Không gian sobolev phụ thuộc thời gian

❇❐ ●■⑩❖ ❉Ư❈ ❱⑨ ✣⑨❖ ❚❸❖ ❚❘×❮◆● ✣❸■ ❍➴❈ ❚❹❨ ❇➁❈ ✣➱ ❱■➌❚ ❨➊◆ ❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P ❑❍➷◆● ●■❆◆ ❙❖❇❖▲❊❱ P❍Ư ❚❍❯❐❈ ❚❍❮■ ●■❆◆ ❈❤✉②➯♥ ♥❣➔♥❤✿ ●✐↔✐ t➼❝❤ ◆❣÷í✐ ữợ ụ rồ ữù ỡ t ♥➠♠ ✷✵✶✸ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ▲í✐ ❝↔♠ ì♥ ❚❤í✐ ❣✐❛♥ trỉ✐ q✉❛ t❤➟t ♥❤❛♥❤✱ ❝❤ỵ♣ ♠➢t ♠➔ ❡♠ ✤➣ ❤♦➔♥ t❤➔♥❤ ❜è♥ ♥➠♠ ✤↕✐ ❤å❝✳ ◆❤ỵ ♥❣➔② ♥➔♦✱ ✤➛✉ ❦❤â❛ ❤å❝ ❜è ♠➭ ❝á♥ ✤÷❛ ✤➳♥ tr÷í♥❣ trữớ ợ ợ t ổ ợ ợ ✈ỵ✐ ❜❛♦ ❜ï ♥❣ï ✈➔ ❧♦ ❧➢♥❣✳ ❱➟② ♠➔ ❝✉è✐ ❝ị♥❣ ❡♠ ❝ơ♥❣ tr↔✐ q✉❛ ❜è♥ ♥➠♠ ❤å❝✳ ❇è♥ ♥➠♠ ❤å❝ t➟♣ ✈ỵ✐ ❜✐➳t ❜❛♦ ❦❤â ❦❤➠♥✱ ✈➜t ✈↔✱ ❝â ♥❤ú♥❣ ✈➜♣ ♥❣➣ ❡♠ t÷ð♥❣ ♥❤÷ ♠➻♥❤ ❦❤ỉ♥❣ t❤➸ ✈÷đt q✉❛✳ ◆❤÷♥❣ ♠♦♥❣ ♠✉è♥ ✤÷đ❝ ❧➔♠ ❦❤â❛ ❧✉➟♥ ❦❤✐ tèt ♥❣❤✐➺♣ ✤➣ t❤ó❝ ✤➞② ❡♠ ♣❤➜♥ ✤➜✉ ♥❤✐➲✉ tr♦♥❣ ❤å❝ t ố ũ ợ t q t ữủ tr ❝→❝ ♥➠♠ ✤➛✉✱ ❡♠ ✤➣ ✤÷đ❝ ✤õ ✤✐➲✉ ❦✐➺♥ ❧➔♠ õ ữợ sỹ ữợ t ụ rồ ▲÷ï♥❣✳ ✣÷đ❝ ❧➔♠ ❦❤â❛ ❧✉➟♥ ❧➔ ♠ët ♥✐➲♠ ✈✉✐✱ ♥✐➲♠ ỹ ợ ố ợ ữ õ ❝ơ♥❣ ❝â ❦❤ỉ♥❣ ➼t ♥é✐ ❧♦ ✈➔ ❦❤ỉ♥❣ ➼t ❦❤â ❦❤➠♥✱ ♥➔♦ ❧➔ ❦❤❛♥ ❤✐➳♠ t➔✐ ❧✐➺✉✱ t❤í✐ ❣✐❛♥ ❤↕♥ tự t ợ tữỡ ố õữ ợ ❦✐➳♥ t❤ù❝ ♠➔ ❡♠ ✤➣ ✤÷đ❝ t❤➛② ❝ỉ ❜ë ♠ỉ♥ tr tr q ũ ợ sỹ ữợ ❞➝♥ ♥❤✐➺t t➻♥❤ ❝õ❛ t❤➛② ❱ơ ❚rå♥❣ ▲÷ï♥❣✱ ❝ơ♥❣ ♥❤÷ sü ✤ë♥❣ ✈✐➯♥✱ ❣✐ó♣ ✤ï ❝õ❛ ❣✐❛ ✤➻♥❤ ✈➔ ❜↕♥ ❜➧ ❝✉è✐ ❝ị♥❣ ❦❤â❛ ❧✉➟♥ ❝ơ♥❣ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤✳ ❊♠ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ ❝❤➙♥ t❤➔♥❤ ♥❤➜t ✤➳♥ t❤➛② ữợ t ổ tr ổ ũ ❣✐❛ ✤➻♥❤ ✈➔ ❜↕♥ ❜➧✳ ❙ì♥ ❧❛✱ t❤→♥❣ ✺ ♥➠♠ ✷✵✶✸ ❙✐♥❤ ✈✐➯♥ ✣é ❱✐➳t ❨➯♥ ✷ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ▼ö❝ ❧ö❝ ▼ð ✤➛✉ ✵✳✶ ỵ õ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✵✳✷ ✣è✐ t÷đ♥❣✱ ♣❤÷ì♥❣ ♣❤→♣✱ ♣❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉ ✳ ✳ ✳ ✳ ✳ ✵✳✷✳✶ ✣è✐ t÷đ♥❣ ♥❣❤✐➯♥ ❝ù✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✵✳✷✳✷ P❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✵✳✷✳✸ P❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✵✳✸ ▼ö❝ ✤➼❝❤✱ ♥❤✐➺♠ ✈ö ✈➔ ♥❤ú♥❣ ✤â♥❣ ❣â♣ ❝õ❛ ❦❤â❛ ❧✉➟♥✳ ✵✳✸✳✶ ▼ö❝ ✤➼❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✵✳✸✳✷ ◆❤✐➺♠ ✈ö ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✵✳✸✳✸ ◆❤ú♥❣ ✤â♥❣ ❣â♣ ❝õ❛ ❦❤â❛ ❧✉➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ▼ët sè ❦✐➳♥ t❤ù❝ ❧✐➯♥ q✉❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶ ❑❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷ ❑❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷✳✶ ❉↕♥❣ ❍❡r♠✐t❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ổ ữợ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✸ ❑❤æ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✸✳✶ ❑❤æ♥❣ ❣✐❛♥ C k (Ω)✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✸✳✷ ❑❤æ♥❣ ❣✐❛♥ Lp(Ω) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✸✳✸ ✣↕♦ ❤➔♠ ②➳✉ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ Wpk (Ω) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✸✳✹ ✣↕♦ ❤➔♠ s✉② rë♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✸✳✺ ❑❤æ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈ Wpk (Ω), (1 ≤ p < ∞), (k ∈ Z+.) ◦ ✶✳✸✳✻ ❑❤æ♥❣ ❣✐❛♥ Wpk (Ω), (1 ≤ p < ∞) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✺ ✺ ✺ ✻ ✻ ✻ ✻ ✻ ✻ ✽ ✽ ✾ ✾ ✶✶ ✶✷ ✶✷ ✶✸ ✶✾ ✷✶ ✷✺ ✸✵ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ✶✳✸✳✼ ❳➜♣ ①➾ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✶✳✸✳✽ ✣à♥❤ ❧➼ t❤→❝ tr✐➸♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✶✳✸✳✾ ❑❤æ♥❣ ❣✐❛♥ ❤➔♠ H −1(Ω) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✷ ❑❤ỉ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈ ♣❤ư t❤✉ë❝ t❤í✐ ❣✐❛♥ ✷✳✶ ✷✳✷ ✷✳✸ ✷✳✹ ❑❤ỉ♥❣ ❣✐❛♥ Lp(0, T ; X) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❑❤æ♥❣ ❣✐❛♥ C([0, T ]; X) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✣↕♦ ❤➔♠ ②➳✉ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ L1(0, T ; X) ❑❤æ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈ Wp1(0, T ; X) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✹✽ ✹✽ ✹✽ ✹✾ ❑➳t ❧✉➟♥ ✺✼ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✺✽ ✹ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ▼ð ✤➛✉ ✵✳✶ ỵ õ Pữỡ tr r r❛ ✤í✐ ✈➔♦ ❦❤♦↔♥❣ t❤➳ ❦➾ t❤ù 17✱ ❞♦ ♥❤✉ ❝➛✉ ❝ì ❤å❝ ✈➔ ❝õ❛ ♥❤✐➲✉ ♥❣❤➔♥❤ ❦❤♦❛ ❤å❝ ❦❤→❝✳ ◆â ♥❣➔② ❝➔♥❣ ❝â ✈❛✐ trá q✉❛♥ trå♥❣ ✈➔ ✤÷đ❝ ù♥❣ ❞ư♥❣ rë♥❣ r➣✐ tr♦♥❣ ❦❤♦❛ ❤å❝ ✈➔ ❝ỉ♥❣ ♥❣❤➺✳ ◆❣➔② ♥❛②✱ ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣ trð t❤➔♥❤ ❜ë ♠ỉ♥ t♦→♥ ❤å❝ ❝ì ❜↔♥ ✈ø❛ ♠❛♥❣ t➼♥❤ ❧➼ t❤✉②➳t ứ t ự rở r rữợ sỹ ♣❤→t tr✐➸♥ ♠↕♥❤ ♠➩ ❝õ❛ ❦❤♦❛ ❤å❝ ✈➔ ❝æ♥❣ ♥❣❤➺✱ ❝❤➢❝ ❝❤➢♥ r➡♥❣ ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣ ❝á♥ ♣❤→t tr✐➸♥ ♠↕♥❤ ♠➩ ❤ì♥ ♥ú❛ tr♦♥❣ t÷ì♥❣ ❧❛✐✱ ♠ð r❛ ♠ët ❝♦♥ ✤÷í♥❣ ❝❤♦ ♥❤ú♥❣ ❛✐ ②➯✉ t❤➼❝❤ ♥❣❤✐➯♥ ❝ù✉ ❚♦→♥ ❤å❝ ù♥❣ ❞ư♥❣✳ P❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣ ✤➣ ✈➔ ✤❛♥❣ ♣❤→t tr✐➸♥ r➜t ♠↕♥❤ ♠➩ ð tr➯♥ t❤➳ ợ ữ ữợ t t ỏ s→❝❤ ♥â✐ ✈➲ ❜ë ♠æ♥ ♥➔②✱ ♥➯♥ ♥â ✈➝♥ ❧➔ ✈➜♥ ✤➲ ❝á♥ ♠ỵ✐ ♠➫✱ ✈➔ ❜➼ ➞♥ ❦➼❝❤ t❤➼❝❤ sü ❦❤→♠ ♣❤→ ❝õ❛ ♥❤ú♥❣ ❛✐ ②➯✉ t❤➼❝❤ ♥â✳ ❍ì♥ ỳ tr q tr t ữủ t ổ ợ t ữợ tổ t rt õ ự tú ✈ỵ✐ ❜ë ♠ỉ♥ ♥➔②✳ ❈❤➼♥❤ ✈➻ ✈➟②✱ ♥❤➡♠ ❣â♣ ♣❤➛♥ ❝❤♦ ♥❤ú♥❣ ❛✐ ②➯✉ t❤➼❝❤ ❜ë ♠æ♥ ♥➔② ♥â✐ ❝❤✉♥❣ ✈➔ ❜↔♥ t❤➙♥ t→❝ ❣✐↔ ♥â✐ r✐➯♥❣ ❤✐➸✉ s➙✉ ❤ì♥ ✈➲ ❜ë ♠æ♥ ♥➔② tæ✐ ♠↕♥❤ ❞↕♥ t➻♠ ❤✐➸✉ ✤➲ t➔✐✿ ✧❑❤ỉ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈ ♣❤ư t❤✉ë❝ t❤í✐ ❣✐❛♥ ✧✳ ✵✳✷ ✣è✐ t÷đ♥❣✱ ♣❤÷ì♥❣ ♣❤→♣✱ ♣❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉ ✵✳✷✳✶ ✣è✐ t÷đ♥❣ ♥❣❤✐➯♥ ❝ù✉ ✣è✐ t÷đ♥❣ ♥❣❤✐➯♥ ❝ù✉ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈ ♣❤ư t❤✉ë❝ t❤í✐ ❣✐❛♥✳ ✺ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ✵✳✷✳✷ P❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉✳ ❱➜♥ ✤➲ ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ tr♦♥❣ ❦❤â❛ ❧✉➟♥ ❧➔ ✈➜♥ ✤➲ ❝á♥ ♠ỵ✐ ♠➫ s♦ ✈ỵ✐ s✐♥❤ ✈✐➯♥ ❜➟❝ ✤↕✐ ❤å❝✱ ✈➻ ✈➟② ♣❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉ sû ❞ư♥❣ ❝❤õ ②➳✉ ❧➔ ♥❣❤✐➯♥ ❝ù✉ ❧➼ t❤✉②➳t ❝ư t❤➸ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ Pữỡ ự ỗ õ sữ t t ❧✐➺✉✱ ✤å❝ ❤✐➸✉ t➔✐ ❧✐➺✉ tr➯♥ ❝ì sð ✤â ♣❤➙♥ t➼❝❤✱ tê♥❣ ❤ñ♣✱ ❞✐➵♥ ❣✐↔✐✱ ❧➔♠ rã ✈➔ tr➻♥❤ ❜➔② t❤➔♥❤ ♠ët ❤➺ t❤è♥❣ ✤➸ ❣✐↔✐ q✉②➳t ❝→❝ ✈➜♥ ✤➲ ✤➦t r❛ ❝õ❛ ❦❤â❛ ❧✉➟♥✳ ✵✳✷✳✸ P❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉ P❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ ❦❤â❛ ❧✉➟♥ ❧➔ ♥❤ú♥❣ ✤à♥❤ ♥❣❤➽❛✱ t➼♥❤ ❝❤➜t✱ ✤à♥❤ ❧➼✱ ✈➔ ❝→❝ ✈➜♥ ✤➲ ❧✐➯♥ q✉❛♥ ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈ ♣❤ư t❤✉ë❝ t❤í✐ ❣✐❛♥✱ ❜❛♦ ỗ ổ Lp(0, T ; X), ổ C([0, T ]; X), ✤↕♦ ❤➔♠ ②➳✉ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ Lp(0, T ; X), ✈➔ ❦❤æ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈ Wp1(0, T ; X)✳ ✵✳✸ ▼ö❝ ✤➼❝❤✱ ♥❤✐➺♠ ✈ö ✈➔ ♥❤ú♥❣ ✤â♥❣ ❣â♣ ❝õ❛ ❦❤â❛ ❧✉➟♥✳ ✵✳✸✳✶ ▼ö❝ ✤➼❝❤ ▼ö❝ ✤➼❝❤ ❝õ❛ ❦❤â❛ ❧✉➟♥ ❧➔ ❧➔♠ rã ❝→❝ ❦❤→✐ ♥✐➺♠✱ t➼♥❤ ❝❤➜t✱ ✤à♥❤ ❧➼✱✳✳✳tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈ ♣❤ư t❤✉ë❝ t❤í✐ ❣✐❛♥✳ ✣â♥❣ ❣â♣ t❤➯♠ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❝❤♦ s✐♥❤ ✈✐➯♥ ✈➔ t➜t ❝↔ ♥❤ú♥❣ ❛✐ ②➯✉ t❤➼❝❤✱ q✉❛♥ t➙♠ ✤➳♥ ❜ë ♠æ♥ ữỡ tr r ợ ✤➼❝❤ ✤➦t r❛ ♥❤✐➺♠ ✈ö ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ ❦❤â❛ ❧✉➟♥ ❧➔ ♥➯✉ r❛ ✈➔ ❝❤ù♥❣ ♠✐♥❤ ❝→❝ ✈➜♥ ✤➲ ❧✐➯♥ q✉❛♥ ✤➳♥ ❦❤ỉ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈ ♣❤ư t❤✉ë❝ t❤í✐ ❣✐❛♥✳ ✵✳✸✳✸ ◆❤ú♥❣ ✤â♥❣ ❣â♣ ❝õ❛ ❦❤â❛ ❧✉➟♥ ✣â♥❣ ❣â♣ ♥ê✐ ❜➟t ❝õ❛ ❦❤â❛ ❧✉➟♥ ❧➔ ❧➔♠ rã r➔♥❣✱ ❝❤✐ t✐➳t ❤ì♥ ❤➺ t❤è♥❣ tr✐ t❤ù❝ ♠ỵ✐✱ ❝❤✉②➯♥ s➙✉ ✈➲ ❜ë ♠ỉ♥ ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣ ❤✐➺♥ ✤↕✐✳ ✣â ✻ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ❧➔ ❝→❝ ❦❤→✐ tự ợ ữ ✤↕♦ ❤➔♠ s✉② rë♥❣✱ ❦❤æ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈ ♥❣♦➔✐ r❛ t❛ ❜✐➳t ❝→❝ t➼♥❤ ❝❤➜t ✈➔ ✈➜♥ ✤➲ ❧✐➯♥ q✉❛♥ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈✱✳✳✳✣➦❝ ❜✐➺t ❦❤â❛ ❧✉➟♥ ❝✉♥❣ ❝➜♣ t❤➯♠ ♠ët ♣❤➛♥ ❦✐➳♥ t❤ù❝ ❝õ❛ ❜ë ♠ỉ♥ ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣ ❤✐➺♥ ✤↕✐✱ ✤â ❧➔ ♥â✐ ✈➲ ❦❤æ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈ ♣❤ư t❤✉ë❝ t❤í✐ ❣✐❛♥ ✈➔ ❝→❝ ✈➜♥ ✤➲ ❧✐➯♥ q✉❛♥✳ ✼ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ❈❤÷ì♥❣ ✶ ▼ët sè ❦✐➳♥ t❤ù❝ ❧✐➯♥ q✉❛♥ ✶✳✶ ❑❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ❈❤♦ E ❧➔ ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ t❤ü❝✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳ ●✐↔ sû E ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì tr➯♥ trữớ ổ ữợ số tỹ R số ự C✳ ❍➔♠ ρ ①→❝ ✤à♥❤ tr➯♥ E ❣å✐ ❧➔ ♠ët ❝❤✉➞♥ tr➯♥ E ♥➳✉ ρ t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ✐✳ ρ(x) ≥ ✈ỵ✐ ∀x ∈ E ✈➔ ρ(x) = t❤➻ x = ✐✐✳ ρ(λx) = |λ| ρ(x) ✐✐✐✳ ρ(x + y) ≤ ρ(x) + ρ(y) ✈ỵ✐ ∀λ ∈ K ✈➔ ∀x ∈ E ✈ỵ✐ ∀x, y ∈ E ❑❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì E ❝ị♥❣ ✈ỵ✐ ♠ët ❝❤✉➞♥ ρ tr➯♥ ♥â ❣å✐ ❧➔ ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ ✤à♥❤ ❝❤✉➞♥ ❤❛② ♥❣➢♥ ❣å♥ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥✳ ▼➺♥❤ ✤➲ ✶✳✶✳ ◆➳✉ ρ ❧➔ ♠ët ❝❤✉➞♥ tr➯♥ E t❤➻ ❝æ♥❣ t❤ù❝ d(x, y) := ρ(x − y), (∀x, y ∈ E) ✭✶✳✶✮ ①→❝ ✤à♥❤ ♠ët ❦❤♦↔♥❣ ❝→❝❤ tr➯♥ E t❤ä❛ ♠➣♥   d(x + y, y + z) = d(x, y)  d(λx, λy) = |λ| d(x, y) ✈ỵ✐ ∀x, y, z ∈ E, ∀λ ∈ K ❑❤♦↔♥❣ ❝→❝❤ d ①→❝ ✤à♥❤ ❜ð✐ ❝ỉ♥❣ t❤ù❝ (1.1) ✤÷đ❝ ❣å✐ ❧➔ ❦❤♦↔♥❣ ❝→❝❤ s✐♥❤ ❜ð✐ ❝❤✉➞♥ ρ ✽ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳ ❑❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ E ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ♠❡tr✐❝ ✈ỵ✐ ❦❤♦↔♥❣ ❝→❝❤ s✐♥❤ ❜ð✐ ❝❤✉➞♥ ①→❝ ✤à♥❤ ❜ð✐ d(x, y) := x − y , ✈ỵ✐ x, y ∈ E ✣à♥❤ ♥❣❤➽❛ ✶✳✸✳ ✐✳ ❉➣② {uk }∞k=1 ⊂ E ữủ tr E ợ ✐✐✳ > 0, ∃N > s❛♦ ❝❤♦ uk − ul < , ✈ỵ✐ k, l ≥ N ❧➔ ✤➛② ✤õ ♥➳✉ ♠é✐ ❞➣② ❈❛✉❝❤② tr♦♥❣ E ✤➲✉ ❤ë✐ tö✱ ❝â ♥❣❤➽❛ ❧➔ ✈ỵ✐ ∞ {uk }∞ k=1 ⊂ E tỗ t u E s {uk }k=1 ❤ë✐ tư ✤➳♥ u E ✐✐✐✳ ❑❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ❧➔ ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ ✤à♥❤ ❝❤✉➞♥ ✤➛② ✤õ✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✹✳ ❑❤ỉ♥❣ ❣✐❛♥ ♠❡tr✐❝ E ✤÷đ❝ ❣å✐ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠❡tr✐❝ ✤➛② ✭❤❛② ✤õ✮ ♥➳✉ ♠å✐ ❞➣② ❈❛✉❝❤② tr♦♥❣ E ✤➲✉ ❤ë✐ tư✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✺✳ ❑❤ỉ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ ✤à♥❤ ❝❤✉➞♥ E ✤÷đ❝ ❣å✐ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ♠❡tr✐❝ ✤➛②✳ ♥➳✉ E ❝ị♥❣ ✈ỵ✐ ♠❡tr✐❝ s✐♥❤ ❜ð✐ ❝❤✉➞♥ tr➯♥ E ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ✣à♥❤ ♥❣❤➽❛ ✶✳✻✳ ❑❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ E ❣å✐ ❧➔ ❦❤↔ ❧② ♥➳✉ E ❝â ♠ët t➟♣ ❝♦♥ ✤➳♠ ✤÷đ❝ trị ♠➟t tr E E tỗ t ởt {xn }n∈N ❝→❝ ♣❤➛♥ tû ❝õ❛ E s❛♦ ❝❤♦ ✈ỵ✐ ♠é✐ x ∈ E ✤➲✉ ❝â ➼t ♥❤➜t ♠ët ❞➣② ❝♦♥ {xn }n∈N ❤ë✐ tö ✤➳♥ x ∗ k ∗ ✣à♥❤ ♥❣❤➽❛ ✶✳✼✳ ❚❛ ♥â✐ r➡♥❣ ❞➣② {uk }∞k=1 ⊂ E ❤ë✐ tö ✤➳♥ u ∈ E ♥➳✉ lim uk − u = k→∞ ❚❛ ❝á♥ ✈✐➳t✱ uk → u ❦❤✐ k → ∞ ✶✳✷ ❑❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt✳ ✶✳✷✳✶ ❉↕♥❣ ❍❡r♠✐t❡✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✽✳ ❈❤♦ E ❧➔ ❦❤æ♥❣ ❣✐❛♥ ✈❡❝tì ✭♣❤ù❝✮✳ ❉↕♥❣ ❍❡r♠✐t❡ tr➯♥ E ❧➔ →♥❤ ①↕ ϕ : E × E −→ C t❤ä❛ ♠➣♥ ✾ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ✈ỵ✐ ❤➡♥❣ sè C ❦❤ỉ♥❣ ♣❤ư t❤✉ë❝ tr➯♥ u✳ ✺✳ ❇➙② ❣✐í t❛ ①➨t tr÷ì♥❣ ❤đ♣ ∂U ❦❤ỉ♥❣ ♣❤➥♥❣ ❣➛♥ x0✳ ❑❤✐ ✤â ✈➻ ∂U ❧➔ C ♥➯♥ t❛ t✐➳♥ ❤➔♥❤ ❧➔♠ ♣❤➥♥❣ ∂U ❣➛♥ x0✳ ❚ù❝ ❧➔ t❛ ①→❝ ✤à♥❤ ✤÷đ❝ tr C ợ ữủ ❧➔ Ψ✱ t❤ä❛ ♠➣♥ Φ ❧➔♠ ♣❤➥♥❣ ∂U ❣➛♥ x0 r❛ ♣❤➼❛ ♥❣♦➔✐✳ ❚❛ ✈✐➳t✿ y = Φ(x), x = Ψ(y), u (y) := u(Ψ(y))✳ ❚✐➳♣ t❤❡♦ t❛ ❧↕✐ ①→❝ ✤à♥❤ ❤➻♥❤ ❝➛✉ By = B(y0, r)✳ ❚÷ì♥❣ tü ♥❤÷ tr t rở ữủ u tứ By+ tợ ❤➔♠ u ①→❝ ✤à♥❤ tr➯♥ t♦➔♥ ❜ë By ♠➔ u ∈ C 1(By ) ❝ơ♥❣ ✤→♥❤ ❣✐→ t÷ì♥❣ tü t❤➻✿ u Wp1 (By ) ≤ Cy u Wp1 (By+ ) ❚r♦♥❣ ✤â✱ Cy ❧➔ ❤➡♥❣ sè ❦❤ỉ♥❣ ♣❤ư t❤✉ë❝ ✈➔♦ u ✳ ❝❤♦ W := Ψ(By )✳ ❙❛✉ ✤â q✉❛② trð ❧↕✐ ❜✐➳♥ x✱ t❛ t❤✉ ✤÷đ❝ ♠ët ❤➔♠ ♠ð rë♥❣ u ❝õ❛ u tr➯♥ W ♠➔ u Wp1 (W ) ≤C u Wp1 (U ) ✭✶✳✶✷✮ ✻✳ ❚ø ∂U t tỗ t ỳ x0i ∂U ✱ Wi ❧➔ ♠ð✱ ✈➔ ui ❧➔ ♠ð rë♥❣ ❝õ❛ u ✤➳♥ Wi (i = 1, N )✱ ♥❤÷ ự trữợ Ni=1 Wi N W0 ⊂⊂ U ❞♦ ✤â U ⊂ N i=1 Wi ✱ ✈➔ t❛ ❣å✐ {ζi }i=0 ❧➔ ♠ët ♣❤➙♥ ❤♦↕❝❤ ✤ì♥ N N tữỡ ự ợ {Wi}i=0 U ✱ t❛ ✈✐➳t u = ζiui✱ tr♦♥❣ ✤â u0 = u✳ i=0 ❙û ❞ö♥❣ (1.12) t❛ ❝â u Wp1 (Rn ) ≤C u Wp1 (U ) ✈ỵ✐ ❤➡♥❣ sè C ❝❤➾ ♣❤ư t❤✉ë❝ ✈➔♦ ✈➔♦ U, p, n ♥❤÷♥❣ ❦❤ỉ♥❣ ♣❤ư t❤✉ë❝ ✈➔♦ u✳ ❍ì♥ ♥ú❛ t❛ ❝â t❤➸ s➢♣ ✤➦t ❝❤♦ ❣✐↔ t❤✐➳t ❝õ❛ u ❧➔ ♥➡♠ tr♦♥❣ V ⊃⊃ U ✼✳ ❇➙② ❣✐í t❛ ①➨t →♥❤ ①↕ E ♠➔ E : Wp1 (U ) −→ Wp1 (Rn ) u −→ Eu = u t❤➻ E ❧➔ t♦→♥ tû t✉②➳♥ t➼♥❤✳ ❇➙② ❣✐í t❛ ①➨t u ∈ Wp1(U )✳ ❉♦ t➼♥❤ trò ♠➟t ❝õ❛ C ∞(U ) ♥➯♥ ổ tỗ t LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ❈❛✉❝❤② ❝→❝ ❤➔♠ {um}∞m=1 ♠➔ um → u tr♦♥❣ Wp1(U ) ❦❤✐ m → ∞ ⑩♣ ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ i ❝ị♥❣ ✈ỵ✐ t➼♥❤ t✉②➳♥ t➼♥❤ ❝õ❛ E t❤➻ t❛ ❝â Eum − Eul Wp1 (Rn ) = E(um − ul ) Wp1 (Rn ) ≤ C um − ul Wp1 (U ) ❇ð✐ {um}∞m=1 ❧➔ ❞➣② ❈❛✉❝❤② ❤ë✐ tö ✈➲ u ❦❤✐ m → ∞✱ ✈➔ E ❧➔ t♦→♥ tû t✉②➳♥ t➼♥❤ ♥➯♥ {Eum}∞m=1 ❝ô♥❣ ❧➔ ❞➣② ❈❛✉❝❤② ❤ë✐ tư tỵ✐ u ✈➔ t❛ ✤➦t✿ Eu := u ❙ü ♠ð rë♥❣ ♥➔② ♥â ❦❤ỉ♥❣ ♣❤ư t❤✉ë❝ ✈➔♦ ✈✐➺❝ ❝❤å♥ ❞➣② ❤➔♠ ①➜♣ ①➾ {um}∞m=1 t❤ä❛ ♠➣♥ ❦➳t ❧✉➟♥ ❝õ❛ ✤à♥❤ ❧➼✳ ✶✳✸✳✾ ❑❤æ♥❣ ❣✐❛♥ ❤➔♠ H −1(Ω) ✣à♥❤ ♥❣❤➽❛ ✶✳✷✷✳ ❑❤æ♥❣ ❣✐❛♥ ✤è✐ ♥❣➝✉ ❝õ❛ H01(Ω) ✤÷đ❝ ❦➼ ❤✐➺✉ ❧➔ H −1(Ω)✳ f ∈ H −1 (Ω) ♥➳✉ f ❧➔ ♠ët ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ ❜à ❝❤➦♥ tr➯♥ H01(Ω)✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✷✸✳ ◆➳✉ f ∈ H −1(Ω)✱ ❝❤✉➞♥ ①→❝ ✤à♥❤ ❜ð✐ f H −1 (Ω) = s✉♣{ f, u |u ∈ H01 (Ω), u H01 (Ω) ≤ 1} ❚❛ ✈✐➳t < f, v > ✤➸ ❦➼ ❤✐➺✉ ❣✐→ trà ❝õ❛ f ∈ H −1(Ω) tr➯♥ u H01() ỵ trú H ✐✳ ●✐↔ sû f ∈ H −1 (Ω) ❑❤✐ ✤â ①✉➜t ❤✐➺♥ ❝→❝ ❤➔♠ f , f , , f n tr♦♥❣ L2 (Ω) s❛♦ ❝❤♦ n f, v = H −1 (Ω) = ✐♥❢ ✭✶✳✶✸✮ i=1 Ω ✐✐✳ ❍ì♥ ♥ú❛✱ f f i vxi dx, (v ∈ H01 (Ω)) f v+ n f i dx | Ω i=0 f t❤ä❛ ♠➣♥ (1) ✈ỵ✐ f , , f n ∈ L2 (Ω) ❚❛ ✈✐➳t✿ f = f − ❈❤ù♥❣ ♠✐♥❤✳ n i i=1 fxi ✶✳ ❱ỵ✐ u, v ∈ H01(Ω) ✤➣ ❝❤♦✱ t❛ ✤à♥❤ ♥❣❤➽❛ t➼❝❤ tr♦♥❣ (u, v) := Du.Dv + uvdx Ω ✹✺ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ❱ỵ✐ ♠é✐ f ∈ H −1 (Ω)✳ ⑩♣ ❞ư♥❣ s r sỹ tỗ t ❤➔♠ u ∈ H01 (Ω) t❤ä❛ ♠➣♥ B[u, v] = f, v ❝❤♦ t➜t ❝↔ ❤➔♠ v ∈ H01 (Ω)✱ ✤â ❧➔ Du.Dv + uvdx = f, v ✭✶✳✶✹✮ Ω ❝❤♦ ♠é✐ v ∈ H01(Ω)✳ ❉♦ u ∈ H01(Ω) s✉② r❛ u ∈ L2(Ω) ✈➔ t❛ ❝ơ♥❣ ❝â ux ✈ỵ✐ i = 1, , n✳ ❈❤♦ ♥➯♥ ♥➳✉ t❛ ✤➦t   f = u i ∈ L2 (Ω) ✭✶✳✶✺✮  f i = ux , (i = 1, , n) i ✈➔ t❤❛② ✈➔♦ (1.14) t❤➻ t❛ ✤÷đ❝ n f, v = f i vxi dx f v+ i=1 Ω ✷✳❇➙② ❣✐í ❣✐↔ sû f ∈ H 1() t (1.13) t tỗ t g0, g1, , gn ∈ L2 (Ω) s❛♦ ❝❤♦ n f, v = ✭✶✳✶✻✮ g i vxi dx g v+ i=1 Ω ✣➦t v = u tr♦♥❣ (1.14) ✈➔ sû ❞ư♥❣ (1.16) ❝❤ó♥❣ t❛ ❝â n 2 g i dx |Du| + |u| dx ≤ Ω ❉♦ ✤â tø (1.15) t❛ t❤➜② n n f ✸✳ ❚ø (1.13) t❛ ❝â i=0 Ω Ω i i=0 Ω i=0 n | f, v | ≤ ♥➳✉ v H (Ω) ≤ 1✳ ✣➦t v = u u tr♦♥❣ (1.14)✱ H (Ω) ▼➔ f Ω ✭✶✳✶✼✮ g i dx dx ≤ i dx i=1 1 f H −1 (Ω) = supp f, v |v ∈ H10 (Ω), u ✹✻ H10 (Ω) ≤1 ✭✶✳✶✽✮ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ♥➯♥ tø (1.17), (1.18) t❛ ❝â f, ⇔ u u H01 (Ω) u = Du.D u u Ω f, v = H01 (Ω) u Ω + u H01 (Ω) u u dx H01 (Ω) |Du|2 + |u|2 dx H01 (Ω) ❍❛② n f H −1 (Ω) = f Ω i dx ✭✶✳✶✾✮ i=1 ❑❤➥♥❣ ✤à♥❤ ✐✐✮ s✉② r❛ tø (1.16) − (1.19) ✣à♥❤ ❧➼ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ✹✼ LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ❈❤÷ì♥❣ ✷ ❑❤ỉ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈ ♣❤ư t❤✉ë❝ t❤í✐ ❣✐❛♥ ❈❤♦ X ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ t❤ü❝ ✈ỵ✐ ❝❤✉➞♥ ✳ ✷✳✶ ❑❤ỉ♥❣ ❣✐❛♥ Lp(0, T ; X) ✣à♥❤ ♥❣❤➽❛ ✷✳✶✳ ❑❤æ♥❣ ❣✐❛♥ Lp (0, T ; X) ỗ tt ữủ u : [0, T ] X ợ T ✐✳ u L (0,T ;X) := ( u(t) p dt) p ✐✐✳ u L∞ (0,T ;X) := ❡ss s✉♣ 0≤t≤T p 0✱ ✈➔ ①→❝ ✤à♥❤ ❤➔♠ uε = ηε ∗ u,✳ ❑❤✐ ✤â t❤❡♦ ❝❤ù♥❣ ♠✐♥❤ ❝õ❛ ✤à♥❤ ❧➼ 1.20 t❤➻ ✈ỵ✐ ♠é✐ sè ε, δ > 0, t❛ ❝â ❈❤ù♥❣ ♠✐♥❤✳ D(η ∗ u) = Du = η ∗ Du ❈❤♦ ♥➯♥ d ε u (t) − uδ (t) dt L2 (U ) = 2(uε (t) − uδ (t), uε (t) − uδ (t))L2 (U ) ❉♦ ✤â uε (t) − uδ (t) L2 (U ) = uε (s) − uδ (s) L2 (U ) ✭✷✳✺✮ t ε δ ε δ u (τ ) − u (τ ), u (τ ) − u (τ ) dτ +2 s ✈ỵ✐ t➜t ❝↔ ≤ s, t ≤ T ❱ỵ✐ ❜➜t ❦ý s ∈ (0, T ) ♠➔ uε (s) → u(s) tr♦♥❣ L2(U ) ✈➔ u (s) → u(s) tr♦♥❣ L2(U ) ❦❤✐ → ⑩♣ ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ❈❛✉❝❤②✲ ❙❝❤✇❛r③✱ t❤➻ tø (2.5) t❛ ❝â ❧✐♠ s✉♣ s✉♣ ε δ u (t) − u (t) ε,δ→0 0≤t≤T L2 (U ) ≤ ❧✐♠ ε,δ→0 T uε (τ ) − uδ (τ ) + uε (τ ) − uδ (τ ) ⑩♣ ❞ö♥❣ ✤à♥❤ ❧➼ 1.20 t❛ ❝â ❦❤✐ , δ → t❤➻ uε (τ ) − uδ (τ ) 2H (U ) → ✈➔ uε(τ ) − uδ (τ ) 2H (U ) → ❱➻ ✈➟② ❧✐♠ s✉♣ s✉♣ uε(t) − uδ (t) H (U ) H01 (U ) dτ 1 ε,δ→0 0≤t≤T ✺✷ L2 (U ) =0 LUAN VAN CHAT LUONG download : add luanvanchat@agmail.com ❉♦ ✤â ❞➣② {uε}0