Phương trình và bất phương trình laplace

✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈ ❚❘❺◆ ❱❿◆ ❚❰■ P❍×❒◆● ❚❘➐◆❍ ❱⑨ ❇❻❚ P❍×❒◆● ❚❘➐◆❍ ▲❆P▲❆❈❊ ❈❤✉②➯♥ ♥❣➔♥❤✿ ❚❖⑩◆ Ù◆● ❉Ö◆● ▼➣ sè✿ ✻✵✳ ✹✻✳ ✵✶✳ ✶✷ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ❍÷î♥❣ ❞➝♥ ❦❤♦❛ ❤å❝ P●❙✳ ❚❙ ❍⑨ ❚■➌◆ ◆●❖❸◆ ❚❤→✐ ◆❣✉②➯♥ ✲ ✷✵✶✹ ▼ö❝ ❧ö❝ ▼ð ✤➛✉ ✶ ✶ ◆❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✈➔ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ▲❛♣❧❛❝❡ ✸ ✶✳✶ ❈→❝ ✤à♥❤ ♥❣❤➽❛✳ ❈æ♥❣ t❤ù❝ t➼❝❤ ♣❤➙♥ tø♥❣ ♣❤➛♥ ✳ ✳ ✳ ✳ ✶✳✶✳✶ ❍➔♠ ✤✐➲✉ ❤á❛✱ ❤➔♠ ❞÷î✐ ✤✐➲✉ ❤á❛ ✈➔ ❤➔♠ tr➯♥ ✤✐➲✉ ❤á❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✳✷ ❈æ♥❣ t❤ù❝ t➼❝❤ ♣❤➙♥ tø♥❣ ♣❤➛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷ ✣➥♥❣ t❤ù❝ ✈➔ ❜➜t ✤➥♥❣ t❤ù❝ ✤è✐ ✈î✐ ❣✐→ trà tr✉♥❣ ❜➻♥❤ ✳ ✶✳✷✳✶ ❈→❝ ✤↕✐ ❧÷ñ♥❣ tr✉♥❣ ❜➻♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷✳✷ ✣à♥❤ ❧þ ✈➲ ❣✐→ trà tr✉♥❣ ❜➻♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✸ ◆❣✉②➯♥ ❧þ ❝ü❝ ✤↕✐ ✈➔ ❝ü❝ t✐➸✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✸✳✶ ◆❣✉②➯♥ ❧þ ❝ü❝ ✤↕✐ ♠↕♥❤ ✈➔ ♥❣✉②➯♥ ❧þ ❝ü❝ t✐➸✉ ♠↕♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✸✳✷ ❚➼♥❤ ❞✉② ♥❤➜t ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❉✐r✐❝❤❧❡t ❝❤♦ ♣❤÷ì♥❣ tr➻♥❤ ▲❛♣❧❛❝❡ ✈➔ ♣❤÷ì♥❣ tr➻♥❤ P♦✐ss♦♥ ✷ ❈→❝ t➼♥❤ ❝❤➜t ❝õ❛ ❤➔♠ ✤✐➲✉ ❤á❛ ✷✳✶ ❇➜t ✤➥♥❣ t❤ù❝ ❍❛r♥❛❝❦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷ ❈æ♥❣ t❤ù❝ ●r❡❡♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷✳✶ ❈æ♥❣ t❤ù❝ ●r❡❡♥ t❤ù ♥❤➜t ✈➔ ❝æ♥❣ t❤ù❝ ●r❡❡♥ t❤ù ❤❛✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷✳✷ ◆❣❤✐➺♠ ❝ì ❜↔♥ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ▲❛♣❧❛❝❡ ✳ ✳ ✳ ✳ ✐ ✸ ✸ ✹ ✺ ✺ ✻ ✼ ✼ ✽ ✶✶ ✶✶ ✶✸ ✶✸ ✶✸ ✷✳✸ ✷✳✹ ✷✳✺ ✷✳✻ ✷✳✼ ✷✳✷✳✸ ❍➔♠ ●r❡❡♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❍➔♠ ●r❡❡♥ ❝õ❛ ❜➔✐ t♦→♥ ❉✐r✐❝❤❧❡t tr♦♥❣ ❤➻♥❤ ❝➛✉✳ ❈æ♥❣ t❤ù❝ P♦✐ss♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✸✳✶ ❍➔♠ ●r❡❡♥ ❝õ❛ ❜➔✐ t♦→♥ ❉✐r✐❝❤❧❡t tr♦♥❣ ❤➻♥❤ ❝➛✉ ✷✳✸✳✷ ❈æ♥❣ t❤ù❝ P♦✐ss♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✣à♥❤ ❧þ ❤ë✐ tö ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✹✳✶ ✣✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ✤➸ ♠ët ❤➔♠ ❧➔ ✤✐➲✉ ❤á❛ ✳ ✳ ✷✳✹✳✷ ❈→❝ ✤à♥❤ ❧þ ❤ë✐ tö ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✣→♥❤ ❣✐→ ❜➯♥ tr♦♥❣ ♠✐➲♥ ✤è✐ ✈î✐ ❝→❝ ✤↕♦ ❤➔♠ ❝õ❛ ❤➔♠ ✤✐➲✉ ❤á❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✺✳✶ ✣→♥❤ ❣✐→ ❜➯♥ tr♦♥❣ ♠✐➲♥ ✤è✐ ✈î✐ ✤↕♦ ❤➔♠ ❝➜♣ ✶ ✷✳✺✳✷ ✣→♥❤ ❣✐→ ❜➯♥ tr♦♥❣ ♠✐➲♥ ✤è✐ ✈î✐ ✤↕♦ ❤➔♠ ❜➜t ❦ý ❇➔✐ t♦→♥ ❉✐r✐❝❤❧❡t✳ P❤÷ì♥❣ ♣❤→♣ ❤➔♠ ✤✐➲✉ ❤á❛ ❞÷î✐ ✳ ✳ ✷✳✻✳✶ ▼ð rë♥❣ ❦❤→✐ ♥✐➺♠ ❤➔♠ ❞÷î✐ ✤✐➲✉ ❤á❛ ✈➔ ❤➔♠ tr➯♥ ✤✐➲✉ ❤á❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✻✳✷ ❈→❝ t➼♥❤ ❝❤➜t ❝õ❛ ❤➔♠ ❞÷î✐ ✤✐➲✉ ❤á❛ ✈➔ ❤➔♠ tr➯♥ ✤✐➲✉ ❤á❛ ♠ð rë♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✻✳✸ P❤÷ì♥❣ ♣❤→♣ P❡rr♦♥ ✭P❤÷ì♥❣ ♣❤→♣ ❤➔♠ ✤✐➲✉ ❤á❛ ❞÷î✐✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✻✳✹ ❍➔♠ ❝❤➢♥ t↕✐ ♠ët ✤✐➸♠ tr➯♥ ❜✐➯♥✱ ❦❤→✐ ♥✐➺♠ ✤✐➸♠ ❝❤➼♥❤ q✉② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✻✳✺ ❚➼♥❤ ❣✐↔✐ ✤÷ñ❝ ❝õ❛ ❜➔✐ t♦→♥ ❉✐r✐❝❤❧❡t ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✻✳✻ ✣✐➲✉ ❦✐➺♥ ❤➻♥❤ ❝➛✉ ♥❣♦➔✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❉✉♥❣ ❧÷ñ♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✶✻ ✶✻ ✶✽ ✶✾ ✶✾ ✷✵ ✷✶ ✷✶ ✷✶ ✷✷ ✷✸ ✷✸ ✷✺ ✷✻ ✷✽ ✸✵ ✸✵ ❑➳t ❧✉➟♥ ✸✷ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✸✸ ✐✐ ▲í✐ ❝❛♠ ✤♦❛♥ ❚æ✐ ①✐♥ ❝❛♠ ✤♦❛♥✱ ▲✉➟♥ ✈➠♥ ♥➔② ❧➔ ❝æ♥❣ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ tæ✐ ❞÷î✐ sü ❤÷î♥❣ ❞➝♥ trü❝ t✐➳♣ ❝õ❛ P●❙✳ ❚❙ ❍➔ ❚✐➳♥ ◆❣♦↕♥✳ ❚r♦♥❣ q✉→ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ ✤➲ t➔✐ ▲✉➟♥ ✈➠♥✱ tæ✐ ✤➣ ❦➳ t❤ø❛ t❤➔♥❤ q✉↔ ❦❤♦❛ ❤å❝ ❝õ❛ ❝→❝ ♥❤➔ ❚♦→♥ ❤å❝ ✈➔ ❝→❝ ♥❤➔ ❑❤♦❛ ❤å❝ ✈î✐ sü tr➙♥ trå♥❣ ✈➔ ❜✐➳t ì♥✳ ❚❤→✐ ◆❣✉②➯♥✱ t❤→♥❣ ✶✵ ♥➠♠ ✷✵✶✹ ❚→❝ ❣✐↔ ❚r➛♥ ❱➠♥ ❚î✐ ✐✐✐ ▲í✐ ❝↔♠ ì♥ ▲✉➟♥ ✈➠♥ ✤÷ñ❝ ❤♦➔♥ t❤➔♥❤ t↕✐ tr÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝✱ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥ ❞÷î✐ sü ❤÷î♥❣ ❞➝♥ t➟♥ t➻♥❤ ❝õ❛ P●❙✳ ❚❙ ❍➔ ❚✐➳♥ ◆❣♦↕♥✳ ❚æ✐ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ ❝❤➙♥ t❤➔♥❤ ✈➔ s➙✉ s➢❝ ✈➲ sü t➟♥ t➙♠ ✈➔ ♥❤✐➺t t➻♥❤ ❝õ❛ ❚❤➛② tr♦♥❣ s✉èt q✉→ tr➻♥❤ tæ✐ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥✳ ❚æ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❇❛♥ ●✐→♠ ❤✐➺✉✱ ♣❤á♥❣ ✣➔♦ t↕♦ ❑❤♦❛ ❤å❝ ✈➔ ◗✉❛♥ ❤➺ q✉è❝ t➳✱ ❑❤♦❛ ❚♦→♥ ✲ ❚✐♥ tr÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝✱ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥ ✈➔ q✉þ t❤➛② ❝æ t❤❛♠ ❣✐❛ ❣✐↔♥❣ ❞↕② ❧î♣ ❝❛♦ ❤å❝ ❦❤â❛ ✻ ✭✷✵✶✷ ✲ ✷✵✶✹✮ ✤➣ q✉❛♥ t➙♠✱ ❣✐ó♣ ✤ï ✈➔ ♠❛♥❣ ✤➳♥ ❝❤♦ tæ✐ ♥❤✐➲✉ ❦✐➳♥ t❤ù❝ ❜ê ➼❝❤ tr♦♥❣ s✉èt t❤í✐ ❣✐❛♥ ❤å❝ t➟♣ t↕✐ tr÷í♥❣✳ ❚æ✐ ❝ô♥❣ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ ✤➳♥ ❣✐❛ ✤➻♥❤✱ ❜↕♥ ❜➧ ✈➔ ❝→❝ ✤ç♥❣ ♥❣❤✐➺♣ ✤➣ ✤ë♥❣ ✈✐➯♥✱ ❣✐ó♣ ✤ï tr♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ ❝õ❛ ♠➻♥❤✳ ❉♦ t❤í✐ ❣✐❛♥ ✈➔ ❦✐➳♥ t❤ù❝ ❝á♥ ❤↕♥ ❝❤➳ ♥➯♥ ❧✉➟♥ ✈➠♥ ❦❤æ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât✳ ❚→❝ ❣✐↔ r➜t ♠♦♥❣ ♥❤➟♥ ✤÷ñ❝ sü ❣â♣ þ ❝õ❛ q✉þ t❤➛② ❝æ ✈➔ ❜↕♥ ✤å❝ ✤➸ ❧✉➟♥ ✈➠♥ ✤÷ñ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ❳✐♥ tr➙♥ trå♥❣ ❝↔♠ ì♥✦ ❚❤→✐ ◆❣✉②➯♥✱ t❤→♥❣ ✶✵ ♥➠♠ ✷✵✶✹ ❚→❝ ❣✐↔ ❚r➛♥ ❱➠♥ ❚î✐ ✐✈ ▼ð ✤➛✉ P❤÷ì♥❣ tr➻♥❤ ▲❛♣❧❛❝❡ ❧➔ ♠ët ♣❤÷ì♥❣ tr➻♥❤ ❝ì ❜↔♥ ✈➔ ❝ê ✤✐➸♥ ❝õ❛ ❧þ t❤✉②➳t ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣✳ ✣➙② ❧➔ ✤↕✐ ❞✐➺♥ q✉❛♥ trå♥❣ ❝õ❛ ❧î♣ ♣❤÷ì♥❣ tr➻♥❤ ❡❧❧✐♣t✐❝✳ ❱✐➺❝ tê♥❣ q✉❛♥ ❝→❝ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ♥❣❤✐➺♠ ♣❤÷ì♥❣ tr➻♥❤ ✈➔ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ▲❛♣❧❛❝❡ ❧➔ ❝➛♥ t❤✐➳t✳ ✣â ❧➔ ❝→❝ ❤➔♠ ✤✐➲✉ ❤á❛✱ tr➯♥ ✤✐➲✉ ❤á❛ ✈➔ ❞÷î✐ ✤✐➲✉ ❤á❛✳ ✣è✐ ✈î✐ ❝→❝ ❤➔♠ ♥➔② ❝â r➜t ♥❤✐➲✉ t➼♥❤ ❝❤➜t✱ ✤à♥❤ ❧þ ✤➣ ✤÷ñ❝ ♥❣❤✐➯♥ ❝ù✉✳ ❈❤➥♥❣ ❤↕♥ ♥❤÷ ♥❣✉②➯♥ ❧þ ❝ü❝ ✤↕✐✱ ❝→❝ ✤à♥❤ ❧þ ✈➲ ❣✐→ trà tr✉♥❣ ❜➻♥❤✱ ✳✳✳ ✣è✐ ✈î✐ ❤➔♠ ✤✐➲✉ ❤á❛✱ ♥❣❤✐➺♠ s✉② rë♥❣ ❝õ❛ ❜➔✐ t♦→♥ ❜✐➯♥ ❉✐r✐❝❤❧❡t ❧✉æ♥ tç♥ t↕✐✳ ◆❤÷♥❣ ð ❧✉➟♥ ✈➠♥ ♥➔② ♥❣❤✐➯♥ ❝ù✉ ♥❣❤✐➺♠ ❝ê ✤✐➸♥ ❝õ❛ ❜➔✐ t♦→♥ ❜✐➯♥ ❉✐r✐❝❤❧❡t✱ ❝ö t❤➸ ①➨t t➼♥❤ ❣✐↔✐ ✤÷ñ❝ ❝õ❛ ❜➔✐ t♦→♥ ❜✐➯♥ ❉✐r✐❝❤❧❡t tr♦♥❣ ♠ët ♠✐➲♥ ❜à ❝❤➦♥✱ ♥❣❤✐➯♥ ❝ù✉ ❦❤✐ ♥➔♦ ❜➔✐ t♦→♥ ❉✐r✐❝❤❧❡t ❧➔ ❣✐↔✐ ✤÷ñ❝ tr♦♥❣ ♠✐➲♥ Ω✳ ❈❤➼♥❤ ✈➻ ✈➟②✱ tr♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② ✤➣ ✤÷❛ ✈➔✐ ❦❤→✐ ♥✐➺♠ ✤✐➸♠ ❝❤➼♥❤ q✉② tr➯♥ ❜✐➯♥ ♠➔ ✤÷ñ❝ ✤à♥❤ ♥❣❤➽❛ t❤æ♥❣ q✉❛ ❦❤→✐ ♥✐➺♠ ❤➔♠ ❝❤➢♥✳ ❑➳t q✉↔ ❝ì ❜↔♥ tr♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② ❧➔ ✤à♥❤ ❧þ ♥â✐ r➡♥❣ ❜➔✐ t♦→♥ ❉✐r✐❝❤❧❡t ❣✐↔✐ ✤÷ñ❝ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ♠å✐ ✤✐➸♠ tr➯♥ ❜✐➯♥ ✤➲✉ ❧➔ ✤✐➸♠ ❝❤➼♥❤ q✉②✳ P❤➛♥ ❝✉è✐ ❝õ❛ ❧✉➟♥ ✈➠♥ ♥❣❤✐➯♥ ❝ù✉ ❦❤✐ ♥➔♦ ♠ët ✤✐➸♠ ❧➔ ❝❤➼♥❤ q✉②✳ ▲✉➟♥ ✈➠♥ ❣ç♠ ✷ ❝❤÷ì♥❣✿ ❈❤÷ì♥❣ ✶ tr➻♥❤ ❜➔② ❝→❝ ❦❤→✐ ♥✐➺♠✱ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ✈➲ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ▲❛♣❧❛❝❡ ✈➔ ❝→❝ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ▲❛♣❧❛❝❡✳ ✣â ❧➔ ❝→❝ ✤à♥❤ ♥❣❤➽❛ ✈➲ ❤➔♠ ✤✐➲✉ ❤á❛✱ ❤➔♠ ❞÷î✐ ✤✐➲✉ ❤á❛✱ tr➯♥ ✤✐➲✉ ❤á❛✱ ❝æ♥❣ t❤ù❝ t➼❝❤ ♣❤➙♥ tø♥❣ ♣❤➛♥✱ ❝→❝ ✤➥♥❣ t❤ù❝ ✈➔ ❜➜t ✤➥♥❣ t❤ù❝ ❣✐→ trà tr✉♥❣ ✶ ❜➻♥❤✱ ♥❣✉②➯♥ ❧þ ❝ü❝ ✤↕✐ ✈➔ ❝ü❝ t✐➸✉✳ ❈❤÷ì♥❣ ✷ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❤➔♠ ✤✐➲✉ ❤á❛✳ ✣â ❧➔ ❜➜t ✤➥♥❣ t❤ù❝ ❍❛r♥❛❝❦✱ ✤÷❛ ✈➔♦ ❝æ♥❣ t❤ù❝ ●r❡❡♥✱ ❤➔♠ ●r❡❡♥ ✤è✐ ✈î✐ ❜➔✐ t♦→♥ ❉✐r✐❝❤❧❡t✱ ♥❣❤✐➯♥ ❝ù✉ ✤à♥❤ ❧þ ❤ë✐ tö ✈➔ ❝→❝ ✤→♥❤ ❣✐→ ❜➯♥ tr♦♥❣ ✤è✐ ✈î✐ ❤➔♠ ✤✐➲✉ ❤á❛✳ P❤➛♥ ❝✉è✐ ♥❣❤✐➯♥ ❝ù✉ ❜➔✐ t♦→♥ ❉✐r✐❝❤❧❡t ❝❤♦ ❤➔♠ ✤✐➲✉ ❤á❛ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❤➔♠ ✤✐➲✉ ❤á❛ ❞÷î✐✳ ❇➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ ♥➔② ✤➣ ✤÷❛ ✈➔♦ ❦❤→✐ ♥✐➺♠ ✤✐➸♠ ❝❤➼♥❤ q✉② tr➯♥ ❜✐➯♥✱ ♣❤→t ❜✐➸✉ ✈➔ ❝❤ù♥❣ ♠✐♥❤ ✤à♥❤ ❧þ ✈➲ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ❝❤♦ t➼♥❤ ❣✐↔✐ ✤÷ñ❝ ❝õ❛ ❜➔✐ t♦→♥ ❜✐➯♥ ❉✐r✐❝❤❧❡t✳ ✣÷❛ ✈➔♦ ✤✐➲✉ ❦✐➺♥ ✤õ ❝❤♦ t➼♥❤ ❝❤➼♥❤ q✉②✱ ✤â ❧➔ ✤✐➲✉ ❦✐➺♥ ❤➻♥❤ ❝➛✉ ♥❣♦➔✐ ❝õ❛ ♠✐➲♥✳ ✣✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ❝❤♦ t➼♥❤ ❝❤➼♥❤ q✉② ❝õ❛ ♠ët ✤✐➸♠ tr➯♥ ❜✐➯♥ ✤÷ñ❝ ♣❤→t ❜✐➸✉ t❤æ♥❣ q✉❛ ❦❤→✐ ♥✐➺♠ ❞✉♥❣ ❧÷ñ♥❣✳ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ ✈➠♥ ❧➔ ❝❤÷ì♥❣ ✷ ❝õ❛ t➔✐ ❧✐➺✉ ❬✷❪✳ ✷ ❈❤÷ì♥❣ ✶ ◆❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✈➔ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ▲❛♣❧❛❝❡ ✶✳✶ ❈→❝ ✤à♥❤ ♥❣❤➽❛✳ ❈æ♥❣ t❤ù❝ t➼❝❤ ♣❤➙♥ tø♥❣ ♣❤➛♥ ✶✳✶✳✶ ❍➔♠ ✤✐➲✉ ❤á❛✱ ❤➔♠ ❞÷î✐ ✤✐➲✉ ❤á❛ ✈➔ ❤➔♠ tr➯♥ ✤✐➲✉ ❤á❛ ❑þ ❤✐➺✉✿ x = (x1 , x2 , ..., xn ) ∈ Rn , x21 + x22 + ... + x2n . ||x|| = ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✳ ❈❤♦ Ω ❧➔ ♠ët ♠✐➲♥ tr♦♥❣ Rn ✈➔ ❤➔♠ sè u t❤✉ë❝ C 2(Ω) ✳ ❚♦→♥ tû ▲❛♣❧❛❝❡ t→❝ ✤ë♥❣ ❧➯♥ u✱ ❦➼ ❤✐➺✉ ❧➔ ∆u✱ ✤÷ñ❝ ①→❝ ✤à♥❤ ❜ð✐✿ n Dj 2 u = divDu, ∆u = j=1 tr♦♥❣ ✤â✱ Du = (D1 u, D2 u, ..., Dn u) ∂u Dj u = ✱ ∂xj ✸ ❧➔ ❣r❛❞✐❡♥t ❝õ❛ u, ✭✶✳✶✮ ∂ 2u ∂ 2u ∂ 2u ∆u = + + ... + = div(Du)✳ ∂x1 2 ∂x2 2 ∂xn 2 ❍➔♠ sè u ✤÷ñ❝ ❣å✐ ❧➔ ❤➔♠ ✤✐➲✉ ❤á❛ ✭❤➔♠ ❞÷î✐ ✤✐➲✉ ❤á❛✱ ❤➔♠ tr➯♥ ✤✐➲✉ ❤á❛ ✮ tr♦♥❣ Ω ♥➳✉ ♥â t❤ä❛ ♠➣♥✿ ∆u(x) = 0 (≥ 0, ≤ 0), ∀x ∈ Ω. ✭✶✳✷✮ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② ❝❤ó♥❣ t❛ ♣❤→t tr✐➸♥ ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❤➔♠ ✤✐➲✉ ❤á❛✱ ❤➔♠ ❞÷î✐ ✤✐➲✉ ❤á❛ ✈➔ ❤➔♠ tr➯♥ ✤✐➲✉ ❤á❛ ❞ò♥❣ ✤➸ ♥❣❤✐➯♥ ❝ù✉ t➼♥❤ ❣✐↔✐ ✤÷ñ❝ ❝õ❛ ❜➔✐ t♦→♥ ❉✐r✐❝❤❧❡t ❝ê ✤✐➸♥ ❝❤♦ ♣❤÷ì♥❣ tr➻♥❤ ▲❛♣❧❛❝❡✱ ∆u = 0✳ P❤÷ì♥❣ tr➻♥❤ ▲❛♣❧❛❝❡ ✈➔ ♣❤÷ì♥❣ tr➻♥❤ ❦❤æ♥❣ t❤✉➛♥ ♥❤➜t t÷ì♥❣ ù♥❣ ❝õ❛ ♥â✱ ♣❤÷ì♥❣ tr➻♥❤ P♦✐ss♦♥ −∆u = f ✱ ❧➔ ♠æ ❤➻♥❤ ❝ì ❜↔♥ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤ ❡❧✐♣t✐❝✳ ✶✳✶✳✷ ❈æ♥❣ t❤ù❝ t➼❝❤ ♣❤➙♥ tø♥❣ ♣❤➛♥ ●✐↔ sû Ω ⊂ Rn ❧➔ ♠✐➲♥ ❜à ❝❤➦♥ tr♦♥❣ Rn ✈î✐ ❜✐➯♥ ∂Ω✱ ❦þ ❤✐➺✉ µ = (µ1 , µ2 , ..., µn ) ❧➔ ✈➨❝tì ♣❤→♣ t✉②➳♥ ♥❣♦➔✐ ✤ì♥ ✈à t↕✐ ✤✐➸♠ x ∈ ∂Ω✱ dS ❧➔ ♣❤➛♥ tû ❞✐➺♥ t➼❝❤ ❝õ❛ ∂Ω✳ ¯ ✱ t❛ ❝â ❝æ♥❣ t❤ù❝ t➼❝❤ ♣❤➙♥ tø♥❣ ♣❤➛♥✿ ❱î✐ u✱v ∈ C 1(Ω) ∩ C 0(Ω) (Dj u)vdx = − Ω u(Dj v)dx + Ω uvµj dS. ✭✶✳✸✮ ∂Ω ❚ø ❝æ♥❣ t❤ù❝ tr➯♥ t❛ s✉② r❛ ✣à♥❤ ❧þ ♣❤➙♥ ❦➻ s❛✉ ✤➙②✳ ❈❤♦ tr÷í♥❣ ¯ ✳ ❑❤✐ ✤â t❛ ❝â ✈➨❝tì ❜➜t ❦➻ w = (w1, w2, ..., wn) tr♦♥❣ C 1(Ω) divwdx = Ω (w, µ)dS, Ω n j ✳ tr♦♥❣ ✤â✱ divw = ∂w ∂x j j=1 ❚❤➟t ✈➟②✱ →♣ ❞ö♥❣ ❝æ♥❣ t❤ù❝ ✭✶✳✸✮ t❛ ❝â✿ ✹ ✭✶✳✹✮ n n Dj wj dx = divwdx = Ω j=1 Ω (Dj wj ).1.dx Ω j=1 n wj .1.µj dS = ∂Ω j=1 = ∂Ω ¯ C 2 (Ω) ✣➦❝ ❜✐➺t ♥➳✉ u ❧➔ ♠ët ❤➔♠ tr♦♥❣ tr♦♥❣ ✭✶✳✹✮ ❝❤ó♥❣ t❛ ❝â✿ ∆udx = Ω tr♦♥❣ ✤â div(Du)dx = Ω (w, µ)dS. ❜➡♥❣ ❝→❝❤ ✤➦t w = Du ∂u dS, ∂µ Du.µ.dS = ∂Ω ✭✶✳✺✮ ∂Ω n ∂u ∂u = µj ✳ ∂µ j=1 ∂xj ✶✳✷ ✣➥♥❣ t❤ù❝ ✈➔ ❜➜t ✤➥♥❣ t❤ù❝ ✤è✐ ✈î✐ ❣✐→ trà tr✉♥❣ ❜➻♥❤ ✶✳✷✳✶ ❈→❝ ✤↕✐ ❧÷ñ♥❣ tr✉♥❣ ❜➻♥❤ ❑þ ❤✐➺✉ ωn ❧➔ t❤➸ t➼❝❤ ❝õ❛ ❤➻♥❤ ❝➛✉ ✤ì♥ ✈à tr♦♥❣ Rn✳ ❑❤✐ ✤â✿ ❚❤➸ t➼❝❤ ❝õ❛ ❤➻♥❤ ❝➛✉ ❜→♥ ❦➼♥❤ R ❧➔ ωnRn✳ ❉✐➺♥ t➼❝❤ ❝õ❛ ♠➦t ❝➛✉ ✤ì♥ ✈à ❧➔✿ nωn✳ ❉✐➺♥ t➼❝❤ ❝õ❛ ♠➦t ❝➛✉ ❜→♥ ❦➼♥❤ R ❧➔ nωnRn−1✳ ✣↕✐ ❧÷ñ♥❣ tr✉♥❣ ❜➻♥❤ ❝õ❛ ❤➔♠ sè u tr➯♥ ♠➦t ❝➛✉ B ❜→♥ ❦➼♥❤ R ❧➔✿ 1 nωn Rn−1 udS. ∂B ✣↕✐ ❧÷ñ♥❣ tr✉♥❣ ❜➻♥❤ ❝õ❛ ❤➔♠ sè u tr♦♥❣ ❤➻♥❤ ❝➛✉ B ❜→♥ ❦➼♥❤ R ❧➔✿ 1 ωn R n udx. B ✺ ✶✳✷✳✷ ✣à♥❤ ❧þ ✈➲ ❣✐→ trà tr✉♥❣ ❜➻♥❤ ✣à♥❤ ❧þ ✤➛✉ t✐➯♥ ❝õ❛ ❝❤ó♥❣ t❛ ✤â ❧➔ ♠ët ❤➺ q✉↔ ❝õ❛ ✤ç♥❣ ♥❤➜t t❤ù❝ ✭✶✳✺✮✱ ❜❛♦ ❣ç♠ ❝→❝ t➼♥❤ ❝❤➜t ♥ê✐ t✐➳♥❣ ✈➲ ❣✐→ trà tr✉♥❣ ❜➻♥❤ ❝õ❛ ❤➔♠ ✤✐➲✉ ❤á❛✱ ❤➔♠ ❞÷î✐ ✤✐➲✉ ❤á❛ ✈➔ ❤➔♠ tr➯♥ ✤✐➲✉ ❤á❛✳ ✣à♥❤ ❧➼ ✶✳✷✳✶✳ ●✐↔ sû u ∈ C 2(Ω) t❤ä❛ ♠➣♥ ∆u = 0 (≥ 0, ≤ 0) tr♦♥❣ Ω✳ ❈❤♦ ❤➻♥❤ ❝➛✉ ❜➜t ❦ý t➙♠ t↕✐ y ✈➔ ❜→♥ ❦➼♥❤ R✿ B = BR(y) ⊂⊂ Ω✱ ❦❤✐ ✤â t❛ ❝â✿ 1 u(y) = (≤, ≥) udS, ✭✶✳✻✮ nω Rn−1 n ∂B u(y) = (≤, ≥) 1 ωn R n ✭✶✳✼✮ udx. B ✣è✐ ✈î✐ ❝→❝ ❤➔♠ ✤✐➲✉ ❤á❛✱ ✣à♥❤ ❧þ ✶✳✷✳✶ ❦❤➥♥❣ ✤à♥❤ r➡♥❣ ❣✐→ trà ❝õ❛ ❤➔♠ t↕✐ t➙♠ ❝õ❛ ❤➻♥❤ ❝➛✉ B ❜➡♥❣ ❣✐→ trà tr✉♥❣ ❜➻♥❤ t➼❝❤ ♣❤➙♥ tr➯♥ ❝↔ ♠➦t ❝➛✉ ∂B ✈➔ tr♦♥❣ ❤➻♥❤ ❝➛✉ B ✳ ◆❤ú♥❣ ❦➳t q✉↔ tr➯♥ ❣å✐ ❧➔ ✣à♥❤ ❧þ ❣✐→ trà tr✉♥❣ ❜➻♥❤✱ tr➯♥ t❤ü❝ t➳ ❝❤ó♥❣ ❝ô♥❣ ♠æ t↔ t➼♥❤ ❝❤➜t ✤➦❝ tr÷♥❣ ❝õ❛ ❤➔♠ ✤✐➲✉ ❤á❛ ✭①❡♠ ✣à♥❤ ❧þ ✷✳✹✳✶ ❞÷î✐ ✤➙②✮✳ ❈❤ù♥❣ ♠✐♥❤✳ ❈❤♦ ρ ∈ (0, R) ✈➔ →♣ ❞ö♥❣ ✤ç♥❣ ♥❤➜t t❤ù❝ ✭✶✳✺✮ ❝❤♦ ❤➻♥❤ ❝➛✉ Bρ = Bρ (y) ❝❤ó♥❣ t❛ t❤✉ ✤÷ñ❝✿ ∂u dS = ∂µ ∆udx = (≥, ≤)0. Bρ ∂Bρ ❉ò♥❣ ♣❤➨♣ ❜✐➳♥ ✤ê✐ tå❛ ✤ë t❤❡♦ ❜→♥ ❦➼♥❤ ✈➔ ❣â❝ x−y ✈➔ ✈✐➳t u(x) = u(y + rω)✱ ❝❤ó♥❣ t❛ ❝â✿ ω= r ∂u dS = ∂µ ∂Bρ ∂u (y + rω)dS = ρn−1 ∂r ∂u (y + rω)dω ∂r |ω|=1 ∂Bρ = ρn−1 r = |x − y|✱ ∂ ∂ρ u(y + rω)dω = ρn−1 |ω=1| ∂ n−1 ρ ∂ρ ∂Bρ = (≥, ≤) 0. ✻ udS ❉♦ ✤â✱ ✈î✐ ρ ∈ (0, R) ❜➜t ❦ý t❛ ❝â✿ ρ1−n udS = (≤, ≥) R1−n udS. ∂BR ∂Bρ ▼➦t ❦❤→❝✱ t❛ ❝â✿ lim ρ1−n ρ→0 udS = nωn u(y), ∂Bρ tr♦♥❣ ✤â ωn ❧➔ ❞✐➺♥ t➼❝❤ ♠➦t ❝õ❛ ♠➦t ❝➛✉ ✤ì♥ ✈à✳ ❚ø ✤â s✉② r❛ ❝æ♥❣ t❤ù❝ ✭✶✳✻✮✳ ✣➸ ♥❤➟♥ ✤÷ñ❝ ❜➜t ✤➥♥❣ t❤ù❝ ❣✐→ trà tr✉♥❣ ❜➻♥❤ tr♦♥❣ ❤➻♥❤ ❝➛✉ t❤➻ t❛ ✈✐➳t ❧↕✐ ✭✶✳✻✮ ❞÷î✐ ❞↕♥❣ s❛✉✿ nωn ρn−1 u(y) = (≤, ≥) udS, ρ ≤ R, ∂Bρ ✈➔ ❧➜② t➼❝❤ ♣❤➙♥ ❤❛✐ ✈➳ ✤è✐ ✈î✐ ρ tø ✵ ✤➳♥ ❘✳ ❚ø ✤â ❝æ♥❣ t❤ù❝ ✭✶✳✼✮ ✤÷ñ❝ s✉② r❛ ♥❣❛② ❧➟♣ tù❝✳ ✶✳✸ ◆❣✉②➯♥ ❧þ ❝ü❝ ✤↕✐ ✈➔ ❝ü❝ t✐➸✉ ✶✳✸✳✶ ◆❣✉②➯♥ ❧þ ❝ü❝ ✤↕✐ ♠↕♥❤ ✈➔ ♥❣✉②➯♥ ❧þ ❝ü❝ t✐➸✉ ♠↕♥❤ ❚ø ✣à♥❤ ❧þ ✶✳✷✳✶ t❛ s✉② r❛ ✤÷ñ❝ ♥❣✉②➯♥ ❧þ ❝ü❝ ✤↕✐ ♠↕♥❤ ❝❤♦ ❤➔♠ ❞÷î✐ ✤✐➲✉ ❤á❛ ✈➔ ♥❣✉②➯♥ ❧þ ❝ü❝ t✐➸✉ ♠↕♥❤ ❝❤♦ ❤➔♠ tr➯♥ ✤✐➲✉ ❤á❛✳ ✣à♥❤ ❧➼ ✶✳✸✳✶✳ ❈❤♦ ∆u ≥ 0 (≤ 0) tr♦♥❣ Ω ✈➔ ❣✐↔ sû r➡♥❣ tç♥ t↕✐ ♠ët ✤✐➸♠ y ∈ Ω ♠➔ u(y) = sup u (inf u) t❤➻ ❤➔♠ u ❧➔ ❤➡♥❣ sè✳ ❉♦ ✤â ♠ët ❤➔♠ ✤✐➲✉ Ω Ω ❤á❛ ❦❤æ♥❣ t❤➸ ♥❤➟♥ ❣✐→ trà ❝ü❝ ✤↕✐ ❤♦➦❝ ❝ü❝ t✐➸✉ tr♦♥❣ ♠✐➲♥ Ω trø ❦❤✐ ♥â ❧➔ ❤➡♥❣ sè✳ ✼ ❈❤ù♥❣ ♠✐♥❤✳ ❈❤♦ ∆u ≥ 0 tr♦♥❣ Ω✱ M = sup u ✈➔ ✤➦t Ω ΩM = {x ∈ Ω | u(x) = M }. ❚❤❡♦ ❣✐↔ t❤✐➳t ΩM ❦❤→❝ ré♥❣✳ ❍ì♥ ♥ú❛ u ❧➔ ❧✐➯♥ tö❝ tr➯♥ ΩM ♠➔ ❧➔ t➟♣ ✤â♥❣ t÷ì♥❣ ✤è✐ tr➯♥ Ω✳ ❈❤♦ z ❧➔ ✤✐➸♠ ❜➜t ❦ý tr♦♥❣ ΩM ✈➔ →♣ ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ❣✐→ trà tr✉♥❣ ❜➻♥❤ ✭✶✳✼✮ ❝❤♦ ❤➔♠ ❞÷î✐ ✤✐➲✉ ❤á❛ u − M tr♦♥❣ ♠ët ❤➻♥❤ ❝➛✉ B = BR (z) ⊂⊂ Ω✳ ❉♦ ✤â ❝❤ó♥❣ t❛ t❤✉ ✤÷ñ❝✿ 0 = u(z) − M ≤ 1 ωn R n (u − M )dx ≤ 0, B s✉② r❛ u = M tr♦♥❣ BR(z)✳ ❉♦ ✤â ΩM ♠ð t÷ì♥❣ ✤è✐ tr♦♥❣ Ω✳ ❚ø ✤â ΩM = Ω✱ ✈➻ ✈➟② u ❧➔ ❤➔♠ ❤➡♥❣ tr➯♥ Ω✳ ❑➳t q✉↔ ❝❤♦ ❝→❝ ❤➔♠ tr➯♥ ✤✐➲✉ ❤á❛ ❝â ✤÷ñ❝ ❜➡♥❣ ❝→❝❤ t❤❛② t❤➳ u ❜ð✐ −u✳ ✣à♥❤ ❧➼ ✶✳✸✳✷✳ ¯ ✈î✐ ∆u ≥ 0 (≤ 0) tr➯♥ Ω✱ ✈î✐ Ω ❧➔ ❜à ❝❤➦♥✱ ❈❤♦ u ∈ C 2(Ω) ∩ C 0(Ω) ❦❤✐ ✤â sup u = sup u (inf u = inf u). ✭✶✳✽✮ Ω ∂Ω Ω ∂Ω ❉♦ ✤â ❝❤♦ ❤➔♠ ✤✐➲✉ ❤á❛ u✱ t❛ ❝â✿ inf u ≤ u(x) ≤ sup u, x ∈ Ω. ∂Ω ∂Ω ❚➼♥❤ ❞✉② ♥❤➜t ♥❣❤✐➺♠ ❝❤♦ ❜➔✐ t♦→♥ ❉✐r✐❝❤❧❡t ❝ê ✤✐➸♥ ❝❤♦ ♣❤÷ì♥❣ tr➻♥❤ ▲❛♣❧❛❝❡ ✈➔ ♣❤÷ì♥❣ tr➻♥❤ P♦✐ss♦♥ tr♦♥❣ ♠✐➲♥ ❜à ❝❤➦♥ ✤÷ñ❝ s✉② r❛ tø ✣à♥❤ ❧þ ✶✳✸✳✷ tr♦♥❣ ♠ö❝ ❞÷î✐ ✤➙②✳ ✶✳✸✳✷ ❚➼♥❤ ❞✉② ♥❤➜t ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❉✐r✐❝❤❧❡t ❝❤♦ ♣❤÷ì♥❣ tr➻♥❤ ▲❛♣❧❛❝❡ ✈➔ ♣❤÷ì♥❣ tr➻♥❤ P♦✐ss♦♥ ❆✳ ❇➔✐ t♦→♥ ❉✐r✐❝❤❧❡t ❝❤♦ ♣❤÷ì♥❣ tr➻♥❤ ▲❛♣❧❛❝❡✿ ✽ ❈❤♦ Ω ❧➔ ♠✐➲♥ ❜à ❝❤➦♥ tr➯♥ Rn✱ ❦❤✐ ✤â ❜➔✐ t♦→♥ ❉✐r✐❝❤❧❡t ❝❤♦ ♣❤÷ì♥❣ tr➻♥❤ ▲❛♣❧❛❝❡ ❧➔✿ t➻♠ ♠ët ❤➔♠ u : Ω → R t❤ä❛ ♠➣♥ ¯ u ∈ C 2 (Ω) ∩ C(Ω), ✈➔ tr♦♥❣ Ω , u = ϕ, tr➯♥ ∂Ω tr♦♥❣ ✤â ϕ ∈ C(∂Ω) ❧➔ ❤➔♠ ❝❤♦ tr÷î❝✳ ❇✳ ❇➔✐ t♦→♥ ❉✐r✐❝❤❧❡t ❝❤♦ ♣❤÷ì♥❣ tr➻♥❤ P♦✐ss♦♥✿ ❈❤♦ Ω ❧➔ ♠✐➲♥ ❜à ❝❤➦♥ tr➯♥ Rn✱ ❦❤✐ ✤â ❜➔✐ t♦→♥ ❉✐r✐❝❤❧❡t ❝❤♦ ♣❤÷ì♥❣ tr➻♥❤ P♦✐ss♦♥ ❧➔✿ t➻♠ ♠ët ❤➔♠ u : Ω → R t❤ä❛ ♠➣♥ ∆u = 0, ¯ u ∈ C 2 (Ω) ∩ C(Ω), ✈➔ tr♦♥❣ Ω , u = ϕ, tr➯♥ ∂Ω tr♦♥❣ ✤â f ∈ C(Ω) ✈➔ ϕ ∈ C(∂Ω) ❧➔ ❤➔♠ ❝❤♦ tr÷î❝✳ −∆u = f, ✣à♥❤ ❧➼ ✶✳✸✳✸✳ ¯ t❤ä❛ ♠➣♥ ∆u = ∆v tr♦♥❣ Ω✱ u = v tr➯♥ ❈❤♦ u, v ∈ C 2(Ω) ∩ C 0(Ω) ∂Ω t❤➻ u = v tr♦♥❣ Ω✳ ❈❤ù♥❣ ♠✐♥❤✳ ✣➦t w = u − v✳ ❚❛ ❝â✿ ∆w = 0 tr➯♥ Ω ♥❣❤➽❛ ❧➔ w ❧➔ ❤➔♠ ✤✐➲✉ ❤á❛ ✈➔ w = 0 tr♦♥❣ ∂Ω ✭❞♦ u = v tr➯♥ ∂Ω✮✳ ❉♦ 0 = inf w ≤ w(x) ≤ sup w = 0, x ∈ Ω, ✭✣à♥❤ ∂Ω ∂Ω ❧þ ✶✳✸✳✷✮. ❙✉② r❛ w = 0 tr♦♥❣ Ω ❤❛② u = v tr♦♥❣ Ω✳ ❈❤ó þ r➡♥❣ ❜➡♥❣ ✣à♥❤ ❧þ ✶✳✸✳✷✱ ❝❤ó♥❣ t❛ ❝â ♥➳✉ u ✈➔ v ❧➔ ❝→❝ ❤➔♠ ✤✐➲✉ ❤á❛ ✈➔ ❤➔♠ ❞÷î✐ ✤✐➲✉ ❤á❛ t÷ì♥❣ ù♥❣✱ u = v tr➯♥ ❜✐➯♥ ∂Ω✱ ❦❤✐ ✤â v ≤ u tr♦♥❣ Ω✳ ❚➼♥❤ ❝❤➜t ♥➔② ❣✐↔✐ t❤➼❝❤ t↕✐ s❛♦ v ✤÷ñ❝ ❣å✐ ❧➔ ❤➔♠ ❞÷î✐ ✤✐➲✉ ❤á❛✳ ◆❤➟♥ ①➨t t÷ì♥❣ ù♥❣ ❝ô♥❣ ✤ó♥❣ ❝❤♦ ❤➔♠ tr➯♥ ✤✐➲✉ ❤á❛✱ ✾ ♥❣❤➽❛ ❧➔ u ❧➔ ❤➔♠ ✤✐➲✉ ❤á❛✱ v ❧➔ ❤➔♠ tr➯♥ ✤✐➲✉ ❤á❛✱ u = v tr➯♥ ∂Ω t❤➻ u ≤ v tr♦♥❣ Ω✳ ❙❛✉ ✤➙②✱ ❝❤ó♥❣ t❛ sû ❞ö♥❣ t➼♥❤ ❝❤➜t ❤➔♠ ❞÷î✐ ✤✐➲✉ ❤á❛ ✈➔ ❤➔♠ tr➯♥ ✤✐➲✉ ❤á❛ ✤➸ ♠ð rë♥❣ ♥❤ú♥❣ ✤à♥❤ ♥❣❤➽❛ ✤è✐ ✈î✐ ❝→❝ ❧î♣ ❤➔♠ rë♥❣ ❤ì♥✳ ✶✵ ❈❤÷ì♥❣ ✷ ❈→❝ t➼♥❤ ❝❤➜t ❝õ❛ ❤➔♠ ✤✐➲✉ ❤á❛ ✷✳✶ ❇➜t ✤➥♥❣ t❤ù❝ ❍❛r♥❛❝❦ ❍➺ q✉↔ t✐➳♣ t❤❡♦ ❝õ❛ ✣à♥❤ ❧þ ✶✳✷✳✶ s➩ ❧➔ ❜➜t ✤➥♥❣ t❤ù❝ ❍❛r♥❛❝❦ ❝❤♦ ❤➔♠ ✤✐➲✉ ❤á❛✳ ✣à♥❤ ❧➼ ✷✳✶✳✶✳ ❈❤♦ u ❧➔ ♠ët ❤➔♠ ✤✐➲✉ ❤á❛ ❦❤æ♥❣ ➙♠ tr♦♥❣ Ω✱ ✈➔ ❝❤♦ ❜➜t ❦ý ♠✐➲♥ ❝♦♥ Ω ⊂⊂ Ω ❜à ❝❤➦♥✱ ❦❤✐ ✤â tç♥ t↕✐ ♠ët ❤➡♥❣ sè C ❝❤➾ ♣❤ö t❤✉ë❝ ✈➔♦ n✱ Ω ✈➔ Ω s❛♦ ❝❤♦✿ sup u ≤ C inf u. ✭✷✳✶✮ Ω Ω ❈❤ù♥❣ ♠✐♥❤✳ ❈❤♦ y ∈ Ω✱ B4R(y) ⊂ Ω✳ ❱î✐ ❤❛✐ ✤✐➸♠ ❜➜t ❦ý x1, x2 ∈ B4R(y)✱ →♣ ❞ö♥❣ ❝æ♥❣ t❤ù❝ ✭✶✳✼✮ t❛ ❝â✿ u(x1 ) = 1 ωn R n udx BR (x1 ) ≤ 1 ωn R n udx, B2R (y) ✈➔ ✶✶ u(x2 ) = 1 ωn (3R)n udx B3R (x2 ) ≥ 1 ωn (3R)n udx. B2R (y) ❉♦ ✤â ❝❤ó♥❣ t❛ ❝â ✤÷ñ❝✿ sup u ≤ 3n . inf u. BR (y) BR (y) ❇➙② ❣✐í ❝❤♦ Ω ⊂⊂ Ω ✭✷✳✷✮ ✈➔ ❝❤å♥ x1, x2 ∈ Ω¯ ✤➸ u(x1 ) = sup u, Ω ✈➔ u(x2 ) = inf u. Ω ❈❤♦ Γ ⊂ Ω¯ ❧➔ ♠ët ❝✉♥❣ ✤â♥❣ x1 ✈➔ x2 ✈➔ ❝❤å♥ R ✤➸ 4R ≤ dist(Γ, ∂Ω). ❚❤❡♦ ✤à♥❤ ❧þ ❍❡✐♥❡✲❇♦r❡❧✱ Γ ❝â t❤➸ ✤÷ñ❝ ❜❛♦ ♣❤õ ❜ð✐ sè ❤ú✉ ❤↕♥ N ✭❝❤➾ ♣❤ö t❤✉ë❝ tr♦♥❣ Ω ✈➔ Ω✮ ❝→❝ ❤➻♥❤ ❝➛✉ ❜→♥ ❦➼♥❤ R✳ ⑩♣ ❞ö♥❣ ❣✐↔ t❤✐➳t ✭✷✳✷✮ tr➯♥ ♠é✐ ❤➻♥❤ ❝➛✉ ✈➔ ❦➳t ❤ñ♣ ❝→❝ ❦➳t q✉↔ ❝õ❛ ❜➜t ✤➥♥❣ t❤ù❝✱ ❝❤ó♥❣ t❛ ❝â✿ u(x1 ) ≤ 3nN u(x2 ). ❉♦ ✤â ❣✐↔ t❤✐➳t ✭✷✳✶✮ ✤÷ñ❝ ❝❤ù♥❣ ♠✐♥❤ ✈î✐ C = 3nN ✳ ❈❤ó þ r➡♥❣ ❤➡♥❣ sè C tr➯♥ ✭✷✳✶✮ ❧➔ ❤➡♥❣ sè ❦❤æ♥❣ ✤ê✐ ✤è✐ ✈î✐ ❝→❝ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ✤ç♥❣ ❞↕♥❣ ✈➔ ❜✐➳♥ ✤ê✐ trü❝ ❣✐❛♦✳ ✶✷ ✷✳✷ ❈æ♥❣ t❤ù❝ ●r❡❡♥ ✷✳✷✳✶ ❈æ♥❣ t❤ù❝ ●r❡❡♥ t❤ù ♥❤➜t ✈➔ ❝æ♥❣ t❤ù❝ ●r❡❡♥ t❤ù ❤❛✐ ◆❤÷ ♠ët sü ♠ð ✤➛✉ ✤➸ ①➨t sü tç♥ t↕✐✱ ❜➙② ❣✐í ❝❤ó♥❣ t❛ s✉② r❛ ♠ët ✈➔✐ ❤➺ q✉↔ ①❛ ❤ì♥ ❝õ❛ ✤à♥❤ ❧þ ♣❤➙♥ ❦➻✱ ✤â ❧➔ ❝æ♥❣ t❤ù❝ ●r❡❡♥✳ ❈❤♦ Ω ❧➔ ♠ët ♠✐➲♥ ♠➔ ð ✤â ✤à♥❤ ❧þ ♣❤➙♥ ❦➻ ❝â t❤➸ →♣ ❞ö♥❣✱ ❣✐↔ ¯ ✳ ❈❤ó♥❣ t❛ ❝❤å♥ w = vDu tr♦♥❣ ❝æ♥❣ sû u ✈➔ v ❧➔ ❤➔♠ sè tr➯♥ C 2(Ω) t❤ù❝ ✭✶✳✹✮ ✤➸ ❝â ✤÷ñ❝ ❝æ♥❣ t❤ù❝ ●r❡❡♥ t❤ù ♥❤➜t✿ v(y)∆u(y)dy + Ω v(y) Du(y)Dv(y)dy = Ω ∂u(y) dSy , ∂µy ✭✷✳✸✮ ∂Ω tr♦♥❣ ✤â µy ❧➔ ✈❡❝tì ♣❤→♣ t✉②➳♥ ♥❣♦➔✐ ✤ì♥ ✈à t↕✐ y ∈ ∂Ω✳ ✣ê✐ ❝❤é u ✈➔ v tr♦♥❣ ✭✷✳✸✮ ✈➔ t❤ü❝ ❤✐➺♥ ♣❤➨♣ trø ❝❤ó♥❣ t❛ ✤÷ñ❝ ❝æ♥❣ t❤ù❝ ●r❡❡♥ t❤ù ❤❛✐✿ v(y)∆u(y) − u(y)∆v(y) dy = Ω v(y) ∂u(y) ∂v(y) − u(y) dSy . ∂µy ∂µy ∂Ω ✭✷✳✹✮ ✷✳✷✳✷ ◆❣❤✐➺♠ ❝ì ❜↔♥ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ▲❛♣❧❛❝❡ P❤÷ì♥❣ tr➻♥❤ ▲❛♣❧❛❝❡ ❝â r2−n ❧➔ ♥❣❤✐➺♠ ✈î✐ n > 2 ✈➔ logr ✈î✐ n = 2✱ tr♦♥❣ ✤â r ❧➔ ❦❤♦↔♥❣ ❝→❝❤ tø ✤✐➸♠ x ✤➳♥ ✤✐➸♠ y ✳ ❚✐➳♣ tö❝ tø ✭✷✳✹✮ ❝❤ó♥❣ t❛ ❝è ✤à♥❤ ✤✐➸♠ x tr♦♥❣ Ω ✈➔ ✤÷❛ ✈➔♦ ❤➔♠ sè s❛✉✿ Γ(x − y) = Γ(|x − y|) =  1  |x − y|2−n , n > 2,    n(2 − n)ωn     1 log|x − y|, 2π ✭✷✳✺✮ n = 2. ❑❤✐ ✤â✱ Γ(x − y) ①→❝ ✤à♥❤ ♥❤÷ tr➯♥ ❣å✐ ❧➔ ♥❣❤✐➺♠ ❝ì ❜↔♥ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ▲❛♣❧❛❝❡✳ ❍➔♠ Γ(x − y) ①→❝ ✤à♥❤ ✈î✐ ♠å✐ x = y✳ ✶✸ ❇➡♥❣ ❝→❝❤ t➼♥❤ t♦→♥ ✤ì♥ ❣✐↔♥ t❛ ❝â✿ ✈î✐ Dj = ∂y∂ Di Γ(x − y) = Dij Γ(x − y) = j 1 (xi − yi )|x − y|−n , nωn 1 |x − y|2 δij − n(xi − yi )(xj − yj ) |x − y|−n−2 . nωn ✭✷✳✻✮ ❘ã r➔♥❣ Γ ❧➔ ❤➔♠ ✤✐➲✉ ❤á❛ ✈î✐ x = y✳ ❈❤ó♥❣ t❛ ❝â ❝→❝ ÷î❝ ❧÷ñ♥❣ s❛✉ ✤è✐ ✈î✐ ✤↕♦ ❤➔♠✿ |Di Γ(x − y)| ≤ 1 |x − y|1−n , nωn |Dij Γ(x − y)| ≤ ✭✷✳✼✮ 1 |x − y|−n . ωn ✣➦❝ ❜✐➺t ❦❤✐ x = y✱ ❝❤ó♥❣ t❛ ❦❤æ♥❣ t❤➸ ❞ò♥❣ Γ tr♦♥❣ ✈✐➺❝ ✤➦t v(y) = Γ(x − y) ✈➔♦ ✤ç♥❣ ♥❤➜t t❤ù❝ ●r❡❡♥ t❤ù ❤❛✐ ✭✷✳✹✮✳ ▼ët ❝→❝❤ ✤➸ ✈÷ñt q✉❛ ✈➜♥ ✤➲ ♥➔② ❧➔ t❤❛② Ω ❜➡♥❣ Ω\B¯ρ✱ tr♦♥❣ ✤â Bρ = Bρ(x) ✈î✐ ρ ✤õ ♥❤ä✳ ❙❛✉ ✤â ❝❤ó♥❣ t❛ ❝â t❤➸ ❦➳t ❧✉➟♥ tø ✭✷✳✹✮ r➡♥❣✿ Γ∆udy = (Γ ∂u ∂Γ − u )dS + ∂µ ∂µ ∂Ω Ω\Bρ (Γ ∂u ∂Γ − u )dS. ∂µ ∂µ ∂Bρ ❱î✐ Γ ∂Bρ ∂u dS = Γ(ρ) ∂µ ∂u dS ∂µ ∂Bρ ≤ nωn ρn−1 Γ(ρ) sup |Du| → 0, Bρ ✈➔ ✶✹ ❦❤✐ ρ → 0, ✭✷✳✽✮ u ∂Γ dS = −Γ (ρ) ∂µ udS, ∂Bρ ∂Bρ ✭❈❤ó þ r➡♥❣ µ ❧➔ ✈➨❝tì ♣❤→♣ t✉②➳♥ ♥❣♦➔✐ ✤ì♥ ✈à ❝õ❛ Ω − Bρ✮ −1 = udS → −u(x), ✈î✐ ρ → 0. nω ρn−1 n ∂Bρ ❉♦ ✤â✱ ❝❤♦ ρ t✐➳♥ ✤➳♥ ✵ ð ❝æ♥❣ t❤ù❝ ✭✷✳✽✮ ❝❤ó♥❣ t❛ ❝â ❝æ♥❣ t❤ù❝ ●r❡❡♥✿ u(y) u(x) = ∂Γ ∂u(y) (x − y)−Γ(x − y) dSy ∂µy ∂µy ✭✷✳✾✮ ∂Ω Γ(x − y)∆u(y)dy, (x ∈ Ω). + Ω ◆➳✉ u ❧➔ ❤➔♠ ✤✐➲✉ ❤á❛ t❤➻ ❝❤ó♥❣ t❛ ♥❤➟♥ ✤÷ñ❝ ❝æ♥❣ t❤ù❝ ❜✐➸✉ ❞✐➵♥ s❛✉ ✤➙② t❤æ♥❣ q✉❛ ♥❣❤✐➺♠ ❝ì ❜↔♥ Γ(x − y) u(x) = u(y) ∂Γ ∂u(y) (x − y) − Γ(x − y) dSy , (x ∈ Ω). ∂µy ∂µy ✭✷✳✶✵✮ ∂Ω ✷✳✷✳✸ ❍➔♠ ●r❡❡♥ ¯ ∩ C 2 (Ω) t❤ä❛ ♠➣♥ ∆h(y) = 0 ❇➙② ❣✐í t❛ ❣✐↔ sû r➡♥❣ h(y) ∈ C 1(Ω) tr♦♥❣ Ω✳ ❑❤✐ ✤â✱ tø ❝æ♥❣ t❤ù❝ ●r❡❡♥ t❤ù ❤❛✐ t❛ ❝â✿ − u(y) ∂u(y) ∂h(y) − h(y) dSy = ∂µy ∂µy h(y)∆u(y)dy. ✭✷✳✶✶✮ Ω ∂Ω ✣➦t G(x, y) = Γ(x − y) + h(y)✱ tø ✭✷✳✾✮ ✈➔ ✭✷✳✶✶✮ ❝❤ó♥❣ t❛ ❝â ✤÷ñ❝ ♠ët ❝æ♥❣ t❤ù❝ tê♥❣ q✉→t ❤ì♥ ✈➲ ❝æ♥❣ t❤ù❝ ✤↕✐ ❞✐➺♥ ●r❡❡♥ u(x) = u(y) ∂G(x, y) ∂u(y) − G(x, y) dSy + ∂µy ∂µy G(x, y)∆u(y)dy. Ω ∂Ω ✶✺ ✭✷✳✶✷✮ ◆➳✉ ❝❤å♥ G(x, y) = 0 ❦❤✐ y ∈ ∂Ω ❝❤ó♥❣ t❛ ❝â ❝æ♥❣ t❤ù❝ ❜✐➸✉ ❞✐➵♥ s❛✉ ✤è✐ ✈î✐ ❤➔♠ ✤✐➲✉ ❤á❛ u(y) u(x) = ∂G(x, y) dSy . ∂µy ✭✷✳✶✸✮ ∂Ω ❍➔♠ G(x, y) ✤÷ñ❝ ❣å✐ ❧➔ ❤➔♠ ●r❡❡♥ ❝õ❛ ❜➔✐ t♦→♥ ❉✐r✐❝❤❧❡t tr➯♥ ♠✐➲♥ Ω✱ ✤æ✐ ❦❤✐ ❝á♥ ✤÷ñ❝ ❣å✐ ❧➔ ❤➔♠ ●r❡❡♥ ❧♦↕✐ ♠ët tr➯♥ Ω✳ ❈æ♥❣ t❤ù❝ ✭✷✳✶✸✮ ❝❤♦ ♣❤➨♣ t➻♠ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❉✐r✐❝❤❧❡t ❝❤♦ ❤➔♠ ✤✐➲✉ ❤á❛ ❦❤✐ ❜✐➳t ❣✐→ trà ❝õ❛ ♥â tr➯♥ ❜✐➯♥✳ ✷✳✸ ❍➔♠ ●r❡❡♥ ❝õ❛ ❜➔✐ t♦→♥ ❉✐r✐❝❤❧❡t tr♦♥❣ ❤➻♥❤ ❝➛✉✳ ❈æ♥❣ t❤ù❝ P♦✐ss♦♥ ✷✳✸✳✶ ❍➔♠ ●r❡❡♥ ❝õ❛ ❜➔✐ t♦→♥ ❉✐r✐❝❤❧❡t tr♦♥❣ ❤➻♥❤ ❝➛✉ ❑❤✐ ♠✐➲♥ Ω ❧➔ ❤➻♥❤ ❝➛✉✱ ❤➔♠ ●r❡❡♥ ❝â t❤➸ ①→❝ ✤à♥❤ ✤÷ñ❝ ♠ët ❝→❝❤ rã r➔♥❣ t❤❡♦ ♣❤÷ì♥❣ ♣❤→♣ ♣❤↔♥ ①↕ ✈➔ ❞➝♥ ✤➳♥ t➼❝❤ ♣❤➙♥ P♦✐ss♦♥ ♥ê✐ t✐➳♥❣ ✤↕✐ ❞✐➺♥ ❝❤♦ ❤➔♠ ✤✐➲✉ ❤á❛ tr➯♥ ♠ët ❤➻♥❤ ❝➛✉✳ ✣à♥❤ ♥❣❤➽❛ ✷✳✸✳✶✳ ●✐↔ sû BR = BR(0) ✈➔ x ∈ BR✱ x = 0✱ t❛ ❣å✐ x¯ = R2 x, |x|2 ✭✷✳✶✹✮ ❧➔ ✤✐➸♠ ✤è✐ ♥❣➝✉ ❝õ❛ x q✉❛ ∂BR✳ ⑩♥❤ ①↕ x → x¯ ✤÷ñ❝ ❣å✐ ❧➔ ♠ët ♣❤➨♣ ♥❣❤à❝❤ ✤↔♦ q✉❛ ♠➦t ❝➛✉ ∂BR✳ ❑þ ❤✐➺✉ ♣❤➛♥ tû ♥❣❤à❝❤ ✤↔♦ ❝õ❛ ✤✐➸♠ ✤➸ ❜ò ❝❤♦ ✤õ BR❀ ◆➳✉ x = 0✱ ❧➜② x¯ = ∞✳ ❙❛✉ ✤â✱ ❞➵ ❞➔♥❣ ①→❝ ✤à♥❤ ✤÷ñ❝ r➡♥❣ ❤➔♠ ●r❡❡♥ ❝❤♦ BR ✶✻ ✤÷ñ❝ ✈✐➳t ❜ð✐✿ G(x, y) = |x|   Γ(|x − y|) − Γ |x − y¯| ,   R     Γ(|x|) − Γ(R), =Γ y=0 y=0 |x|2 + |y|2 − 2xy − Γ ( |x||y| 2 ) + R2 − 2xy , R ✭✷✳✶✺✮ ✈î✐ ♠å✐ x, y ∈ BR, x = y✳ ❍➔♠ ●r❡❡♥ ①→❝ ✤à♥❤ ❜ð✐ ✭✷✳✶✺✮ ❝â t➼♥❤ ❝❤➜t ¯R . G(x, y) = G(y, x), G(x, y) ≤ 0, ✈î✐ x, y ∈ B ✭✷✳✶✻✮ ❍ì♥ ♥ú❛✱ ❜➡♥❣ ❝→❝❤ t➼♥❤ t♦→♥ trü❝ t✐➳♣ t↕✐ x ∈ ∂BR ✤↕♦ ❤➔♠ t❤æ♥❣ t❤÷í♥❣ ❝õ❛ ❤➔♠ G ✤÷ñ❝ ✈✐➳t ❜ð✐✿ ∂G ∂G R2 − |y|2 = = |x − y|−n ≥ 0. ∂µ ∂|x| nωn R ¯R ) ❧➔ ❤➔♠ ✤✐➲✉ ❤á❛✱ ❝❤ó♥❣ u ∈ C 2 (BR ) ∩ C 1 (B ❚ø ✤â ♥➳✉ ✭✷✳✶✸✮ t➼❝❤ ♣❤➙♥ P♦✐ss♦♥✿ R2 − |y|2 u(y) = nωn R udsx . |x − y|n ✭✷✳✶✼✮ t❛ ❝â ❜ð✐ ✭✷✳✶✽✮ ∂BR ❱➳ ♣❤↔✐ ❝õ❛ ❝æ♥❣ t❤ù❝ ✭✷✳✶✽✮ ✤÷ñ❝ ❣å✐ ❧➔ t➼❝❤ ♣❤➙♥ P♦✐ss♦♥ ❝õ❛ ❤➔♠ u✳ ▼ë✐ ❧➟♣ ❧✉➟♥ ✤ì♥ ❣✐↔♥ ❝❤♦ t❤➜② r➡♥❣ ❝æ♥❣ t❤ù❝ t➼❝❤ ♣❤➙♥ P♦✐ss♦♥ t✐➳♣ tö❝ ✤ó♥❣ ✈î✐ u ∈ C 2(BR) ∩ C 0(B¯R)✳ ❈❤ó þ r➡♥❣ ❜➡♥❣ ❝→❝❤ ❝❤♦ y = 0✱ ❝❤ó♥❣ t❛ ❧↕✐ ❝â ❝æ♥❣ t❤ù❝ ❝→❝ ✤à♥❤ ❧þ ✈➲ ❣✐→ trà tr✉♥❣ ❜➻♥❤ ❝❤♦ ❝→❝ ❤➔♠ ✤✐➲✉ ❤á❛✳ ❚r♦♥❣ t❤ü❝ t➳ t➜t ❝↔ ❝→❝ ✤à♥❤ ❧þ tr÷î❝ ❝õ❛ ❝❤÷ì♥❣ ♥➔② ❝â t❤➸ ✤÷ñ❝ s✉② r❛ ❧➔ ❤➺ q✉↔ ❝õ❛ ✭✷✳✶✸✮ ✈î✐ Ω = BR(0)✳ ❚❤✐➳t ❧➟♣ sü tç♥ t↕✐ ❝õ❛ ❜➔✐ t♦→♥ ❉✐r✐❝❤❧❡t ❝ê ✤✐➸♥ tr➯♥ ❤➻♥❤ ❝➛✉ ❝❤ó♥❣ t❛ ❝➛♥ ❦➳t q✉↔ ♥❣÷ñ❝ ❧↕✐ ❝õ❛ ❝æ♥❣ t❤ù❝ ✭✷✳✶✽✮✱ ✈➔ ❜➙② ❣✐í ❝❤ó♥❣ t❛ ❝❤ù♥❣ ♠✐♥❤ ✤✐➲✉ ♥➔②✳ ✶✼ ✷✳✸✳✷ ❈æ♥❣ t❤ù❝ P♦✐ss♦♥ ✣à♥❤ ❧➼ ✷✳✸✳✶✳ ❈❤♦ B = BR(0) ✈➔ ϕ ❧➔ ♠ët ❤➔♠ ❧✐➯♥ tö❝ tr➯♥ ∂B ✳ ❑❤✐ ✤â ❤➔♠ u ✤÷ñ❝ ①→❝ ✤à♥❤ ❜ð✐ ❝æ♥❣ t❤ù❝ s❛✉ u(x) =       ϕ(y)dy R2 − |x|2 ,x ∈ B nωn R ∂B |x − y|n      ϕ(x), ✭✷✳✶✾✮ x ∈ ∂B ¯ t❤ä❛ ♠➣♥ ∆u = 0 tr➯♥ B ✈➔ ✈î✐ ♠å✐ x0 ∈ ∂Ω s➩ t❤✉ë❝ C 2(B) ∩ C 0(B) t❤➻ u(x) → u(x0) ❦❤✐ x → x0✳ ❈❤ù♥❣ ♠✐♥❤✳ ❘ã r➔♥❣ u t❤ä❛ ♠➣♥ ✭✷✳✶✾✮ ❧➔ ❤➔♠ ✤✐➲✉ ❤á❛ tr➯♥ B ✳ ❚❤➟t ✈➟②✱ ❞♦ ❤➔♠ G ✈➔ ∂G ❧➔ ✤✐➲✉ ❤á❛ tr➯♥ x✱ ❤♦➦❝ ♥â ❝â t❤➸ ①→❝ ∂µ ✤à♥❤ ❜ð✐ t➼♥❤ t♦→♥ trü❝ t✐➳♣✳ ❚❤✐➳t ❧➟♣ sü ❧✐➯♥ tö❝ ❝õ❛ u tr➯♥ ∂B ✱❝❤ó♥❣ t❛ sû ❞ö♥❣ ❝æ♥❣ t❤ù❝ P♦✐ss♦♥ ✭✷✳✶✽✮ ❝❤♦ tr÷í♥❣ ❤ñ♣ ✤➦❝ ❜✐➺t u = 1 ✤➸ ✤÷ñ❝ ✤ç♥❣ ♥❤➜t t❤ù❝✿ K(x, y)dsy = 1, ✈î✐ ♠å✐ x ∈ B, ✭✷✳✷✵✮ ∂B tr♦♥❣ ✤â K ❧➔ ❤↕t ♥❤➙♥ P♦✐ss♦♥✱ K(x, y) = R2 − |x|2 , x ∈ B, y ∈ ∂B. nωn R|x − y|n ✭✷✳✷✶✮ ❉➽ ♥❤✐➯♥ t➼❝❤ ♣❤➙♥ tr➯♥ ✭✷✳✷✵✮ ❝â t❤➸ ✤→♥❤ ❣✐→ trü❝ t✐➳♣ ♥❤÷♥❣ ♥â ❧➔ ♠ët t➼♥❤ t♦→♥ ♣❤ù❝ t↕♣✳ ❇➙② ❣✐í ❝❤ó♥❣ t❛ ❝❤♦ x0 ∈ ∂B ✈➔ ❧➔ ♠ët sè ❞÷ì♥❣ tò② þ✱ ❝❤å♥ δ > 0 ✤➸ |ϕ(x) − ϕ(x0 )| < ♥➳✉ |x − x0 | < δ ✈➔ ❝❤♦ |ϕ| ≤ M tr♦♥❣ ✶✽ ∂B ✳ ❑❤✐ ✤â ♥➳✉ |x − x0| < 2δ ✱ t❤❡♦ ✭✷✳✶✾✮ ✈➔ ✭✷✳✷✵✮ ❝❤ó♥❣ t❛ ❝â✿ K(x, y)(ϕ(y) − ϕ(x0 ))dsy |u(x) − u(x0 )| = ∂B K(x, y)|ϕ(y) − ϕ(x0 )|dsy ≤ |y−x0 |≤δ K(x, y)|ϕ(y) − ϕ(x0 )|dsy + ≤ + |y−x0 |>δ 2M (R2 − |x|2 )Rn−2 (δ/2)n . ◆➳✉ ❜➙② ❣✐í |x − x0| ❧➔ ✤õ ♥❤ä✱ rã r➔♥❣ ❧➔ |u(x) − u(x0)| < 2 ✈➔ ¯ ❧➔ ✤✐➲✉ ❝➛♥ t➻♠✳ ❞♦ ✤â u ❧➔ ❤➔♠ ❧✐➯♥ tö❝ t↕✐ x0✳ ❉♦ ✤â u ∈ C 0(B) ❈❤ó þ r➡♥❣ ✤è✐ sè tr÷î❝ ✤â ❧➔ ✤à❛ ♣❤÷ì♥❣✱ ❝â ♥❣❤➽❛ ❧➔ ♥➳✉ ϕ ❝❤➾ ❣✐î✐ ❤↕♥ ✈➔ ❦❤↔ t➼❝❤ tr➯♥ ∂B ✈➔ ❧✐➯♥ tö❝ t↕✐ x0 t❤➻ u(x) → u(x0 ) ❦❤✐ x → x0 . ✷✳✹ ✣à♥❤ ❧þ ❤ë✐ tö ✷✳✹✳✶ ✣✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ✤➸ ♠ët ❤➔♠ ❧➔ ✤✐➲✉ ❤á❛ ❇➙② ❣✐í ❝❤ó♥❣ t❛ ①❡♠ ①➨t ♠ët sè ❤➺ q✉↔ trü❝ t✐➳♣ ❝õ❛ ❝æ♥❣ t❤ù❝ t➼❝❤ ♣❤➙♥ P♦✐ss♦♥✳ ❚✉② ♥❤✐➯♥ ❜❛ ✤à♥❤ ❧þ ❞÷î✐ ✤➙② s➩ ❦❤æ♥❣ ✤÷ñ❝ ②➯✉ ❝➛✉ ❝❤♦ sü ♣❤→t tr✐➸♥ s❛✉✳ ✣➛✉ t✐➯♥ ❝❤ó♥❣ t❛ ❝â t❤➸ t❤➜② ❤➔♠ ✤✐➲✉ ❤á❛ ✤÷ñ❝ ✤➦❝ tr÷♥❣ ❜ð✐ ❣✐→ ❣✐→ trà tr✉♥❣ ❜➻♥❤ ❝õ❛ ♥â✳ ✣à♥❤ ❧➼ ✷✳✹✳✶✳ ▼ët ❤➔♠ u tr➯♥ C 0(Ω) ❧➔ ❤➔♠ ✤✐➲✉ ❤á❛ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ tr♦♥❣ ♠å✐ ❤➻♥❤ ❝➛✉ B ⊂ BR(y) ⊂⊂ Ω ♥â t❤ä❛ ♠➣♥ ❝æ♥❣ t❤ù❝ ❣✐→ trà tr✉♥❣ ❜➻♥❤✿ u(y) = 1 nωn Rn−1 uds. ∂B ✶✾ ✭✷✳✷✷✮ ❈❤ù♥❣ ♠✐♥❤✳ ❚❤❡♦ ✣à♥❤ ❧þ ✷✳✸✳✶✱ tç♥ t↕✐ ♠ët ❤➔♠ ✤✐➲✉ ❤á❛ h tr➯♥ ❜➜t ❦ý ❤➻♥❤ ❝➛✉ B ⊂⊂ Ω✱ ♥❤÷ ✈➟② h = u tr♦♥❣ ∂B ✳ ❳➨t ❤✐➺✉ w = u − h✱ s❛✉ ✤â s➩ ✤÷ñ❝ ♠ët ❤➔♠ ✤→♣ ù♥❣ ✤÷ñ❝ ❝→❝ ❣✐→ trà tr✉♥❣ ❜➻♥❤ tr♦♥❣ ❜➜t ❦ý ❤➻♥❤ ❝➛✉ tr➯♥ B ✳ ❉♦ ✤â ♥❣✉②➯♥ ❧þ ❝ü❝ ✤↕✐ ✈➔ ❦➳t q✉↔ ❞✉② ♥❤➜t ❝õ❛ ✣à♥❤ ❧þ ✶✳✸✳✶✱ ✶✳✸✳✷ ✈➔ ✶✳✸✳✸ →♣ ❞ö♥❣ ✈î✐ w✱ tø ❜➜t ✤➥♥❣ t❤ù❝ ❣✐→ trà tr✉♥❣ ❜➻♥❤ ✤➣ ❝❤➾ r❛ t➼♥❤ ❝❤➜t ❝õ❛ ❤➔♠ ✤✐➲✉ ❤á❛ ✤÷ñ❝ sû ❞ö♥❣ tr➯♥ ♣❤➨♣ ❧➜② ✤↕♦ ❤➔♠ ❝õ❛ ❝❤ó♥❣✳ ❉♦ ✤â w = 0 tr➯♥ B ✈➔ ❞♦ ✤â ❤➔♠ u ♣❤↔✐ ✤✐➲✉ ❤á❛ tr➯♥ Ω✳ ✷✳✹✳✷ ❈→❝ ✤à♥❤ ❧þ ❤ë✐ tö ✣à♥❤ ❧➼ ✷✳✹✳✷✳ ●✐î✐ ❤↕♥ ❝õ❛ ❞➣② ❤ë✐ tö ✤➲✉ ❝→❝ ❤➔♠ ✤✐➲✉ ❤á❛ ❧➔ ❤➔♠ ✤✐➲✉ ❤á❛✳ ❚❤❡♦ ✣à♥❤ ❧þ ✷✳✹✳✷✱ ♥➳✉ {un} ❧➔ ❞➣② ❝õ❛ ❤➔♠ ✤✐➲✉ ❤á❛ tr➯♥ ♠✐➲♥ ❜à ❝❤➦♥ Ω✱ ✈î✐ ❣✐→ trà {ϕn} ❝â ❣✐î✐ ❤↕♥ ❧✐➯♥ tö❝ ♠➔ ❤ë✐ tö ✤➲✉ tr♦♥❣ ∂Ω ✤➳♥ ❤➔♠ ϕ✱ s❛✉ ✤â ❞➣② {un } ❤ë✐ tö ✤➲✉ ✭ t❤❡♦ ♥❣✉②➯♥ ❧þ ❝ü❝ ✤↕✐✮ ✤➳♥ ❤➔♠ ✤✐➲✉ ❤á❛ ❝â ❣✐î✐ ❤↕♥ ❧➔ ❣✐→ trà ϕ tr♦♥❣ ∂Ω✳ ❚❤❡♦ ❜➜t ✤➥♥❣ t❤ù❝ ❣✐→ trà tr✉♥❣ ❜➻♥❤ ❝õ❛ ❍❛r♥❛❝❦✱ ✣à♥❤ ❧þ ✷✳✶✳✶✱ ❝❤ó♥❣ t❛ ❝ô♥❣ ❝â t❤➸ s✉② r❛ ✣à♥❤ ❧þ ✷✳✹✳✷✱ ✤à♥❤ ❧þ ❤ë✐ tö ❍❛r♥❛❝❦✳ ✣à♥❤ ❧➼ ✷✳✹✳✸✳ ❈❤♦ ❞➣② {un} ❧➔ ❞➣② ✤ì♥ ✤✐➺✉ t➠♥❣ ❝→❝ ❤➔♠ ✤✐➲✉ ❤á❛ tr➯♥ ♠ët ♠✐➲♥ Ω ✈➔ ❣✐↔ sû t↕✐ ♠ët ✤✐➸♠ y ∈ Ω✱ ❞➣② {un (y)} ❧➔ ❜à ❝❤➦♥✳ ❑❤✐ ✤â ❞➣② ❤ë✐ tö ✤➲✉ tr➯♥ ♠✐➲♥ ❝♦♥ ❜➜t ❦ý ❜à ❝❤➦♥ Ω ⊂⊂ Ω ✤➳♥ ♠ët ❤➔♠ ✤✐➲✉ ❤á❛✳ ❈❤ù♥❣ ♠✐♥❤✳ ❉➣② {un(y)} ❤ë✐ tö✳ ❚❤➟t ✈➟②✱ ❝❤♦ tò② þ > 0✱ tç♥ t↕✐ ♠ët sè N s❛♦ ❝❤♦ 0 ≤ um (y) − un (y) < ✈î✐ ♠å✐ N < n ≤ m. ◆❤÷♥❣ ❦❤✐ ✤â t❤❡♦ ✣à♥❤ ❧þ ✷✳✶✳✶✱ ❝❤ó♥❣ t❛ ❝â✿ sup |um (x) − un (x)| < C. , ✷✵ ✈î✐ ❤➡♥❣ sè C ♣❤ö t❤✉ë❝ tr♦♥❣ Ω ✈➔ Ω✳ ❉♦ ✤â {un} ❤ë✐ tö ✤➲✉ ✈➔ t❤❡♦ ✣à♥❤ ❧þ ✷✳✹✳✷✱ ❣✐î✐ ❤↕♥ ❝õ❛ ❤➔♠ ❧➔ ✤✐➲✉ ❤á❛✳ ✷✳✺ ✣→♥❤ ❣✐→ ❜➯♥ tr♦♥❣ ♠✐➲♥ ✤è✐ ✈î✐ ❝→❝ ✤↕♦ ❤➔♠ ❝õ❛ ❤➔♠ ✤✐➲✉ ❤á❛ ✷✳✺✳✶ ✣→♥❤ ❣✐→ ❜➯♥ tr♦♥❣ ♠✐➲♥ ✤è✐ ✈î✐ ✤↕♦ ❤➔♠ ❝➜♣ ✶ ❇➡♥❣ ❝→❝❤ ❧➜② ✤↕♦ ❤➔♠ trü❝ t✐➳♣ ❝õ❛ t➼❝❤ ♣❤➙♥ P♦✐ss♦♥ ❝â t❤➸ t❤✉ ✤÷ñ❝ ÷î❝ t➼♥❤ ❜➯♥ tr♦♥❣ ❝õ❛ ❝→❝ ❞➝♥ s✉➜t ❝❤♦ ❤➔♠ ✤✐➲✉ ❤á❛✳ ◆❣♦➔✐ r❛ ❝→❝ ÷î❝ t➼♥❤ ♥❤÷ ✈➟② ❝ô♥❣ t❤❡♦ ✤à♥❤ ❧þ ❣✐→ trà tr✉♥❣ ❜➻♥❤✳ ❈❤♦ u ❧➔ ✤✐➲✉ ❤á❛ tr➯♥ Ω ✈➔ B = BR(y) ⊂⊂ Ω✳ ●r❛❞✐❡♥ ❝õ❛ ❤➔♠ u✱ Du ❝ô♥❣ ❧➔ ✤✐➲✉ ❤á❛ tr➯♥ Ω t❤❡♦ ✤à♥❤ ❧þ ❣✐→ trà tr✉♥❣ ❜➻♥❤ ✈➔ ✤à♥❤ ❧þ ♣❤➙♥ ❦ý ♠➔✿ Du(y) = 1 ωn R n Dudx = 1 ωn R n B |Du(y)| ≤ uvds, B n sup |u|, R ∂B ✈➔ ❞♦ ✤â t❛ ♥❤➟♥ ✤÷ñ❝ ❝æ♥❣ t❤ù❝ s❛✉ ✤➙② ✤→♥❤ ❣✐→ ❜➯♥ tr♦♥❣ ♠✐➲♥ ✤è✐ ✈î✐ ✤↕♦ ❤➔♠ ❝➜♣ ♠ët |Du(y)| ≤ n sup |u|, dy Ω ✭✷✳✷✸✮ tr♦♥❣ ✤â dy = dist(y, ∂Ω)✳ ❇➡♥❣ ❝→❝❤ ❧➜② t➼❝❤ ♣❤➙♥ ❧✐➯♥ t✐➳♣ t÷ì♥❣ ù♥❣ ❝õ❛ ÷î❝ t➼♥❤ ✭✷✳✷✸✮ tr➯♥ ❝→❝ ❤➻♥❤ ❝➛✉ ❧ç♥❣ ♥❤❛✉✱ ❝→❝❤ ✤➲✉ ♥❤❛✉ ❝❤ó♥❣ t❛ ✤÷ñ❝ ❝→❝ ÷î❝ t➼♥❤ ❝❤♦ ❝→❝ ❞➝♥ s✉➜t ❜➟❝ ❝❛♦✳ ✷✳✺✳✷ ✣→♥❤ ❣✐→ ❜➯♥ tr♦♥❣ ♠✐➲♥ ✤è✐ ✈î✐ ✤↕♦ ❤➔♠ ❜➜t ❦ý ✣à♥❤ ❧➼ ✷✳✺✳✶✳ ✷✶ ❈❤♦ u ❧➔ ❤➔♠ ✤✐➲✉ ❤á❛ tr♦♥❣ Ω ✈➔ ❝❤♦ Ω ❧➔ t➟♣ ❝♦♥ ❝♦♠♣❛❝t ❜➜t ❦ý ❝õ❛ Ω✳ ❑❤✐ ✤â ❝❤♦ ♠ët ❝❤➾ sè α ❜➜t ❦ý ❝❤ó♥❣ t❛ ❝â sup |Dα u| ≤ Ω n|α| d |α| sup |u|, Ω ✭✷✳✷✹✮ tr♦♥❣ ✤â d = dist(Ω , ∂Ω)✱ α = (α1 , α2 , ..., αn )✱ αj ∈ N✱ Dα = D1α1 D2α2 ...Dnαn ✱ |α| = α1 + α2 + ... + αn ✳ ▼ët ❤➺ q✉↔ trü❝ t✐➳♣ ❝õ❛ r➔♥❣ ❜✉ë❝ ✭✷✳✷✹✮ ❧➔ sü ❧✐➯♥ tö❝ ✤➲✉ tr➯♥ ♠✐➲♥ ❝♦♥ ❝õ❛ ❝→❝ ❞➝♥ s✉➜t ❜➜t ❦ý ❜à ❝❤➦♥ ❝õ❛ ❤➔♠ ✤✐➲✉ ❤á❛✳ ❉♦ ✤â t❤❡♦ ✤à♥❤ ❧þ ❆r③❡❧❛✱ ❝❤ó♥❣ t❛ t❤➜② ❝→❝ ❤➔♠ ✤✐➲✉ ❤á❛ ❜à ❝❤➦♥ ❤➻♥❤ t❤➔♥❤ ♠ët ❤å✳ ✣à♥❤ ❧➼ ✷✳✺✳✷✳ ▼ët ❞➣② ❜➜t ❦ý ❝õ❛ ❝→❝ ❤➔♠ ✤✐➲✉ ❤á❛ tr♦♥❣ ♠ët ♠✐➲♥ Ω ❝❤ù❛ ♠ët ❞➣② ❝♦♥ ❤ë✐ tö ✤➲✉ tr♦♥❣ ♠✐➲♥ ❝♦♥ ❝♦♠♣❛❝t ❝õ❛ Ω ✤➳♥ ♠ët ❤➔♠ ✤✐➲✉ ❤á❛✳ ✣à♥❤ ❧þ ✷✳✺✳✷ ✤÷ñ❝ s✉② r❛ trü❝ t✐➳♣ tø ✣à♥❤ ❧þ ✷✳✹✳✷✱ ✤à♥❤ ❧þ ❤ë✐ tö✳ ✷✳✻ ❇➔✐ t♦→♥ ❉✐r✐❝❤❧❡t✳ P❤÷ì♥❣ ♣❤→♣ ❤➔♠ ✤✐➲✉ ❤á❛ ❞÷î✐ ❚❛ ①➨t ❜➔✐ t♦→♥ ❉✐r✐❝❤❧❡t ❝❤♦ ♣❤÷ì♥❣ tr➻♥❤ ▲❛♣❧❛❝❡✿ ❈❤♦ Ω ❧➔ ♠✐➲♥ ❜à ❝❤➦♥ tr➯♥ Rn✱t➻♠ ♠ët ❤➔♠ u : Ω → R t❤ä❛ ♠➣♥ ¯ u ∈ C 2 (Ω) ∩ C(Ω), ✈➔ tr♦♥❣ Ω u = ϕ, tr➯♥ ∂Ω tr♦♥❣ ✤â ϕ ∈ C(∂Ω) ❧➔ ❤➔♠ ❝❤♦ tr÷î❝✳ ∆u = 0, ✷✷ , ✷✳✻✳✶ ▼ð rë♥❣ ❦❤→✐ ♥✐➺♠ ❤➔♠ ❞÷î✐ ✤✐➲✉ ❤á❛ ✈➔ ❤➔♠ tr➯♥ ✤✐➲✉ ❤á❛ ❇➙② ❣✐í✱ ❝❤ó♥❣ t❛ ✤➦t r❛ ♠ët ✈➜♥ ✤➲ ❧➔ ✤➸ t✐➳♣ ❝➟♥ ✈î✐ ❝➙✉ ❤ä✐ sü tç♥ t↕✐ ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ ❝õ❛ ❜➔✐ t♦→♥ ❉✐r✐❝❤❧❡t ❝ê ✤✐➸♥ tr➯♥ ♠✐➲♥ tò② þ ❜à ❝❤➦♥✳ ✣➸ ❣✐↔✐ ✈➜♥ ✤➲ tr➯♥ ❝❤ó♥❣ t❛ sû ❞ö♥❣ ♣❤÷ì♥❣ ♣❤→♣ P❡rr♦♥ ❝õ❛ ❝→❝ ❤➔♠ ✤✐➲✉ ❤á❛ ❞÷î✐ [P E] ♠➔ ❝❤õ ②➳✉ ❞ü❛ tr➯♥ ♥❣✉②➯♥ ❧þ ❝ü❝ ✤↕✐ ✈➔ ❦❤↔ ♥➠♥❣ ❣✐↔✐ ✤÷ñ❝ ❝õ❛ ❜➔✐ t♦→♥ ❉✐r✐❝❤❧❡t tr➯♥ ❤➻♥❤ ❝➛✉✳ P❤÷ì♥❣ ♣❤→♣ ♥➔② ❝â ♠ët sè ✤➦❝ ✤✐➸♠ ❤➜♣ ❞➝♥ ❧➔ ✤ì♥ ❣✐↔♥✱ ♣❤➙♥ t➼❝❤ ❝→❝ ✈➜♥ ✤➲ tç♥ t↕✐ ❜➯♥ tr♦♥❣ ❝õ❛ ❝→❝❤ ①û ❧þ ✤✐➸♠ ❜✐➯♥ ❝õ❛ ❝→❝ ♥❣❤✐➺♠ ✈➔ ❝â t❤➸ ❞➵ ❞➔♥❣ ♠ð rë♥❣ ✤➳♥ ❧î♣ t❤ù ❤❛✐ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❊❧✐♣t✐❝✳ ❈â ❝→❝❤ t✐➳♣ ❝➟♥ ❦❤→❝ ❝ô♥❣ ✤÷ñ❝ ❜✐➳t ✤➳♥ ✈➲ ✤à♥❤ ❧þ sü tç♥ t↕✐ ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ ♥❤÷ ♣❤÷ì♥❣ ♣❤→♣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥✱ ✈➼ ❞ö ♥❤÷ tr♦♥❣ ❝→❝ ❝✉è♥ s→❝❤ [KE2] [GU ]✱ ✈➔ ♣❤÷ì♥❣ ♣❤→♣ ❜✐➳♥ ♣❤➙♥ ❤♦➦❝ ♣❤➨♣ ①➜♣ ①➾ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt✳ ✣à♥❤ ♥❣❤➽❛ C 0(Ω) ❤➔♠ ✤✐➲✉ ❤á❛ ❞÷î✐ ✈➔ ❤➔♠ ✤✐➲✉ ❤á❛ tr➯♥ ✤÷ñ❝ ❦❤→✐ q✉→t ♥❤÷ s❛✉✳ ✣à♥❤ ♥❣❤➽❛ ✷✳✻✳✶✳ ▼ët ❤➔♠ u tr➯♥ C 0(Ω) ✤÷ñ❝ ❣å✐ ❧➔ ❤➔♠ ❞÷î✐ ✤✐➲✉ ❤á❛ ✭❤➔♠ tr➯♥ ✤✐➲✉ ❤á❛✮ tr➯♥ Ω ♥➳✉ ✈î✐ ♠å✐ ❤➻♥❤ ❝➛✉ B ⊂⊂ Ω ✈➔ ♠å✐ ❤➔♠ ✤✐➲✉ ❤á❛ h tr➯♥ B t❤ä❛ ♠➣♥ u ≤ (≥) h tr➯♥ ∂B ✱ ❝❤ó♥❣ t❛ ❝â u ≤ (≥) h tr♦♥❣ B✳ ❈→❝ t➼♥❤ ❝❤➜t ❝õ❛ C 0(Ω) ❝→❝ ❤➔♠ ❞÷î✐ ✤✐➲✉ ❤á❛ ❞➵ ❞➔♥❣ ✤÷ñ❝ t❤✐➳t ❧➟♣✳ ✷✳✻✳✷ ❈→❝ t➼♥❤ ❝❤➜t ❝õ❛ ❤➔♠ ❞÷î✐ ✤✐➲✉ ❤á❛ ✈➔ ❤➔♠ tr➯♥ ✤✐➲✉ ❤á❛ ♠ð rë♥❣ ❆✳❚➼♥❤ ❝❤➜t ✶✿ ◆➳✉ u ❧➔ ❤➔♠ ❞÷î✐ ✤✐➲✉ ❤á❛ tr➯♥ ♠ët ♠✐➲♥ Ω✱ t❤➻ ♥â t❤ä❛ ♠➣♥ ♥❣✉②➯♥ ❧þ ❝ü❝ ✤↕✐ tr➯♥ Ω❀ ✈➔ ♥➳✉ v ❧➔ tr➯♥ ✤✐➲✉ ❤á❛ tr➯♥ ♠✐➲♥ ❜à ❝❤➦♥ Ω ✈î✐ v ≥ u tr♦♥❣ ∂Ω✱ t❤➻ ❤♦➦❝ ❧➔ v > u ❦❤➢♣ Ω ❤♦➦❝ v ≡ u✳ ✷✸ ✣➸ ❝❤ù♥❣ ♠✐♥❤ sü ❦❤➥♥❣ ✤à♥❤ tr➯♥✱ ❣✐↔ sû ♥❣÷ñ❝ ❧↕✐✱ t↕✐ ♠é✐ ✤✐➸♠ x0 ∈ Ω ❝❤ó♥❣ t❛ ❝â✿ (u − v)(x0 ) = sup(u − v) = M ≥ 0. Ω ❱➔ ❝❤ó♥❣ t❛ ❝â t❤➸ ❣✐↔ t❤✐➳t ❝â ❤➻♥❤ ❝➛✉ B = B(x0) ❦➼♥ s❛♦ ❝❤♦ u − v = M tr➯♥ ∂B. ❈❤♦ u¯✱ v¯ ❧➔ ❝→❝ ❤➔♠ ✤✐➲✉ ❤á❛ ✈➔ ❧➛♥ ❧÷ñt ❜➡♥❣ u✱ v tr➯♥ ∂B ✭✣à♥❤ ❧þ ✷✳✸✳✶✮✱ t❛ t❤➜② r➡♥❣✿ M ≥ sup(¯ u − v¯) ≥ (¯ u − v¯)(x0 ) ≥ (u − v)(x0 ) = M. ∂B ❱➔ ❞♦ ✤â ✤➥♥❣ t❤ù❝ ①↔② r❛ ❦❤➢♣ ♥ì✐✳ ❚❤❡♦ ♥❣✉②➯♥ ❧þ ❝ü❝ ✤↕✐ ✤è✐ ✈î✐ ❝→❝ ❤➔♠ ✤✐➲✉ ❤á❛ ✭✣à♥❤ ❧þ ✶✳✸✳✶✮ t❛ ❝â u¯ − v¯ = M tr♦♥❣ B ✈➔ ❞♦ ✤â u − v ≡ M tr♦♥❣ ∂B ✱ ✤✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈î✐ ❝→❝❤ ❝❤å♥ ❝õ❛ B ✳ ❇✳❚➼♥❤ ❝❤➜t ✷✿ ❈❤♦ u ❧➔ ❤➔♠ ❞÷î✐ ✤✐➲✉ ❤á❛ tr➯♥ Ω ✈➔ B ❧➔ ❤➻♥❤ ❝➛✉ ❝❤ù❛ trå♥ tr♦♥❣ Ω✳ ❑þ ❤✐➺✉ u¯ ❧➔ ❤➔♠ ✤✐➲✉ ❤á❛ tr➯♥ B ✭✤➣ ❝❤♦ ❜ð✐ t➼❝❤ ♣❤➙♥ P♦✐ss♦♥ ❝õ❛ u tr➯♥ ∂B ✮ t❤ä❛ ♠➣♥ u¯ = u tr➯♥ ∂B ✳ ❈❤ó♥❣ t❛ ✤à♥❤ ♥❣❤➽❛ tr♦♥❣ Ω ❤➔♠ ♥➙♥❣ ✤✐➲✉ ❤á❛ ❝õ❛ u ✭tr➯♥ B ✮ ❜➡♥❣✿ u¯(x), x ∈ B, U (x) = ✭✷✳✷✺✮ u(x), x ∈ Ω − B. ❑❤✐ ✤â ❤➔♠ U ❝ô♥❣ ❧➔ ❤➔♠ ❞÷î✐ ✤✐➲✉ ❤á❛ tr♦♥❣ Ω✳ ❳➨t ♠ët ❤➻♥❤ ❝➛✉ tò② þ B ⊂⊂ Ω ✈➔ ❝❤♦ h ❧➔ ❤➔♠ ✤✐➲✉ ❤á❛ tr♦♥❣ B t❤ä❛ ♠➣♥ h ≥ U tr➯♥ ∂B ✳ ❚ø u ≤ U tr♦♥❣ B ❝❤ó♥❣ t❛ ❝â u ≤ h tr♦♥❣ B ✈➔ ❞♦ ✤â U ≤ h tr♦♥❣ B − B ✳ ❈ô♥❣ tø U ❧➔ ❤➔♠ ✤✐➲✉ ❤á❛ tr♦♥❣ B ✱ ❝❤ó♥❣ t❛ ❝â t❤❡♦ ♥❣✉②➯♥ ❧þ ❝ü❝ ✤↕✐ U ≤ h tr♦♥❣ B ∩ B ✳ ❉♦ ✤â U ≤ h tr♦♥❣ B ✈➔ U ❧➔ ❤➔♠ ✤✐➲✉ ❤á❛ ❞÷î✐ tr♦♥❣ Ω✳ ❈✳❚➼♥❤ ❝❤➜t ✸✿ ❈❤♦ ❝→❝ ❤➔♠ u1, u2, ..., uN ❧➔ ❝→❝ ❤➔♠ ❞÷î✐ ✤✐➲✉ ❤á❛ tr♦♥❣ Ω✳ ❚❤➻ ❤➔♠ u(x) = max{u1(x), ..., uN (x)} ❝ô♥❣ ❧➔ ❤➔♠ ❞÷î✐ ✤✐➲✉ ❤á❛ tr♦♥❣ Ω✳ ✣➙② ❧➔ ♠ët ❤➺ q✉↔ ❝õ❛ ✤à♥❤ ♥❣❤➽❛ ❤➔♠ ❞÷î✐ ✤✐➲✉ ❤á❛✳ ✷✹ ❚÷ì♥❣ ù♥❣ ❦➳t q✉↔ ❝❤♦ ❤➔♠ tr➯♥ ✤✐➲✉ ❤á❛ ❝â ✤÷ñ❝ ❜➡♥❣ ❝→❝❤ t❤❛② u ❜ð✐ −u tr♦♥❣ ❝→❝ t➼♥❤ ❝❤➜t ✭✶✮✱ ✭✷✮ ✈➔ ✭✸✮✳ ✷✳✻✳✸ P❤÷ì♥❣ ♣❤→♣ P❡rr♦♥ ✭P❤÷ì♥❣ ♣❤→♣ ❤➔♠ ✤✐➲✉ ❤á❛ ❞÷î✐✮ ❇➙② ❣✐í ❝❤♦ Ω ❧➔ ❜à ❝❤➦♥ ✈➔ ϕ ❧➔ ❤➔♠ ❜à ❝❤➦♥ tr➯♥ ∂Ω✳ ▼ët ❤➔♠ ¯ ✤÷ñ❝ ❣å✐ ❧➔ ❤➔♠ ❞÷î✐ ✤✐➲✉ ❤á❛ ✤è✐ ✈î✐ ϕ ♥➳✉ ❞÷î✐ ✤✐➲✉ ❤á❛ u ∈ C0(Ω) ¯ ♥â t❤ä❛ ♠➣♥ u ≤ ϕ tr➯♥ ∂Ω✳ ❚÷ì♥❣ tü ❤➔♠ tr➯♥ ✤✐➲✉ ❤á❛ ❝õ❛ C0(Ω) ✤÷ñ❝ ❣å✐ ❧➔ ❤➔♠ tr➯♥ ✤✐➲✉ ❤á❛ ✤è✐ ✈î✐ ϕ ♥➳✉ ♥â t❤ä❛ ♠➣♥ u ≥ ϕ tr➯♥ ∂Ω✳ ❚❤❡♦ ♥❣✉②➯♥ ❧þ ❝ü❝ ✤↕✐ ♠å✐ ❤➔♠ ❞÷î✐ ✤➲✉ ♥❤ä ❤ì♥ ❤♦➦❝ ❜➡♥❣ ♠å✐ ❤➔♠ tr➯♥✳ ✣➦❝ ❜✐➺t✱ ❤➔♠ ❤➡♥❣ ≤ inf ϕ (≥ sup ϕ) ❧➔ ♥❤ú♥❣ ❤➔♠ ❞÷î✐ ∂Ω ∂Ω ✭❤➔♠ tr➯♥✮✳ ❑þ ❤✐➺✉ Sϕ ❧➔ t➟♣ ❤ñ♣ ❝→❝ ❤➔♠ ❞÷î✐ ✤✐➲✉ ❤á❛ ✤è✐ ✈î✐ ϕ✳ ❈ì sð ❝õ❛ ❦➳t q✉↔ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ P❡rr♦♥ ❝❤ù❛ tr♦♥❣ ✤à♥❤ ❧þ s❛✉✿ ✣à♥❤ ❧➼ ✷✳✻✳✶✳ ❍➔♠ sè u(x) = sup v(x), v∈Sϕ ❧➔ ❤➔♠ ✤✐➲✉ ❤á❛ tr♦♥❣ Ω✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❤❡♦ ♥❣✉②➯♥ ❧þ ❝ü❝ ✤↕✐ ❜➜t ❦➻ ❤➔♠ v ∈ Sϕ ✤➲✉ t❤ä❛ ♠➣♥ v ≤ sup ϕ, ❞♦ ✤â u ❧➔ ①→❝ ✤à♥❤✳ ❈❤♦ y ❧➔ ✤✐➸♠ tò② þ ❝è ✤à♥❤ ❝õ❛ Ω✳ ❚❤❡♦ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ ❤➔♠ u✱ tç♥ t↕✐ ♠ët ❞➣② {vn} ⊂ Sϕ s❛♦ ❝❤♦ vn (y) → u(y). ❇➡♥❣ ❝→❝❤ t❤❛② t❤➳ vn ❜➡♥❣ max(vn, inf ϕ)✱ ❝❤ó♥❣ t❛ ❝â t❤➸ ❣✐↔ sû r➡♥❣ ❞➣② {vn} ❧➔ ❜à ❝❤➦♥✳ ❇➙② ❣✐í ❝❤å♥ R s❛♦ ❝❤♦ ❤➻♥❤ ❝➛✉ B = BR(y) ⊂⊂ Ω ✈➔ ①→❝ ✤à♥❤ Vn tø ❤➔♠ ♥➙♥❣ ✤✐➲✉ ❤á❛ ❝õ❛ vn tr♦♥❣ B t❤❡♦ ✭✷✳✷✺✮✳ ❑❤✐ ✤â Vn ∈ Sϕ✱ ✷✺ ✈➔ t❤❡♦ ✣à♥❤ ❧þ ✷✳✺✳✷ ❞➣② {Vn} ❝❤ù❛ ❞➣② ❝♦♥ {Vn } ❤ë✐ tö ✤➲✉ tr➯♥ ♠å✐ ❤➻♥❤ ❝➛✉ Bρ(y) ✈î✐ ρ < R ✤➳♥ ♠ët ❤➔♠ v ❧➔ ❤➔♠ ✤✐➲✉ ❤á❛ tr♦♥❣ B ✳ ❘ã r➔♥❣ v ≤ u tr♦♥❣ B ✈➔ v(y) = u(y)✳ ❈❤ó♥❣ t❛ ❦❤➥♥❣ ✤à♥❤ r➡♥❣ tr➯♥ t❤ü❝ t➳ u = v tr♦♥❣ B ✳ ●✐↔ sû v(z) < u(z) t↕✐ ♠ët sè z ∈ B ✳ ❑❤✐ ✤â tç♥ t↕✐ ♠ët ❤➔♠ u¯ ∈ Sϕ s❛♦ ❝❤♦ Vn (y) → u(y) k v(z) < u¯(z). ❳→❝ ✤à♥❤ wk = max(¯u, Vn ) ✈➔ ❝ô♥❣ ❣✐è♥❣ ❤➔♠ ♥➙♥❣ ✤✐➲✉ ❤á❛ Wk ♥❤÷ tr♦♥❣ ✭✷✳✷✺✮✱ ❝❤ó♥❣ t❛ t❤✉ ✤÷ñ❝ ♠ët ❞➣② ❝♦♥ ❝õ❛ ❞➣② {Wk } ❤ë✐ tö ✤➳♥ ♠ët ❤➔♠ ✤✐➲✉ ❤á❛ w t❤ä❛ ♠➣♥ v ≤ w ≤ u tr♦♥❣ B ✈➔ v(y) = w(y) = u(y)✳ ◆❤÷♥❣ t❤❡♦ ♥❣✉②➯♥ ❧þ ❝ü❝ ✤↕✐ ❝❤ó♥❣ t❛ ❝â v = u tr♦♥❣ B ✳ ✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈î✐ sü ①→❝ ✤à♥❤ ❝õ❛ u¯ ✈➔ ❞♦ ✤â u ❧➔ ❤➔♠ ✤✐➲✉ ❤á❛ tr♦♥❣ Ω✳ ❈→❝ ❦➳t q✉↔ ✤➣ ✤÷ñ❝ tr➻♥❤ ❜➔② ð tr➯♥ ❝õ❛ ♠ët ❤➔♠ ✤✐➲✉ ❤á❛ ❧➔ t➼♥❤ ❝❤➜t ♥❣❤✐➺♠ ✭✤÷ñ❝ ❣å✐ ❧➔ ♥❣❤✐➺♠ P❡rr♦♥✮ ❝õ❛ ❜➔✐ t♦→♥ ❉✐r✐❝❤❧❡t ❝ê ✤✐➸♥✿ ∆u = 0✱ u = ϕ tr♦♥❣ ∂Ω✳ ❚❤➟t ✈➟②✱ ♥➳✉ ❜➔✐ t♦→♥ ❉✐r✐❝❤❧❡t ❣✐↔✐ ✤÷ñ❝✱ ❧í✐ ❣✐↔✐ ❝õ❛ ♥â ❧➔ ✤ç♥❣ ♥❤➜t ✈î✐ ❧í✐ ❣✐↔✐ P❡rr♦♥✳ ●✐↔ sû ❝❤♦ w ❧➔ ♥❣❤✐➺♠✳ ❘ã r➔♥❣ w ∈ Sϕ ✈➔ t❤❡♦ ♥❣✉②➯♥ ❧þ ❝ü❝ ✤↕✐ t❤➻ w ≥ u ✈î✐ ♠å✐ u ∈ Sϕ✳ ❈❤ó♥❣ t❛ ❝❤ó þ r➡♥❣ ❝→❝❤ ❝❤ù♥❣ ♠✐♥❤ ❝õ❛ ✣à♥❤ ❧þ ✷✳✻✳✶ ❝â t❤➸ ❞ü❛ tr➯♥ ✤à♥❤ ❧þ ❤ë✐ tö ❍❛r♥❛❝❦✱ ✣à♥❤ ❧þ ✷✳✹✳✸✱ t❤❛② ❝❤♦ ✤à♥❤ ❧þ ❝♦♠♣❛❝t✱ ✣à♥❤ ❧þ ✷✳✺✳✶ ✭①❡♠ ❜➔✐ t♦→♥ ✷✳✶✵✮✳ k ✷✳✻✳✹ ❍➔♠ ❝❤➢♥ t↕✐ ♠ët ✤✐➸♠ tr➯♥ ❜✐➯♥✱ ❦❤→✐ ♥✐➺♠ ✤✐➸♠ ❝❤➼♥❤ q✉② ❚r♦♥❣ ♣❤÷ì♥❣ ♣❤→♣ P❡rr♦♥ ♥❣❤✐➯♥ ❝ù✉ ❝→❝❤ ①û ❧þ ❜✐➯♥ ❝õ❛ ♥❣❤✐➺♠ t❤ü❝ ❝❤➜t ✤÷ñ❝ t→❝❤ tø sü tç♥ t↕✐ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥✳ ❚✐➳♣ tö❝ ❣✐↔ t❤✐➳t ❝õ❛ ❣✐→ trà ❜✐➯♥ ❧➔ sü ❧✐➯♥ t❤æ♥❣ ✤➳♥ þ ♥❣❤➽❛ ❤➻♥❤ ❤å❝ ❝õ❛ ❜✐➯♥ t❤æ♥❣ q✉❛ ❦❤→✐ ♥✐➺♠ ❝õ❛ ❤➔♠ ❝❤➢♥✳ ✷✻ ✣à♥❤ ♥❣❤➽❛ ✷✳✻✳✷✳ ¯ ✱ ❈❤♦ ξ ❧➔ ♠ët ✤✐➸♠ ❝õ❛ ∂Ω✳ ❑❤✐ ✤â ❝❤♦ ♠ët ❤➔♠ w t❤✉ë❝ C 0(Ω) w = wξ ✤÷ñ❝ ❣å✐ ❧➔ ❤➔♠ ❝❤➢♥ t↕✐ ξ t÷ì♥❣ ✤è✐ ✤➳♥ Ω ♥➳✉✿ ✭✐✮ w ❧➔ tr➯♥ ✤✐➲✉ ❤á❛ tr♦♥❣ Ω✱ ¯ ✱ w(ξ) = 0✳ ✭✐✐✮ w > 0 tr♦♥❣ Ω\ξ ▼ët ✤à♥❤ ♥❣❤➽❛ tê♥❣ q✉→t ❤ì♥ ✈➲ ❤➔♠ ❝❤➢♥ ❝❤➾ ②➯✉ ❝➛✉ ❤➔♠ tr➯♥ ✤✐➲✉ ❤á❛ w ❧✐➯♥ tö❝ ✈➔ ♠❛♥❣ ❞➜✉ ❞÷ì♥❣ tr➯♥ Ω✱ ✈➔ w(x) → 0 ✈î✐ x → ξ ✳ ▼ët ✤➦❝ ✤✐➸♠ q✉❛♥ trå♥❣ ❝õ❛ ❦❤→✐ ♥✐➺♠ ❤➔♠ ❝❤➢♥ ❧➔ ♠ët t➼♥❤ ❝❤➜t ✤à❛ ♣❤÷ì♥❣ tr➯♥ ❜✐➯♥ ❝õ❛ ∂Ω✳ ❈ö t❤➸ ❧➔✱ t❛ ✤à♥❤ ♥❣❤➽❛ w ❧➔ ♠ët ❤➔♠ ❝❤➢♥ ✤à❛ ♣❤÷ì♥❣ t↕✐ ξ ∈ ∂Ω ♥➳✉ ❝â ♠ët sè N ❝õ❛ ξ s❛♦ ❝❤♦ w t❤ä❛ ♠➣♥ ✤à♥❤ ♥❣❤➽❛ ð tr➯♥ tr♦♥❣ Ω ∩ N ✳ ❑❤✐ ✤â ♠ët ❤➔♠ ❝❤➢♥ t↕✐ ξ t÷ì♥❣ ✤è✐ ✤➳♥ Ω ✤÷ñ❝ ✤à♥❤ ♥❣❤➽❛ ♥❤÷ s❛✉✿ ✣à♥❤ ♥❣❤➽❛ ✷✳✻✳✸✳ ❈❤♦ B ❧➔ ♠ët ❤➻♥❤ ❝➛✉ t❤ä❛ ♠➣♥ ξ ∈ B ⊂⊂ N ✈➔ m = Ninf w > 0✳ −B ❍➔♠ w(x) ¯ = min(m, w(x)), m, ¯ ∩ B, x∈Ω ¯ − B. x∈Ω ❧➔ ♠ët ❤➔♠ ❝❤➢♥ t↕✐ ξ t÷ì♥❣ ✤è✐ ✤➳♥ Ω✱ t❤❡♦ ❝→❝ t➼♥❤ ❝❤➜t ✭✐✮ ✈➔ ✭✐✐✮✳ ❚❤➟t ✈➟②✱ w¯ ❧➔ ❧✐➯♥ tö❝ tr♦♥❣ Ω¯ ✈➔ ❧➔ ❤➔♠ tr➯♥ ✤✐➲✉ ❤á❛ tr♦♥❣ Ω t❤❡♦ t➼♥❤ ❝❤➜t ✸ ❝õ❛ ❝→❝ ❤➔♠ tr➯♥ ✤✐➲✉ ❤á❛❀ t➼♥❤ ❝❤➜t ✭✐✐✮ ✤÷ñ❝ s✉② trü❝ t✐➳♣✳ ✣à♥❤ ♥❣❤➽❛ ✷✳✻✳✹✳ ✣✐➸♠ ❜✐➯♥ ✤÷ñ❝ ❣å✐ ❧➔ ✤✐➸♠ ❝❤➼♥❤ q✉② ✭✤è✐ ✈î✐ t♦→♥ tû ▲❛♣❧❛❝❡✮ ♥➳✉ tç♥ t↕✐ ♠ët ❤➔♠ ❝❤➢♥ t↕✐ ✤✐➸♠ ✤â✳ ❑➳t ❤ñ♣ ❣✐ú❛ ❤➔♠ ❝❤➢♥ ✈➔ ❝→❝❤ ①û ❧þ ✤✐➸♠ ❜✐➯♥ ❝õ❛ ❧í✐ ❣✐↔✐ ❝❤ù❛ tr♦♥❣ ❝→❝ ✤à♥❤ ❧þ s❛✉✳ ✷✼ ✣à♥❤ ❧➼ ✷✳✻✳✷✳ ❈❤♦ u ❧➔ ❤➔♠ ✤✐➲✉ ❤á❛ ✤➣ ✤÷ñ❝ ✤à♥❤ ♥❣❤➽❛ tr➯♥ Ω t❤❡♦ ♣❤÷ì♥❣ ♣❤→♣ P❡rr♦♥ ✭✣à♥❤ ❧þ ✷✳✻✳✶✮✳ ◆➳✉ ξ ❧➔ ♠ët ✤✐➸♠ ❜✐➯♥ ❝❤➼♥❤ q✉② ❝õ❛ Ω ✈➔ ϕ ❧➔ ❧✐➯♥ tö❝ t↕✐ ξ ✱ ❦❤✐ ✤â✿ u(x) → ϕ(ξ) ✈î✐ x → ξ. ❈❤ù♥❣ ♠✐♥❤✳ ❈❤å♥ > 0 ✈➔ ❝❤♦ M = sup |ϕ|✳ ❚ø ξ ❧➔ ♠ët ✤✐➸♠ ❜✐➯♥ ❝❤➼♥❤ q✉②✱ tç♥ t↕✐ ♠ët ❤➔♠ ❝❤➢♥ w t↕✐ ξ ✈➔ ❞♦ t➼♥❤ ❧✐➯♥ tö❝ ❝õ❛ ϕ✱ tç♥ t↕✐ ❝→❝ ❤➡♥❣ sè δ ✈➔ k s❛♦ ❝❤♦ |ϕ(x) − ϕ(ξ)| < ♥➳✉ |x − ξ| < δ, ✈➔ k.w(x) ≥ 2M ♥➳✉ |x − ξ| ≥ δ. ❈→❝ ❤➔♠ ϕ(ξ) + + kw✱ ϕ(ξ) − − kw t÷ì♥❣ ù♥❣ ❧➔ ❤➔♠ tr➯♥ ✈➔ ❤➔♠ ❞÷î✐ t÷ì♥❣ ✤è✐ ✤➳♥ ϕ✳ ◆❤÷ ✈➟② tø ✤à♥❤ ♥❣❤➽❛ ❝õ❛ ❤➔♠ u ✈➔ t❤ü❝ t➳ ❧➔ ♠å✐ ❤➔♠ tr➯♥ trë✐ ❤ì♥ ♠å✐ ❤➔♠ ❞÷î✐✱ tr♦♥❣ Ω ❝❤ó♥❣ t❛ ❝â✿ ϕ(ξ) − − kw(x) ≤ u(x) ≤ ϕ(ξ) + + kw(x), ❤♦➦❝ |u(x) − ϕ(ξ)| ≤ + kw(x). ❚ø x → ξ✳ w(x) → 0 ✈î✐ x → ξ✱ ❝❤ó♥❣ t❛ t❤✉ ✤÷ñ❝ u(x) → ϕ(ξ) ✈î✐ ✣✐➲✉ ♥➔② ❞➝♥ ♥❣❛② ❧➟♣ tù❝ ✤➳♥ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ✤➸ ❜➔✐ t♦→♥ ❉✐r✐❝❤❧❡t ❧➔ ❣✐↔✐ ✤÷ñ❝✳ ✷✳✻✳✺ ❚➼♥❤ ❣✐↔✐ ✤÷ñ❝ ❝õ❛ ❜➔✐ t♦→♥ ❉✐r✐❝❤❧❡t ✣à♥❤ ❧➼ ✷✳✻✳✸✳ ❇➔✐ t♦→♥ ❉✐r✐❝❤❧❡t ❝ê ✤✐➸♥ tr♦♥❣ ♠✐➲♥ ❜à ❝❤➦♥ ❧➔ ❣✐↔✐ ✤÷ñ❝ ✈î✐ ❤➔♠ ❝❤♦ tr÷î❝ tr➯♥ ❜✐➯♥ tò② þ ❧✐➯♥ tö❝ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ❝→❝ ✤✐➸♠ ❜✐➯♥ ✤➲✉ ❧➔ ❝❤➼♥❤ q✉②✳ ✷✽ ❈❤ù♥❣ ♠✐♥❤✳ ◆➳✉ ❝→❝ ❣✐→ trà ❝õ❛ ϕ tr➯♥ ❜✐➯♥ ❧➔ ❧✐➯♥ tö❝ ✈➔ ❜✐➯♥ ∂Ω ❜❛♦ ❣ç♠ ❝→❝ ✤✐➸♠ ❝❤➼♥❤ q✉②✱ t❤❡♦ ✣à♥❤ ❧þ ✷✳✻✳✷ ❤➔♠ ✤✐➲✉ ❤á❛ ❝❤♦ ❜ð✐ ♣❤÷ì♥❣ ♣❤→♣ P❡rr♦♥ ❧➔ ❣✐↔✐ ✤÷ñ❝ ✤è✐ ✈î✐ ❜➔✐ t♦→♥ ❉✐r✐❝❤❧❡t✳ ◆❣÷ñ❝ ❧↕✐✱ ❣✐↔ sû r➡♥❣ ❜➔✐ t♦→♥ ❉✐r✐❝❤❧❡t ❧➔ ❣✐↔✐ ✤÷ñ❝ ✈î✐ ♠å✐ ❜✐➯♥ ❧✐➯♥ tö❝✳ ❈❤♦ ξ ∈ ∂Ω✳ ❑❤✐ ✤â ❤➔♠ ϕ(x) = |x − ξ| ❧➔ ❧✐➯♥ tö❝ tr➯♥ ∂Ω ✈➔ ❤➔♠ ✤✐➲✉ ❤á❛ ❣✐↔✐ ❜➔✐ t♦→♥ ❉✐r✐❝❤❧❡t tr♦♥❣ Ω ✈î✐ ❣✐→ trà ❝õ❛ ♥â tr➯♥ ❜✐➯♥ ❧➔ ϕ ❤✐➸♥ ♥❤✐➯♥ ❧➔ ❜à ❝❤➦♥ t↕✐ ξ ✳ ❉♦ ✤â ξ ❧➔ ❝❤➼♥❤ q✉②✱ ❤❛② ♠å✐ ✤✐➸♠ ❝õ❛ ∂Ω ✤➲✉ ❧➔ ❝❤➼♥❤ q✉②✳ ❚rð ❧↕✐ ❝➙✉ ❤ä✐ q✉❛♥ trå♥❣✿ ◆❤ú♥❣ ♠✐➲♥ ♥➔♦ ❝â ❜✐➯♥ ❧➔ ❝→❝ ✤✐➸♠ ❝❤➼♥❤ q✉②❄ ◆â ♠ð r❛ ✤✐➲✉ ❦✐➺♥ ✤õ tê♥❣ q✉→t ❝â t❤➸ ✤÷ñ❝ ❜➢t ✤➛✉ tø t➼♥❤ ❝❤➜t ❤➻♥❤ ❤å❝ ✤à❛ ♣❤÷ì♥❣ ❝õ❛ ❜✐➯♥✳ ❈❤ó♥❣ t❛ ♥❤➢❝ ❧↕✐ ♠ët sè ✤✐➲✉ ❦✐➺♥ s❛✉✳ ◆➳✉ n = 2✱ ①➨t ♠ët ✤✐➸♠ ❜✐➯♥ z0 ❝õ❛ ♠ët ♠✐➲♥ ❜à ❝❤➦♥ Ω ✈➔ ✤➦t ❣è❝ t↕✐ z0 ✈î✐ tå❛ ✤ë ❝ü❝ r✱ θ ✳ ●✐↔ sû ❝â ♠ët ❧➙♥ ❝➟♥ N ❝õ❛ z0 s❛♦ ❝❤♦ ♠ët ♥❤→♥❤ ❞✉② ♥❤➜t ❝õ❛ θ ❝â ❣✐→ trà ✤÷ñ❝ ①→❝ ✤à♥❤ tr➯♥ Ω ∩ N ✱ ❤♦➦❝ tr➯♥ ♠ët t❤➔♥❤ ♣❤➛♥ ❝õ❛ Ω ∩ N ❝â z0 tr➯♥ ❜✐➯♥ ❝õ❛ ♥â✳ ❚❛ t❤➜② r➡♥❣✿ w = −Re logr 1 =− 2 , logz log r + θ2 ❧➔ ♠ët ❤➔♠ ❝❤➢♥ ❝ö❝ ❜ë ✭ ❝❤➢♥ ②➳✉✮ t↕✐ z0✳ ❉♦ ✤â z0 ❧➔ ✤✐➸♠ ❝❤➼♥❤ q✉②✳ ✣➦❝ ❜✐➺t✱ z0 ❧➔ ✤✐➸♠ ❜✐➯♥ ❝❤➼♥❤ q✉② ♥➳✉ ♥â ❧➔ ✤✐➸♠ ❦➳t t❤ó❝ ❝õ❛ ♠ët ❝✉♥❣ ✤ì♥ ♥➡♠ ð ♣❤➼❛ ♥❣♦➔✐ ❝õ❛ Ω✳ ❉♦ ✤â ❜➔✐ t♦→♥ ❉✐r✐❝❤❧❡t tr➯♥ ♠➦t ♣❤➥♥❣ ❧➔ ❧✉æ♥ ❧✉æ♥ ❣✐↔✐ ✤÷ñ❝ ✤è✐ ✈î✐ ❜✐➯♥ ❝â ❣✐→ trà ❧✐➯♥ tö❝ tr➯♥ ♠ët ♠✐➲♥ ✭♠✐➲♥ ❜à ❝❤➦♥✮ ❝â ❝→❝ ✤✐➸♠ ❜✐➯♥ t❤✉ ✤÷ñ❝ tø ❜➯♥ ♥❣♦➔✐ ❝õ❛ ❝✉♥❣ ✤ì♥✳ ❚ê♥❣ q✉→t ❤ì♥✱ ❣✐è♥❣ ♥❤÷ ❤➔♠ ❝❤➢♥ r➡♥❣ ❜➔✐ t♦→♥ ❣✐→ trà ❜✐➯♥ ❧➔ ❣✐↔✐ ✤÷ñ❝ ♥➳✉ ♠å✐ t❤➔♥❤ ♣❤➛♥ ❝õ❛ ♠✐➲♥ ❜❛♦ ❣ç♠ ♥❤✐➲✉ ❤ì♥ ♠ët ✤✐➸♠✳ ❱➼ ❞ö ✈➲ ❝→❝ ♠✐➲♥ ♥❤÷ ✈➟② ❧➔ ❝→❝ ♠✐➲♥ ❜à ❝❤➦♥ ❜ð✐ ♠ët sè ❤ú✉ ❤↕♥ ❝→❝ ✤÷í♥❣ ❝♦♥❣ ❦❤➨♣ ❦➼♥✳ ▼ët ✈➼ ❞ö ❦❤→❝ ❧➔ ♥❤→t ❝➢t ✤ì♥ ✈à ❝ò♥❣ ♠ët ✷✾ ✈á♥❣ ❝✉♥❣❀ tr♦♥❣ tr÷í♥❣ ❤ñ♣ ♥➔② ❣✐→ trà ❜✐➯♥ ❝â t❤➸ ❧➔ ❣✐❛♦ tr➯♥ ❝→❝ ❝↕♥❤ ✤è✐ ❞✐➺♥ ❝õ❛ ♥❤→t ❝➢t✳ ✣è✐ ✈î✐ sè ❝❤✐➲✉ ❝❛♦ ❤ì♥ t❤ü❝ ❝❤➜t ❧➔ ❦❤→❝ ♥❤❛✉ ✤→♥❣ ❦➸ ✈➔ ❜➔✐ t♦→♥ ❉✐r✐❝❤❧❡t ❦❤æ♥❣ t❤➸ ❣✐↔✐ ✤÷ñ❝ tr➯♥ tê♥❣ q✉→t t÷ì♥❣ ù♥❣✳ ❉♦ ✤â✱ ♠ët ✈➼ ❞ö ❞♦ ▲❡❜❡s❣✉❡ tr➻♥❤ ❜➔② ❝❤♦ t❤➜② ♠ët ♠➦t ❦➼♥ tr♦♥❣ ❜❛ ❝❤✐➲✉ ✈î✐ ♠ët ✤➾♥❤ ❝â ✤✐➸♠ ❧ò✐ ❤÷î♥❣ ✈➔♦ tr♦♥❣ ❝â ♠ët ✤✐➸♠ ❦❤æ♥❣ ❝❤➼♥❤ q✉② t↕✐ ✤➛✉ ❝õ❛ ✤✐➸♠ ❧ò✐ ✤â✳ ✷✳✻✳✻ ✣✐➲✉ ❦✐➺♥ ❤➻♥❤ ❝➛✉ ♥❣♦➔✐ ▼ët ✤✐➲✉ ❦✐➺♥ ✤õ ✤ì♥ ❣✐↔♥ ❝❤♦ t➼♥❤ ❝❤➼♥❤ q✉② ❝õ❛ ✤✐➸♠ ξ ∈ ∂Ω ❧➔ ✤✐➲✉ ❦✐➺♥ ❤➻♥❤ ❝➛✉ ♥❣♦➔✐❀ tù❝ ❧➔✱ tç♥ t↕✐ ♠ët ❤➻♥❤ ❝➛✉ ♥❣♦➔✐ B = BR(y) t❤ä❛ ♠➣♥ B¯ ∩ Ω¯ = ξ ✳ ◆➳✉ ✤✐➲✉ ❦✐➺♥ ♥❤÷ ✈➟② ✤÷ñ❝ t❤ä❛ ♠➣♥✱ t❤➻ ❤➔♠ w ✤÷ñ❝ ①→❝ ✤à♥❤ ❜ð✐   R2−n − |x − y|2−n , n ≥ 3, w(x) =  log |x − y| , n = 2, R ✭✷✳✷✻✮ ❧➔ ♠ët ❤➔♠ ❝❤➢❝ ❝❤➢♥ t↕✐ ξ ✳ ❉♦ ✤â ❝→❝ ✤✐➸♠ ❜✐➯♥ ❝õ❛ ♠ët ♠✐➲♥ t❤✉ë❝ ❧î♣ C 2 ✤➲✉ ❧➔ ❝→❝ ✤✐➸♠ ❝❤➼♥❤ q✉② ✭①❡♠ ♠ö❝ ✷✳✽ tr♦♥❣ ❬✷❪✮✳ ✷✳✼ ❉✉♥❣ ❧÷ñ♥❣ ✣✐➲✉ ❦✐➺♥ ❝❤♦ t➼♥❤ ❝❤➼♥❤ q✉② ❝õ❛ ♠ët ✤✐➸♠ ❜✐➯♥ ❝â t❤➸ ♠æ t↔ q✉❛ ❦❤→✐ ♥✐➺♠ ❞✉♥❣ ❧÷ñ♥❣ ❝õ❛ ♠✐➲♥✳ ❈→❝ ❦❤→✐ ♥✐➺♠ ✈➟t ❧þ ❝õ❛ ❞✉♥❣ ❧÷ñ♥❣ ❝✉♥❣ ❝➜♣ ♠ët t➼♥❤ ❝❤➜t ✤➦❝ tr÷♥❣ ❝õ❛ ✤✐➸♠ ❜✐➯♥✳ ❈❤♦ Ω ❧➔ ♠ët ♠✐➲♥ ❜à ❝❤➦♥ tr♦♥❣ Rn(n ≥ 3) ✈î✐ ❜✐➯♥ ∂Ω✱ ✈➔ ❝❤♦ u ❧➔ ❤➔♠ ✤✐➲✉ ❤á❛ ①→❝ ✤à♥❤ tr➯♥ ♣❤➛♥ ❜ò ❝õ❛ Ω¯ ✈➔ t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❜✐➯♥ u = 1 tr➯♥ ∂Ω ✈➔ u = 0 t↕✐ ✈æ ❝ü❝✳ ✣↕✐ ❧÷ñ♥❣ |Du|2 dx, cap Ω = Rn −Ω ✸✵ ✭✷✳✷✼✮ ✤÷ñ❝ ✤à♥❤ ♥❣❤➽❛ ❧➔ ❞✉♥❣ ❧÷ñ♥❣ ❝õ❛ Ω✳ ❚r♦♥❣ t➽♥❤ ✤✐➺♥ ❤å❝✱ cap Ω ❧➔ ✤✐➺♥ t➼❝❤ tr♦♥❣ ✈➟t ❞➝♥ ∂Ω ✤÷ñ❝ ❤➻♥❤ t❤➔♥❤ ❦❤✐ ♥â ❝â ✤✐➺♥ t❤➳ ❜➡♥❣ ✶ s♦ ✈î✐ ð ♥❣♦➔✐ ✈æ ❝ü❝✳ ✣➦❝ ❜✐➺t✱ ❝❤ó♥❣ ❝â ✤➦❝ t➼♥❤ ❜✐➳♥ ♣❤➙♥ cap Ω = inf v∈K |Dv|2 , ✭✷✳✷✽✮ tr♦♥❣ ✤â✱ tr♦♥❣ Ω}, ✈î✐ C01(Rn) ❧➔ tê♥❣ ❤ñ♣ ❝→❝ ❤➔♠ ❦❤↔ ✈✐ ❧✐➯♥ tö❝ tr♦♥❣ Rn ✈➔ ❜➡♥❣ ❦❤æ♥❣ ð ♥❣♦➔✐ ♠ët ❤➻♥❤ ❝➛✉ ♥➔♦ ✤â✳ ✣➸ ❦✐➸♠ tr❛ t➼♥❤ ❝❤➼♥❤ q✉② ❝õ❛ ♠ët ✤✐➸♠ x0 ∈ ∂Ω✱ ①➨t ❜➜t ❦ý λ ∈ (0, 1)✱ λ ❝è ✤à♥❤ ✈➔ ✤➦t K = {v ∈ C01 (Rn ) | v = 1 Cj = cap{x ∈ / Ω | |x − x0 | ≤ λj }. ❚✐➯✉ ❝❤✉➞♥ ❲✐❡♥❡r ♥â✐ r➡♥❣ x0 ❧➔ ✤✐➸♠ ❜✐➯♥ ❝❤➼♥❤ q✉② ❝õ❛ ∂Ω ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ❝❤✉é✐ ∞ Cj /λj(n−2) , ✭✷✳✷✾✮ i=0 ♣❤➙♥ ❦ý ✭①❡♠ ♠ö❝ ✷✳✾ ❝õ❛ ❬✷❪✮✳ ✸✶ ❑➳t ❧✉➟♥ ◆ë✐ ❞✉♥❣ ❝❤➼♥❤ ✤÷ñ❝ tr➻♥❤ ❜➔② tr♦♥❣ ❧✉➟♥ ✈➠♥ ❜❛♦ ❣ç♠✿ ✲ ❚r➻♥❤ ❜➔② ❦❤→✐ ♥✐➺♠ ♣❤÷ì♥❣ tr➻♥❤ ✈➔ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ▲❛♣❧❛❝❡✱ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ✤➥♥❣ t❤ù❝✱ ❜➜t ✤➥♥❣ t❤ù❝ ✈➲ ❣✐→ trà tr✉♥❣ ❜➻♥❤✱ ♥❣✉②➯♥ ❧þ ❝ü❝ ✤↕✐ ✈➔ ❝ü❝ t✐➸✉ ✤è✐ ✈î✐ ❤➔♠ ✤✐➲✉ ❤á❛✱ tr➯♥ ✤✐➲✉ ❤á❛ ✈➔ ❞÷î✐ ✤✐➲✉ ❤á❛ ✳ ✲ ◆❣❤✐➯♥ ❝ù✉ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ❤➔♠ ✤✐➲✉ ❤á❛✿ ❜➜t ✤➥♥❣ t❤ù❝ ❍❛r✲ ♥❛❝❦✱ ❦❤→✐ ♥✐➺♠ ❤➔♠ ●r❡❡♥ ✤è✐ ✈î✐ ❜➔✐ t♦→♥ ❉✐r✐❝❤❧❡t✱ ✤à♥❤ ❧þ ❤ë✐ tö✱ ❝→❝ ✤→♥❤ ❣✐→ ❜➯♥ tr♦♥❣ ♠✐➲♥ ✤è✐ ✈î✐ ❤➔♠ ✤✐➲✉ ❤á❛✳ ✲ ◆❣❤✐➯♥ ❝ù✉ ❜➔✐ t♦→♥ ❉✐r✐❝❤❧❡t ✤è✐ ✈î✐ ❤➔♠ ✤✐➲✉ ❤á❛✱ tr➻♥❤ ❜➔② ❦❤→✐ ♥✐➺♠ ✤✐➸♠ ❝❤➼♥❤ q✉② tr➯♥ ❜✐➯♥✱ ♣❤→t ❜✐➸✉ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ❝❤♦ t➼♥❤ ❣✐↔✐ ✤÷ñ❝ ❝õ❛ ❜➔✐ t♦→♥✱ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ✤➸ ♠ët ✤✐➸♠ tr➯♥ ❜✐➯♥ ❧➔ ❝❤➼♥❤ q✉②✳ ✸✷ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❚➔✐ ❧✐➺✉ ❚✐➳♥❣ ❱✐➺t ❬✶❪ ❚r➛♥ ✣ù❝ ❱➙♥ ✭✷✵✵✹✮✱ ▲þ t❤✉②➳t ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣✱ ◆❤➔ ①✉➜t ❜↔♥ ✣↕✐ ❤å❝ ◗✉è❝ ❣✐❛ ❍➔ ◆ë✐✳ ❚➔✐ ❧✐➺✉ ❚✐➳♥❣ ❆♥❤ ❬✷❪ ❉✳ ●✐❧❜❛r❣ ❛♥❞ ◆✳ ❚r✉❞✐♥❣❡r ✭✷✵✵✶✮✱ ❊❧❧✐♣t✐❝ ❉✐❢❢❡r❡♥t✐❛❧ ❊q✉❛✲ t✐♦♥s ♦❢ ❙❡❝♦♥❞ ❖r❞❡r✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ❇❡r❧✐♥ ❍❡✐❞❡❧❜❡r❣ ◆❡✇ ❨♦r❦✳ ✸✸ ❳⑩❈ ◆❍❾◆ ❈❍➓◆❍ ❙Û❆ ▲❯❾◆ ❱❿◆ ❳→❝ ♥❤➟♥ ❧✉➟♥ ✈➠♥ t❤↕❝ s➽ ❝õ❛ ❤å❝ ✈✐➯♥ ❝❛♦ ❤å❝ ❚r➛♥ ❱➠♥ ❚î✐✳ ❚➯♥ ✤➲ t➔✐ ❧✉➟♥ ✈➠♥ P❤÷ì♥❣ tr➻♥❤ ✈➔ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ▲❛♣❧❛❝❡ ❈❤✉②➯♥ ♥❣➔♥❤✿ ❚♦→♥ ù♥❣ ❞ö♥❣ ▼➣ sè✿ ✻✵✳✹✻✳✵✶✳✶✷ ❇↔♦ ✈➺ ♥❣➔② ✶✶✳✶✵✳✷✵✶✹ ✣➣ ❝❤➾♥❤ sû❛ t❤❡♦ ♥❤÷ ❦➳t ❧✉➟♥ ❝õ❛ ❍ë✐ ✤ç♥❣ ❜↔♦ ✈➺ t↕✐ tr÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ✲ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥ ●✐→♦ ✈✐➯♥ ❤÷î♥❣ ❞➝♥ P●❙✳ ❚❙✳ ❍➔ ❚✐➳♥ ◆❣♦↕♥ ✸✹  ... tỡ t ỡ t x dS tỷ t t õ ổ tự t tứ ợ uv C 1() C 0() (Dj u)vdx = u(Dj v)dx + uvàj dS ứ ổ tự tr t s r ỵ s trữớ õ t õ tỡ t w = (w1, w2, , wn) tr C 1() divwdx = (w,