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LUẬN văn THẠC sĩ HAY bài toán stick slip và một số phương pháp tìm nghiệm gần đúng

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✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆
❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈

✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖

◆●❯❨➍◆ ❚❍➚ ❑❍❯❨➊◆

❇⑨■ ❚❖⑩◆ ❙❚■❈❑✲❙▲■P ❱⑨ ▼❐❚ ❙➮
P❍×❒◆● P❍⑩P ❚➐▼ ◆●❍■➏▼ ●❺◆ ✣Ĩ◆●

▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈

❚❤→✐ ◆❣✉②➯♥ ✲ ◆➠♠ ✷✵✶✺

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✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆
❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈

✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖

◆●❯❨➍◆ ❚❍➚ ❑❍❯❨➊◆

❇⑨■ ❚❖⑩◆ ❙❚■❈❑✲❙▲■P ❱⑨ ▼❐❚ ❙➮
P❍×❒◆● P❍⑩P ❚➐▼ ◆●❍■➏▼ ●❺◆ ✣Ĩ◆●
❈❤✉②➯♥ ♥❣➔♥❤✿ ❚❖⑩◆ Ù◆● ❉Ö◆●
▼➣ sè✿ ✻✵✳✹✻✳✵✶✳✶✷

▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆


ữớ ữợ ồ


◆➠♠ ✷✵✶✺

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ừ trữ ổ

ừ ữớ ữợ ❦❤♦❛ ❤å❝

❚❙✳ ❱ô ❱✐♥❤ ◗✉❛♥❣

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✐✐

▲í✐ ❝↔♠ ì♥
✣➸ ❤♦➔♥ t❤➔♥❤ ✤÷đ❝ ❧✉➟♥ ✈➠♥ ♠ët ❝→❝❤ tổ ổ
ữủ sỹ ữợ ú ✤ï ♥❤✐➺t t➻♥❤ ❝õ❛ P●❙✳❚❙ ❱ơ ❱✐♥❤ ◗✉❛♥❣
✭❚r÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❍å❝✮✳ ❚æ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉
s➢❝ ✤➳♥ t❤➛② ✈➔ ①✐♥ ❣û✐ ❧í✐ tr✐ ➙♥ ♥❤➜t ❝õ❛ tỉ✐ ✤è✐ ✈ỵ✐ ♥❤ú♥❣ ✤✐➲✉ t❤➛②
✤➣ ❞➔♥❤ ❝❤♦ tỉ✐✳

❚ỉ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❜❛♥ ❧➣♥❤ ✤↕♦ ♣❤á♥❣ s ồ qỵ t
ổ ợ ồ ❑✼❈ ✭✷✵✶✹✲ ✷✵✶✻✮ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❍å❝
✲ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥ ✤➣ t➟♥ t➻♥❤ tr✉②➲♥ ✤↕t ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ qỵ
ụ ữ t tổ t ❦❤â❛ ❤å❝✳
❚ỉ✐ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ ❝❤➙♥ t❤➔♥❤ ♥❤➜t tợ ỳ
ữớ ổ ở ❤é trđ ✈➔ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ ❝❤♦ tỉ✐ tr♦♥❣ s✉èt
q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥✳ ❳✐♥ tr➙♥ trå♥❣ ❝↔♠ ì♥✦

❚❤→✐ ♥❣✉②➯♥✱ t❤→♥❣ ✶✷ ♥➠♠ ✷✵✶✺
◆❣÷í✐ ✈✐➳t ❧✉➟♥ ✈➠♥

◆❣✉②➵♥ ❚❤à ❑❤✉②➯♥

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✐✐✐

▼ư❝ ❧ư❝
▲í✐ ❝↔♠ ì♥

✐✐

▼ư❝ ❧ư❝

✐✐✐

▼ð ✤➛✉




▼ët sè ❦➼ ❤✐➺✉ ✈✐➳t t➢t



✶ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚



✶✳✶

✶✳✷

❑❤æ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳



✶✳✶✳✶

¯
❑❤æ♥❣ ❣✐❛♥ C k Ω

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳



✶✳✶✳✷

❑❤æ♥❣ ❣✐❛♥ Lp (Ω) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳




✶✳✶✳✸

❑❤æ♥❣ ❣✐❛♥ ❲

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳



✶✳✶✳✹

❑❤æ♥❣ ❣✐❛♥ H01 (Ω) ✈➔ ❦❤→✐ ♥✐➺♠ ✈➳t ❝õ❛ ❤➔♠ ✳ ✳



✶✳✶✳✺

❈æ♥❣ t❤ù❝ ●r❡❡♥✱ ❜➜t ✤➥♥❣ t❤ù❝ P♦✐♥❝❛r❡ ✳ ✳ ✳ ✳



✶✳✶✳✻

❑❤ỉ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈ ✈ỵ✐ ❝❤➾ sè ➙♠ H −1 (Ω) ✈➔ H − 2 (∂Ω) ✶✵

1,p

(Ω)


1

P❤÷ì♥❣ tr➻♥❤ ❊❧❧✐♣t✐❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✶

✶✳✷✳✶

❑❤→✐ ♥✐➺♠ ♥❣❤✐➺♠ ②➳✉ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✳ ✳ ✳ ✳ ✳

✶✶

✶✳✷✳✷

✣à♥❤ ♥❣❤➽❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✷

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✐✈
✶✳✷✳✸
✶✳✸

✶✳✹

▼➺♥❤ ✤➲ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳




tự sỡ ỗ ❧➦♣ ❝ì ❜↔♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳





ữủ ỗ ợ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳





ữủ ỗ ứ ỵ ỡ sỹ ở tử ừ
ữỡ ❧➦♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳



ỵ tt ✈➲ s❛✐ ♣❤➙♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳





Pữỡ ữợ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✹

✶✳✹✳✷


❇➔✐ t♦→♥ s❛✐ ♣❤➙♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✺

✶✳✹✳✸

❑➳t ❧✉➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✻

✷ ❇➔✐ t♦→♥ st✐❝❦✲s❧✐♣ ✈➔ ♣❤÷ì♥❣ ♣❤→♣ t➻♠ ♥❣❤✐➺♠ ❞↕♥❣ t✐➺♠
❝➟♥

✶✼

✷✳✶

▼ỉ ❤➻♥❤ ❜➔✐ t♦→♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✼

✷✳✷

▼ët sè ♣❤÷ì♥❣ ♣❤→♣ t➻♠ ♥❣❤✐➺♠ ❞↕♥❣ ❦❤❛✐ tr✐➸♥

✳ ✳ ✳ ✳

✶✾

✷✳✸


P❤÷ì♥❣ ♣❤→♣ ❙❋❇■▼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✵

❑➳t ❧✉➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✻

✷✳✸✳✶

✸ P❤÷ì♥❣ ♣❤→♣ ❧➦♣ ❣✐↔✐ ❜➔✐ t♦→♥ st✐❝❦ ✕ s❧✐♣ tê♥❣ q✉→t
✸✳✶

✷✼

❈ì sð ỵ tt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✼

✸✳✶✳✶

❈ì sð ♣❤÷ì♥❣ ♣❤→♣ ❝❤✐❛ ♠✐➲♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳





ỡ ỗ ừ t tỷ






ỡ ỗ ❦➳t ❤ñ♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✷

✸✳✸

▼ët sè ❦➳t q✉↔ t❤ü❝ ♥❣❤✐➺♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✹

❑➳t ❧✉➟♥

✸✼

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❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦

✸✽

P❍❺◆ P❍Ö ▲Ö❈

✹✵


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▼ð ✤➛✉
❇➔✐ t♦→♥ ❙t✐❝❦✲❙❧✐♣ ❧➔ ♠ët ❞↕♥❣ ❜➔✐ t♦→♥ ♠➝✉ ỹ ừ ữỡ tr
s ỏ ợ t ♥❤➜t✳ ✣✐➸♠ ✤➦❝ ❜✐➺t ❝õ❛ ❜➔✐ t♦→♥ ♥➔②
❧➔ ❤➺ ✤✐➲✉ ❦✐➺♥ ❜✐➯♥ ❝õ❛ ❜➔✐ t♦→♥ ❧➔ ❞↕♥❣ ❦➻ ❞à tù❝ ❧➔ tr➯♥ ♠ët ✤♦↕♥ ❜✐➯♥
trì♥ ♥↔② s✐♥❤ ❤✐➺♥ t÷đ♥❣ t❤✐➳✉ ố ợ ỗ
tớ ♣❤→t s✐♥❤ ♥❤ú♥❣ ✤✐➸♠ ❦➻ ❞à ❧➔ ❝→❝ ✤✐➸♠ ❣✐❛♦ ❣✐ú❛ ✤✐➲✉ ❦✐➺♥ ❤➔♠ ✈➔
✤✐➲✉ ❦✐➺♥ ✤↕♦ ❤➔♠✳ ✣➙② ❧➔ ♠ët ♠æ ❤➻♥❤ ♠æ t↔ sü ❞❛♦ ✤ë♥❣ ❝õ❛ ❝→❝ t
ỗ õ q ❞↕♥❣ ♥❣➔♠✱ ❣è✐ tü❛ ✈➔ ❜✐➯♥
tü ❞♦ ❤é♥ ❤ñ♣✳ ✣➙② ❧➔ ♠ët ♠ỉ ❤➻♥❤ ❜➔✐ t♦→♥ ✤÷đ❝ ❝→❝ t→❝ ❣✐↔ tr➯♥ t❤➳
❣✐ỵ✐ r➜t q✉❛♥ t➙♠✱ ❝â t➼♥❤ ù♥❣ ❞ư♥❣ ❝→♦✳ ❱➻ t➼♥❤ ❝❤➜t ❦➻ ❞à ♥➯♥ ✈✐➺❝ t➻♠
♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❦❤ỉ♥❣ t❤➸ t❤ü❝ ❤✐➺♥ ❜➡♥❣ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ t❤ỉ♥❣
t❤÷í♥❣✳ t tr t ợ tữớ t
t t ữợ s
t ♣❤→t tø ❝→❝ ✤✐➸♠ ❦➻ ❞à ❧➔ ✤✐➸♠ ❣✐❛♦ ❣✐ú❛ ❝→❝ ❧♦↕✐ ✤✐➲♥ ❦✐➺♥
❜✐➯♥✱ ♥❣÷í✐ t❛ t➻♠ ❝→❝❤ ①➙② ❞ü♥❣ r ữợ
tồ ỹ tọ ✤✐➲✉ ❦✐➺♥ ❝õ❛ ❜➔✐ t♦→♥ ✈➔ tø ✤â ♥❣❤✐➺♠ ①➜♣ ①➾ ❝õ❛
❜➔✐ t♦→♥ ✤÷đ❝ ①→❝ ✤à♥❤ ❜➡♥❣ ❝→❝ ❝ỉ♥❣ t❤ù❝ ❦❤❛✐ tr✐➸♥ ❞↕♥❣ ❝❤✉é✐
❤➔♠ t❤æ♥❣ q✉❛ ❝→❝ ❤➺ ❤➔♠ r✐➯♥❣✳ ❚ø ✤â ❜➔✐ t♦→♥ ✤÷❛ ✈➲ ✈✐➺❝ ①→❝
✤à♥❤ ❝→❝ ❤➺ sè ❝õ❛ ❦❤❛✐ tr✐➸♥ ❜➡♥❣ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ✤↕✐ sè✳
✷✳ ỷ ử ỵ tt t tỷ ỹ sỡ ỗ
tr t tr➯♥ ❜✐➯♥ ✤➸ ❝❤✉②➸♥ ❜➔✐ t♦→♥ ❝â ❝❤ù❛ ❝→❝ ✤✐➸♠
❦➻ ❞à ✈➲ ❝→❝ ❜➔✐ t♦→♥ ❝♦♥ ❦❤æ♥❣ ❝❤ù❛ ✤✐➸♠ ❦➻ t ủ ợ ữỡ

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♣❤→♣ ♣❤➙♥ r➣ ♣❤÷ì♥❣ tr➻♥❤ ❝➜♣ ❜è♥ ✈➲ ❤❛✐ ♣❤÷ì♥❣ tr➻♥❤ ❝➜♣ ❤❛✐✳
❚ø ✤â →♣ ❞ư♥❣ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ s❛✐ ♣❤➙♥ ✤➸ ❣✐↔✐ q✉②➳t ❝→❝ ❜➔✐ t♦→♥
❝♦♥ q✉❛ ✤â ①➙② ❞ü♥❣ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❣è❝ ❜❛♥ ✤➛✉✳
❳✉➜t ♣❤→t tø ♣❤➙♥ t➼❝❤ ✤â✱ ♠ö❝ t✐➯✉ ♥❣❤✐➯♥ ❝ù✉ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ ✈➠♥ ❧➔
t➻♠ ❤✐➸✉ ✈➲ ♠æ ❤➻♥❤ ❜➔✐ t♦→♥ ❙t✐❝❦✲❙❧✐♣✱ ♥❣❤✐➯♥ ❝ù✉ ❝ì sð ❝õ❛ ♣❤÷ì♥❣
♣❤→♣ ❦❤❛✐ tr✐➸♥ t➻♠ ♥❣❤✐➺♠ ①➜♣ ừ t t ỗ tớ
ự ỡ s ừ ỵ tt t tỷ ũ ữỡ
r ❝❤✉②➸♥ ❜➔✐ t♦→♥ ❙t✐❝❦✲❙❧✐♣ ✈➲ ❝→❝ ❜➔✐ t♦→♥ ❡❧❧✐♣t✐❝ ❝➜♣ ❤❛✐✱ sû ❞ư♥❣
♣❤÷ì♥❣ ♣❤→♣ s❛✐ ♣❤➙♥ ✤➸ ①→❝ ✤à♥❤ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❣è❝✳ ❙♦ s→♥❤ ❦➳t
q✉↔ t❤ü❝ ♥❣❤✐➺♠ ❝õ❛ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣✳ ❈→❝ ❦➳t q✉↔ t❤ü❝ ♥❣❤✐➺♠ ✤÷đ❝
t❤ü❝ ❤✐➺♥ tr➯♥ ♠→② t➼♥❤ ✤✐➺♥ tû✳
◆ë✐ ❞✉♥❣ ❝õ❛ ❜↔♥ ❧✉➟♥ ✈➠♥ ✤÷đ❝ tr➻♥❤ ❜➔② tr♦♥❣ ✸ ❝❤÷ì♥❣✳
❈❤÷ì♥❣ ✶✿ ❚r➻♥❤ ❜➔② ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ❝ì sð ✈➲ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ❤➔♠✱
❧➼ t❤✉②➳t ữỡ tr s ỏ ỵ tt t tỷ sỡ ỗ
ừ t tỷ sỹ ở tử tt s ổ
ữợ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ s❛✐ ♣❤➙♥ ✤↕♦ ❤➔♠✱ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ữợ
ữỡ r ổ t t ữỡ ♣❤→♣ ❦❤❛✐
tr✐➸♥ t❤ỉ♥❣ q✉❛ ❝→❝ ❤➺ ❤➔♠ r✐➯♥❣✱ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ t➻♠ ♥❣❤✐➺♠ ①➜♣ ①➾✳
❈❤÷ì♥❣ ✸ ❚r➻♥❤ ❜➔② ♠ët sè ❦➳t q✉↔ t❤ü❝ ♥❣❤✐➺♠ ✤è✐ ✈ỵ✐ ❜➔✐ t♦→♥
❙t✐❝❦✲❙❧✐♣✳
▲✉➟♥ ✈➠♥ ♥➔② ữủ t ữợ sỹ ữợ t t ừ ❚❙
❱ơ ❱✐♥❤ ◗✉❛♥❣✱ ❡♠ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ ❝❤➙♥ t❤➔♥❤ ❝õ❛ ♠➻♥❤ ✤è✐
✈ỵ✐ t❤➛②✳ ❊♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❝→❝ t❤➛②✱ ❝ỉ ❣✐→♦ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝
✲ ✣↕✐ ❤å❝ ❚❤→✐ ♥❣✉②➯♥ ✤➣ t❤❛♠ ❣✐❛ ❣✐↔♥❣ ❞↕②✱ ❣✐ó♣ ✤ï ❡♠ tr♦♥❣ s✉èt
q✉→ tr➻♥❤ ❤å❝ t➟♣ ♥➙♥❣ ❝❛♦ tr➻♥❤ ✤ë ❦✐➳♥ t❤ù❝✳ ❚✉② ♥❤✐➯♥ ✈➻ ✤✐➲✉ ❦✐➺♥
t❤í✐ ❣✐❛♥ ✈➔ ❦❤↔ ♥➠♥❣ ❝â ❤↕♥ ♥➯♥ ❧✉➟♥ ✈➠♥ ❦❤æ♥❣ t❤➸ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣
t❤✐➳✉ sât✳ ❊♠ ❦➼♥❤ ♠♦♥❣ ❝→❝ t❤➛② ❝æ ❣✐→♦ õ õ ỵ


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✤➸ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳

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▼ët sè ❦➼ ❤✐➺✉ ✈✐➳t t➢t
L

❚♦→♥ tû ❡❧❧✐♣t✐❝ ✳

Rn

❑❤æ♥❣ ❣✐❛♥ ❊✉❝❧✐❞ n ❝❤✐➲✉✳



▼✐➲♥ ❣✐ỵ✐ ♥ë✐ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ Rn ✳

∂Ω

❇✐➯♥ trì♥ ▲✐♣s❝❤✐t③✳

C k (Ω)


❑❤ỉ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ❝â ✤↕♦ ❤➔♠ ❝➜♣ k ❧✐➯♥ tư❝✳

L2 (Ω)

❑❤ỉ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ✤♦ ✤÷đ❝ ữỡ t

W 1,p ()

ổ ợ sè p✳

H 1/2 (∂Ω)

❑❤æ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈ ❝→❝ ❤➔♠ ❝â ✈➳t ❜➡♥❣ ❦❤æ♥❣ tr➯♥ ∂Ω ✳

H01 (Ω)

❑❤æ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ❝â ✈➳t ❜➡♥❣ ❦❤æ♥❣ tr➯♥ ∂Ω ✳

H −1 (Ω)

❑❤æ♥❣ ❣✐❛♥ ✤è✐ ♥❣➝✉ ✈ỵ✐ H01 (Ω)✳

H −1/2 (∂Ω) ❑❤ỉ♥❣ ❣✐❛♥ ✤è✐ ♥❣➝✉ ợ H 1/2 ()
SF BIM

Pữỡ t



tỷ ▲❛♣❧❛❝❡✳




❚♦→♥ tû ●r❛❞✐❡♥t✳

Dα u

✣↕♦ ❤➔♠ r✐➯♥❣ ❝õ❛ u ❝➜♣ |α|✳

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❈❤÷ì♥❣ ✶
❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚
❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② ❧✉➟♥ ✈➠♥ tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ❝ì sð ✈➲ ❝→❝
❦❤ỉ♥❣ ❣✐❛♥ ❤➔♠✱ ❧➼ t❤✉②➳t ♣❤÷ì♥❣ tr➻♥❤ s♦♥❣ ✤✐➲✉ ❤á❛✱ ❧➼ t❤✉②➳t t tỷ
ỵ tt sỡ ỗ ✈➔ ♣❤÷ì♥❣ ♣❤→♣ s❛✐ ♣❤➙♥✳ ❈→❝ ❦✐➳♥
t❤ù❝ ❝ì ❜↔♥ ✤÷đ❝ t❤❛♠ ❦❤↔♦ tr♦♥❣ ❝→❝ t➔✐ ❧✐➺✉ ❬✷❪✱ ❬✻❪✱ ❬✼❪✳

✶✳✶ ❑❤æ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈✳
✶✳✶✳✶

¯
❑❤æ♥❣ ❣✐❛♥ C k Ω

¯
●✐↔ sû Ω ❧➔ ♠ët ♠✐➲♥ ❜à ❝❤➦♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❊✉❝❧✐❞ n ❝❤✐➲✉ Rn ✈➔ Ω
¯ , (k = 0, 1, 2...) ❧➔ t➟♣ ❝→❝ ❤➔♠ ❝â

❧➔ ❜❛♦ ✤â♥❣ ❝õ❛ Ω✳ ❚❛ ❦➼ ❤✐➺✉ C k Ω
¯ ✳ ❚❛ ✤÷❛ ✈➔♦ C k Ω
¯
✤↕♦ ❤➔♠ ✤➳♥ ❝➜♣ k ❦➸ ❝↔ k tr♦♥❣ Ω✱ ❧✐➯♥ tö❝ tr♦♥❣ Ω
❝❤✉➞♥✿

u

¯)
C k (Ω

max |Dα u (x)| ,

=
|α|=k

¯
x∈Ω

tr♦♥❣ ✤â α = (α1 , α2 , ..., αn ) ✤÷đ❝ ❣å✐ ❧➔ ✤❛ ❝❤➾ sè ✈❡❝tì ✈ỵ✐ ❝→❝ tå❛ ✤ë
♥❣✉②➯♥ ❦❤æ♥❣ ➙♠✱ |α| = α1 + α2 + ... + αn ✿

∂ α1 +...+αn u
D u = α1
.
∂x1 ...∂xαnn
α

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¯ ❝õ❛ ❝→❝ ❤➔♠ ✈➔
❙ü ❤ë✐ tö t❤❡♦ ❝❤✉➞♥ ✤➣ ❝❤♦ ❧➔ sü ❤ë✐ tư ✤➲✉ tr♦♥❣ Ω
¯ ✈ỵ✐ ❝❤✉➞♥ ✤➣
t➜t ❝↔ ✤↕♦ ❤➔♠ ❝õ❛ ❝❤ó♥❣ ✤➳♥ ❝➜♣ k ✳ ❘ã r➔♥❣ t➟♣ C k Ω
❝❤♦ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳

✶✳✶✳✷

❑❤æ♥❣ ❣✐❛♥ Lp (Ω)

●✐↔ sû Ω ❧➔ ♠ët ♠✐➲♥ tr♦♥❣ Rn ✈➔ p ❧➔ ♠ët sè t❤ü❝ ❞÷ì♥❣✳ ❚❛ ❦➼ ❤✐➺✉

Lp () ợ ữủ f tr➯♥ Ω s❛♦ ❝❤♦✿
|f (x)|p dx < ∞.


❚r♦♥❣ Lp (Ω) t ỗ t tr Ω✳ ◆❤÷
✈➟② ❝→❝ ♣❤➛♥ tû ❝õ❛ Lp (Ω) ❧➔ ❝→❝ ợ tữỡ ữỡ ữủ
tọ ✈➔ ❤❛✐ ❤➔♠ t÷ì♥❣ ✤÷ì♥❣ ♥➳✉ ❝❤ó♥❣ ❜➡♥❣ ♥❤❛✉ ❤➛✉
❦❤➢♣ tr➯♥ Ω✳ ❱➻ ✿

|f (x) + g (x)|p

(|f (x) + g (x)|)p

2p (|f (x)|p + |g (x)|p ) ,

♥➯♥ rã r➔♥❣ Lp (Ω) ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì✳

❚❛ ✤÷❛ ✈➔♦ Lp (Ω) ♣❤✐➳♠ ❤➔♠ .

u

p

=





✶✳✶✳✸

❑❤ỉ♥❣ ❣✐❛♥ ❲

1,p

p

✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐✿

|u (x)|p dx


1
p

.




(Ω)

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✳ ❈❤♦ Ω ❧➔ ♠ët ♠✐➲♥ tr♦♥❣ Rn ✳ ❍➔♠ u(x)✤÷đ❝ ❣å✐ ❧➔
❦❤↔ t➼❝❤ ✤à❛ ♣❤÷ì♥❣ tr♦♥❣ Ω ♥➳✉ u(x) ❧➔ ♠ët ❤➔♠ tr♦♥❣ Ω ✈➔ ✈ỵ✐ ộ

x0 tỗ t ởt ❝õ❛ x0 ✤➸ u(x) ❦❤↔ t➼❝❤ tr♦♥❣ ω ✳

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✷✳ ❈❤♦ Ω ❧➔ ♠ët ♠✐➲♥ tr♦♥❣ Rn ✳ ❍➔♠ u(x), v(x)✤÷đ❝

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❣å✐ ❧➔ ❦❤↔ t➼❝❤ ✤à❛ ♣❤÷ì♥❣ tr♦♥❣ Ω s❛♦ ❝❤♦ t❛ ❝â ❤➺ t❤ù❝✿



∂kϕ
k
u
dx
=
(−1)
k
∂x1 k1 ...∂xnn




∂ku
ϕdx,
∂xk11 ...∂xknn

✤è✐ ✈ỵ✐ ♠å✐ ϕ (x) ∈ C0k (Ω) , k = k1 + ... + kn , ki
✤â✱

∂ku
k
∂x11 ...∂xknn

0 (i = 1, 2, ..., n)✳ ❑❤✐

✤÷đ❝ ❣å✐ ❧➔ ✤↕♦ ❤➔♠ s✉② rë♥❣ ❝➜♣ k ❝õ❛ u(x)✳

❑➼ ❤✐➺✉✿

∂ku
v (x) = k1
.
∂x1 ...∂xknn

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✸✳ ●✐↔ sû p ❧➔ ♠ët sè t❤ü❝✱ 1

p < ∞, Ω ❧➔ ♠ët ♠✐➲♥

tr♦♥❣ Rn ✳ ❑❤æ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈ ❲1,p (Ω) ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ♥❤÷ s❛✉✿
❲1,p (Ω) =

u|u ∈ Lp (Ω) ,


∂u
∈ Lp (Ω) , i = 1, ..., n .
∂xi

❚r♦♥❣ ✤â ❝→❝ ✤↕♦ ❤➔♠ tr➯♥ ❧➔ ❝→❝ ✤↕♦ ❤➔♠ s✉② rë♥❣✳
❱ỵ✐ p = 2 ✱ t❛ ❦➼ ❤✐➺✉
❲1,2 (Ω) = H 1 (Ω) ,
♥❣❤➽❛ ❧➔✿

H 1 (Ω) =

✶✳✶✳✹

u|u ∈ L2 (Ω) ,

∂u
∈ L2 (Ω) , i = 1, 2, ..., n .
∂xi

❑❤æ♥❣ ❣✐❛♥ H01 (Ω) ✈➔ ❦❤→✐ ♥✐➺♠ ✈➳t ❝õ❛ ❤➔♠

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✹✳ ❱ỵ✐ ❜➜t ❦➻ 1

p < ∞✱ ❦❤ỉ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈ ❲1,p
0 (Ω)

✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ♥❤÷ ❝→❝ ❜❛♦ ✤â♥❣ ❝õ❛ D (Ω) ✭❦❤æ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ❦❤↔
✈✐ ✈æ ❤↕♥ õ t tr tữỡ ự ợ ừ ❲1,p
0 (Ω) ✳

❑❤ỉ♥❣ ❣✐❛♥ H01 (Ω) ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐✿

H01 (Ω) = ❲1,2
0 (Ω) .

✣à♥❤ ❧➼ ✶✳✶✳✺✳ ●✐↔ sû ∂Ω ❧➔ ❧✐➯♥ tö❝ ▲✐♣s❝❤✐t③ t❤➻✿ ✐✮ ◆➳✉ 1

❲01,p (Ω) ⊂ Lq (Ω) ❧➔✿

p
t❤➻

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✲ ◆❤ó♥❣ ❈♦♠♣❛❝t ✤è✐ ✈ỵ✐ q ∈ [1, p∗] tr♦♥❣ ✤â p∗1 = p1 − n1 ✱
✲ ◆❤ó♥❣ ❧✐➯♥ tư❝ ✤è✐ ✈ỵ✐ q = p∗✳
✐✐✮ ◆➳✉ p = n t❤➻
❲01,n (Ω) ⊂ Lq (Ω)
❧➔ ♥❤ó♥❣ ❈♦♠♣❛❝t ♥➳✉ q ∈ [1, +∞]✳
✐✐✐✮ ◆➳✉ p > n t❤➻
❲01,p (Ω) ⊂ C 0
ú t
ỵ t
ỗ t t ởt →♥❤ ①↕ t✉②➳♥ t➼♥❤ ❧✐➯♥ tư❝ ✤÷đ❝ ❣å✐ ❧➔ ✈➳t
✣à♥❤ ❧➼ ✶✳✶✳✻✳



γ : H 1 Rn−1 × R+
→ L2 Rn−1

s❛♦ ợ t u H 1 Rn1 ì R+∗ ∩ C 0 Rn−1 × R+ ✱ t❛ ❝â✳
✐✐✮ ●✐↔ sû Ω ❧➔ ♠ët t➟♣ ♠ð tr♦♥❣ Rn s❛♦ tử st t
tỗ t t ♠ët →♥❤ ①↕ t✉②➳♥ t➼♥❤ ❧✐➯♥ tö❝✿
γ : H 1 (Ω) → L2 (∂Ω)

s❛♦ ❝❤♦ ✈ỵ✐ ❜➜t ❦➻ u ∈ H 1 (Ω) ∩ C 0 Ω¯ t❛ ❝â γ (u) = u|∂Ω✳
❍➔♠ γ (u) ✤÷đ❝ ❣å✐ ❧➔ ✈➳t ❝õ❛ u tr➯♥ ∂Ω✳
✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✼✳ ●✐↔ sû ❜✐➯♥ ∂Ω ❧➔ ❧✐➯♥ tư❝ ▲✐♣s❝❤✐t③✱ ❦❤ỉ♥❣ ❣✐❛♥
1

H 2 (∂Ω) ✤÷đ❝ ❣å✐ ❧➔ ♠✐➲♥ ❣✐→ trà ❝õ❛ →♥❤ ①↕ ✈➳t γ ✱ tù❝ ❧➔✿
1

H 2 (∂Ω) = γ H 1 (Ω) .

●✐↔ sû ∂Ω ❧➔ ❧✐➯♥ tư❝ ▲✐♣s❝❤✐t③ t❤➻✿
(∂Ω) ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✈ỵ✐ ❝❤✉➞♥✿

✣à♥❤ ❧➼ ✶✳✶✳✽✳

✐✮ H

1
2

u


2

1/
H 2 (∂Ω)

2

|u (x)| dsx +

=
∂Ω

∂Ω ∂Ω

|u (x) − u (y)|2
dsx dsy .
|x − y|n+1

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ỗ t ởt số C () s
(u)

Cγ (Ω) u

1

H 2 (∂Ω)


H 1 (Ω) , ∀u

∈ H 1 (Ω)

❑❤✐ ✤â Cγ (Ω) ✤÷đ❝ ❣å✐ ❧➔ ❤➡♥❣ sè ✈➳t✳
❇ê ✤➲ ✶✳✶✳✾✳ ●✐↔ sû ∂Ω ❧➔ ❧✐➯♥ tö❝ ▲✐♣s❝❤✐t③✱ ❦❤æ♥❣ ❣✐❛♥H 2 (∂Ω) ❝â ❝→❝
1

t➼♥❤ ❝❤➜t s❛✉✿
✐✮ ❚➟♣ {u|∂Ω, u ∈ C ∞ (Rn)} ❧➔ trò ♠➟t tr♦♥❣ H
✐✐✮ ú H () L2 ()
ỗ t ởt ①↕ t✉②➳♥ t➼♥❤ ❧✐➯♥ tö❝✿

1
2

(∂Ω)✳

1
2

1

g ∈ H 2 (∂Ω) → ug ∈ H 1 (Ω) .

❱ỵ✐ γ (ug ) = g tỗ t ởt số C1 () ♣❤ö t❤✉ë❝ ♠✐➲♥ Ω s❛♦
❝❤♦✿
ug


✶✳✶✳✺

H 1 (Ω)

C1 (Ω) g

1

1
H 2 (∂Ω)

, ∀g ∈ H 2 (Ω) .

❈æ♥❣ t❤ù❝ ●r❡❡♥✱ ❜➜t ✤➥♥❣ t❤ù❝ P♦✐♥❝❛r❡

✣à♥❤ ❧➼ ✶✳✶✳✶✵✳ ✭❈æ♥❣ t❤ù❝ ●r❡❡♥✮✳ ●✐↔ sû ∂Ω ❧➔ ❧✐➯♥ tö❝ ▲✐♣s❝❤✐t③✱ ❝❤♦
u, v ∈ H 1 (Ω)
u

❦❤✐ ✤â✿

∂u
dx = −
∂xi



v

∂u

dx +
∂xi



γ (u) γ (v) ni ds, 1

i

n,

∂Ω

tr♦♥❣ ✤â n = (n1, ..., nn) ❧➔ ✈❡❝tì ♣❤→♣ t✉②➳♥ ♥❣♦➔✐ ❝õ❛ Ω ✳
❚➼♥❤ ❝❤➜t ✶✳✶✳✶✶✳

●✐↔ sû ❜✐➯♥ ∂Ω ❧➔ ❧✐➯♥ tö❝ ▲✐♣s❝❤✐t③✳ ❑❤✐ ✤â✿

H01 (Ω) = u|u ∈ H 1 (Ω) , γ (u) = 0 .

❚➼♥❤ ❝❤➜t t tự Pr ỗ t ởt sè CΩ

s❛♦ ❝❤♦✿

u

L2 (Ω)

CΩ ∇u


L2 (Ω) , ∀u

∈ H01 (Ω) .

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✶✵
❚r♦♥❣ ✤â ❤➡♥❣ sè CΩ ♣❤ư t❤✉ë❝ ✈➔♦ ✤÷í♥❣ ❦➼♥❤ ❝õ❛ Ω ✤÷đ❝ ❣å✐ ❧➔ ❤➡♥❣
sè P♦✐♥❝❛r❡✳ ❇➜t ✤➥♥❣ t❤ù❝ Pr õ ỵ r u = u L ()
♠ët ❝❤✉➞♥ tr➯♥ H 1 (Ω) ✤➣ ①→❝ ✤à♥❤✳
2

✣à♥❤ ❧➼ ✶✳✶✳✶✸✳ ✭❇➜t ✤➥♥❣ t❤ù❝ P♦✐♥❝❛r❡ ♠ð rë♥❣✮ ●✐↔ sû ❜✐➯♥ ∂Ω ❧✐➯♥

tö❝ ▲✐♣s❝❤✐t③✱ ∂Ω = Γ1 ∪ Γ2✱ tr♦♥❣ ✤â Γ1, Γ2 ❧➔ ❝→❝ t➟♣ ✤â♥❣✱ rí✐ ♥❤❛✉✱
Γ1 ❝â ✤ë ữỡ õ tỗ t số C () s❛♦ ❝❤♦ ✿
u
∀u ∈ H 1 (Ω) , γ (u) = 0

✶✳✶✳✻

CΩ ∇u

L2 (Ω)

L2 (Ω) ,

tr➯♥ Γ1.


❑❤ỉ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈ ✈ỵ✐ ❝❤➾ sè ➙♠ H −1 (Ω) ✈➔ H − 2 (∂Ω)
1

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✹✳ ❚❛ ❦➼ ❤✐➺✉ H −1 (Ω) ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✤÷đ❝
①→❝ ✤à♥❤ ❜ð✐✿

H −1 (Ω) = H01 (Ω) ,
✈ỵ✐ ❝❤✉➞♥✿

F, u
F
❚r♦♥❣ ✤â F, u

H −1 (Ω)

=

sup

u

H01 (Ω)\{0}

H −1 (Ω),H01 (Ω)

H −1 (Ω),H01 (Ω)

.

H01 (Ω)


❧➔ t➼❝❤ ♥➠♥❣ ❧÷đ♥❣ tr➯♥ ❝➦♣ ❦❤ỉ♥❣ ❣✐❛♥ ✤è✐

♥❣➝✉✳

❇ê ✤➲ ✶✳✶✳✶✺✳ ❈❤♦ F H 1 () t tỗ t n + 1 ❤➔♠ f0 , f1 , ..., fn tr♦♥❣
L2 (Ω)

s❛♦ ❝❤♦✿

n

F = f0 +

ố ỗ tớ
F

2
H 1 (Ω)

i=1

= ✐♥❢

∂fi
.
∂xi

n


fi

2
L2 (Ω) ,

i=0

tr♦♥❣ ✤â ✐♥❢ ❧➜② tr➯♥ t➜t ❝↔ ❝→❝ ✈❡❝tì (f0, f1, ..., fn) tr♦♥❣

L2 (Ω)

n+1



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✶✶
✶✳✷ P❤÷ì♥❣ tr➻♥❤ ❊❧❧✐♣t✐❝
●✐↔ sû Ω ∈ Rn ❧➔ ♠✐➲♥ ợ ở ợ = t ữỡ tr ✤↕♦
❤➔♠ r✐➯♥❣ t✉②➳♥ t➼♥❤ ❝➜♣ 2m ❝õ❛ ➞♥ ❤➔♠ u (x) , x ∈ Ω

aα (x) Dα u = f (x).

Au =

✭✶✳✶✮

|α| 2m


❚r♦♥❣ ✤â aα (x) , f (x) ❧➔ trữợ A ởt t tỷ
t t t õ
ợ t ữỡ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣ ❝➜♣ ❤❛✐✳
✐✐✮ ❱ỵ✐ ♠❂✷ t❤➻ ✭✶✳✶✮ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣ ❝➜♣ ❜è♥✳
❇➔✐ t♦→♥ t➻♠ ♥❣❤✐➺♠ ❝õ❛ ✭✶✳✶✮✤÷đ❝ ❣å✐ ❧➔ ❜➔✐ t♦→♥ ❜✐➯♥ ♥➳✉ tr➯♥ ❜✐➯♥ Γ
♥❣❤✐➺♠ u(x) t❤ä❛ ♠➣♥ ♠ët sè ✤✐➲✉ ❦✐➺♥ ❜✐➯♥✿

Bi (u) = gi , i = 0, 1, ..., m − 1.
❚r♦♥❣ ✤â Bi (u) , i = 0, 1, ..., m − 1 ❧➔ ❝→❝ t♦→♥ tû ❜✐➯♥✳

✶✳✷✳✶

❑❤→✐ ♥✐➺♠ ♥❣❤✐➺♠ ②➳✉ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤

❳➨t ♣❤÷ì♥❣ tr➻♥❤✿



✭✶✳✷✮

u = f.

●✐↔ sû u ∈ C 2 (Ω) , f ∈ C (Ω) ✈➔ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✷✮ t❤ä❛ ♠➣♥ tr♦♥❣
♠✐➲♥ Ω✳ ❑❤✐ ✤â✱ u(x) ✤÷đ❝ ❣å✐ ❧➔ ♥❣❤✐➺♠ ❝ê ✤✐➸♥ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✷✮✳
▲➜② ❤➔♠ ϕ ❜➜t ❦➻ t❤✉ë❝ D (Ω) = C0∞ () ợ ừ rỗ
t t❛ ✤÷đ❝✿




uϕdx =


f ϕdx.

✭✶✳✸✮



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✶✷
⑩♣ ❞ư♥❣ ❝ỉ♥❣ t❤ù❝ ●r❡❡♥ ✈➔♦ ✭✶✳✸✮ ✈➔ ❦➳t ❤đ♣ ✈ỵ✐ ✤✐➲♥ ❦✐➺♥ ϕ|∂Ω = 0 t❛
❝â ✿
n



i=1

∂ϕ ∂u
dx =
∂xi ∂xi

f ϕdx

✭✶✳✹✮




❤❛②

∇u∇f dx =


f ϕdx.


◆❤÷ ✈➟②✱ ♥➳✉ u ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✷✮ t❤➻ ❝â ✭✶✳✹✮✳ ◆❤÷♥❣

¯ C (Ω) t❤➻ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✷✮ ❦❤ỉ♥❣ ❝â ♥❣❤✐➺♠ ❝ê ✤✐➸♥✳ ❱➟②✱ t❛
♥➳✉ f ∈
❝➛♥ ♠ð rë♥❣ ❦❤→✐ ♥✐➺♠ ❦❤✐ f ∈ L2 (Ω)✳

✶✳✷✳✷

✣à♥❤ ♥❣❤➽❛

✣à♥❤ ♥❣❤➽❛
●✐↔ sû u ∈ H 1 (Ω) , f ∈ L2 (Ω) , u ✤÷đ❝ ❣å✐ ❧➔ ♥❣❤✐➺♠ ②➳✉ ❝õ❛ ♣❤÷ì♥❣
tr➻♥❤ ✭✶✳✶✮ ♥➳✉ ✭✶✳✸✮ ✤÷đ❝ t❤ä❛ ♠➣♥✳

✶✳✷✳✸

▼➺♥❤ ✤➲

▼➺♥❤ ✤➲
◆➳✉ u ❧➔ ♥❣❤✐➺♠ ②➳✉ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✷✮ ✈➔ u ∈ C 2 (Ω) , f ∈ C (Ω)
t❤➻ u ❧➔ ♥❣❤✐➺♠ ❝ê ✤✐➸♥✱ tù❝ ❧➔ −


u=f ✳

✶✳✸ tự sỡ ỗ ỡ


ữủ ỗ ❧➦♣ ❤❛✐ ❧ỵ♣

❳➨t ❜➔✐ t♦→♥✿

Ay = f

✭✶✳✺✮

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✶✸
tr♦♥❣ ✤â A : H → H ❧➔ t♦→♥ tû t✉②➳♥ t➼♥❤ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt
t❤ü❝ ❤ú✉ ❤↕♥ ❝❤✐➲✉ H ✳ ●✐↔ sû A ❧➔ t♦→♥ tû ✤è✐ ①ù♥❣✱ ①→❝ ữỡ

f H tỡ tũ ỵ
r ộ ữỡ ♣❤→♣ ❧➦♣✱ ①✉➜t ♣❤→t tø y0 ❜➜t ❦➻ t❤✉ë❝ H ✱ ♥❣÷í✐
t❛ ✤÷❛ r❛ ❝→❝❤ ①→❝ ✤à♥❤ ♥❣❤✐➺♠ ①➜♣ ①➾ y1, y2 , ..., yk , ... ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤
✭✶✳✺✮✳ ❈→❝ ①➜♣ ①➾ ♥❤÷ ✈➟② ✤÷đ❝ ❜✐➳t ♥❤÷ ❧➔ ❝→❝ ❝➦♣ ❣✐→ trà ❧➦♣ ✈ỵ✐ ❝❤➾ sè
❧➦♣ k = 1, 2, ...✱ ❜↔♥ ❝❤➜t ❝õ❛ ♥❤ú♥❣ ♣❤÷ì♥❣ ♣❤→♣ ♥➔② ❧➔ ❣✐→ trà yk+1 ❝â
t❤➸ ✤÷đ❝ t➼♥❤ t❤ỉ♥❣ q✉❛ ❝→❝ ❣✐→ tr trữợ yk , yk1 , ...
Pữỡ ữủ ồ ữỡ ởt ữợ
ữợ ♥➳✉ ①➜♣ ①➾ yk+1 ❝â t❤➸ ✤÷đ❝ t➼♥❤ t❤ỉ♥❣ q✉❛ ởt tr
trữợ õ t ừ ữủ ỗ ợ


Bk

yk+1 yk
+ Ayk = f, k = 0, 1, 2, ...
k+1



ữủ ỗ ❝❤♦ t❛ ①➜♣ ①➾ ❝❤➼♥❤ ①→❝ ♥❣❤✐➺♠ y ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤
✭✶✳✺✮ ✈ỵ✐ ❜➜t ❦➻ t♦→♥ tû Bk ✈➔ ❝→❝❤ ❝❤å♥ t số k+1
Bk = E t ữủ ỗ ữủ ồ ữủ ỗ

yk+1 yk
+ Ayk = f, k = 0, 1, 2, ...
θk+1

✭✶✳✼✮

❚r♦♥❣ tr÷í♥❣ ❤đ♣ θk = θ ❧➔ ❤➡♥❣ sè t❤➻ ❧÷đ❝ ỗ ỏ ồ ữủ
ỗ ỡ
Bk = E t ữủ ỗ ữủ ồ ữủ ỗ



ữủ ỗ ứ ỵ ỡ sỹ ở tử ừ ữỡ


ữủ ỗ ✭✶✳✻✮ ✈ỵ✐ t♦→♥ tû Bk = B ✱ t❤❛♠ sè θk+1 = θ ❦❤æ♥❣ ✤ê✐


(k = 0, 1, 2, ...) ỏ ữủ ồ ữủ ỗ ứ õ
B

yk+1 − yk
+ Ayk = f, k = 0, 1, 2...
θ

✭✶✳✽✮

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✶✹
✣à♥❤ ❧➼ ✶✳✸✳✶✳

◆➳✉ A ❧➔ t♦→♥ tû ✤è✐ ①ù♥❣ ✱ ①→❝ ✤à♥❤ ❞÷ì♥❣ t❤➻✿

1
B > θA
2

❤❛②

1
(Bx, x) > θ (❆x, x) , ∀x ∈ H,
2

✭✶✳✾✮

❧➔ ✤✐➲✉ ❦✐➺♥ ✤õ ❝❤♦ sü ở tử ừ ữủ ỗ tr ổ HA ✈ỵ✐

ρ < 1 tè❝ ✤ë ❤ë✐ tư ❝➜♣ sè ♥❤➙♥✳
zk+1

A

ρ zk

A, k

= 0, 1, 2, ....



ỵ tt s


Pữỡ ữợ

ữợ s t t

u = f, x ∈ Ω,
 u = g,
x ∈ ∂Ω.

tr♦♥❣ ✤â Ω = (x, y) ∈ R2 , a

x

b, c


y

✭✶✳✶✶✮

d ✱ ❝❤å♥ ✷ sè ♥❣✉②➯♥ N >

1 ✈➔ M > 1✱ t h = (ba)/N ồ ữợ ữợ t x k = (dc)/M ồ
ữợ ữợ t y ✣➦t xi = a + ih, yj = c + jk, i = ✵✱.., N, j = ✵✱.., M.
▼é✐ ✤✐➸♠ (xi , yj ) ồ ởt út ữợ ỵ ❤✐➺✉ ❧➔ ♥ót (i, j)✳ ❚➟♣ t➜t ❝↔ ❝→❝
♥ót tr♦♥❣ ỵ hk út tr ❣å✐ ❧➔ ♥ót ❜✐➯♥✱ t➟♣ t➜t ❝↔
¯ hk = Ωhk hk ồ ởt ữợ s
út ỵ hk t


tr

ữợ ộ số t út ừ ữợ ồ ởt
ữợ tr ừ ữợ u(x, y) t út ữợ (i, j) t tt uij ộ
t r ữợ u
u(x, y) ①→❝ ✤à♥❤ t↕✐ ♠å✐ u(x, y) ∈ Ω

uij ✳

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✶✺
✶✳✹✳✷

❇➔✐ t♦→♥ s❛✐ ♣❤➙♥


¯ = Ω ∪ Γ✱ ❳➨t ❜➔✐ t♦→♥ Lu = f ✱ ❣✐↔ sû ❜➔✐ t♦→♥ ❝â ♥❣❤✐➺♠
❑➼ ❤✐➺✉ Ω
¯ ✈➔ ❣✐↔ sû
u ∈ C 4 (Ω)
∂ 4u
(x, y)
max
¯ ∂x4
(x,y)∈Ω

∂ 4u
(x, y)
C1 = const, max
¯ ∂y 4
(x,y)∈Ω

C2 = const ✭✶✳✶✷✮

❉♦ ✤â t❤❡♦ ❝æ♥❣ t❤ù❝ ❚❛②❧♦r t❛ ❝â✿

u(xi+1 , yj ) = u(xi + h, yj )
∂u h2 ∂ 2 u h3 ∂ 3 u
+

+ O(h4 ),
= u(xi , yj ) − h
2
3
∂x 2! ∂x

3! ∂x
❤❛②

u(xi+1 , yj ) − 2u(xi , yj ) + u(xi−1 , yj ) ∂ 2 u
= 2 + O(h2 ).
2
h
∂x
❚÷ì♥❣ tü t❛ ❝â✿
u(xi , yj+1 ) = u(xi , yj + k) = u(xi , yj ) + k

∂u k 2 ∂ 2 u k 3 ∂ 3 u
+
+
+ O(k 4 ),
2
3
∂y
2! ∂y
3! ∂y

∂u k 2 ∂ 2 u k 3 ∂ 3 u
u(xi , yj−1 ) = u(xi , yj − k) = u(xi , yj ) − k
+

+ O(k 4 ).
2
3
∂y
2! ∂y

3! ∂y
❉♦ ✤â✿

u(xi , yj+1 ) − 2u(xi , yj ) + u(xi , yj−1 ) ∂ 2 u
= 2 + O(k 2 ).
2
k
∂y
❱➟② t❛ ❝â✿

u(xi+1 , yj ) − 2u(xi , yj ) + u(xi−1 , yj ) u(xi , yj+1 ) − 2u(xi , yj ) + u(xi , yj−1 )
+
h2
k2
= ∆u + O(h2 + k 2 ).
✣➦t✿ ∆hk u ≡

ui+1,j −✷ui,j +ui−1,j
h2

+

ui,j+1 −2ui,j +ui,j−1
.
k2

❑❤✐ ✤â ❝❤ù♥❣ tä✿

∆kh u = ∆u + O(h2 + k 2 ).
❙è ❤↕♥❣ O h2 + k 2 ❧➔ ♠ët ✈ỉ ❝ị♥❣ ❜➨ ❜➟❝ ❤❛✐✳ ❚❛ ♥â✐ t♦→♥ tû ∆kh ①➜♣ ①➾

t♦→♥ tû ∆ ✱ ✤✐➲✉ ✤â ❝❤♦ ♣❤➨♣ ∆ t❤❛② ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❜➡♥❣ ♣❤÷ì♥❣

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✶✻
tr➻♥❤ s❛✐ ♣❤➙♥✿ ∆hk u = fij ,

fij = f (xi , yj ),

(xi , yj ) ∈ Ωhk . tù❝ ❧➔✿

ui+1,j − 2ui,j + ui−1j ui,j+1 − 2ui,j + ui,j−1
+
= f (xi , yj ), (xi , yj ) hk .
h2
k2

ỗ tớ t ❦✐➺♥✿

uij = g(xi , yj ),

(xi , yj ) ∈ Γhk .

✭✶✳✶✹✮

❚❛ ✤÷đ❝ ❜➔✐ t♦→♥ s❛✐ ♣❤➙♥ ❤♦➔♥ ❝❤➾♥❤✿ ❚➻♠ ữợ u t út (i, j)
t ữỡ tr s ợ ◆❤÷
✈➟② ✈✐➺❝ t➻♠ ♥❣❤✐➺♠ ①➜♣ ①➾ ❝õ❛ ❜➔✐ t♦→♥ ✈✐ ợ ở
ữủ ữ ✈✐➺❝ ❣✐↔✐ ❜➔✐ t♦→♥ s❛✐ ♣❤➙♥ ✭✶✳✶✸✮ ✈ỵ✐ ✤✐➲✉ ❦✐➺♥

✭✶✳✶✹✮ ❜➡♥❣ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ✤↕✐ sè✳

✶✳✹✳✸

❑➳t ❧✉➟♥

◆ë✐ ❞✉♥❣ ❝❤÷ì♥❣ ✶ ✤➣ tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ✈➲ ổ
ữỡ tr t ỵ tt sỡ ỗ ữỡ
s tự tr➯♥ ✤➣ ✤÷đ❝ t❤❛♠ ❦❤↔♦ tø ❝→❝ t➔✐ ❧✐➺✉ ❬✻❪✱
❬✼❪✱ ❬✽❪✳

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✶✼

❈❤÷ì♥❣ ✷
❇➔✐ t♦→♥ st✐❝❦✲s❧✐♣ ✈➔ ♣❤÷ì♥❣ ♣❤→♣
t➻♠ ♥❣❤✐➺♠ ❞↕♥❣ t✐➺♠ ❝➟♥
✷✳✶ ▼ỉ ❤➻♥❤ ❜➔✐ t♦→♥
❳➨t ♠ỉ ❤➻♥❤ ❜➔✐ t♦→♥ tr÷đt ❝õ❛ t➜♠ tr♦♥❣ ♠ỉ✐ tr÷í♥❣ ❝❤➜t ❧ä♥❣✱ ❞↕♥❣
❤➻♥❤ ❤å❝ ❝õ❛ ❞á♥❣ ❝❤↔② ✤÷đ❝ ♠ỉ t↔ tr♦♥❣ ❤➻♥❤ ✶ ❤♦➦❝ ❤➻♥❤ ✷✳ ❉♦ t➼♥❤
✤è✐ ①ù♥❣✱ ❝❤➾ ❝â ♥û❛ tr➯♥ ❝õ❛ ♠✐➲♥ ỏ ữủ t tự
ợ SD ✳ P❤➛♥ ❜✐➯♥ SA ✈➔ SE ✤↕✐ ❞✐➺♥ ❝❤♦ ❝→❝ ❜ù❝ t÷í♥❣ ✈➔ ❝→❝ ❜➲
♠➦t ♣❤➥♥❣ t÷ì♥❣ ù♥❣ SC ✈➔ SE t÷ì♥❣ ù♥❣ ❧➔ ❝→❝ ❜✐➯♥ tü ❞♦✳
❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔②✱ ♠ỉ ❤➻♥❤ tr➯♥ ✤÷đ❝ ♠ỉ t↔ ❜ð✐ ♣❤÷ì♥❣ tr➻♥❤
s♦♥❣ ✤✐➲✉ ❤á❛

∇4 ψ = 0 tr♦♥❣ Ω,


✭✷✳✶✮

✈ỵ✐ ψ ❧➔ ❤➔♠ ❞á♥❣ ❝❤↔② ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐

ux ≡

∂ψ
∂ψ
✈➔ uy ≡ − ,
∂y
∂x

✭✷✳✷✮

ux ✈➔ uy ❧➔ ❝→❝ t❤➔♥❤ ♣❤➛♥ ✈➟♥ tè❝ t❤❡♦ ữợ x y tữỡ ự
t t t ỵ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❜✐➯♥ ❝õ❛ ❜➔✐ t♦→♥ t÷ì♥❣ ù♥❣ ✤÷đ❝
♠ỉ t↔ tr♦♥❣ ❝→❝ ❤➻♥❤ ✭✷✳✶✮ ❤♦➦❝ ❤➻♥❤ ✭✷✳✷✮ ✳

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t ữợ ỏ ❝❤↔② ψ

❍➻♥❤ ✷✳✷✿ ❇➔✐ t♦→♥ ❣✐→♥ ✤♦↕♥ ♣❤➥♥❣ ❜à ❜✐➳♥ ờ ữợ u = 1
sỷ ❞ö♥❣ ❝→❝ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ψ = u + 1 ✱ ❜➔✐ t♦→♥ tr♦♥❣ ❤➻♥❤
✭✷✳✶✮ ✤÷đ❝ ♠ỉ ❤➻♥❤ ❤â❛ ♥❤÷ s❛✉✿

∇4 u = 0 tr♦♥❣ Ω,


✭✷✳✸✮

∂u
= 0 tr➯♥ SA ,
∂y

✭✷✳✹✮

✈ỵ✐ ❤➺ ✤✐➲✉ ❦✐➺♥ ❜✐➯♥

u = 0,

u = 0, ∇2 u = 0 tr➯♥ SB ,
∂∇2 u
∂u
= 0,
= 0 tr➯♥ SC ,
∂x
∂x
u = −1, ∇2 u = 0 tr➯♥ SD ,
1
∂u
u = y(3 − y 2 ) − 1,
tr➯♥ SD .
2
∂x
❇➔✐ t♦→♥ ✤÷đ❝ ♠ỉ t↔ tr♦♥❣ ❤➻♥❤ ✭✷✳✷✮ ❤♦➔♥ t♦➔♥ t÷ì♥❣ tü ❝ơ♥❣ ✤÷đ❝ ✤÷❛
✈➲ ♠ỉ ❤➻♥❤ t♦→♥ ❤å❝ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ s♦♥❣ ✤✐➲✉ ❤á❛ ✈ỵ✐ ❤➺ ✤✐➲✉ ❦✐➺♥


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