Dạng chuẩn tắc của phương trình đạo hàm riêng tuyến tính cấp hai trên mặt phẳng

✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ✖✖✖✖✖✖✖✖✖✖✖✖ ✣➄◆● ❚❍➚ ▲❖❆◆ ❉❸◆● ❈❍❯❽◆ ❚➁❈ ❈Õ❆ P❍×❒◆● ❚❘➐◆❍ ✣❸❖ ❍⑨▼ ❘■➊◆● ❚❯❨➌◆ ❚➑◆❍ ❈❻P ❍❆■ ❚❘➊◆ ▼➄❚ P❍➃◆● ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ✲ ✷✵✷✵ ✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ✣➦♥❣ ❚❤à ▲♦❛♥ ❉❸◆● ❈❍❯❽◆ ❚➁❈ ❈Õ❆ P❍×❒◆● ❚❘➐◆❍ ✣❸❖ ❍⑨▼ ❘■➊◆● ❚❯❨➌◆ ❚➑◆❍ ❈❻P ❍❆■ ❚❘➊◆ ▼➄❚ P❍➃◆● ❈❤✉②➯♥ ♥❣➔♥❤✿ ❚♦→♥ ●✐↔✐ ❚➼❝❤ ▼➣ sè✿ ✽ ✹✻ ✵✶ ✵✷ ▲❯❾◆ ❱❿◆ ữợ ❚❙✳ ❚❘➚◆❍ ❚❍➚ ❉■➏P ▲■◆❍ ✐ ❚❍⑩■ ◆●❯❨➊◆ ✲ ✷✵✷✵ ✐✐ ▲í✐ ❝❛♠ ✤♦❛♥ ✧❉↕♥❣ ❝❤✉➞♥ t➢❝ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣ t✉②➳♥ t➼♥❤ ❝➜♣ ❤❛✐ tr➯♥ ♠➦t ♣❤➥♥❣✧ ❧➔ tr➻♥❤ ❜➔② ♥❣❤✐➯♥ ❝ù✉ ❦❤♦❛ ❤å❝ ❝õ❛ r✐➯♥❣ tæ✐ ữợ sỹ ữợ trỹ t r ❉✐➺♣ ▲✐♥❤✳ ❚æ✐ ①✐♥ ❝❛♠ ✤♦❛♥ ▲✉➟♥ ✈➠♥ ◆❣♦➔✐ r❛✱ tr♦♥❣ ❧✉➟♥ ✈➠♥ tỉ✐ ❝á♥ sû ❞ư♥❣ ♠ët sè ❦➳t q✉↔✱ ♥❤➟♥ ①➨t ❝õ❛ ♠ët sè t→❝ ❣✐↔ ❦❤→❝ ✤➲✉ õ ú t tr ỗ ốr q tr ♥❣❤✐➯♥ ❝ù✉✱ tæ✐ ✤➣ ❦➳ t❤ø❛ t❤➔♥❤ q✉↔ ❦❤♦❛ ❤å❝ ❝õ❛ ❝→❝ ♥❤➔ ❦❤♦❛ ❤å❝ ✈ỵ✐ sü tr➙♥ trå♥❣ ✈➔ ❜✐➳t ì♥✳ ◆➳✉ ♣❤→t ❤✐➺♥ ❜➜t ❦ý sü ❣✐❛♥ ❧➟♥ ♥➔♦ tæ✐ ①✐♥ ❤♦➔♥ t♦➔♥ ❝❤à✉ tr→❝❤ ♥❤✐➺♠ ✈➲ ♥ë✐ ❞✉♥❣ ❧✉➟♥ ✈➠♥ ❝õ❛ ♠➻♥❤✳ ❚❤→✐ ◆❣✉②➯♥✱ ♥❣➔② ✶✺ t❤→♥❣ ✾ ♥➠♠ ✷✵✷✵ ❚→❝ ❣✐↔ ✣➦♥❣ ❚❤à ▲♦❛♥ ✐✐ ▲í✐ ❝↔♠ ì♥ ▲✉➟♥ ✈➠♥ ♥➔② ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ t↕✐ tr÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ❚❤→✐ ◆❣✉②➯♥✳ ❚→❝ ❣✐↔ ①✐♥ ❜➔② tä ❧á♥❣ ❦➼♥❤ trå♥❣ ✈➔ ❜✐➳t ì♥ s➙✉ s➢❝ ✤➳♥ ❚❙✳ ❚rà♥❤ ❚❤à ❉✐➺♣ ▲✐♥❤ ♥❣÷í✐ t❤➛② ✤➣ trü❝ t✐➳♣ ữợ t t t ❣✐↔ tr♦♥❣ s✉èt t❤í✐ ❣✐❛♥ ♥❣❤✐➯♥ ❝ù✉ ✈ø❛ q✉❛✳ ❚→❝ ❣✐↔ tr➙♥ trå♥❣ ❣û✐ ❧í✐ ❝↔♠ ì♥ ✤➳♥ ❝→❝ t❤➛②✱ ❝æ ❣✐→♦ ❑❤♦❛ ❚♦→♥✱ P❤á♥❣ ✣➔♦ t↕♦ ❙❛✉ ✤↕✐ ❤å❝✱ ❝→❝ ❜↕♥ ❤å❝ ✈✐➯♥ ❧ỵ♣ ❈❛♦ ❤å❝ ❑✷✻ ❚♦→♥ ❣✐↔✐ t➼❝❤ tr÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ❚❤→✐ ◆❣✉②➯♥ ✤➣ ❧✉ỉ♥ ❣✐ó♣ ✤ï✱ t↕♦ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧đ✐ ❝❤♦ t→❝ ❣✐↔ tr♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉ t↕✐ tr÷í♥❣✳ ❚→❝ ❣✐↔ ❝ơ♥❣ ①✐♥ ❜➔② tä ❜✐➳t ì♥ s➙✉ s➢❝ tợ ữớ t ổ ✤ë♥❣ ✈✐➯♥ t→❝ ❣✐↔ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ❧➔♠ ❧✉➟♥ ✈➠♥✳ ❚→❝ ❣✐↔ ♠♦♥❣ ♥❤➟♥ ✤÷đ❝ ♥❤ú♥❣ þ ❦✐➳♥ ✤â♥❣ ❣â♣ q✉þ ❜→✉ ❝õ❛ ❝→❝ t❤➛② ❝æ ✈➔ ❜↕♥ ✤å❝ ✤➸ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ❚❤→✐ ◆❣✉②➯♥✱ ♥❣➔② ✶✺ t❤→♥❣ ✾ ♥➠♠ ✷✵✷✵ ◆❣÷í✐ t❤ü❝ ❤✐➺♥ ✣➦♥❣ ❚❤à ▲♦❛♥ ✐✐✐ ▼ö❝ ❧ö❝ ❚r❛♥❣ ❜➻❛ ♣❤ö ▲í✐ ❝❛♠ ✤♦❛♥ ▲í✐ ❝↔♠ ì♥ ▲í✐ ♥â✐ ✤➛✉ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✐ ✐✐ ✐✐✐ ✶ ✹ ✶✳✶ ▼ët sè ❦❤→✐ ♥✐➺♠ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✷ P❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣ ❝➜♣ ❤❛✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✷✳✶ P❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✷✳✷ ❉↕♥❣ ❝❤✉➞♥ t➢❝ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❤②♣❡r❜♦❧✐❝ ✳ ✳ ✳ ✳ ✷✵ ✶✳✷✳✸ ❉↕♥❣ ❝❤✉➞♥ t➢❝ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ♣❛r❛❜♦❧✐❝ ✳ ✳ ✳ ✳ ✷✸ ✶✳✷✳✹ ❉↕♥❣ ❝❤✉➞♥ t➢❝ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❡❧✐♣t✐❝ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷ ❉↕♥❣ ❝❤✉➞♥ t➢❝ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣ t✉②➳♥ t➼♥❤ ❝➜♣ ❤❛✐ tr➯♥ ♠➦t ♣❤➥♥❣ ✷✽ ✷✳✶ P❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣ t✉②➳♥ t➼♥❤ ❝➜♣ ❤❛✐ ✈ỵ✐ ❤❛✐ ❜✐➳♥ ✤ë❝ ❧➟♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✷✳✷ ❉↕♥❣ ❝❤✉➞♥ t➢❝ ❦❤ỉ♥❣ ✤à❛ ♣❤÷ì♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✷✳✸ ❉↕♥❣ ❝❤✉➞♥ t➢❝ trì♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✷✳✸✳✶ ✣à♥❤ ❧➼ rót ❣å♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✷✳✸✳✷ ❉↕♥❣ ❝❤✉➞♥ t➢❝ trì♥ ❝❤♦ ❝→❝ ✤✐➸♠ ❦➻ ❞à ❣➜♣ ✳ ✳ ✳ ✳ ✹✼ ✐✈ ❑➳t ❧✉➟♥ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✺✶ ✺✷ ✈ ▲í✐ ♥â✐ ✤➛✉ ❙ü ❦❤ð✐ ✤➛✉ ❝õ❛ ỵ tt t ữỡ tr ✤↕♦ ❤➔♠ r✐➯♥❣ t✉②➳♥ t➼♥❤ ❝➜♣ ❤❛✐ tr➯♥ ♠➦t ♣❤➥♥❣ ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ ✈➔♦ ❦❤♦↔♥❣ ❣✐ú❛ t❤➳ ❦✛ ✶✽✳ ❱➔♦ t❤í✐ ✤✐➸♠ ✤â ❞✬❆❧❡♠❜❡rt ✈➔ ❊✉❧❡r ✤➣ ✤➲ ①✉➜t ♣❤÷ì♥❣ tr➻♥❤ sâ♥❣ ✈➔ ♣❤÷ì♥❣ tr➻♥❤ ▲❛♣❧❛❝❡ ✤➸ ♠ỉ t↔ sü ❝❤✉②➸♥ ✤ë♥❣ ❝õ❛ ❞➙② ✈➔ sü t❤❛② t❤➳ ✈➟♥ tè❝ ❝õ❛ ❝❤➜t ❧ä♥❣ ❦❤ỉ♥❣ ♥➨♥ ✤÷đ❝ t÷ì♥❣ ù♥❣✳ ❙❛✉ ❦❤✐ ①✉➜t ❤✐➺♥ ♥❤ú♥❣ ❞↕♥❣ ❝❤✉➞♥ t➢❝ ♠➔ ✤↕✐ ❞✐➺♥ ❝❤♦ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ❧♦↕✐ ❡❧✐♣t✐❝ ✈➔ ♣❤÷ì♥❣ tr➻♥❤ ❧♦↕✐ ❤②♣❡r❜♦❧✐❝✱ ✤÷đ❝ sû ❞ư♥❣ ♥❤✐➲✉ tr♦♥❣ ❣✐↔✐ t➼❝❤ ✤➸ →♣ ❞ư♥❣ tr♦♥❣ ✈✐➺❝ ❣✐↔✐ q✉②➳t ❝→❝ ❜➔✐ t♦→♥ ❦❤→❝ ♥❤❛✉✳ ◆❣➔② ♥❛② ✈➜♥ ✤➲ ♥➔② ✤÷đ❝ ♥❤✐➲✉ ♥❣÷í✐ q✉❛♥ t➙♠ ✈➔ t❤÷í♥❣ ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ tr♦♥❣ ❧➽♥❤ ✈ü❝ ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣✳ ❳➨t ♣❤÷ì♥❣ tr➻♥❤ tê♥❣ q✉→t a(x, y)uxx + b(x, y)uxy + c(x, y)uyy = 0, ✭✵✳✶✮ ✈ỵ✐ a, b, c ❧➔ ❝→❝ ❤➺ sè trì♥✱ ❝â t❤➸ ✤÷đ❝ ✤÷❛ ✈➲ ❝→❝ ❞↕♥❣ ✤à❛ ♣❤÷ì♥❣ ❣➛♥ ✤✐➸♠ ❜➜t ❦ý ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❤②♣❡r❜♦❧ ✈➔ ❡❧✐♣ t÷ì♥❣ ù♥❣✱ tù❝ ❧➔ ①➨t ❜✐➺t t❤ù❝ D ✈ỵ✐ D = b2 − 4ac ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✵✳✶✮ t❤❡♦ t❤ù tü ❧➔ ❞÷ì♥❣ ✈➔ ➙♠✱ ❜➡♥❣ ❝→❝❤ t❤❛② ✤ê✐ ❝→❝ tå❛ ✤ë trì♥ ✈➔ t❤ü❝ ❤✐➺♥ ♣❤➨♣ ♥❤➙♥ tr➯♥ ♠ët ❤➔♠ trì♥ ❜➜t ❜✐➳♥ t❤➼❝❤ ❤đ♣ ✭①❡♠❬✹❪✮✳ ✣è✐ ✈ỵ✐ ♠ët ❜ë ❜❛ tê♥❣ q✉→t trì♥ ❤♦➦❝ trì♥ ✤➛② ✤õ tr♦♥❣ tỉ♣ỉ ❲❤✐t♥❡② ❜✐➺t t❤ù❝ ❧➔ ré♥❣ ❤♦➦❝ ❧➔ ♠ët ✤÷í♥❣ ❝♦♥❣ trì♥ ✤÷đ❝ ♥❤ó♥❣ tr♦♥❣ ♠➦t ♣❤➥♥❣✳ ◆❤÷ ✈➟② ❝❤♦ ♠ët ♣❤÷ì♥❣ tr➻♥❤ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ sâ♥❣ ❞↕♥❣ uxx − uyy = ✈➔ ♣❤÷ì♥❣ tr➻♥❤ ▲❛♣❧❛❝❡ ❞↕♥❣ uxx + uyy = ❝→❝ ❞↕♥❣ ❝❤✉➞♥ t➢❝ ❤✐➺♥ ♥❛② ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✵✳✶✮ ❝❤➼♥❤ ❧➔ ❣➛♥ ♠ët ✤✐➸♠ ♥➡♠ ♥❣♦➔✐ ✤÷í♥❣ t❤➥♥❣ ♥➔②✳ ✣÷í♥❣ t❤➥♥❣ ♥➔② ✤÷đ❝ ❣å✐ ❧➔ ❞↕♥❣ ✤÷í♥❣ t❤❛② ✤ê✐ ✈➻ ❜➜t ❦ý ✤✐➸♠ ♥➔♦ ❣➛♥ ♥â ♣❤÷ì♥❣ tr➻♥❤ ỗ õ r Pữỡ ✶ tr➻♥❤ ✭✵✳✶✮ t❤❛② ✤ê✐ ❞↕♥❣ tr♦♥❣ ♠✐➲♥ ✤÷đ❝ ❣å✐ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ❞↕♥❣ ❤é♥ ❤đ♣✳ ❚r♦♥❣ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ ❚r✐❝♦♠✐ ✭❬①❡♠ ✻❪✮ ✤➣ ①➨t ♠ët ♣❤÷ì♥❣ tr➻♥❤ ❣➛♥ ✤✐➸♠ P ❝õ❛ ❞↕♥❣ ✤÷í♥❣ t❤❛② ✤ê✐✱ ✤â ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ❦❤æ♥❣ s✉② ❜✐➳♥ ❝õ❛ ❜✐➺t t❤ù❝✱ tù❝ ❧➔ D(P ) = ✈➔ dD(P ) = ✈➔ t↕✐ ✤â ♣❤÷ì♥❣ ✤➦❝ tr÷♥❣ dy : dx ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ♣❤÷ì♥❣ tr➻♥❤ a(x, y)dy − b(x, y)dxdy + c(x, y)dx2 = ✭✵✳✷✮ P❤÷ì♥❣ tr➻♥❤ ✭✵✳✷✮ ❦❤ỉ♥❣ t✐➳♣ t✉②➳♥ ợ ữớ t ởt ữ r ✤÷❛ r❛ ❝❤♦ ✭✵✳✶✮ ❞↕♥❣ ❝❤✉➞♥ t➢❝ ✤÷đ❝ ❦➼ ❤✐➺✉ uyy + yuxx = ✭✵✳✸✮ ❙❛✉ ❦❤✐ t❤❛② ✤ê✐ ❝→❝ tå❛ ✤ë trì♥ ✈➔ t❤ü❝ ❤✐➺♥ ♥❤➙♥ tr➯♥ ♠ët ❤➔♠ trì♥ ❜➜t ❜✐➳♥✳ Ð ❞↕♥❣ ♣❤÷ì♥❣ tr➻♥❤ t❤❛② ✤ê✐ tr➯♥ trư❝ ❤♦➔♥❤ ✈➔ ♥â t❤✉ë❝ ♣❤÷ì♥❣ tr➻♥❤ ❧♦↕✐ ❡❧✐♣t✐❝ tr♦♥❣ ♠✐➲♥ y > ✈➔ ❤②♣❡r❜♦❧✐❝ tr♦♥❣ ♠✐➲♥ y < 0✳ ❍ì♥ ♥ú❛✱ ♥❣÷í✐ t❛ ✤➣ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ ♣❤÷ì♥❣ tr➻♥❤ ð ❞↕♥❣ ❝❤✉➞♥ t➢❝ ❝â ❤❛✐ ✤➦❝ t➼♥❤ t↕✐ ♠é✐ ✤✐➸♠ x0 ❝õ❛ trö❝ ❤♦➔♥❤✳ ❈→❝ ✤➦❝ t➼♥❤ ♥➡♠ tr♦♥❣ ♠✐➲♥ y ≤ ✈➔ ❝â ❞↕♥❣ 9(x x0 )2 = 4y ố ợ ữỡ tr➻♥❤ ✭✵✳✸✮✱ ❚r✐❝♦♠✐ ✭❬①❡♠ ✻❪✮ tr➻♥❤ ❜➔② ❞↕♥❣ ♠ỵ✐ ✈➲ ❧♦↕✐ ❜➔✐ t♦→♥ ❣✐→ trà ❜✐➯♥ tr♦♥❣ ♠✐➲♥ ❜à ❝❤➦♥ ❜ð✐ ❝→❝ ✤➦❝ tr÷♥❣ ❣✐❛♦ ♥❤❛✉ ✤✐ tø ❤❛✐ ✤✐➸♠ ❝õ❛ ❞↕♥❣ ✤÷í♥❣ t❤❛② ✤ê✐ ✈➔ ❜ð✐ ♠ët ❝✉♥❣ trì♥ ♥➡♠ tr♦♥❣ ♠✐➲♥ y > ✈➔ ♥è✐ ❝→❝ ✤✐➸♠ ♥➔② ❧↕✐✳ ✣è✐ ✈ỵ✐ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❜✐➯♥ ❝õ❛ ❉✐r✐❝❤❧❡t ✤÷đ❝ ①→❝ ✤à♥❤ tr➯♥ ❝✉♥❣ ♥➔② ✈➔ tr➯♥ ♠ët tr♦♥❣ trữ ổ ự ỵ sỹ tỗ t t t ◆❣➔② ♥❛② ✈➜♥ ✤➲ ♥➔② ✤÷đ❝ ✤➦t t➯♥ ❧➔ ❚r✐❝♦♠✐ ■✳ ❚r♦♥❣ ♥❣❤✐➯♥ ❝ù✉ ❚r✐❝♦♠✐ ✭❬①❡♠ ✻❪✮ ❝ô♥❣ ❝✉♥❣ ❝➜♣ ♥➲♥ t↔♥❣ ❝❤♦ ❞↕♥❣ ❝❤✉➞♥ t➢❝ ✭✵✳✸✮ ♥❤÷♥❣ ❝❤ù♥❣ ♠✐♥❤ ❝õ❛ ỉ♥❣ ❝❤÷❛ ✤➛② ✤õ✳ ❙❛✉ ✤â ❝❤ù♥❣ ♠✐♥❤ ✤ó♥❣ ❝❤♦ ❞↕♥❣ ♥➔② ✤÷đ❝ t❤ü❝ ❤✐➺♥ ❜ð✐ ❈✐❜r❛r✐♦✳ ◆❤÷ ❧÷✉ þ ð tr➯♥✱ ♥❤ú♥❣ ❦➳t q✉↔ ❝õ❛ ❚r✐❝♦♠✐ ✤➣ ✤÷đ❝ sỷ t ỹ tr ự ỵ tt ❝→❝ ❞↕♥❣ ❝❤✉➞♥ t➢❝ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣ tr t ữợ t t tr ỵ tt ✈➲ ❝→❝ ❞↕♥❣ ❝❤✉➞♥ t➢❝ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣ ❤é♥ ❤ñ♣ tê♥❣ q✉→t tr➯♥ ♠➦t ♣❤➥♥❣ ❝❤õ ②➳✉ ✤÷đ❝ t❤ü❝ ❤✐➺♥ s❛✉ ✤â✳ ▲✉➟♥ ✈➠♥ ♥➔② ✤÷đ❝ tr➻♥❤ ❜➔② t❤❡♦ t➔✐ ❧✐➺✉ ❬✸❪✱ ❬✹❪✱ ❬✼❪ ♥❤ú♥❣ ❦➳t q✉↔ ữủ ỵ rút ❜✐➳♥ t❤ù❝ ❦❤→❝ ♥❤❛✉ ✤÷đ❝ sû ❞ư♥❣ ✤➸ t❤✉ ✤÷đ❝ ❝→❝ ❦➳t q✉↔ ♥➔②✳ ✸ t÷ì♥❣ ù♥❣ s✐♥❤ r❛ ❜ð✐ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ x˙ = −1 x, y˙ = −2 y ✈➔ ❈→❝ ❤➔♠ h(x) = x ✈➔ α(t, x) = 2t ❝❤➾ r❛ r➡♥❣ ❤❛✐ ❞á♥❣ ♥➔② ❧➔ t÷ì♥❣ ✤÷ì♥❣ tỉ♣ỉ✳ ❚✉② ♥❤✐➯♥✱ ❤❛✐ ❞á♥❣ ♥➔② ❦❤ỉ♥❣ ❧✐➯♥ ❤đ♣ tỉ♣ỉ ❞♦ ♥➳✉ ✤➦t t = π t❛ t❤➜② ❤➔♠ h : R2 → R2 t❤ä❛ ♠➣♥ x˙ = f (x) s➩ t❤ä❛ ♠➣♥ h(x) = h(−x) ✈ỵ✐ ♠å✐ x✱ ✤✐➲✉ ✤â ❝â ♥❣❤➽❛ ❧➔ h ❦❤æ♥❣ t❤➸ ❦❤↔ ♥❣❤à❝❤✳ ✷✳✸✳✶ ✣à♥❤ rút ỵ rút tr ✤➙② ❧➔ tr♦♥❣ tr÷í♥❣ ❤đ♣ t❤❛♠ sè✳ ❚❤❛② ✈➔♦ ✤â✱ ❝❤♦ ♠ët ❞↕♥❣ r➩ ♥❤→♥❤ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥✱ ①➨t ❤å trì♥ ❝õ❛ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ t❤❛♠ sè ❤â❛ ❜ð✐ t❤❛♠ sè ❤ú✉ ❤↕♥ ❝❤✐➲✉✳ ❚➜t ❝↔ ❣✐↔ t❤✐➳t ❧➔ ✤à❛ ♣❤÷ì♥❣ ❣➛♥ ✤✐➸♠ ✤❛♥❣ ①➨t ❝õ❛ ✤÷í♥❣ trá♥ ❜✐➺t t❤ù❝✱ tr♦♥❣ ✤â ✈✐ ♣❤➙♥ ❝õ❛ ❜✐➺t t❤ù❝ ❧➔ ❦❤→❝ ✈ỵ✐ ❣✐→ trà ✤➣ ❝❤♦ ❜➡♥❣ ❝õ❛ t số ữỡ tr trữớ ữợ t t ❝õ❛ ✤÷í♥❣ trá♥✳ ◆ë✐ ❞✉♥❣ ♥➔② ✤÷đ❝ tr➻♥❤ ❜➔② t❤❡♦ ❜➔✐ ❜→♦ sè ✭❬✹❪✱ ❬✼❪✮✳ ▼➺♥❤ ✤➲ ✷✳✸✳✸✳ ▼ët ❤å trì♥ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ a (x, y, ε) dy − 2b (x, y, ε) dxdy + c (x, y, ε) dx2 = 0, ✭✷✳✶✷✮ ❝â t❤❛♠ sè ε ❤ú✉ ❤↕♥ ❝❤✐➲✉ ❣➛♥ ✤✐➸♠ P ❝õ❛ ✤÷í♥❣ ❝♦♥❣ ❜✐➺t t❤ù❝ D(P ) = 0, dD(P ) = 0✳ ❑❤✐ ✤â trữớ ữợ t t ữớ õ dy + c (x, y, ε) dx2 = ✭✷✳✶✸✮ ❣➛♥ ✤✐➸♠ ❣è❝ t↕✐ c✱ ✈➔ c(O) = = cx(O) = cy (O) ❧➔ ♠ët ❤➔♠ trì♥ ♠ỵ✐ s❛✉ ❦❤✐ t❤❛② ✤ê✐ sü ❧ü❛ ❝❤å♥ tå❛ ✤ë✳ ✸✾ ❈❤ù♥❣ ♠✐♥❤✳ ✣➛✉ t✐➯♥✱ ❝❤å♥ tå❛ ✤ë trì♥ ✤à❛ ♣❤÷ì♥❣ ❣➛♥ ✤✐➸♠ P ✈ỵ✐ ✤✐➸♠ ❣è❝ tå❛ ✤ë✱ t↕✐ ✤✐➸♠ s ữợ trử tồ õ tr ữỡ tr trữớ ữợ õ ữỡ tr ữủ t ữợ a (x, y, ε) p2 − 2b (x, y, ε) p + c (x, y, ε) = 0, t↕✐ p = dy dx ✈➔ a, b, c ❧➔ ❝→❝ ❤➔♠ trì♥❀ a(O) = = b(O) = c(O), cy (O) = ❞♦ tå❛ ✤ë ✤➣ ❝❤å♥ ✈➔ ✤✐➲✉ ❦✐➺♥ D(O) = 0, |Dx (O)| + |Dy (O)| = 0✳ Ð ✤➙②✱ O = (0, 0, 0)✳ ✣è✐ ✈ỵ✐ ♠ët tå❛ ✤ë ♠ỵ✐ y, y = Y (x, y, ε) t❛ ❝â dy dy = Yx (x, y, ε) + Yy (x, y, ε) dx dx ❚❤➳ ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ ❝â ✤÷đ❝ s❛✉ ❦❤✐ ❜✐➳♥ ✤ê✐ ✤ì♥ ❣✐↔♥ ♣❤÷ì♥❣ tr➻♥❤ Yy2 dy dx + 2Yy [Yx − b(x, Y, ε)] dy + c(x, Y, ε) + Yx2 − 2b(x, Y, ε)Yx = 0, dx ✭✷✳✶✹✮ tr♦♥❣ ✤â sè ❤↕♥❣ t❤ù ❤❛✐ ❧➔ ♥➳✉ ❜✐➸✉ t❤ù❝ tr♦♥❣ ♥❣♦➦❝ ✈✉ỉ♥❣ ❧➔ 0✱ ♥❣❤➽❛ ❧➔ Y ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ Yx = b(x, Y, ε) ✣è✐ ✈ỵ✐ ❜➜t ❦➻ ✤✐➲✉ ❦✐➺♥ trì♥ ✤➣ ❝❤♦ tr➯♥ ♠➦t ♣❤➥♥❣ x = 0✱ ♣❤÷ì♥❣ tr➻♥❤ ❝✉è✐ ❝ị♥❣ ❝â ❞✉② ♥❤➜t ♥❣❤✐➺♠ trì♥ ❜ð✐ ✈➻ ♠➦t ♣❤➥♥❣ x = ❦❤æ♥❣ ❝â ✤✐➸♠ ✤➦❝ tr÷♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤✳ ❈❤å♥ Y (0, y, ε) = y tữỡ ự ữợ Y (x, y, ε) = y + xB(x, y, ε), t↕✐ B ❧➔ ♠ët ❤➔♠ trì♥✱ ❞♦ ❇ê ✤➲ ❍❛❞❛♠❛r❞✳ ❚❤❛② t❤➳ ♥❣❤✐➺♠ ❝❤♦ ✭✷✳✶✹✮ ✈➔ ♥❤➙♥ ❦➳t q✉↔ ✈ỵ✐ Yy−2 ❝â ❞↕♥❣ ữỡ tr tt ợ ởt ợ c, c(O) = = cx (O) = cy (O)✳ ❈❤♦ ✭✷✳✶✸✮ ♣❤➨♣ ♥➙♥❣ ❧➯♥ ❧ô② t❤ø❛ ❣➜♣ σ ❝â ❞↕♥❣ (x, p) → (x, −p) , ✹✵ tr♦♥❣ tå❛ ✤ë x ✈➔ p = dy dx tr➯♥ ♣❤÷ì♥❣ tr➻♥❤ ♠➦t ữỡ tr trữớ ữợ õ t ữủ tr➯♥ ♣❤÷ì♥❣ tr➻♥❤ ♠➦t ♣❤➥♥❣ ❜ð✐ tr÷í♥❣ ✈❡❝tì v := (−2p, cx + pcy ) ❑❤✐ ✤â ✤✐➸♠ ❣è❝ ❧➔ ✤✐➸♠ ❦➻ ❞à ❝õ❛ tr÷í♥❣ ✈❡❝tì ♥➔②✱ ♥❣♦➔✐ r❛ tr➯♥ ✤÷í♥❣ t❤➥♥❣ p = ❝õ❛ ❝→❝ ✤✐➸♠ ❝è ✤à♥❤ ❝õ❛ ♣❤➨♣ ♥➙♥❣ ❧➯♥ ❧ơ② t❤ø❛ ❣➜♣✱ tr÷í♥❣ ♥➔② õ ữợ trử r❛✱ t↕✐ ✤✐➸♠ (x, p) ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ♠➦t ♣❤➥♥❣✱ trữớ ữợ ụ tứ ❣➜♣ ❧➔ σ∗ v (x, p) = (2p, −cx + pcy ) ❉♦ ✤â ✤à♥❤ t❤ù❝ ❝õ❛ ♠❛ tr➟♥ ✈ỵ✐ ❝→❝ ❝ët v ✈➔ σ∗ v t↕✐ ♠ët ✤✐➸♠ (x, p) ❝â ❣✐→ trà 4p2 cy ✳ ❉♦ cy (O) = ♥➯♥ ✤à♥❤ t❤ù❝ ❝õ❛ ♠❛ tr➟♥ ♥➔② ❝â ❝❤➼♥❤ ①→❝ sè ❜➟❝ ❤❛✐ tr➯♥ ✤÷í♥❣ t❤➥♥❣ ❝õ❛ ❝→❝ ✤✐➸♠ ❝è ✤à♥❤ ❝õ❛ ♣❤➨♣ ♥➙♥❣ ❧➯♥ ❧ô② t❤ø❛ ❣➜♣✳ ✣➦❝ ❜✐➺t tr÷í♥❣ v ✈➔ σ∗ v ❝❤➾ t❤➥♥❣ ❤➔♥❣ tr➯♥ ❞á♥❣✳ ❚r➯♥ ❝ì sð ✤â ❣✐ỵ✐ t❤✐➺✉ ❝→❝ ❦❤→✐ ♥✐➺♠ ✈➲ t➼♥❤ t÷ì♥❣ t❤➼❝❤✳ ❚r➯♥ ♠➦t ♣❤➥♥❣✱ tr÷í♥❣ ✈❡❝tì ✈➔ ♣❤➨♣ ♥➙♥❣ ❧➯♥ ❧ơ② t❤ø❛ ❦❤↔ ✈✐ ✈ỵ✐ ✤÷í♥❣ t❤➥♥❣ ❝õ❛ ❝→❝ ✤✐➸♠ ❝è ✤à♥❤ ❣å✐ ❧➔ t÷ì♥❣ t❤➼❝❤ t↕✐ ♠ët ✤✐➸♠ ❝õ❛ ✤÷í♥❣✱ ♥➳✉ ❣➛♥ ✤✐➸♠ ♥➔② ✤à♥❤ t❤ù❝ ❝õ❛ ♠❛ tr➟♥ ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ tr÷í♥❣ ữợ ụ tứ õ số r t trữớ ữợ ụ tứ ợ ữớ ✤✐➸♠ ❝è ✤à♥❤ ✤÷đ❝ ❣å✐ ❧➔ t÷ì♥❣ t❤➼❝❤ t↕✐ ✤✐➸♠ ❝õ❛ ✤÷í♥❣ ♥➳✉ tr÷í♥❣ ❝â t❤➸ ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ trữớ tỡ tữỡ t ợ ụ tứ t↕✐ ✤✐➸♠ ♥➔②✳ ❚➼♥❤ t÷ì♥❣ t❤➼❝❤ ❝õ❛ ♣❤ỉ✐ ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ t÷ì♥❣ tü✳ ❱➼ ❞ư ✷✳✸✳✹✳ ❚r♦♥❣ ♠➦t ♣❤➥♥❣✱ tr÷í♥❣ ✈❡❝tì (x, αy) ✈ỵ✐ α > ✈➔ ♣❤➨♣ ♥➙♥❣ ❧➯♥ ❧ô② t❤ø❛ (x, y) → (α+1)x−2αy 2x−(α+1)y , α−1 α−1 ❧➔ t÷ì♥❣ t❤➼❝❤✳ ❍❛✐ ✤è✐ t÷đ♥❣ ✭❤➔♠ ❤♦➦❝ ♣❤ỉ✐ ❝õ❛ ❤➔♠✱ ①↕ ↔♥❤✱ ✳✳✳✮ ✤÷đ❝ ❣å✐ ❧➔ C r ✲t÷ì♥❣ ✤÷ì♥❣ ❞å❝ t❤❡♦ tr÷í♥❣ ✈❡❝tì ❦❤↔ ✈✐ v ❣å✐ t➢t ❧➔ C r ✲t÷ì♥❣ ✤÷ì♥❣ ♥➳✉ ❝❤ó♥❣ ❝â t❤➸ ❜✐➳♥ ✤ê✐ t❤➔♥❤ ❜ð✐ →♥❤ ①↕ C r ✲ ✈✐ ỗ ổ ữớ t trữớ ❝❤ó♥❣✳ ✣è✐ ✈ỵ✐ ❤å t❤❛♠ sè ❤ú✉ ❤↕♥ ❝õ❛ Cvr tữỡ ữỡ C r ỗ ổ tỗ tợ tỹ tr t sè ε ✈➔ →♥❤ ①↕ ❝→❝ ✤÷í♥❣ ❝♦♥❣ t➼❝❤ ♣❤➙♥ ❝õ❛ tr÷í♥❣ (v, ε˙ = 0) ✈➔♦ ❝❤➼♥❤ ♥â❀ C r tữỡ ữỡ õ tỗ t số ỵ ổ ✤✐➸♠ ❣è❝ ❝õ❛ ❤å trì♥ (v, σ1) ✈➔ ❝→❝ ❝➦♣ t÷ì♥❣ t❤➼❝❤ ❝õ❛ tr÷í♥❣ ✈❡❝tì ✈➔ ♣❤➨♣ ♥➙♥❣ ❧➯♥ ❧ơ② t❤ø❛ ❝ị♥❣ ✈ỵ✐ t❤❛♠ sè ❤ú✉ ❤↕♥ ❝❤✐➲✉ ✈➔ ♠➦t ♣❤➥♥❣ ❝õ❛ ❝→❝ ✤✐➸♠ ❝è ✤à♥❤ ❝õ❛ ♣❤➨♣ ♥➙♥❣ ❧➯♥ ❧ơ② t❤ø❛✱ ♠➔ ✤✐ q✉❛ ✤✐➸♠ ❣è❝ ❧➔ Cv∞✲ t÷ì♥❣ ✤÷ì♥❣✱ ♥➳✉ ❝❤♦ ❜➜t ❦➻ ❣✐→ trà t❤❛♠ sè ❝è ✤à♥❤ ❣➛♥ ❜➡♥❣ 0✱ tr÷í♥❣ v ❧➔ ✤÷í♥❣ ♥➡♠ ♥❣❛♥❣ ❝→❝ ✤✐➸♠ ❝è ✤à♥❤ ❝õ❛ ♣❤➨♣ ❧ô② t❤ø❛ ❤➛✉ ❦❤➢♣ ♥ì✐✳ (v, σ2 ) ●å✐ ♣❤→t ❜✐➸✉ ♥➔② ❧➔ ✤à♥❤ ỵ rút ỵ rút ❝ù✉ ❞↕♥❣ ❝❤✉➞♥ t➢❝ ✤à❛ ♣❤÷ì♥❣ ❝õ❛ ❞↕♥❣ ❝õ❛ ♣❤÷ì♥❣ tr tr trữớ ủ tờ qt ỵ t❤✉②➳t ✈➲ ❞↕♥❣ ❝❤✉➞♥ t➢❝ ✤à❛ ♣❤÷ì♥❣ ❝õ❛ ❝→❝ ❝➦♣ tữỡ t ợ trữớ tỡ ụ tứ ữợ ự ỵ ❣✐è♥❣ ♥❤÷ tr♦♥❣ ❜➔✐ ❜→♦ sè ❬✺❪✳ ●ë♣ ❝❤ù♥❣ ♠✐♥❤ ỵ ổ ❝ö ❝❤➼♥❤ ✤➸ s✉② r❛ ❝→❝ ❞↕♥❣ ❝❤✉➞♥ t➢❝ ❝❤♦ ❞↕♥❣ r➩ ♥❤→♥❤ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ tr➯♥ ♠ët trữớ tỡ ự ỵ ❣è❝ tå❛ ✤ë ❝→❝ tå❛ ✤ë ✤à❛ ♣❤÷ì♥❣ trì♥ x✱ y ✈➔ ε ❞↕♥❣ ❧→ tr➯♥ t❤❛♠ sè ♥❤÷ ✈➟② ❤å σ1 ✈➔ ♠➦t ♣❤➥♥❣ ❝õ❛ ❝→❝ ✤✐➸♠ ❝è ✤à♥❤ ❝õ❛ ♣❤➨♣ ♥➙♥❣ ❧➯♥ ❧ô② t❤ø❛ ❝â ❞↕♥❣ (x, y, ε) → (x, −y, ε) ✈➔ y = t÷ì♥❣ ù♥❣✳ ❱✐➺❝ ❧ü❛ ❝❤å♥ ♥❤÷ ✈➟② ❧➔ ✤õ ♠å✐ ❤➔♠ trì♥ f ✱f (O) = 0✱ ♠➔ ✈✐ ♣❤➙♥ t↕✐ ✤✐➸♠ ❣è❝ ❝â ❣✐→ trà ❦❤→❝ tr➯♥ ✈❡❝tì r✐➯♥❣ ❝õ❛ ✤↕♦ ❤➔♠ σ1,∗ (O) ✈➔ ❝❤å♥ tå❛ ✤ë ✤à❛ ữỡ ợ x = F + F, y = F − σ1∗ F ✱ ✈➔ ❝ò♥❣ tå❛ ✤ë ε ✳ ❚❤❡♦ t➼♥❤ t÷ì♥❣ t❤➼❝❤ ❝õ❛ v = (v1 , v2 ) ✈➔ σ1 ✱ t➻♠ ❤➔♠ sè v1 (x, y, ε) v2 (x, y, ε) = −v1 (x, y, ε)v2 (x, −y, ε)−v1 (x, −y, ε)v2 (x, y, ε), v1 (x, −y, ε) −v2 (x, −y, ε) ❝â ❝❤➼♥❤ ①→❝ sè ❜➟❝ ❤❛✐ tr➯♥ ♠➦t ♣❤➥♥❣ y = ❝õ❛ ❝→❝ ✤✐➸♠ ❝è ✤à♥❤ ❝õ❛ ♣❤➨♣ ♥➙♥❣ ❧➯♥ ❧ô② t❤ø❛✳ ✹✷ ✣➦❝ ❜✐➺t tr➯♥ ♠➦t ♣❤➥♥❣ ♥➔② v1 v2 ≡ 0✳ ◆❤÷♥❣ tr÷í♥❣ (v, ε˙ = 0) ❧➔ ✤÷í♥❣ ♥➡♠ ♥❣❛♥❣ ✤➳♥ ♠➦t ♣❤➥♥❣ ❤➛✉ ❦❤➢♣ ♥ì✐ ❞♦ t sỷ ỵ õ t❤ù❝ ❝✉è✐ ❝ò♥❣ ❜❛♦ ❤➔♠ v1 (x, 0, ε) ≡ 0✳ ◆❣♦➔✐ r❛ (0, 1, 0) ❧➔ sü ①✉➜t ❤✐➺♥ ❝õ❛ ✤↕♦ ❤➔♠ σ1∗ , σ2∗ ✈ỵ✐ ❣✐→ trà r✐➯♥❣ −1 t↕✐ ✤✐➸♠ ❜➜t ❦ý ❝õ❛ ♠➦t ♣❤➥♥❣✳ ❉♦ ✤â ❝→❝ ✤↕♦ ❤➔♠ ♥➔② ❧➔ ❣✐è♥❣ ♥❤❛✉ t↕✐ ✤✐➸♠ ❜➜t ❦ý ❝õ❛ ♠➦t ♣❤➥♥❣✱ ♥➯♥ tr♦♥❣ ❝→❝ tå❛ ✤ë ✤÷đ❝ ố õ t t ữợ ❞↕♥❣ (x, y, ε) → x + y r (x, y, ε) , −y + y s (x, y, ε) , ε , ✈ỵ✐ ♠ët sè ❤➔♠ trì♥ r s tỗ t tồ = (x + y R (x, y, ε) , η = y + y S (x, y, ε) , ✈ỵ✐ ♠ët sè ❤➔♠ trì♥ R ✈➔ S ✈➔ ❣✐è♥❣ ε✱ tr♦♥❣ ✤â ❤å σ2 ❝õ❛ ♣❤➨♣ ♥➙♥❣ ❧➯♥ ❧ô② t❤ø❛ ❝â ❞↕♥❣ (ζ, η, ε) → (ζ, −η, ε) P❤➛♥ ❝á♥ ❧↕✐ ❝õ❛ ❝❤ù♥❣ ♠✐♥❤ ❞ü❛ tr➯♥ ữỡ ỗ t số ữủ t ❚❤♦♠✳ ❚↕✐ ✤à❛ ♣❤÷ì♥❣ ❣➛♥ ✤✐➸♠ ❣è❝ ①➨t sü ❜✐➳♥ ❞↕♥❣ trì♥ ❝õ❛ tå❛ ✤ë γt : (ζt , ηt , ε) → (ζt , −ηt , ε) , tr♦♥❣ ✤â ζt = x + ty R (x, y, ε) , ηt = y + ty S (x, y, ε) , ✈➔ ❜✐➳♥ ❞↕♥❣ t÷ì♥❣ ù♥❣ ❝õ❛ ❤å ♣❤➨♣ ♥➙♥❣ ❧➯♥ ❧ô② t❤ø❛ σ1 ✈➔ σ2 ✿ (ζt , ηt , ε) → (ζt , −ηt , ε) ❙✉② r❛ tø Vt ✱ tr÷í♥❣ ❜✐➳♥ ❞↕♥❣ ✈✐ ♣❤➙♥ t÷ì♥❣ ù♥❣ ❧➔ ✈➟♥ tè❝ ❝❤✉②➸♥ ✤ë♥❣ ❝õ❛ ✤✐➸♠ ữợ ởt trỡ ❧ô② t❤ø❛✳ ❉➵ t❤➜② Vt ❧➔ ❜✐➳♥ ❞↕♥❣ ✤❛♥❣ ①➨t ❦❤ỉ♥❣ ❝â ♣❤➛♥ tû ❞å❝ t❤❡♦ trư❝ t❤❛♠ sè✳ ❇ê ✤➲ ✷✳✸✳✻✳ ❚r÷í♥❣ V ❧➔ tr÷í♥❣ ❜✐➳♥ ❞↕♥❣ ✈✐ ♣❤➙♥ ❝õ❛ ❜✐➳♥ ❞↕♥❣ trì♥ ❤å σ✱ ♥â ❧➔ t❤ỵ tr➯♥ t❤❛♠ sè ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ σ∗V ✹✸ = −V ❇ê ✤➲ ✷✳✸✳✼✳ ✣è✐ ✈ỵ✐ ❜✐➳♥ ❞↕♥❣ g ❝õ❛ ỗ t tự õ tợ tr t số ợ ❤å ✈➟♥ tè❝ h ❝õ❛ ♣❤➨♣ ♥➙♥❣ ❧➯♥ ❧ô② t❤ø❛ σ : (x, y, ε) → (x, −y, ε) ❜à ❜✐➳♥ ❞↕♥❣ ✈ỵ✐ ✈➟♥ tè❝ h − σ∗h ◆❤ú♥❣ ❜ê ✤➲ ♥➔② t÷ì♥❣ tü ♥❤÷ ♥❤ú♥❣ ♣❤→t ❜✐➸✉ t÷ì♥❣ ù♥❣ tø ❜➔✐ ❜→♦ sè ✭❬✺❪✱ ❬✻❪✮✳ ❈❤ù♥❣ ♠✐♥❤ ❜ð✐ ♣❤➨♣ t♦→♥ trü❝ t✐➳♣✳ ❉♦ ❇ê ✤➲ ✷✳✸✳✻ ✤➣ ❝❤ù♥❣ ♠✐♥❤ ỵ ữỡ [0, 1] ❝õ❛ t✲ trö❝✱ tr➻♥❤ ❜➔② ❜✐➳♥ ❞↕♥❣ ✈➟♥ tè❝ Vt = ft v − (γt∗ ft ) γ∗ v ✭✷✳✶✺✮ ❚r♦♥❣ ✤â v ❧➔ tr÷í♥❣ ✈❡❝tì ✈➔ ft ❧➔ ❤➔♠ trì♥ tr➯♥ t✱ x✱ y ✈➔ ε✳ ❑❤↔ ♥➠♥❣ ❣✐↔✐ ữủ ữỡ tr ố ợ ft ỹ tr sỹ tữỡ t ợ v ♠➔ ♥❣❛② ❧➟♣ tù❝ ❜❛♦ ❤➔♠ sü t÷ì♥❣ t❤➼❝❤ ❝õ❛ ❤å v ✈➔ γt ❝❤♦ ♠å✐ t ∈ [0, 1] ❞♦ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ γt ✳ ❇✐➳♥ ❞↕♥❣ ✈➟♥ tè❝ ✭❜ä q✉❛ ❝❤➾ sè t✮ V tr➯♥ ♠➦t ♣❤➥♥❣ y = ✭❤♦➦❝ η = 0✮ ❝â ❦❤æ♥❣ ➼t ♥❤➜t ❝õ❛ ❝➜♣ ❤❛✐✳ ❉♦ ✤â✱ ♥â ❝â t❤➸ ✤÷đ❝ tr➻♥❤ ữợ V = h(, , ) r(, η, ε) ✭✷✳✶✻✮ , ✈ỵ✐ ♠ët sè ❤➔♠ trì♥ h ✈➔ r✳ ◆❣♦➔✐ r❛✱ ❞♦ ❇ê ✤➲ ✷✳✸✳✻ ❜✐➳♥ ❞↕♥❣ ✈➟♥ tè❝ ♣❤↔✐ t❤ä❛ ♠➣♥ ✤➥♥❣ t❤ù❝ γ∗ V = −V ✳ ❚❤❛② ✈➔♦ ❜✐➸✉ t❤ù❝ ✭✷✳✶✻✮ ❝â η h(ζ, −η, ε) −η r(ζ, −η, ε) =− η h(ζ, η, ε) η r(ζ, η, ε) ❉♦ ✤â h(ζ, −η, ε) = −h(ζ, η, ε) ✈➔ r(ζ, −η, ε) = r(ζ, η, ε) ✈ỵ✐ ❝→❝ ❤➔♠ h ✈➔ r ❧➛♥ ❧÷đt ❧➔ ❝❤➤♥ ✈➔ ❧➫✳ ❉♦ õ t t ữỡ ố õ ữợ ❞↕♥❣ h(ζ, η, ε) = ηp(ζ, η , ε) ✈➔ r(ζ, η, ε) = q(ζ, η , ε) ✈ỵ✐ ♠ët sè ❤➔♠ trì♥ p ✈➔ q ✳ ❉♦ ✤â ✈➟♥ tè❝ V ❝â ❞↕♥❣ V (ζ, η, ε) = η p(ζ, η , ε) ✹✹ ∂ ∂ + η q(ζ, η , ε) ∂ζ ∂η ✭✷✳✶✼✮ ❍ì♥ ♥ú❛ tr➯♥ ♠➦t ♣❤➥♥❣ η = ❝â γ∗ v = −v ❞♦ ✤✐➲✉ ❣✐↔ sû ỵ t tữỡ t t t ♥➡♠ ♥❣❛♥❣✳ ❉♦ ✤â ❣➛♥ ✤✐➸♠ ❣è❝ ❝õ❛ tr÷í♥❣ v õ t ữủ t ữợ v(, , ) = ηl(ζ, η, ε) ∂ ∂ + m(ζ, η, ε) , ∂ζ ∂η ✭✷✳✶✽✮ ✈ỵ✐ ♠ët sè ❤➔♠ trì♥ l ✈➔ m✳ ▲ó❝ ♥➔② ❤➔♠ f ❧➔ tê♥❣ ❝õ❛ ❝→❝ ♣❤➛♥ ❝❤➤♥ ✈➔ ❧➫ ❝õ❛ ♥â ✈ỵ✐ ❜✐➳♥ sè ❦❤ỉ♥❣ ✤ê✐ η ✱ tù❝ ❧➔ f (ζ, η, ε) = u(ζ, η , ε) + ηω(ζ, η , ε) t↕✐ u ✈➔ ω ❧➔ ❝→❝ ❤➔♠ trì♥✳ ❚❤❛② t❤➳ ❜✐➸✉ t❤ù❝ ♥➔② ❝❤♦ f ✈➔ ❜✐➸✉ t❤ù❝ ✭✷✳✶✼✮ ✈➔ ✭✷✳✶✽✮ t❤➔♥❤ ❝→❝ sè ❤↕♥❣ ❜➯♥ ♣❤↔✐ ❝õ❛ ✭✷✳✶✺✮ ❝â ✤÷đ❝ ∂ ∂ + m(ζ, η, ε) ∂ζ ∂η ✭✷✳✶✾✮ (γt∗ ft ) (ζ, η, ε) = f (ζ, −η, ε) = u(ζ, η , ε) − ηω(ζ, η , ε) (ft v)(ζ, η, ε) = [u(ζ, η , ε) + ηω(ζ, η , ε)] ηl(ζ, η, ε) ∂ ∂ (γt∗ v) (ζ, η, ε) = − ηl(ζ, −η, ε) ∂ζ + m(ζ, −η, ε) ∂η ∂ ∂ + m(ζ, −η, ε) ∂η ((γt∗ ft ) γt∗ v) (ζ, η, ε) = −[u(ζ, η , ε) − ηω(ζ, η , ε)] ηl(ζ, −η, ε) ∂ζ ✭✷✳✷✵✮ ▲ó❝ ♥➔② t❤❛② t❤➳ ❝→❝ ❜✐➸✉ t❤ù❝ ✭✷✳✶✼✮✱ ữủt ợ V ft v ✈➔ (γt∗ ft ) γt∗v ✮ ✈➔♦ ❜✐➸✉ t❤ù❝ ✭✷✳✶✺✮ t❛ ❝â ∂ ∂ η p(ζ, η , ε) ∂ζ + η q(ζ, η , ε) ∂η ∂ ∂ = [u(ζ, η , ε) + ηω(ζ, η , ε)] ηl(ζ, η, ε) ∂ζ + m(ζ, η, ε) ∂η ∂ ∂ +[u(ζ, η , ε) − ηω(ζ, η , ε)](ηl(ζ, −η, ε) ∂ζ + m(ζ, −η, ε) ∂η ) ∂ = [uη(l(ζ, η, ε) + l(ζ, −η, ε)) + ωη (l(ζ, η, ε) − l(ζ, −η, ε))] ∂ζ ∂ +[u(m(ζ, η, ε) + m(ζ, −η, ε)) + ωη(m(ζ, η, ε) − m(ζ, −η, ε))] ∂η ❈➙♥ ❜➡♥❣ ❝→❝ ♣❤➛♥ tû ♣❤➼❛ tr→✐ ✈➔ ✈➳ ♣❤↔✐ ❝õ❛ ❜✐➸✉ t❤ù❝ t✐➳♣✱ t❤❡♦ ✤✐ ✤➳♥ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤ tr➯♥ u ✈➔ ω ✿ uη(l(ζ, η, ε) + l(ζ, −η, ε)) + ωη (l(ζ, η, ε) − l(ζ, −η, ε)) = η p(ζ, η , ε) u(m(ζ, η, ε) + m(ζ, −η, ε)) + ωη(m(ζ, η, ε) − m(ζ, −η, ε)) = η q(ζ, η , ε) ✹✺ Ð ✤➙② ❝❤✐❛ ♣❤÷ì♥❣ tr➻♥❤ ✤➛✉ t✐➯♥ ❝❤♦ η rỗ rút ữỡ tr õ u(l(, , ε) + l(ζ, −η, ε)) + ωη(l(ζ, η, ε) − l(ζ, −η, ε)) = η p(ζ, η , ε) u(m(ζ, η, ε) + m(ζ, −η, ε)) + ωη(m(ζ, η, ε) − m(ζ, −η, ε) = η q(ζ, η , ε) ✭✷✳✷✶✮ ✣à♥❤ t❤ù❝ ❝õ❛ ♠❛ tr➟♥ ❝õ❛ ❤➺ ♥➔② ❧➔ l(ζ, η, ε) + l(ζ, −η, ε) η(l(ζ, η, ε) − l(ζ, −η, ε)) m(ζ, η, ε) + m(ζ, −η, ε) η(m(ζ, η, ε) − m(ζ, −η, ε)) = η [L+Q(ζ, η, ε)], t↕✐ L = 2[l(0, 0, 0)mη (0, 0, 0) − lη (0, 0, 0)m(0, 0, 0)] ✈➔ Q ❧➔ ❤➔♠ trì♥ tr✐➺t t✐➯✉ t↕✐ ✤✐➸♠ ❣è❝✳ ❉♦ ✤â ✭✷✳✷✶✮ ❧➔ trì♥ ❣✐↔✐ ữủ ố ợ u ố L = ❜ð✐ ✈➻ ✈➳ ♣❤↔✐ ❝õ❛ ❤➺ ❝❤✐❛ ❝❤♦ η ✳ ◆❤÷♥❣ L ❦❤ỉ♥❣ ❜➡♥❣ ❞♦ ✤✐➲✉ ❦✐➺♥ t➼♥❤ t÷ì♥❣ t❤➼❝❤ ❝õ❛ ❤å ♣❤➨♣ ♥➙♥❣ ❧➯♥ ❧✉ÿ t❤ø❛ ✈➔ tr÷í♥❣✳ ❚❤➟t ✈➟② ❞♦ t➼♥❤ t÷ì♥❣ t❤➼❝❤✱ ❞✐➺♥ t➼❝❤ ❝õ❛ ❤➻♥❤ ❜➻♥❤ ❤➔♥❤ ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ ❣✐→ trà ❝õ❛ tr÷í♥❣ v ✈➔ γ∗ v ❝â sè ❜➟❝ ❤❛✐ tr➯♥ ♠➦t ♣❤➥♥❣ η = ❝õ❛ ❝→❝ ✤✐➸♠ ❝è ✤à♥❤ ❝õ❛ ♣❤➨♣ ♥➙♥❣ ❧➯♥ ❧ô② t❤ø❛✳ ❉♦ ✤â ❤➔♠ v ηl(ζ, η, ε) m(ζ, η, ε) (ζ, η, ε) = γ∗ v −ηl(ζ, −η, ε) −m(ζ, −η, ε) = −η[l(ζ, η, ε)m(ζ, −η, ε) − l(ζ, −η, ε)m(ζ, η, ε)] = 2η [L + H(ζ, η, ε)], ✈ỵ✐ H ❧➔ ❤➔♠ trì♥ ❜➜t ❜✐➳♥ t↕✐ ✤✐➸♠ ❣è❝ ❝â sè ❜➟❝ ❤❛✐ tr➯♥ ♠➦t ♣❤➥♥❣ ♥➔②✳ ❱➻ ✈➟② L = 0✱ ✈➔ t➼♥❤ ❝❤➜t ✤à❛ ♣❤÷ì♥❣ ❣➛♥ ✤✐➸♠ ❣è❝✱ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✷✶✮ ❞➵ ❞➔♥❣ ❣✐↔✐ ✤÷đ❝ ❦❤✐ ❣✐ú ♥❣✉②➯♥ u ✈➔ ω ✳ ❉♦ ✤â ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✺✮ ❞➵ ❞➔♥❣ ❣✐↔✐ ✤÷đ❝ ✈➔ ♣❤ỉ✐ ❝õ❛ ❤å ♣❤➨♣ ♥➙♥❣ ❧➯♥ ❧ô② t❤ø❛ σ1 ✈➔ σ2 t↕✐ ✤✐➸♠ ❣è❝ ❧➔ C∞ v ✲ t÷ì♥❣ ✤÷ì♥❣ ♠↕♥❤✳ ỵ rút ữủ ự ú ỵ ❉↕♥❣ ❝❤✉➞♥ t➢❝ urr + (1 − r)uφφ = ụ sỷ ữỡ tr ợ t sè ❤ú✉ ❤↕♥ ❝❤✐➲✉✳ ❈ö t❤➸✱ ♥➳✉ ❝❤♦ ♠ët sè ❣✐→ trà ✹✻ ❝õ❛ t❤❛♠ sè ❝â ❝→❝ ✤✐➲✉ ❦✐➺♥ tr ú ỵ trữợ ữủ tọ t t➢❝ ❣➛♥ ✤✐➸♠ ❦➻ ❞à ❣➜♣ ✤❛♥❣ ①➨t ❝â ❞↕♥❣ urr + (1 r)u = tr ỗ ♣❤ỉ✐ ❝õ❛ ♠➦t ♣❤➥♥❣✱ ♠➔ ❤✐➺♥ ♥❛② t➼♥❤ ❧✐➯♥ tư❝ tở t số ữ ữ ỵ tr ♣❤÷ì♥❣ tr➻♥❤ ❝õ❛ ✭✷✳✷✷✮ ❞ị♥❣ ✤➸ ♣❤➙♥ t➼❝❤ ❝→❝ ✤✐➸♠ ❦ý ❞à ❝õ❛ ✤✐➺♥ tr÷í♥❣ sâ♥❣ ✤✐➺♥ tø✳ ◆❤÷♥❣ ❞➵ t r ỗ ổ ổ tỗ t t ỡ ữỡ tr r ỳ tỗ ữ ♠ët ❝ì sð ❜ê s✉♥❣✳ ❱➻ ✈➟②✱ ❝→❝ ❤➻♥❤ t❤ù❝ ❝❤✉➞♥ t➢❝ tæ♣æ ❝❤♦ ❝→❝ ✤✐➸♠ ❦➻ ❞à ❣➜♣ ❦❤æ♥❣ t❤➸ →♣ ❞ö♥❣ trü❝ t✐➳♣ ✤➸ ♣❤➙♥ t➼❝❤ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣ ❦✐➸✉ ❤é♥ ❤đ♣✳ ✷✳✸✳✷ ❉↕♥❣ ❝❤✉➞♥ t➢❝ trì♥ ❝❤♦ ❝→❝ ✤✐➸♠ ❦➻ ❞à ❣➜♣ ❙è ♠ơ ❝õ❛ ♠ët ✤✐➸♠ ❦➻ ❞à ❤②♣❡r❜♦❧ ❝õ❛ tr÷í♥❣ ✈❡❝tì tr➯♥ ♠➦t ♣❤➥♥❣ ✤÷đ❝ ①→❝ ✤à♥❤ ❧➔ t✛ ❧➺ ❝õ❛ ❣✐→ trà r✐➯♥❣ ♠ỉ✤✉♥ ❧ỵ♥ ♥❤➜t ❝õ❛ ♥â t✉②➳♥ t➼♥❤ ❤â❛ t↕✐ ✤✐➸♠ ♥❤ä ♥❤➜t ❝❤♦ ✤✐➸♠ ②➯♥ ♥❣ü❛✱ ✤✐➸♠ ♥ót ✈ỵ✐ ♠ỉ✤✉♥ t✛ ❧➺ ❝õ❛ ♣❤➛♥ ↔♦ ❣✐→ trà r ợ tỹ t ởt trữớ ữợ số ụ ởt ý r ữủ ✤à♥❤ ♥❣❤➽❛ ❧➔ ♠ët tr♦♥❣ ❝→❝ tr÷í♥❣ ✈❡❝tì t÷ì♥❣ ù♥❣ ợ r ố ụ ữủ t ổ ỵ ữỡ tr trì♥ tê♥❣ q✉→t a(x, y)uxx + b(x, y)uxy + c(x, y)uyy = 0, ♠é✐ ✤✐➸♠ ❦➻ ❞à ❣➜♣ P ❧➔ ❤②♣❡r❜♦❧✐❝✳ P❤÷ì♥❣ tr➻♥❤ ✤➦❝ tr÷♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ♥➔② ❧➔ tữỡ ữỡ tổổ ợ ữỡ tr uxx + (y + Kx2 )uyy = 0, ✭✷✳✷✷✮ ❣➛♥ ❣è❝ tå❛ ✤ë ✈ỵ✐ K = −1, 201 ✈➔ ❧➔ ✤✐➸♠ ②➯♥ ♥❣ü❛✱ ✤✐➸♠ ♥ót ✈➔ t✐➯✉ ✤✐➸♠ t÷ì♥❣ ù♥❣✳ ✹✼ ❱➼ ❞ư ✷✳✸✳✾✳ ❚r➯♥ ♠➦t ♣❤➥♥❣ ❝õ❛ ❜✐➳♥ sè u ✈➔ v ♣❤æ✐ ❝õ❛ C k ✲ t✉②➳♥ t➼♥❤ ❤â❛ tê♥❣ q✉→t ợ k trữớ ữợ õ ữ ✤✐➸♠ ❦➻ ❞à ✈ỵ✐ sè ♠ơ α ❝õ❛ ✤✐➸♠ ②➯♥ ♥❣ü❛✱ ✤✐➸♠ ♥ót ❤♦➦❝ t✐➯✉ ✤✐➸♠ ❧➔ C k ✲ q✉ÿ ✤↕♦ t÷ì♥❣ ✤÷ì♥❣ tø ♣❤ỉ✐ tr➯♥ ✤✐➸♠ ❣è❝ ❝õ❛ trữớ ữợ trữớ tỡ k ✈ỵ✐ k = u ω ❝❤♦ ✤✐➸♠ ②➯♥ ♥❣ü❛ ✈➔ ✤✐➸♠ ♥ót✱ ✈➔ ✈ỵ✐ k = α 2(α+1) α2 +1 ❝❤♦ t✐➯✉ ✤✐➸♠✳ ❱❡❝tì ♥➔② ✈➔ ♣❤➨♣ ♥➙♥❣ ❧➯♥ ❧ô② t❤ø❛ (u, ω) → (u, −ω) ❧➔ tữỡ t ỵ Pổ ữỡ tr➻♥❤ ✤➦❝ tr÷♥❣ ✭✵✳✶✮ C k ✲ t✉②➳♥ t➼♥❤ ❤â❛✳ ❑❤✐ ✤â ✤✐➸♠ ❦➻ ❞à ❣➜♣ ✈ỵ✐ sè ♠ơ α ❧➔ ♣❤ỉ✐ t↕✐ ✤✐➸♠ ❣è❝ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ uxx + −y + tr♦♥❣ ❤➺ tå❛ ✤ë trì♥ t❤➼❝❤ ❤đ♣✳ kx uxx = 0, ❈→❝ ♣❤→t ❜✐➸✉ s❛✉ ✤➙② tr➻♥❤ ❜➔② ❝→❝ ❞↕♥❣ ❝❤✉➞♥ t➢❝ t÷ì♥❣ ù♥❣ ❝❤♦ ♠↕♥❣ ữợ trữ õ ỵ ❬✺❪✱❬✻❪✮ P❤ỉ✐ ❝õ❛ ❤å ❝→❝ ✤➦❝ tr÷♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✵✳✶✮ ❧➔ ❤②♣❡r❜♦❧✐❝ C k ✲ t✉②➳♥ t➼♥❤ ❤â❛ ✈➔ ✤✐➸♠ ❦➻ ❞à ❣➜♣ ✈ỵ✐ sè ♠ơ α✳ ❑❤✐ ✤â C ỗ ổ ợ ổ t ố ❤å ❝→❝ ✤÷í♥❣ ❝♦♥❣ |x ± √ y|−α x √ ± y = c, (c ∈ R), α ❤♦➦❝ (|x ± ❤♦➦❝ √ y|−α x √ √ ± y = c) ∪ (x ± y = 0), c ∈ R, α √ ±α−1 y = R sin(α−1 ln R + c) , ≤ c ≤ 2π, √ x ± y = R cos(α−1 ln R + c) ✈ỵ✐ ✤✐➸♠ ②➯♥ ♥❣ü❛✱ ✤✐➸♠ ♥ót ✈➔ t✐➯✉ ✤✐➸♠ t❤➼❝❤ ❤đ♣✱ t↕✐ R = ✹✽ (x ± √ y)2 + α−2 y ✳ ✣✐➲✉ ❦✐➺♥ ❝õ❛ C ∞ ✲ t✉②➳♥ t➼♥❤ ❤â❛ tr ỵ ổ ❈ư t❤➸✱ t✐➯✉ ✤✐➸♠ ❦❤ỉ♥❣ s✉② ❜✐➳♥ ❧✉ỉ♥ ❧➔ C ∞ ✲ t✉②➳♥ t➼♥❤ ❤â❛✱ ✈➔ ♠ët ♥ót ❧➔ C ∞ ✲ t✉②➳♥ t➼♥❤ ❤â❛ ♥➳✉ sè ♠ô ❝õ❛ ♥â ❦❤ỉ♥❣ ♣❤↔✐ ❧➔ sè tü ♥❤✐➯♥✳ ❈✉è✐ ❝ị♥❣✱ t❤❡♦ ✤à♥❤ ỵ ỹ ợ số ụ ❧➔ C ∞ ✲ t✉②➳♥ t➼♥❤ ❤â❛ ♥➳✉ ✤✐➸♠ (1, α) ❧➔ ✤✐➸♠ ❝â ❞↕♥❣ (M, ν) tù❝ ❧➔ min{|1 − m1 − m2 α| , |α − m1 − m2 α| } ≥ M v |m| ❝❤♦ ♠å✐ sè ♥❣✉②➯♥ ✈❡❝tì m = (m1 , m2 ) ✈ỵ✐ m1 , m2 ❦❤ỉ♥❣ ➙♠ ✈➔ ✈ỵ✐ m1 + m2 ≥ t r ữợ t ủ ✈ỵ✐ M > ❧➔ ❝→❝ ✤✐➸♠ ❝â ❞↕♥❣ (M, ν) ❧➔ ❜➡♥❣ 0✱ ♥➳✉ ν > 1✳ ❉♦ ✤â✱ ❝â ✤✐➲✉ ❦✐➺♥ C ∞ ✲ t✉②➳♥ t➼♥❤ ❤â❛ ❝❤♦ t➟♣ ❝♦♥ ♠ð trò ♠➟t ♠ð ð ❦❤➢♣ ♠å✐ ❝õ❛ ❝→❝ ♥ót ✈➔ t✐➯✉ ✤✐➸♠✱ ✈➔ ❝❤➾ ❝❤♦ ❝→❝ t➟♣ ❝♦♥ trị ♠➟t ð ❦❤➢♣ ♠å✐ ♥ì✐ ❝õ❛ ✤✐➸♠ ②➯♥ ♥❣ü❛✳ ❇➔✐ t♦→♥ ❧➔ ❝ë♥❣ ❤÷ð♥❣ ❝❤♦ ✤✐➸♠ ②➯♥ ♥❣ü❛ ð ❦❤➢♣ ♠å✐ ♥ì✐ trị ♠➟t ✈➔ ❝ë♥❣ ❤÷ð♥❣ ✤✐➸♠ ②➯♥ ♥❣ü❛ ♥â✐ ❝❤✉♥❣ ❧➔ ❦❤æ♥❣ C ∞ ✲ t✉②➳♥ t➼♥❤ ❤â❛✳ ❉↕♥❣ ❝❤✉➞♥ t➢❝ ❝❤♦ ❝ë♥❣ ❤÷ð♥❣ ❣➜♣ tê♥❣ q✉→t ✈➔ ❝→❝ ✤✐➸♠ ❦➻ ❞à ❣➜♣ ❝➜♣ ✤➲✉ ✤↕t ✤÷đ❝ tr♦♥❣ ❜➔✐ ❜→♦ sè ❬✷❪✳❱✐➺❝ ✤÷❛ r❛ ❝→❝ t q tữỡ ự tữỡ tỹ ợ ỵ tr é tr trữớ ❤đ♣ ❝ë♥❣ ❤÷ð♥❣ ✤✐➸♠ ②➯♥ ♥❣ü❛ ❣➜♣ ✤➸ ❤♦➔♥ t❤➔♥❤ ữợ ✤➦❝ tr÷♥❣ ❝❤♦ ♣❤÷ì♥❣ tr➻♥❤ ❤é♥ ❤đ♣ tê♥❣ q✉→t ❣➛♥ ❝→❝ ✤✐➸♠ ❦➻ ❞à ❣➜♣✱ ♥❣❤➽❛ ❧➔ ❝❤♦ t➟♣ ❝♦♥ ♠ð trị ♠➟t ❦❤➢♣ ❝õ❛ ✭✵✳✶✮ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❝õ❛ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ tr♦♥❣ ✤ó♥❣ trì♥ ❤♦➦❝ trì♥ ✤➛② ✤õ tổổ t ỵ Pổ ❝→❝ ✤➦❝ tr÷♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✵✳✶✮ tr➯♥ ✤✐➸♠ ②➯♥ ♥❣ü❛ ❣➜♣ ✈ỵ✐ sè ♠ơ α = − qr ✱ tr♦♥❣ ✤â r ✈➔ q ❧➔ sè tü ♥❤✐➯♥ ✈➔ qr ❧➔ ♣❤➙♥ sè tè✐ ❣✐↔♥✳ ❑❤✐ ✤â C ∞✲ ✈✐ ♣❤æ✐ ❝❤♦ ♣❤æ✐ t↕✐ ✤✐➸♠ ❣è❝ ❝õ❛ ❤å ✤➦❝ tr÷♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ k uxx + −y − x2 ± xr+q+2 +Ax2(r+q)+2 uyy = 0, α tr♦♥❣ ✤â k = 2(α+1) ✈➔ A ❧➔ t❤❛♠ sè t❤ü❝✳ ú ỵ ữ ỵ t ✤↕t ✤÷đ❝ ❦❤ỉ♥❣ ❧➔ ❞↕♥❣ ❝❤✉➞♥ t➢❝ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤✳ Pữỡ tr tữỡ ự ợ t ✹✾ ♥❤÷ ♥❤❛✉ ❧➯♥ ❤➔♠ trì♥ ♠➔ ❜➡♥❣ tr♦♥❣ ♠✐➲♥ D > 0✳ ❈❤➥♥❣ ❤↕♥ ♥❤÷✱ ♥➳✉ ❝❤♦ ♣❤÷ì♥❣ tr tờ qt ổ ữợ t ♥â trð t❤➔♥❤ ♠ët tr♦♥❣ ♥❤ú♥❣ ♣❤÷ì♥❣ tr➻♥❤ k uxx + −y − x2 uyy = 0, ❣➛♥ ✤✐➸♠ ố tồ õ õ ỵ ❝❤➼♥❤ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣ t÷ì♥❣ ù♥❣ ❜à ❣✐↔♠ ❝â ❞↕♥❣ k (1 + a(x, y))uxx + b(x, y)uxy + −y − x2 (1 + c(x, y))uyy = 0, ✈ỵ✐ ♠ët sè ❤➔♠ trì♥ a✱ b ✈➔ c ❜➡♥❣ tr♦♥❣ ♠✐➲♥ D > 0✱ ❜➡♥❣ ❝→❝❤ t❤❛② ✤ê✐ trì♥ tå❛ ✤ë ✈➔ ♥❤➙♥ tr➯♥ ❝→❝ ❤➔♠ trì♥ ❦❤ỉ♥❣ ❜➜t ❜✐➳♥✳ ▲➔♠ t❤➳ ♥➔♦ ✤➸ ❧♦↕✐ ❜ä ♥❤ú♥❣ a✱ b ✈➔ c ❧➔ ♠ët ✈➜♥ ✤➲ ♠ð ❝➛♥ ❝â t❤í✐ ❣✐❛♥ ♥❣❤✐➯♥ ❝ù✉ t✐➳♣✳ ✺✵ ❑➳t ❧✉➟♥ ❉↕♥❣ ❝❤✉➞♥ t➢❝ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣ t✉②➳♥ t➼♥❤ ❝➜♣ ❤❛✐ tr➯♥ ♠➦t ♣❤➥♥❣ ❧➔ ♠ët ❜➔✐ t♦→♥ ✤÷đ❝ ✤÷❛ r❛ s❛✉ ❦❤✐ ①✉➜t ❤✐➺♥ ♥❤ú♥❣ ❞↕♥❣ ❝❤✉➞♥ t➢❝ ♠➔ ✤↕✐ ❞✐➺♥ ❝❤♦ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ❧♦↕✐ ❡❧✐♣t✐❝ ✈➔ ♣❤÷ì♥❣ tr➻♥❤ ❧♦↕✐ ❤②♣❡r❜♦❧✐❝✱ ✤÷đ❝ sû ❞ư♥❣ ♥❤✐➲✉ tr♦♥❣ ❣✐↔✐ t➼❝❤ ✤➸ →♣ ❞ö♥❣ tr♦♥❣ ✈✐➺❝ ❣✐↔✐ q✉②➳t ❝→❝ ❜➔✐ t♦→♥ ❦❤→❝ ♥❤❛✉✱ ♠æ t↔ sü ❝❤✉②➸♥ ✤ë♥❣ ❝õ❛ ❞➙② ✈➔ sü t❤❛② t❤➳ ✈➟♥ tè❝ ❝õ❛ ❝❤➜t ❧ä♥❣ ❦❤ỉ♥❣ ♥➨♥ ✤÷đ❝ t÷ì♥❣ ù♥❣✳ ◆ë✐ ❞✉♥❣ ❝❤➼♥❤ ✤÷đ❝ tr➻♥❤ tr ỗ r ởt sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ✈➲ ❦❤→✐ ♥✐➺♠ ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣✱ ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣ ❝➜♣ ❤❛✐✱ ❞↕♥❣ ❝❤✉➞♥ t➢❝ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❤②♣❡r❜♦❧✐❝✱ ♣❛r❛❜♦❧✐❝✱ ❡❧✐♣t✐❝✳ ✷✳ ❚r➻♥❤ ❜➔② ❞↕♥❣ ❝❤✉➞♥ t➢❝ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣ t✉②➳♥ t➼♥❤ ❝➜♣ ❤❛✐ ✈ỵ✐ ❤❛✐ ❜✐➳♥ ✤ë❝ ❧➟♣✱ ❞↕♥❣ ❝❤✉➞♥ t➢❝ trì♥✱ ❞↕♥❣ ❝❤✉➞♥ t➢❝ ❦❤ỉ♥❣ ✤à❛ ♣❤÷ì♥❣✳ ❙û ❞ư♥❣ ❝→❝ ❦➳t q✉↔ ✤↕t ✤÷đ❝ ✈➔♦ ✈✐➺❝ ❝❤ù♥❣ ♠✐♥❤ sü ✤à♥❤ ❧➼ rót ❣å♥✳ ✺✶ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✶✳ ❇r✉❝❡ ❏ ✳❲✱ ❚❛r✐ ❋✱ ❋❧❡t❝❤❡r ● ❏✳✭ ✷✵✵✵✮✱ ❇✐❢✉r❝❛t✐♦♥s ♦❢ ❜✐♥❛r② ❞✐❢❢❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✱ Pr♦❝ ❘♦② ❙♦❝ ❊❞✐♥❜✉r❣❤ ❙❡❝t ❆✱ ✶✸✵✿ ✹✽✺✕✺✵✻✳ ✷✳ ❉❛✈②❞♦✈ ❆✳ ❆✳✱ ❘♦s❛❧❡s✲●♦♥③❛❧❡s ❊✳ ✭✶✾✾✻✮✱ ❈♦♠♣❧❡t❡ ❝❧❛ss✐❢✐❝❛t✐♦♥ ♦❢ ❣❡♥❡r✐❝ ❧✐♥❡❛r s❡❝♦♥❞✲♦r❞❡r ♣❛rt✐❛❧ ❞✐❢❢❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ✐♥ t❤❡ ♣❧❛♥❡✱ ❉♦❦❧ ▼❛t❤✱ ✸✺✵✿ ✶✺✶✕✶✺✹✳ ✸✳ ❉❛✈②❞♦✈ ❆✳ ❆✳ ✭✷✵✶✽✮✱ ◆♦r♠❛❧ ❢♦r♠s ♦❢ ❧✐♥❡❛r s❡❝♦♥❞ ♦r❞❡r ♣❛r✲ t✐❛❧ ❞✐❢❢❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ♦♥ t❤❡ ♣❧❛♥❡✱ ❙❝✐ ❈❤✐♥❛ ▼❛t❤✱ ✻✶✱ ❤tt♣s✿✴✴❞♦✐✳♦r❣✴✶✵✳✶✵✵✼✴s✶✶✹✷✺✲✵✶✼✲✾✸✵✸✲✵ ✹✳ ❉❛✈②❞♦✈ ❆✳ ❆✱ ❉✐❡♣ ▲✳ ❚✳ ❚✳ ✭✷✵✶✵✮✱ ◆♦r♠❛❧ ❢♦r♠s ❢♦r ❢❛♠✐❧✐❡s ♦❢ ❧✐♥✲ ❡❛r ❡q✉❛t✐♦♥s ♦❢ ♠✐①❡❞ t②♣❡ ♥❡❛r ♥♦♥✲r❡s♦♥❛♥t ❢♦❧❞❡❞ s✐♥❣✉❧❛r ♣♦✐♥ts✱ ❘✉ss✐❛♥ ▼❛t❤ ❙✉r✈❡②s✱ ✻✺✿ ✾✽✹✕✾✽✻✳ ✺✳ ❉❛✈②❞♦✈ ❆✳ ❆✳ ✭✶✾✽✺✮✱ ❚❤❡ ♥♦r♠❛❧ ❢♦r♠ ♦❢ ❛ ❞✐❢❢❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ t❤❛t ✐s ♥♦t s♦❧✈❡❞ ✇✐t❤ r❡s♣❡❝t t♦ ❞❡r✐✈❛t✐✈❡✱ ✐♥ t❤❡ ♥❡✐❣❤❜♦✉r❤♦♦❞ ♦❢ ✐ts s✐♥❣✉❧❛r ♣♦✐♥t✱ ❋✉♥❝t ❆♥❛❧ ❆♣♣❧✱ ✶✾✿ ✽✶✕✽✾✳ ✻✳ ❉❛✈②❞♦✈ ❆ ❆✳✭✶✾✾✹✮✱ ◗✉❛❧✐t❛t✐✈❡ ❚❤❡♦r② ♦❢ ❈♦♥tr♦❧ ❙②st❡♠s✳ ❚r❛♥s✲ ❧❛t✐♦♥s ♦❢ ▼❛t❤❡♠❛t✐❝❛❧ ▼♦♥♦❣r❛♣❤s✱ ✈♦❧✳ ✶✹✶✳ Pr♦✈✐❞❡♥❝❡✱ ❆♠❡r ▼❛t❤ ❙♦❝✱ ✳ ✼✳ ❉❛✈②❞♦✈ ❆✳ ❆✱ ❉✐❡♣ ▲✳ ❚✳ ❚✳ ✭✷✵✶✶✮✱ ❘❡❞✉❝t✐♦♥ t❤❡♦r❡♠ ❛♥❞ ♥♦r♠❛❧ ❢♦r♠s ♦❢ ❧✐♥❡❛r s❡❝♦♥❞ ♦r❞❡r ♠✐①❡❞ t②♣❡ P❉❊ ❢❛♠✐❧✐❡s ✐♥ t❤❡ ♣❧❛♥❡✱ ❚❲▼❙ ❏ P✉r❡ ❆♣♣❧ ▼❛t❤✱ ✷✿ ✹✹✕✺✸✳ ✺✷ ✽✳ ❑♦♥❞r❛t✐❡✈ ❱ ❆✱ ▲❛♥❞✐s ❊ ▼✳ ✭✶✾✽✽✮✱ ◗✉❛❧✐t❛t✐✈❡ t❤❡♦r② ♦❢ s❡❝♦♥❞ ♦r❞❡r ❧✐♥❡❛r ♣❛rt✐❛❧ ❞✐❢❢❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✱ ■t♦❣✐ ◆❛✉❦✐ ✐ ❚❡❦❤♥✐❦✐ ❙❡r ❙♦✈r❡♠ Pr♦❜❧ ▼❛t ❋✉♥❞ ◆❛♣r✱ ✸✷✿ ✾✾✕✷✶✺✳ ✾✳ ❨✳ P✐♥❝❤♦✈❡r✱ ❏✳ ❘✉❜❡♥st❡✐♥ ✭✷✵✵✺✮✱ ❆♥ ■♥tr♦❞✉❝t✐♦♥ t♦ P❛rt✐❛❧ ❉✐❢✲ ❢❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s✱ ❈❛♠❜r✐❞❣❡✳ ✺✸