Bài toán stick slip và một số phương pháp tìm nghiệm gần đúng

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

◆●❯❨➍◆ ❚❍➚ ❑❍❯❨➊◆

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

▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈

❚❤→✐ ◆❣✉②➯♥ ✲ ◆➠♠ ✷✵✶✺


✣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
ễ

◆➠♠ ✷✵✶✺

ừ trữ ổ


ừ ữớ ữợ ❦❤♦❛ ❤å❝

❚❙✳ ❱ô ❱✐♥❤ ◗✉❛♥❣


✐✐

▲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➯♥


✐✐✐

▼ư❝ ❧ư❝
▲í✐ ❝↔♠ ì♥

✐✐

▼ư❝ ❧ư❝

✐✐✐

▼ð ✤➛✉

✶

▼ët sè ❦➼ ❤✐➺✉ ✈✐➳t t➢t

✹

✶ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚

✺

✶✳✶ ❑❤æ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺
✶✳✶✳✶ ❑❤æ♥❣ ❣✐❛♥ C k Ω¯ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺
✶✳✶✳✷ ❑❤æ♥❣ ❣✐❛♥ Lp (Ω) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻
✶✳✶✳✸ ❑❤æ♥❣ ❣✐❛♥ ❲ 1,p (Ω) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻

✶✳✶✳✹ ❑❤æ♥❣ ❣✐❛♥ H01 (Ω) ✈➔ ❦❤→✐ ♥✐➺♠ ✈➳t ❝õ❛ ❤➔♠ ✳ ✳ ✼
✶✳✶✳✺ ❈æ♥❣ t❤ù❝ ●r❡❡♥✱ ❜➜t ✤➥♥❣ t❤ù❝ P♦✐♥❝❛r❡ ✳ ✳ ✳ ✳ ✾
✶✳✶✳✻ ❑❤ỉ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈ ✈ỵ✐ ❝❤➾ sè ➙♠ H −1 (Ω) ✈➔ H − (∂Ω) ✶✵
✶✳✷ P❤÷ì♥❣ tr➻♥❤ ❊❧❧✐♣t✐❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶
✶✳✷✳✶ ❑❤→✐ ♥✐➺♠ ♥❣❤✐➺♠ ②➳✉ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✳ ✳ ✳ ✳ ✳ ✶✶
✶✳✷✳✷ ✣à♥❤ ♥❣❤➽❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷
1
2


✐✈

✶✳✷✳✸

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

✶✷

✶✳✸ ❑✐➳♥ t❤ù❝ ✈➲ ❝→❝ sì ỗ ỡ





ữủ ỗ ❧ỵ♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳



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 tờ qt
ỡ s ỵ tt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✼
✷✼

✸✳✶✳✶

❈ì sð ♣❤÷ì♥❣ ♣❤→♣ ❝❤✐❛ ♠✐➲♥ ✳ ✳ ✳

ỡ ỗ ❝õ❛ t♦→♥ tû ❜✐➯♥ ♠✐➲♥ ✳ ✳ ✳ ✳ ✳



ỡ ỗ t ủ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✷

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

✸✹

❑➳t ❧✉➟♥

✸✼


✈

❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦

✸✽

P❍❺◆ P❍Ö ▲Ö❈

✹✵

✶

▼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


✷

♣❤→♣ ♣❤➙♥ 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


✸

✤➸ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳


✹

▼ët sè ❦➼ ❤✐➺✉ ✈✐➳t t➢t
L
Rn

Ω
∂Ω
C k (Ω)
L2 (Ω)
W 1,p (Ω)
H 1/2 (∂Ω)
H01 (Ω)
H −1 (Ω)
H −1/2 (∂Ω)
SF BIM
∆
∇
Dα u

❚♦→♥ tû ❡❧❧✐♣t✐❝ ✳
❑❤ỉ♥❣ ❣✐❛♥ ❊✉❝❧✐❞ n ❝❤✐➲✉✳
▼✐➲♥ ❣✐ỵ✐ ♥ë✐ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ Rn✳
❇✐➯♥ trì♥ ▲✐♣s❝❤✐t③✳
❑❤ỉ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ❝â ✤↕♦ ❤➔♠ ❝➜♣ k ❧✐➯♥ tư❝✳
❑❤ỉ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ✤♦ ✤÷đ❝ ữỡ t
ổ ợ số p

ổ ❣✐❛♥ ❙♦❜♦❧❡✈ ❝→❝ ❤➔♠ ❝â ✈➳t ❜➡♥❣ ❦❤æ♥❣ tr➯♥ ∂Ω ✳
❑❤æ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ❝â ✈➳t ❜➡♥❣ ❦❤æ♥❣ tr➯♥ ∂Ω ✳
❑❤ỉ♥❣ ❣✐❛♥ ✤è✐ ♥❣➝✉ ✈ỵ✐ H01(Ω)✳
❑❤ỉ♥❣ ❣✐❛♥ ✤è✐ ♥❣➝✉ ✈ỵ✐ H 1/2(∂Ω)✳
P❤÷ì♥❣ ♣❤→♣ t➼❝❤ ♣❤➙♥ ❜✐➯♥✳
❚♦→♥ tû ▲❛♣❧❛❝❡✳
❚♦→♥ tû ●r❛❞✐❡♥t✳
✣↕♦ ❤➔♠ r✐➯♥❣ ❝õ❛ u ❝➜♣ |α|✳


✺

❈❤÷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♥❣ ❣✐❛♥

¯
Ck 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➯♥ ❦❤ỉ♥❣ ➙♠✱ |α| = α1 + α2 + ... + αn ✿
∂ α1 +...+αn u
D u = α1
.
∂x1 ...∂xαnn
α

✻

¯ ❝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 ✤à♥❤ tr➯♥ Ω s❛♦ ❝❤♦✿
|f (x)|p dx < ∞.
Ω

❚r♦♥❣ Lp () 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

=





✶✳✶✳✸ ❑❤æ♥❣ ❣✐❛♥ ❲ 1,p (Ω)

p

✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐✿

|u (x)|p dx
Ω

1
p

.



✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✳ ❈❤♦ Ω ❧➔ ♠ët ♠✐➲♥ tr♦♥❣ Rn✳ ❍➔♠ u(x)✤÷đ❝ ❣å✐ ❧➔
❦❤↔ t➼❝❤ ✤à❛ ♣❤÷ì♥❣ tr♦♥❣ Ω ♥➳✉ u(x) ❧➔ ♠ët ❤➔♠ tr♦♥❣ ợ ộ
x0 tỗ t ởt ❧➙♥ ❝➟♥ ω ❝õ❛ x0 ✤➸ u(x) ❦❤↔ t➼❝❤ tr♦♥❣ ω ✳

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

✼

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

Ω

∂kϕ
k
u
dx
=
(−1)
k
∂x1 k1 ...∂xnn

Ω

∂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♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ❦❤↔
✈✐ ổ õ t tr tữỡ ự ợ ❝❤✉➞♥ ❝õ❛ ❲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
❲0

1,p

(Ω) ⊂ Lq (Ω)

❧➔✿

p
t❤➻

✽

✲ ◆❤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 ❞✉② ♥❤➜t ♠ët →♥❤ ①↕ t✉②➳♥ t➼♥❤ ❧✐➯♥ tư❝ ✤÷đ❝ ❣å✐ ❧➔ ✈➳t
∗
γ : H 1 Rn−1 × R+
→ L2 Rn−1

s❛♦ ❝❤♦ ✈ỵ✐ ❜➜t ❦➻ u ∈ H 1 Rn−1 × 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♥❣ ❣✐❛♥
H 2 (∂Ω) ✤÷đ❝ ❣å✐ ❧➔ ♠✐➲♥ ❣✐→ trà ❝õ❛ →♥❤ ①↕ ✈➳t γ ✱ tù❝ ❧➔✿
1

1

H 2 (∂Ω) = γ H 1 (Ω) .

✣à♥❤ ❧➼ ✶✳✶✳✽✳ ●✐↔ sû ∂Ω ❧➔ ❧✐➯♥ tö❝ ▲✐♣s❝❤✐t③ t❤➻✿
✐✮ H

1
2

(∂Ω)
u

❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✈ỵ✐ ❝❤✉➞♥✿

2

1/
H 2 (∂Ω)

2

|u (x)| dsx +

=
∂Ω

∂Ω ∂Ω

|u (x) − u (y)|2

dsx dsy .
|x − y|n+1




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

1
2

(∂Ω) ❝â ❝→❝

t➼♥❤ ❝❤➜t s❛✉✿

1

✐✮ ❚➟♣ {u|∂Ω , u ∈ C ∞ (Rn )} ❧➔ trò ♠➟t tr♦♥❣ H 2 ()
1

ú H 2 () L2 ()
ỗ t↕✐ ♠ët →♥❤ ①↕ t✉②➳♥ t➼♥❤ ❧✐➯♥ tö❝✿
1

g ∈ H 2 (∂Ω) → ug ∈ H 1 (Ω) .
❱ỵ✐ γ (ug ) = g tỗ t ởt số C1 (Ω) ❝❤➾ ♣❤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) ❦❤✐ ✤â✿
u

Ω

∂u
dx = −
∂xi

v
Ω

∂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 số C
s ❝❤♦✿

u

L2 (Ω)

CΩ ∇u

L2 (Ω) , ∀u

∈ H01 (Ω) .


✶✵

❚r♦♥❣ ✤â ❤➡♥❣ sè CΩ ♣❤ư t❤✉ë❝ ✈➔♦ ✤÷í♥❣ ❦➼♥❤ ❝õ❛ Ω ✤÷đ❝ ❣å✐ ❧➔ ❤➡♥❣
sè P♦✐♥❝❛r❡✳ ❇➜t ✤➥♥❣ t❤ù❝ Pr õ ỵ r u = u

L2 ()

ởt ❝❤✉➞♥ tr➯♥ H 1 (Ω) ✤➣ ①→❝ ✤à♥❤✳

✣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 ở ữỡ õ tỗ t số C (Ω) s❛♦ ❝❤♦ ✿
u

CΩ ∇u

L2 (Ω)

L2 (Ω) ,

∀u ∈ H 1 (Ω) , γ (u) = 0 tr➯♥ Γ1 .

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

1
2

(∂Ω)

✣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 +
i=1

fi
.
xi

ố ỗ tớ
n

F

2
H −1 (Ω)

= ✐♥❢

fi

2
L2 (Ω) ,

i=0

tr♦♥❣ ✤â ✐♥❢ ❧➜② tr➯♥ t➜t ❝↔ ❝→❝ ✈❡❝tì (f0 , f1 , ..., fn ) tr♦♥❣ L2 (Ω)

n+1

✳





Pữỡ tr t

sỷ 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 =

Ω

f ϕdx.
Ω

✭✶✳✸✮


✶✷

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

Ω

i=1

∂ϕ ∂u
dx =
∂xi ∂xi

f ϕdx

✭✶✳✹✮

Ω

❤❛②
∇u∇f dx =

Ω

f ϕdx.
Ω

◆❤÷ ✈➟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 ✳

✶✳✸ ❑✐➳♥ t❤ù❝ ✈➲ ❝→❝ sỡ ỗ ỡ

ữủ ỗ ợ
t ❜➔✐ t♦→♥✿
Ay = f

✭✶✳✺✮

✶✸

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

✭✶✳✽✮


✶✹

✣à♥❤ ❧➼ ✶✳✸✳✶✳ ◆➳✉ A ❧➔ t♦→♥ tû ✤è✐ ①ù♥❣ ✱ ①→❝ ✤à♥❤ ❞÷ì♥❣ t❤➻✿
1
1
B > θA ❤❛② (Bx, x) > θ (❆x, x) , ∀x ∈ H,
2
2

✭✶✳✾✮

❧➔ ✤✐➲✉ ❦✐➺♥ ừ sỹ ở tử ừ ữủ ỗ tr ❦❤ỉ♥❣ ❣✐❛♥ HA ✈ỵ✐

ρ < 1 tè❝ ✤ë ❤ë✐ tư số
zk+1



A

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 = (b−a)/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 ) ❣å✐ ởt út ữợ ỵ út (i, j) tt
út tr ỵ hk ◆ót ð tr➯♥ ❜✐➯♥ Γ ❣å✐ ❧➔ ♥ót ❜✐➯♥✱ t➟♣ tt
út ỵ hk t hk = hk hk ồ ởt ữợ s
tr
ữợ ộ số t út ừ ữợ ồ ởt
ữợ tr ừ ữợ u(x, y) t út ữợ (i, j) ✈✐➳t t➢t ❧➔ uij ✳ ▼é✐
❤➔♠ u(x, y) ①→❝ ✤à♥❤ t↕✐ ♠å✐ u(x, y) ∈ Ω¯ t↕♦ r❛ ❤➔♠ ữợ u
uij


✶✺
✶✳✹✳✷

❇➔✐ t♦→♥ s❛✐ ♣❤➙♥

❑➼ ❤✐➺✉ Ω¯ = Ω ∪ Γ✱ ❳➨t ❜➔✐ t♦→♥ Lu = f ✱ ❣✐↔ sû ❜➔✐ t♦→♥ ❝â ♥❣❤✐➺♠
¯ ✈➔ ❣✐↔ sû
u ∈ C 4 (Ω)
∂ 4u
max
(x, y)
¯ ∂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 + k2 ❧➔ ♠ët ✈ỉ ❝ị♥❣ ❜➨ ❜➟❝ ❤❛✐✳ ❚❛ ♥â✐ t♦→♥ tû ∆kh ①➜♣ ①➾
t♦→♥ tû ∆ ✱ ✤✐➲✉ ✤â ❝❤♦ ♣❤➨♣ ∆ t❤❛② ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❜➡♥❣ ♣❤÷ì♥❣


✶✻

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➔✐ ❧✐➺✉ ❬✻❪✱

❬✼❪✱ ❬✽❪✳


✶✼

❈❤÷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❛ ♠✐➲♥ ❞á♥❣ 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 tố t ữợ x y tữỡ ự

t t t ỵ ✤✐➲✉ ❦✐➺♥ ❜✐➯♥ ❝õ❛ ❜➔✐ t♦→♥ t÷ì♥❣ ù♥❣ ✤÷đ❝
♠ỉ t↔ tr♦♥❣ ❝→❝ ❤➻♥❤ ✭✷✳✶✮ ❤♦➦❝ ❤➻♥❤ ✭✷✳✷✮ ✳




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 ,
∂u
1
tr➯♥ SD .
u = y(3 − y 2 ) − 1,
2
∂x

❇➔✐ t♦→♥ ✤÷đ❝ ♠ỉ t↔ tr♦♥❣ ❤➻♥❤ ✭✷✳✷✮ ❤♦➔♥ t♦➔♥ t÷ì♥❣ tü ❝ơ♥❣ ✤÷đ❝ ✤÷❛
✈➲ ♠ỉ ❤➻♥❤ t♦→♥ ❤å❝ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ s♦♥❣ ✤✐➲✉ ❤á❛ ✈ỵ✐ ❤➺ ✤✐➲✉ ❦✐➺♥