✐r ♦ ❝❛s♦ ❞❡ ✉♠❛ ♣❛rtí❝✉❧❛ q✉â♥t✐❝❛ ❧✐✈r❡ s❡ ♠♦✈❡♥❞♦ ❡♠ ✉♠ t✉❜♦ Ω❡♠ R3✳ ▼❛✐s ♣r❡❝✐s❛♠❡♥t❡✱ ♦ ▲❛♣❧❛❝✐❛♥♦ ❞❡ ❉✐r✐❝❤❧❡t ❡♠ Ω✳
❉❛❞♦Ω✱ ✉♠ ♣❛râ♠❡tr♦ εé ❛❝r❡s❝❡♥t❛❞♦ ❞❡ ♠♦❞♦ q✉❡ ❛ s❡q✉ê♥❝✐❛ ❞❡ t✉❜♦s Ωε
s❡ r❡❞✉③❛ ❛ ✉♠❛ ❝✉r✈❛ ❡♠ R3 q✉❛♥❞♦ ε →0✳ P♦r ❡①❡♠♣❧♦✱ s❡ Ω ={(x, y, z)∈R3 :
y2+z2 = 1} é ✉♠ ❝✐❧✐♥❞r♦ ❞❡ R3✱ ❛ s❡q✉ê♥❝✐❛ Ω
ε ={(x, εy, εz)∈R3 : (x, y, z)∈Ω}
s❡ ❛♣r♦①✐♠❛ ❞❡ ✉♠❛ r❡t❛ ❡♠ R3 ✭❡✐①♦ x✮ q✉❛♥❞♦ ε → 0✳ ❖ ♦❜❥❡t✐✈♦ é ❛♥❛❧✐s❛r ♦
❝♦♠♣♦rt❛♠❡♥t♦ ❞♦ ❍❛♠✐❧t♦♥✐❛♥♦ ❞❛ ♣❛rtí❝✉❧❛ ♥♦ ❧✐♠✐t❡ ε → 0✳ ❉❡s❞❡ q✉❡ ♥❡ss❡
❧✐♠✐t❡ ❛s r❡❣✐õ❡sΩε s❡ ❛♣r♦①✐♠❛♠ ❞♦ ❡✐①♦x✱ ✐❞❡♥t✐✜❝❛♠♦s ❡st❡ ú❧t✐♠♦ ❝♦♠ ♦ ❡s♣❛ç♦
✉♥✐❞✐♠❡♥s✐♦♥❛❧ R ❡ ❝❤❛♠❛♠♦s ❡st❡ ❛❝♦♥t❡❝✐♠❡♥t♦ ❞❡ r❡❞✉çã♦ ❞❡ ❞✐♠❡♥sã♦✳
◆❛ ♣r✐♠❡✐r❛ s❡çã♦ ✈❛♠♦s ❝♦♥str✉✐r ❞❡t❛❧❤❛❞❛♠❡♥t❡ ❛ r❡❣✐ã♦ ❡♠ q✉❡ ❛ ♣❛rtí❝✉❧❛ s❡ ❡♥❝♦♥tr❛ ❝♦♥✜♥❛❞❛✳ ❉❡ ❢❛t♦✱ ❛s r❡❣✐õ❡s ♣♦❞❡♠ s❡r ❜❡♠ ♠❛✐s ❣❡r❛✐s ❞♦ q✉❡ ❝✐❧✐♥✲ ❞r♦s ❞❡R3✳ ◆❛ s❡❣✉♥❞❛ s❡çã♦ ✈❛♠♦s ❞✐s❝✉t✐r ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❛ ♣❛rtí❝✉❧❛ q✉❛♥❞♦
ε→0✳ ◆❛ t❡r❝❡✐r❛ s❡çã♦ ✈❛♠♦s ❞❡✜♥✐r ❛s ❢♦r♠❛s q✉❛❞rát✐❝❛s ❛ss♦❝✐❛❞❛s ❛♦ ▲❛♣❧❛✲
❝✐❛♥♦ ❞❡ ❉✐r✐❝❤❧❡t ❡♠Ωε✳ ❱❡r❡♠♦s q✉❡ ❡ss❛s ❢♦r♠❛s ❡stã♦ ❞✐r❡t❛♠❡♥t❡ r❡❧❛❝✐♦♥❛❞❛s
❝♦♠ ❛ ❣❡♦♠❡tr✐❛ ❞❡ Ωε✳ ◆❛ ú❧t✐♠❛ s❡çã♦ ✈❛♠♦s ❛♣r❡s❡♥t❛r r❡s✉❧t❛❞♦s ❥á ❝♦♥❤❡❝✐❞♦s
s♦❜r❡ ♦ ❛ss✉♥t♦ ❡ q✉❡ ✉s❛r❡♠♦s ❡♠ ♦✉tr♦s ❝❛♣ít✉❧♦s✳
❖s r❡s✉❧t❛❞♦s ❛♣r❡s❡♥t❛❞♦s ♥❡st❡ ❝❛♣ít✉❧♦ ❥á sã♦ ❝♦♥❤❡❝✐❞♦s ❡ s❡ ❜❛s❡✐❛♠✱ ♣r✐♥✲ ❝✐♣❛❧♠❡♥t❡✱ ♥❛s r❡❢❡rê♥❝✐❛s ❬✶❪ ❡ ❬✺❪✳ ❚❛✐s r❡s✉❧t❛❞♦s ❢♦r♠❛♠ ♦ ♣♦♥t♦ ❞❡ ♣❛rt✐❞❛ ♣❛r❛
♥♦ss♦s ♣ró♣r✐♦s r❡s✉❧t❛❞♦s ❡ ♣r♦❜❧❡♠❛s q✉❡ r❡s♦❧✈❡♠♦s ♣❡sq✉✐s❛r❀ ❞❡st❛ ♠❛♥❡✐r❛✱ ❡st❡ ❝❛♣ít✉❧♦ ♣♦❞❡ s❡r ✈✐st♦ ❝♦♠♦ ✉♠❛ ♣r❡♣❛r❛çã♦ ♣❛r❛ ♦ q✉❡ s❡❣✉❡ ♥❡st❛ t❡s❡✳
✸✳✶ ●❡♦♠❡tr✐❛ ❞♦ ❞♦♠í♥✐♦
❙❡❥❛ I = [a, b]✱ −∞ ≤ a < b ≤ ∞✱ ✉♠ ✐♥t❡r✈❛❧♦ ❞❡ R ❡ r :I ⊆ R → R3 ✉♠❛
❝✉r✈❛ s✐♠♣❧❡s C2 ❡♠ R3 ♣❛r❛♠❡tr✐③❛❞❛ ♣❡❧♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ❛r❝♦ s✳ ❆ ❝✉r✈❛t✉r❛ ❞❡ r ♥♦ ♣♦♥t♦ s é ❞❡♥♦t❛❞❛ ♣♦r k(s)✳ ❖s ✈❡t♦r❡s
T(s) = r′(s), N(s) = 1
k(s)T
′(s), B(s) =T(s)
×N(s),
❞❡♥♦t❛♠✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♦s ✈❡t♦t❡s t❛♥❣❡♥t❡✱ ♥♦r♠❛❧ ❡ ❜✐♥♦r♠❛❧ ❛ ❝✉r✈❛ ♥♦ ♣♦♥t♦
r(s)✳ ❱❛♠♦s ❛ss✉♠✐r q✉❡ ❛s ❡q✉❛çõ❡s ❞❡ ❋r❡♥❡t sã♦ s❛t✐s❢❡✐t❛s✿
⎛ ⎜ ⎜ ⎜ ⎝
T′
N′
B′
⎞ ⎟ ⎟ ⎟ ⎠
=
⎛ ⎜ ⎜ ⎜ ⎝
0 k 0
−k 0 τ
0 −τ 0
⎞ ⎟ ⎟ ⎟ ⎠
⎛ ⎜ ⎜ ⎜ ⎝
T
N
B
⎞ ⎟ ⎟ ⎟ ⎠
,
❡♠ q✉❡ τ(s) é ❛ t♦rçã♦ ❞❛ ❝✉r✈❛ ♥♦ ♣♦♥t♦ s✳
❙❡❥❛ S ✉♠ s✉❜❝♦♥❥✉♥t♦ ❛❜❡rt♦✱ ❧✐♠✐t❛❞♦✱ s✐♠♣❧❡s♠❡♥t❡ ❝♦♥❡①♦ ❡ ♥ã♦✲✈❛③✐♦ ❞❡
R2✳ ❖ ❝♦♥❥✉♥t♦
Ω ={x∈R3 :x=r(s) +y1N(s) +y2B(s), s∈I, y= (y1, y2)∈S}
é ♦❜t✐❞♦ tr❛♥s❧❛❞❛♥❞♦✲s❡ ❛ r❡❣✐ã♦ S ❛♦ ❧♦♥❣♦ ❞❛ ❝✉r✈❛✳
Ω ❛✐♥❞❛ ♣♦❞❡ s❡r ❞❡❢♦r♠❛❞❛ ❞❡ ♠♦❞♦ q✉❡ ❡♠ ❝❛❞❛ ♣♦♥t♦ r(s) ❛ r❡❣✐ã♦ S ❢❛③
✉♠❛ r♦t❛çã♦ ❞❡ â♥❣✉❧♦ α(s)✳ ❆ ♥♦✈❛ r❡❣✐ã♦ é ❞❛❞❛ ♣♦r
Ωα ={x∈R3 :x=r(s) +y1Nα(s) +y2Bα(s), s∈I, y= (y1, y2)∈S},
❡♠ q✉❡
Nα(s) := cosα(s)N(s) + senα(s)B(s),
Bα(s) := −sinα(s)N(s) + cosα(s)B(s).
❈♦♠ ❛ ❝♦♥str✉çã♦ ❛❝✐♠❛✱ ✈❛♠♦s ❝♦♥s✐❞❡r❛r ♦ ♣❛râ♠❡tr♦ ε >0❡♠S ❡ ❛ r❡❣✐ã♦
Ωαε ={x∈R3 :x=r(s) +εy1Nα(s) +εy2Bα(s), s∈I, y= (y1, y2)∈S}. ✭✸✳✶✮
❆ss✐♠✱ Ωα
ε é ✏❡s♣r✐♠✐❞❛✑ à ❝✉r✈❛ r(s)q✉❛♥❞♦ ε→0✳
✸✳✷ P❛rtí❝✉❧❛ ❧✐✈r❡
❱❛♠♦s ❝♦♥s✐❞❡r❛r ♦ ❝❛s♦ ❞❡ ✉♠❛ ♣❛rtí❝✉❧❛ q✉â♥t✐❝❛ s❡ ♠♦✈❡♥❞♦ ❧✐✈r❡♠❡♥t❡ ♥❛ r❡❣✐ã♦ Ωα
ε ❞❡✜♥✐❞❛ ♥❛ s❡çã♦ ❛♥t❡r✐♦r✳ ❱❛♠♦s ✐♠♣♦r ❛ ❝♦♥❞✐çã♦ ❞❡ ❉✐r✐❝❤❧❡t ♥❛
❢r♦♥t❡✐r❛ ❞❡ Ωα
ε✳ ▼❛✐s ♣r❡❝✐s❛♠❡♥t❡✱ s❡ −∆αε ❞❡♥♦t❛ ♦ ▲❛♣❧❛❝✐❛♥♦ ❞❡ ❉✐r✐❝❤❧❡t ❡♠
Ωα
ε✱ ✈❛♠♦s ❡st✉❞❛r ❛s ❢♦r♠❛s q✉❛❞rát✐❝❛s ❛ss♦❝✐❛❞❛s ❛ ❡ss❡s ♦♣❡r❛❞♦r❡s✱ ♦✉ s❡❥❛✱
ϕ→
Ωα ε
|∇ϕ|2dx, ϕ∈H01(Ωαε),
❡♠ q✉❡ ∇ ❞❡♥♦t❛ ♦ ❣r❛❞✐❡♥t❡ ❞❡ ϕ ♥❛s ❝♦♦r❞❡♥❛❞❛s ✉s✉❛✐s ❞❡ R3✳ P❛r❛ ❝❛❞❛ ε > 0
❝♦♥s✐❞❡r❡♠♦s ♦ s❡❣✉✐♥t❡ ♣r♦❜❧❡♠❛ ❞❡ ❛✉t♦✈❛❧♦r❡s✿
⎧ ⎨ ⎩
−∆uε =λεuε
uε∈H01(Ωαε).
✭✸✳✷✮
❈♦♥s✐❞❡r❡♠♦s ♦ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❡♠ q✉❡ I é ✉♠ ✐♥t❡r✈❛❧♦ ❧✐♠✐t❛❞♦✳ ❆ss✐♠✱
❝♦♠♦Ωα
ε é ❧✐♠✐t❛❞♦✱ ♦ ❡s♣❡❝tr♦ σε ❞♦ ♣r♦❜❧❡♠❛ ✭✸✳✷✮ é ❞✐s❝r❡t♦✳ ▼❛✐s ♣r❡❝✐s❛♠❡♥t❡✱
σε = {λεi;i ∈ N} ❡♠ q✉❡ 0 < λε0 < λε1 ≤ λε2 ≤ · · · sã♦ ♥ú♠❡r♦s r❡❛✐s✱ ♣♦s✐t✐✈♦s ❡ ♣♦❞❡♠ s❡r ❛rr✉♠❛❞♦s ❞❡ ♠♦❞♦ q✉❡ ❢♦r♠❡♠ ✉♠❛ s❡q✉ê♥❝✐❛ ❝r❡s❝❡♥t❡✳ ◗✉❛♥❞♦ I =R
♦ ❡s♣❡❝tr♦ ❞♦ ♣r♦❜❧❡♠❛ ✭✸✳✷✮ ♥ã♦ é ♥❡❝❡ss❛r✐❛♠❡♥t❡ ❞✐s❝r❡t♦✳ ◗✉❛♥❞♦ ε→ 0 ❛ r❡❣✐ã♦ Ωα
ε t♦r♥❛✲s❡ ❝❛❞❛ ✈❡③ ♠❛✐s ❡str❡✐t❛✳ ❙❡ ✉♠❛ ♣❛rtí❝✉❧❛
❧✐✈r❡ é ♦❜r✐❣❛❞❛ ❛ ♣❡r♠❛♥❡❝❡r ♥✉♠❛ r❡❣✐ã♦ ❡s♣❛❝✐❛❧ ♠✉✐t♦ ♣❡q✉❡♥❛ s✉❛ ❡♥❡r❣✐❛ t❡♥❞❡ ❛ ✐♥✜♥✐t♦✳ ❖ q✉❡ ♦❝♦rr❡ sã♦ ♦s❝✐❧❛çõ❡s tr❛♥s✈❡rs❛✐s ♠✉✐t♦ rá♣✐❞❛s q✉❛♥❞♦ ε → 0✳
❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ❛ ❡♥❡r❣✐❛ t♦t❛❧ ❞♦ s✐st❡♠❛ ❞✐✈❡r❣❡ ❡ ✈❛✐ ❛ ✐♥✜♥✐t♦ q✉❛♥❞♦ ε→0✳
P♦r ❡st❡ ♠♦t✐✈♦✱ ❞❡✈❡♠♦s r❡t✐r❛r ✏❛❧❣♦✑ ❞❛ ❢♦r♠❛ q✉❛❞rát✐❝❛ ❝♦♠♦ ✉♠❛ ❢♦r♠❛ ❞❡ ✐♠♣❡❞✐r ❡st❛ ✏❡①♣❧♦sã♦✑ ♥♦ ❧✐♠✐t❡ ε→0✳
❆ ♣r✐♥❝í♣✐♦ ♥ã♦ é ❝❧❛r♦ q✉❛❧ ❡①♣r❡ssã♦ ❞❡✈❡ s❡r s✉❜tr❛í❞❛ ❞❛ ❢♦r♠❛ q✉❛❞rát✐❝❛✳ ❆♥❛❧✐s❡♠♦s ♦ s❡❣✉✐♥t❡ ❝❛s♦✿ s❡❥❛ λ0 ♦ ♣r✐♠❡✐r♦ ❛✉t♦✈❛❧♦r ✭♦✉ s❡❥❛✱ ♦ ♠❡♥♦r✮ ❞♦ ♦♣❡r❛❞♦r ❞❡ ▲❛♣❧❛❝❡ ❝♦♠ ❝♦♥❞✐çõ❡s ❞❡ ❉✐r✐❝❤❧❡t ♥❛ r❡❣✐ã♦ S✱ ♦✉ s❡❥❛✱
−∆u0 =λ0u0, u0 ∈H01(S), λ0 >0,
S|
u0|2ds = 1.
❆ ❛✉t♦❢✉♥çã♦ u0 ❡stá ❞✐r❡t❛♠❡♥t❡ r❡❧❛❝✐♦♥❛❞❛ às ♦s❝✐❧❛çõ❡s tr❛♥s✈❡rs❛✐s✳ ❉❡✲ ✈✐❞♦ ❛ ❡st❡ ❢❛t♦✱ ✈❛♠♦s s✉❜tr❛✐r ♦s t❡r♠♦s ❞❛ ❢♦r♠❛ λ0/ε2 ❝♦♠♦ ✉♠ ❝❛♠✐♥❤♦ ❞❡ ❝♦♥t♦r♥❛r ♦ ♣r♦❜❧❡♠❛✳
P❛ss❛♠♦s ❛❣♦r❛ ❛ ♦❜s❡r✈❛r ❛ s❡q✉ê♥❝✐❛ ❞❡ ❢♦r♠❛s q✉❛❞rát✐❝❛s
Fε(ϕ) :=
Ωα ε
|∇ϕ|2− λ0 ε2|ϕ|
2
dx, ✭✸✳✸✮
❞❡✜♥✐❞❛s ❡♠ H1
0(Ωαε)✳ ▲❡♠❜r❡♠♦s q✉❡✱ ❞❛ t❡♦r✐❛ ❞❡ ♦♣❡r❛❞♦r❡s ❧✐♥❡❛r❡s✱ λ0 é ✉♠ ❛✉t♦✈❛❧♦r s✐♠♣❧❡s✳
❘❡s✉♠✐❞❛♠❡♥t❡✱ ♦ q✉❡ q✉❡r❡♠♦s é s❛❜❡r ♦ q✉❡ ♦❝♦rr❡ q✉❛♥❞♦ ε → 0 ❝♦♠
❛ s❡q✉ê♥❝✐❛ ❞❡ ❢♦r♠❛s q✉❛❞rát✐❝❛s ❛❝✐♠❛✳ P♦❞❡♠♦s ❞✐③❡r q✉❡ ♦ q✉❡ ✜③❡♠♦s✱ ❛♦ s✉❜tr❛✐r ♦s t❡r♠♦s ❞❛ ❢♦r♠❛ λ0/ε2✱ ❢♦✐ s❡♣❛r❛r ❛ ❞✐♥â♠✐❝❛ rá♣✐❞❛ ✭♦s❝✐❧❛❝õ❡s✮ ❞❛ ❞✐♥â♠✐❝❛ ❧❡♥t❛ ✭♥❛ ❝✉r✈❛✮✳ ■st♦ ❡stá ❞✐r❡t❛♠❡♥t❡ r❡❧❛❝✐♦♥❛❞♦ ❝♦♠ ♦ ♣r✐♥❝í♣✐♦ ❞❛ ✐♥❝❡rt❡③❛ ❡♠ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛✳
✸✳✸ ▼✉❞❛♥ç❛ ❞❡ ✈❛r✐á✈❡✐s
❈♦♠♦ ❡♠ ✭✸✳✸✮ ❛ r❡❣✐ã♦ ❞❡ ✐♥t❡❣r❛çã♦ ❞❡♣❡♥❞❡ ❞♦ ♣❛râ♠❡tr♦ ε✱ ♦ ♦❜❥❡t✐✈♦
❞❡st❛ s❡çã♦ é ❢❛③❡r ✉♠❛ ♠✉❞❛♥ç❛ ❞❡ ✈❛r✐á✈❡✐s ❞❡ ♠♦❞♦ q✉❡ ❛ r❡❣✐ã♦ ❞❡ ✐♥t❡❣r❛çã♦ ✐♥❞❡♣❡♥❞❛ ❞❡ε >0✳
❈♦♥s✐❞❡r❡♠♦s ❛ ❛♣❧✐❝❛çã♦
fα
ε : I×S → Ωαε
(s, y1, y2) → r(s) +εy1Nα(s) +εy2Bα(s).
❱❛♠♦s s✉♣♦r q✉❡ ||k||∞,||τ||∞,||α′||∞<∞✳ ❊ss❛s ❝♦♥❞✐çõ❡s ✈ã♦ ❣❛r❛♥t✐r q✉❡✱
♣❛r❛ ε s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥♦✱ fα
ε s❡❥❛ ✉♠ ❞✐❢❡♦♠♦r✜s♠♦✳
❈♦♠♦ ❛♥t❡s✱ ∇ ❞❡♥♦t❛ ♦ ❣r❛❞✐❡♥t❡ ✉s✉❛❧ ♥❛s ❝♦♦r❞❡♥❛❞❛s ❝❛rt❡s✐❛♥❛s (x, y, z)✳
❱❛♠♦s ❢❛③❡r ✉♠❛ ♠✉❞❛♥ç❛ ❞❡ ✈❛r✐á✈❡✐s✱ ❛ ♣❛rt✐r ❞❡ fα
ε✱ ❡ ♣❛ss❛r ❛ ❢♦r♠❛ q✉❛❞rát✐❝❛
✉s✉❛❧ ♥❛s ❝♦♦r❞❡♥❛❞❛s (x, y, z)✱
bαε(ψ) =
Ωα ε
|∇ψ|2dx, dombαε =H01(Ωαε),
♣❛r❛ ❛s ❝♦♦r❞❡♥❛❞❛s(s, y1, y2)❞❡I×S✳ ❆ss✐♠✱ ♣❛ss❛♠♦s ❛ tr❛❜❛❧❤❛r ❡♠ ✉♠ ❞♦♠í♥✐♦ ✜①♦ ♣❛r❛ t♦❞♦ ε > 0✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ♦ ♣r❡ç♦ ❛ ♣❛❣❛r é ✉♠❛ ♠étr✐❝❛ ❘✐❡♠❛♥♥✐❛♥❛
G=Gα
ε ♥ã♦✲tr✐✈✐❛❧ ❛ q✉❛❧ é ✐♥❞✉③✐❞❛ ♣❡❧♦ ❞✐❢❡♦♠♦r✜s♠♦ fεα✱ ♦✉ s❡❥❛✱
G= (Gij), Gij =ei, ej=Gji, 1≤i, j ≤3,
❡♠ q✉❡
e1 =
∂fα ε
∂s , e2 =
∂fα ε
∂y1
, e3 =
∂fα ε
∂y2
.
❖ r❡❢❡rê♥❝✐❛❧ ❞❡ ❋r❡♥❡t é ❞❛❞♦ ♣♦r
J = ⎛ ⎜ ⎜ ⎜ ⎝ e1 e2 e3 ⎞ ⎟ ⎟ ⎟ ⎠ = ⎛ ⎜ ⎜ ⎜ ⎝
βε −ε(τ +α′)z⊥α, y ε(τ +α′)zα, y
0 εcosα εsenα
0 −εsenα εcosα
⎞ ⎟ ⎟ ⎟ ⎠ ❡♠ q✉❡
βε(s, y) = 1−εk(s)zα, y, zα := (cosα,−senα), zα⊥ := (senα,cosα).
❆ ♠❛tr✐③ ✐♥✈❡rs❛ ❞❡ J é
J−1 =
⎛ ⎜ ⎜ ⎜ ⎜ ⎝ 1 βε
(τ+α′)y
2
βε −
(τ +α′)y
1
βε
0 cosα
ε −
senα ε
0 senα
ε cosα ε ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ .
◆♦t❡♠♦s q✉❡ JJt = G ❡ detJ =|detG|1/2 =ε2β
ε(s, y)✳ ❉❡s❞❡ q✉❡ k é ✉♠❛
❢✉♥çã♦ ❧✐♠✐t❛❞❛✱ ♣❛r❛ ε s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥♦✱ βε ♥ã♦ s❡ ❛♥✉❧❛ ❡♠ I×S✳ ❆ss✐♠✱
βε>0❡ fεα é ✉♠ ❞✐❢❡♦♠♦r✜s♠♦ ❧♦❝❛❧✳ ❊①✐❣✐♥❞♦ q✉❡ fεα s❡❥❛ ✐♥❥❡t♦r❛ ✭♦✉ s❡❥❛✱ q✉❡ ♦
t✉❜♦ ♥ã♦ s❡ ❛✉t♦✲✐♥t❡r❝❡♣t❛✮ ♦❜t❡♠♦s ✉♠ ❞✐❢❡♦♠♦r✜s♠♦ ❣❧♦❜❛❧ ♣❛r❛ ε ♣❡q✉❡♥♦✳
■♥tr♦❞✉③✐♥❞♦ ❛ ♥♦t❛çã♦
||ψ||2G =
I×S|
ψ(s, y1, y2)|2ε2βε(s, y)dsdy,
♦❜t❡♠♦s ❛ s❡q✉ê♥❝✐❛ ❞❡ ❢♦r♠❛s q✉❛❞rát✐❝❛s
˜
bα ε(ψ) :=
I×S|
J−1∇ψ|2ε2β
ε(s, y)dyds, dom˜bαε =H01(I×S, G),
❡ ❛❣♦r❛ ∇ = (∂s,∇y) ✭∂s é ❞❛ ❞❡r✐✈❛❞❛ ❝♦♠ r❡s♣❡✐t♦ ❛ s ❡ ∇y = (∂/∂y1, ∂/∂y2)✮✳ ▼❛✐s ♣r❡❝✐s❛♠❡♥t❡✱ ❛ ♠✉❞❛♥ç❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❛❝✐♠❛ ❢♦✐ ♦❜t✐❞❛ ♣❡❧❛ tr❛♥s❢♦r♠❛çã♦ ✉♥✐tár✐❛
Uα
ε : L2(Ωαε) → L2(I×S, G)
φ → φ◦fα
ε
◆♦ ❡♥t❛♥t♦✱ ❝♦♥t✐♥✉❛r❡♠♦s ❞❡♥♦t❛♥❞♦ Uα
εψ ♣♦r ψ✳
❈♦♠♦ ❞✐s❝✉t✐❞♦ ♥❛ s❡çã♦ ❛♥t❡r✐♦r✱ ✈❛♠♦s s✉❜tr❛✐r ♦ t❡r♠♦ (λ0/ε2)||ψ||2G ❡ ♣❛s✲
s❛r ❛ ❡st✉❞❛r ❛ s❡q✉ê♥❝✐❛ ❞❡ ❢♦r♠❛s q✉❛❞rát✐❝❛s
ˆ
aα ε(ψ) :=
1 ε2
J−1∇ψ 2 G− λ0
ε2||ψ|| 2
G+c||ψ||2G
, ✭✸✳✹✮
❡♠ q✉❡c:=||k(s)2/4||
∞.❈♦♠♦ ✈❡r❡♠♦s ♥❛ ♣ró①✐♠❛ s❡çã♦✱ s♦♠❛r ❛ ❝♦♥st❛♥t❡ c > 0✱
❛ss✐♠ ❞❡✜♥✐❞❛✱ ✐♠♣❧✐❝❛ q✉❡ ❛s ❢♦r♠❛s q✉❛❞rát✐❝❛s s❡❥❛♠ t♦❞❛s ♣♦s✐t✐✈❛s✳ ❆❧❣✉♥s ❝á❧❝✉❧♦s ♠♦str❛♠ q✉❡
ˆ
aαε(ψ) =
I×S
1
βε |
ψ′+ (∇yψ·Ry)(τ +α′)ψ|2
+βε
|∇yψ|2
ε2 −λ0
|ψ|2
ε2
+βεc|ψ|2
dyds, ✭✸✳✺✮
❡♠ q✉❡ dom ˆaα
ε = H01(I ×S)✳ ❉❡♥♦t❛♠♦s ♣♦r ψ′ ❛ ❞❡r✐✈❛❞❛ ❞❡ ψ ❡♠ r❡❧❛çã♦ ❛ s✱
∇yψ ♦ ❣r❛❞✐❡♥t❡ ❞❡ ψ ❡♠ r❡❧❛çã♦ ❛ y= (y1, y2)❡ R é ❛ ♠❛tr✐③ ❞❡ r♦t❛çã♦
⎛ ⎝
0 1
−1 0
⎞ ⎠.
❖❜s❡r✈❛çã♦ ✸✳✶ ➱ ♠✉✐t♦ ✐♠♣♦rt❛♥t❡ q✉❡ βε(s, y)→1✉♥✐❢♦r♠❡♠❡♥t❡ ♣❛r❛ ε→0✳
❉❡ ❢❛t♦✱ s❡❥❛ || · || ❞❡♥♦t❛♥❞♦ ❛ ♥♦r♠❛ ✉s✉❛❧ ❡♠ L2(R×S)✳ ❊①✐st❡♠ ❢✉♥çõ❡s β−
ε ❡
β+
ε ❞❡ ♠♦❞♦ q✉❡
βε−|| · || ≤ || · ||G ≤βε+|| · ||,
❡β−
ε (s, y)→1✱βε+(s, y)→1q✉❛♥❞♦ε→0✳ ❇❛st❛ t♦♠❛r♠♦sβε−(s, y) = inf
(s,y)∈R×Sβε(s, y)
❡βε+(s, y) = sup
(s,y)∈R×S
βε(s, y)✳ ❊st❛ ♣r♦♣r✐❡❞❛❞❡ ✐♠♣❧✐❝❛ q✉❡ ♦s ❡s♣❛ç♦s L2(R×S, βε(s, y))
❝♦✐♥❝✐❞❡♠ ❛❧❣❡❜r✐❝❛♠❡♥t❡ ❝♦♠ L2(R×S)✳ ❆ss✐♠✱ ♣♦❞❡♠♦s tr❛❜❛❧❤❛r ❧✐✈r❡♠❡♥t❡ ❡♠
L2(R×S)✳
✸✳✹ ❆❧❣✉♥s r❡s✉❧t❛❞♦s
❊♠ ❬✶❪ é ❝♦♥s✐❞❡r❛❞♦ ♦ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❡♠ q✉❡ I é ✉♠ ✐♥t❡r✈❛❧♦ ❧✐♠✐t❛❞♦✳ ❖
♣r✐♥❝✐♣❛❧ r❡s✉❧t❛❞♦ ❞❛q✉❡❧❡ ❛rt✐❣♦ é✿ ❚❡♦r❡♠❛ ✸✳✶ ❙❡❥❛ Ωα
ε ❞❡✜♥✐❞♦ ♣♦r ✭✸✳✶✮ ❡♠ q✉❡ I é ✉♠ ✐♥t❡r✈❛❧♦ ❧✐♠✐t❛❞♦✳ ❙❡❥❛
{λj(ε)}∞j=1 ❛ s❡q✉ê♥❝✐❛ ♥ã♦✲❞❡❝r❡s❝❡♥t❡ ❞❡ ❛✉t♦✈❛❧♦r❡s ❞♦ ▲❛♣❧❛❝✐❛♥♦ ❞❡ ❉✐r✐❝❤❧❡t ❡♠ H1
0(Ωαε)✳ ❊♥tã♦✱
λj(ε) =
λ0
ε2 +μj + o(1), q✉❛♥❞♦ ε →0,
❡♠ q✉❡ {μj}∞j=1 ❞❡♥♦t❛ ❛ s❡q✉ê♥❝✐❛ ♥ã♦✲❞❡❝r❡s❝❡♥t❡ ❞❡ ❛✉t♦✈❛❧♦r❡s ❞♦ ♦♣❡r❛❞♦r ❞❡ ❙❝❤rö❞✐♥❣❡r
−w′′(s) +
C(S)(τ(s) +α′(s))− k(s)
2
4
w(s) ❡♠ L2(I). ✭✸✳✻✮
❆q✉✐ C(S)é ✉♠❛ ❝♦♥st❛♥t❡ ♥ã♦✲♥❡❣❛t✐✈❛ ❞❡♣❡♥❞❡♥❞♦ s♦♠❡♥t❡ ❞❛ r❡❣✐ã♦ S✳
❱❡r❡♠♦s ❛❣♦r❛ q✉❡ ❛ ✐♥✢✉ê♥❝✐❛ ❞❛ ❝✉r✈❛t✉r❛ ♥♦ ♦♣❡r❛❞♦r ❡❢❡t✐✈♦ ✭✸✳✻✮ ♥ã♦ é ❛♣❡♥❛s ❞❡✈✐❞♦ ❛♦ t❡r♠♦ βε(s, y) = 1−εk(s)(zα·y)✳
❈♦♠♦ ❡♠ ❬✶❪✱ ♣❛r❛ ❝❛❞❛ ξ∈R2✱ ❝♦♥s✐❞❡r❡♠♦s ♦ s❡❣✉✐♥t❡ ♣r♦❜❧❡♠❛ ♣❡rt✉r❜❛❞♦✿
−div[(1−(ξ·y))∇yu] =λ(1−(ξ·y))u, u∈H01(S).
❚♦♠❛♥❞♦ ξ = εk(s)zα✱ ♣❛r❛ ε s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥♦✱ ♦ ♦♣❡r❛❞♦r ♣❡rt✉r✲
❜❛❞♦ é ♣♦s✐t✐✈♦ ❡ ❝♦♠ r❡s♦❧✈❡♥t❡ ❝♦♠♣❛❝t♦✳ ❉❡♥♦t❛♠♦s ♣♦r λ(ξ)> 0 s❡✉ ♣r✐♠❡✐r♦
❛✉t♦✈❛❧♦r✱ ♦✉ s❡❥❛✱
λ(ξ) = inf
{u∈H1
0(S):u=0}
S(1−(ξ·y))|∇yu|
2dy
S(1−(ξ·y))|u|2dy
.
