Luận văn điều khiển tối ưu bất đẳng thức biến phân elliptic

❇❐ ●■⑩❖ ❉Ư❈ ❱⑨ ✣⑨❖ ❚❸❖ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❍⑨ ◆❐■ ✷ P❍❆◆ ◗❯➮❈ ❱❿◆ ✣■➋❯ ❑❍■➎◆ ❚➮■ ×❯ ❇❻❚ ✣➃◆● ❚❍Ù❈ ❇■➌◆ P❍❹◆ ❊▲▲■P❚■❈ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ❍⑨ ◆❐■✱ ◆❿▼ ✷✵✶✽ ❇❐ ●■⑩❖ ❉Ö❈ ❱⑨ ✣⑨❖ ❚❸❖ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❍⑨ ◆❐■ ✷ P❍❆◆ ◗❯➮❈ ❱❿◆ ✣■➋❯ ❑❍■➎◆ ❚➮■ ×❯ ❇❻❚ ✣➃◆● ❚❍Ù❈ ❇■➌◆ P❍❹◆ ❑■➎❯ ❊▲▲■P❚■❈ ❈❤✉②➯♥ ♥❣➔♥❤✿ ❚♦→♥ ❣✐↔✐ t➼❝❤ ▼➣ sè✿ ✽ ✹✻ ✵✶ ✵✷ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ◆●×❮■ ❍×❰◆● ❉❼◆ ❑❍❖❆ ❍➴❈ ❚❙✳ ❚❘❺◆ ❱❿◆ ❇➀◆● ❍⑨ ◆❐■✱ ◆❿▼ ✷✵✶✽ ▲í✐ ❝↔♠ ì♥ ❚ỉ✐ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ tỵ✐ ❚❙✳ ❚r➛♥ ữớ ữợ t t t ữợ tổ õ t t ❧✉➟♥ ✈➠♥ ♥➔②✳ ❚ỉ✐ ❝ơ♥❣ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ ❝❤➙♥ t❤➔♥❤ tỵ✐ ❝→❝ t❤➛② ❝ỉ ♣❤á♥❣ ❙❛✉ ✤↕✐ ❤å❝✱ ❝ò♥❣ ❝→❝ t❤➛② ❝ỉ ❣✐→♦ ❞↕② ❧ỵ♣ t❤↕❝ sÿ ❝❤✉②➯♥ ♥❣➔♥❤ ❚♦→♥ ❣✐↔✐ t➼❝❤✱ tr÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ❍➔ ◆ë✐ ✷ ✤➣ ❣✐ó♣ ✤ï tỉ✐ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣✳ ◆❤➙♥ ❞à♣ ♥➔② tỉ✐ ❝ơ♥❣ ①✐♥ ✤÷đ❝ ❣û✐ ❧í✐ ❝↔♠ ì♥ ❝❤➙♥ t❤➔♥❤ tỵ✐ ❣✐❛ ✤➻♥❤✱ ❜↕♥ ❜➧ ✤➣ ❧✉ỉ♥ ✤ë♥❣ ✈✐➯♥✱ ❝ê ✈ơ✱ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧đ✐ ❝❤♦ tỉ✐ tr♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥✳ ❍➔ ◆ë✐✱ t❤→♥❣ ✼ ♥➠♠ ✷✵✶✽ ❚→❝ ❣✐↔ P❤❛♥ ◗✉è❝ ❱➠♥ ✶ ▲í✐ ❝❛♠ ✤♦❛♥ ❚æ✐ ①✐♥ ❝❛♠ ✤♦❛♥ ❧✉➟♥ ✈➠♥ ❚❤↕❝ s➽ ❝❤✉②➯♥ ♥❣➔♥❤ t ợ t tố ữ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❡❧❧✐♣t✐❝✑ ❧➔ ❦➳t q✉↔ ❝õ❛ q tr t ự t ữợ sỹ ữợ r r q tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥✱ t→❝ ❣✐↔ ✤➣ ❦➳ t❤ø❛ ♥❤ú♥❣ t❤➔♥❤ tü✉ ❝õ❛ ❝→❝ ♥❤➔ ❦❤♦❛ ❤å❝ ✈ỵ✐ sü tr➙♥ trå♥❣ ✈➔ ❜✐➳t ì♥✳ ❍➔ ◆ë✐✱ t❤→♥❣ ✼ ♥➠♠ ✷✵✶✽ ❚→❝ ❣✐↔ P❤❛♥ ◗✉è❝ ❱➠♥ ✷ ▼ö❝ ❧ö❝ ▼ð ✤➛✉ ✹ ✶ ✻ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✶✳✶ ❚♦→♥ tû ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✷ ❚♦→♥ tû ♣❤✐ t✉②➳♥ t➠♥❣ tr÷ð♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✸ ❇➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❡❧❧✐♣t✐❝ trø✉ t÷đ♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✷ ✣✐➲✉ ❦❤✐➸♥ tè✐ ÷✉ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❡❧❧✐♣t✐❝ ✶✹ ✷✳✶ ❇➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❡❧❧✐♣t✐❝✳ ❙ü tỗ t t ✈➟t ❝↔♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ t ỗ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✳✸ ❇➔✐ t♦→♥ ❡❧❧✐♣t✐❝ ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ❜✐➯♥ ♠ët ♣❤➼❛ ✳ ✳ ✳ ✳ ✳ ✷✳✷ ✣✐➲✉ ❦❤✐➸♥ tè✐ ÷✉ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❡❧❧✐♣t✐❝ ✳ ✳ ✳ ✳ ✳ ✷✳✷✳✶ ❇➔✐ t♦→♥ ✤✐➲✉ ❦❤✐➸♥ tè✐ ÷✉ tê♥❣ q✉→t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷✳✷ ❈→❝❤ t✐➳♣ ❝➟♥ ❝❤✉♥❣ ✤è✐ ✈ỵ✐ ỵ ỹ tè✐ ÷✉ ♣❤÷ì♥❣ tr➻♥❤ ❡❧❧✐♣t✐❝ ♥û❛ t✉②➳♥ t➼♥❤ ✷✳✷✳✹ ✣✐➲✉ ❦❤✐➸♥ tè✐ ÷✉ ❜➔✐ t♦→♥ ✈➟t ❝↔♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✶✹ ✷✺ ✷✼ ✸✵ ✸✶ ✸✹ ✹✻ ✺✽ ❑➳t ❧✉➟♥ ✻✹ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✻✺ ✸ ▼ð ✤➛✉ ✶✳ ▲➼ ❞♦ t ỵ tt tố ữ ♠ët ♣❤➛♥ ♠ð rë♥❣ ❝õ❛ ♣❤➨♣ t➼♥❤ ❜✐➳♥ ♣❤➙♥✱ ❧➔ ởt ữỡ tố ữ ỵ tt t s Pữỡ ợ ❞♦ ❝æ♥❣ ❧❛♦ ✤â♥❣ ❣â♣ ❝õ❛ ♥❤➔ t♦→♥ ❤å❝ ▲✐➯♥ ❳æ P♦♥tr②❛❣✐♥ ✈➔ ♥❤➔ t♦→♥ ❤å❝ ▼ÿ ❘✐❝❤❛r❞ ❇❡❧❧♠❛♥✳ ✣➳♥ ỵ tt ữủ sỹ q t ❝ù✉ ❝õ❛ r➜t ♥❤✐➲✉ ♥❤➔ ❦❤♦❛ ❤å❝ tr♦♥❣ ✈➔ ♥❣♦➔✐ ữợ tố ữ tr ❦❤ỉ♥❣ ❣✐❛♥ ❤ú✉ ❤↕♥ ❝❤✐➲✉ ✤➣ ✤↕t ✤÷đ❝ ♥❤✐➲✉ ❦➳t q✉↔ q✉❛♥ trå♥❣✱ t✉② ♥❤✐➯♥ ✤è✐ ✈ỵ✐ ❤➺ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ✈æ ❤↕♥ ❝❤✐➲✉ t❤➻ ✈➝♥ ❝á♥ ❦❤→ ❤↕♥ ❝❤➳✳ ❱ỵ✐ ♠♦♥❣ ♠✉è♥ t➻♠ ❤✐➸✉ s➙✉ ❤ì♥ ✈➲ ✈➜♥ ✤➲ ữủ sỹ ữợ r ❇➡♥❣✱ tæ✐ ✤➣ ❝❤å♥ ✤➲ t➔✐✿ ✧ ✣✐➲✉ ❦❤✐➸♥ tè✐ ÷✉ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❡❧❧✐♣t✐❝ ✧ ❧➔♠ ❧✉➟♥ ✈➠♥ ❝❛♦ ❤å❝ ❝õ❛ ♠➻♥❤✳ ✷✳ ▼ö❝ ✤➼❝❤ ♥❣❤✐➯♥ ❝ù✉ ❚➻♠ ❤✐➸✉ ✈➲ ❜➔✐ t♦→♥ ✤✐➲✉ ❦❤✐➸♥ tè✐ ÷✉ ①→❝ ✤à♥❤ ❜ð✐ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❡❧❧✐♣t✐❝✳ ✣➦❝ ❜✐➺t ❧➔ ✈✐➺❝ ♠ỉ t↔ ✤✐➲✉ ❦✐➺♥ ❝➛♥ tè✐ ÷✉ ✈➔ ❦❤↔ ♥➠♥❣ ù♥❣ ❞ư♥❣ ❝õ❛ ❝❤ó♥❣✳ ✸✳ ◆❤✐➺♠ ✈ư ♥❣❤✐➯♥ ❝ù✉ ❚➻♠ ❤✐➸✉ ✈➲✿ ✹ ✰ ❚♦→♥ tû ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐✱ t♦→♥ tû t➠♥❣ tr÷ð♥❣✱ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❡❧❧✐♣t✐❝❀ ✰ ❇➔✐ t♦→♥ ✤✐➲✉ ❦❤✐➸♥ tè✐ ÷✉ ✤è✐ ợ t tự t ố tữủ ✈➔ ♣❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉ ✰✣è✐ t÷đ♥❣✿ ❇➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❡❧❧✐♣t✐❝ ✈➔ ❜➔✐ t♦→♥ ✤✐➲✉ ❦❤✐➸♥ tè✐ ÷✉✳ ✰P❤↕♠ ✈✐✿ ◆❣❤✐➯♥ ❝ù✉ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ❦❤↔ ♥➠♥❣ ù♥❣ ❞ư♥❣ ❝õ❛ ❝❤ó♥❣✳ ✺✳ P❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉ ❙û ❞ư♥❣ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❝õ❛ ●✐↔✐ t➼❝❤ ❜✐➳♥ ♣❤➙♥✱ ❚è✐ ữ ổ ỗ ố ữ ổ trỡ trú trú ỗ ❤❛✐ ❝❤÷ì♥❣✿ ❈❤÷ì♥❣ ✶✿ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ❈❤÷ì♥❣ ✷✿ ✣✐➲✉ ❦❤✐➸♥ tè✐ ÷✉ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❡❧❧✐♣t✐❝✳ ✺ ❈❤÷ì♥❣ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ◆ë✐ ❞✉♥❣ ❝õ❛ ❝❤÷ì♥❣ ♥➔② ❝❤õ ②➳✉ ✤÷đ❝ t❤❛♠ ❦❤↔♦ tø ❬✹❪✱ ❝❤÷ì♥❣ ✶ ✈➔ ❝❤÷ì♥❣ ✷✱ t→❝ ❣✐↔ tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ❝ì sð ❧✐➯♥ q✉❛♥ tỵ✐ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ❜➔✐ t♦→♥ ✣✐➲✉ ❦❤✐➸♥ tè✐ ÷✉ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❡❧❧✐♣t✐❝✱ t♦→♥ tû ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐✱ t♦→♥ tû t➠♥❣ tr÷ð♥❣✱ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❡❧❧✐♣t✐❝ trø✉ t÷đ♥❣✳ ✶✳✶ ❚♦→♥ tû ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐ ◆➳✉ X ✈➔ Y ❧➔ ❤❛✐ ❦❤ỉ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤✱ X × Y ❧➔ t➼❝❤ ✤➲ ❝→❝ ❝õ❛ ❝❤ó♥❣✳ ❈→❝ ♣❤➛♥ tû ❝õ❛ X × Y ✈✐➳t ❧➔ [x, y] ð ✤➙② x ∈ X ✈➔ y ∈ Y ◆➳✉ A ❧➔ ♠ët t♦→♥ tû ✤❛ trà tø X ✈➔♦ Y, ❝❤ó♥❣ t õ t ỗ t õ ợ ỗ t ♥â tr♦♥❣ X × Y ✿ {[x, y] ∈ X × Y ; y ∈ Ax} ◆❣÷đ❝ ❧↕✐✱ ♠é✐ t➟♣ A ⊂ X × Y, ①→❝ ✤à♥❤ ♠ët t♦→♥ tû A t❤❡♦ ❝→❝❤ Ax = {y ∈ X; [x, y] ∈ A} , D (A) = {x ∈ X; Ax = Ø} Ax, A−1 = {[y, x] ; [x, y] ∈ A} R (A) = x∈D(A) ❚ø ✤➙② ✈➲ s ú t s ỗ t t tỷ tứ X Y ợ ỗ t ú tr X × Y ✈➔ ❞♦ ✤â t❛ ❝â t❤➸ ♥â✐ ♠ët tữỡ ữỡ t X ì Y t❤❛② ❝❤♦ t♦→♥ tû tø X tỵ✐ Y ✻ ◆➳✉ A, B ⊂ X × Y ✈➔ λ ❧➔ ♠ët sè t❤ü❝✱ t❛ ✤➦t✿ λA = {[x, λy] ; [x, y] ∈ A} ; A + B = {[x, y + z] ; [x, y] ∈ A, [x, z] ∈ B} ; ✈ỵ✐ ♠ët y ∈ Y ❚r♦♥❣ ♠ư❝ ♥➔②✱ X ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ t❤ü❝ ✈ỵ✐ ❦❤ỉ♥❣ ❣✐❛♥ ✤è✐ ♥❣➝✉ X ∗ ◆➳✉ X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt✱ trứ trữớ ủ ró t ỗ t X ợ ❦❤æ♥❣ ❣✐❛♥ ✤è✐ ♥❣➝✉ ❝õ❛ ♥â✳ AB = [x, z] ; [x, y] ∈ B, [y, z] ∈ A, ❚➟♣ A X ì X tữỡ ữỡ t tỷ A : X → X ∗✮ ✤÷đ❝ ❣å✐ ❧➔ ✤ì♥ ✤✐➺✉ ♥➳✉✿ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳ (x1 − x2 , y1 − y2 ) ≥ 0, ∀ [xi , yi ] ∈ A, i = 1, ❚➟♣ ✤ì♥ ✤✐➺✉ A X ì X ữủ ỡ ❝ü❝ ✤↕✐ ♥➳✉ ♥â ❦❤æ♥❣ t❤ü❝ sü ❝❤ù❛ tr♦♥❣ ❜➜t ❦➻ t➟♣ ✤ì♥ ✤✐➺✉ ♥➔♦ ❝õ❛ X × X ∗ ú ỵ r A t tỷ ỡ tr tø X ✈➔♦ X ∗, t❤➻ A ❧➔ ✤ì♥ ✤✐➺✉ ♥➳✉ (x1 − x2 , Ax1 − Ax2 ) ≥ 0, ∀ (x1 , x2 ) ∈ D (A) ▼ët ✈➼ ❞ư ✤ì♥ ❣✐↔♥ ✈➲ t➟♣ ✤ì♥ ✤✐➺✉ ❝õ❛ X × X ∗ ❧➔ →♥❤ ①↕ ✤è✐ ♥❣➝✉ J ❝õ❛ X ❈❤♦ A ❧➔ t♦→♥ tû ✤ì♥ trà tø X ✈➔♦ X ∗ ✈ỵ✐ D (A) = X ❚♦→♥ tỷ A ữủ ỷ tử ợ ∀x, y ∈ X ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳ w∗ − lim A (x + λy) = Ax λ→0 A ✤÷đ❝ ❣å✐ ❧➔ ♠↕♥❤✲②➳✉✯ tù❝ ❧➔ ❧✐➯♥ tö❝ ♥➳✉ ♥â ❧➔ ❤➔♠ ❧✐➯♥ tö❝ tø X ✈➔♦ Xw∗ , w∗ − lim Axn = Ax xn →x ✼ A ✤÷đ❝ ❣å✐ ❧➔ ❝÷ï♥❣ ❜ù❝ ♥➳✉ lim xn − x0 , yn n→∞ xn −1 =∞ ✭✶✳✶✮ ✈ỵ✐ ♠ët x0 ∈ A ✈➔ ♠å✐ [xn, yn] ∈ A s❛♦ ❝❤♦ limn→∞ xn = ∞ A ✤÷đ❝ ❣å✐ ❧➔ ❜à ❝❤➦♥ ♥➳✉ ♥â ❜à ❝❤➦♥ tr➯♥ ♠é✐ t➟♣ ❝♦♥ ❜à ❝❤➦♥✳ ❈❤♦ X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ t❤ü❝ ✈➔ ϕ : X → R ỗ ỷ tử ữợ õ ❧➔ ♠ët t➟♣ ❝♦♥ ✤ì♥ ✤✐➺✉ ❝ü❝ ✤↕✐ ❝õ❛ X ì X ỵ t q ♠➔ ❝❤ó♥❣ t❛ ❝â t❤➸ ù♥❣ ❞ư♥❣ ♥❣❤✐➯♥ ❝ù✉ sü tỗ t t t ợ ❦✐➺♥ ❜✐➯♥ t❤➼❝❤ ❤đ♣✳ ❱ỵ✐ Ω ❧➔ t➟♣ ❝♦♥ ♠ð ❜à ❝❤➦♥ ❝õ❛ RN , ✈➔ g : R → R tữớ ỗ ỷ tử ữợ s❛♦ ❝❤♦ ∈ D(∂g) ❚❛ ✤à♥❤ ♥❣❤➽❛ ❤➔♠ ϕ : L2 (Ω) → R ❜ð✐ ϕ(y) = Ω | y|2 + g(y) dx +∞ ♥➳✉ y ∈ H01(Ω) ✈➔ g(y) ∈ L1(Ω), ♥➳✉ tr→✐ ❧↕✐ ✭✶✳✷✮ ❍➔♠ ỷ tử ữợ ỗ + ❍ì♥ ♥ú❛✱ ♥➳✉ ❜✐➯♥ ∂Ω ✤õ trì♥ ✭❝❤➥♥❣ ❤↕♥ ❧ỵ♣ C ỗ t L2 () ì L2 () ữủ ✶✳✶✳ ∂ϕ = {[y, w] ; w ∈ L2 (Ω); y ∈ H01 (Ω) ∩ H (Ω), ❤✳❦✳♥✳ , x ∈ Ω} ❉♦ ✤â✱ ✈ỵ✐ ♠é✐ f ∈ L2(Ω), ❜➔✐ t♦→♥ ❉✐r✐❝❤❧❡t − y + ∂g(y) f ❤✳❦✳♥✳ tr♦♥❣ Ω, y=0 tr➯♥ ∂Ω, ❝â ♥❣❤✐➺♠ ❞✉② ♥❤➜t y ∈ H01(Ω) × H 2(Ω) w(x) + y(x) ∈ ∂g(y(x)), ✽ t ỵ r ợ ộ δ > ❝â t➟♣ ❤đ♣ ❝♦♥ ✤♦ ✤÷đ❝ Eδ ⊂ Ω s❛♦ ❝❤♦ m (Eδ ) ≤ δ ✈➔ ❝→❝ ❞➣② pε , vε → v ∗ , yε ❜à ❝❤➦♥ tr♦♥❣ L∞ (Ω\Eδ ) ✈➔ ❤ë✐ tö ✤➲✉ tr➯♥ Ω\Eδ ◆❤➙♥ ✭✷✳✶✵✾✮ ✈ỵ✐ ςλ (pε) tr♦♥❣ ✤â ςλ ❧➔ ①➜♣ ①➾ trì♥ ❝õ❛ ❤➔♠ ❞➜✉✱ ♥❣❤➽❛ ❧➔✱ ✈ỵ✐ λ > ✵✱  ♥➳✉ r ≥ λ, 1 ςλ (r) = λ r ♥➳✉ − λ < r < λ,  −1 ♥➳✉ r ≤ −λ ❚❛ ❝â n n n A0 pε ςλ (pε ) dx ≥ (Apε , ςλ (pε )) = Ω ✈➔ ❞♦ ✤â ε−1 (β (yε ) − vε ) ςλ (pε + ε (v ∗ − vε )) dx Ω ≤ ε−1 (β (yε ) − vε ) (ςλ (pε + ε (v ∗ − vε )) − ςλ (pε )) dx Ω + ξε ∀ε > L1 (Ω) , sỷ > tý ỵ ✈➔ Eδ ⊂ Ω s❛♦ ❝❤♦ m (Eδ ) ≤ δ ✈➔ sü ❤ë✐ tö tr♦♥❣ ✭✷✳✹✼✮ ❧➔ ✤➲✉ tr➯♥ Ω\Eδ ✳ ❑❤✐ ✤â✱ ❝❤♦ ε = εn → tr♦♥❣ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥✱ t❛ t❤➜② lim sup ε−1 n εn →0 Ω (β (yεn ) − vεn ) ςλ (pεn + εn (v ∗ − vεn )) dx ≤ C + lim sup ε−1 n εn →0 Eδ , |β (yεn ) − vεn | dx ✈➔ ✤✐➲✉ ♥➔② s✉② r❛ lim sup ε−1 n εn →0 Ω |β (yεn ) − vεn | dx ≤ C + lim sup ε−1 n εn →0 ✭✷✳✶✶✻✮ Eδ |β (yεn ) − vεn | dx , tr♦♥❣ ✤â C ✤ë❝ ❧➟♣ ✈ỵ✐ δ ✭δ ✤õ ♥❤ä✮✳ ▼➦t ❦❤→❝✱ ✈➻ {β (yε ) − vε } ❜à ❝❤➦♥ tr♦♥❣ L2 (Ω) , ♥➯♥ t❛ ❝â n ε−1 n |β (yεn ) − vεn | dx ≤ Cδ , Eδ ✺✶ n tr♦♥❣ ✤â C ✤ë❝ ❧➟♣ ✈ỵ✐ n ✈➔ δ ❑❤✐ ✤â✱ ❝❤♦ δ = ε2n t❤➻ ✭✷✳✻✵✮ ❝❤♦ t❛ lim sup ε−1 n εn →0 |β (yεn ) − vεn | dx ≤ C, Ω ❤❛② lim sup Apεn εn →0 L1 (Ω) ✭✷✳✶✶✼✮ ✭✷✳✶✶✽✮ ≤C ❉♦ {ξε } ❜à ❝❤➦♥ tr♦♥❣ L2 (Ω) , t❛ ❝â t❤➸ ❣✐↔ sû r➡♥❣ n ②➳✉ tr♦♥❣ {ξεn } → ξ L2 (Ω) ✈➔ t❤❡♦ ❬✹❪✲▼➺♥❤ ✤➲ ✷✳✶✸✱ ξ ∈ ∂g (y∗) ❚÷ì♥❣ tü✱ ❝❤♦ εn → tr t ữủ tỗ t↕✐ ♠ët ❞➣② ❝♦♥ s✉② rë♥❣ ❝õ❛ εn ❧➔ ελ s❛♦ ❝❤♦ Apε → µ ②➳✉✯ tr♦♥❣ (L∞ (Ω))∗ , ✭✷✳✶✶✾✮ (β (yε ) − vε ) → η ②➳✉✯ tr♦♥❣ (L∞ (Ω))∗ , ε λ λ λ λ tr♦♥❣ ✤â µ = Ap tr➯♥ L∞ (Ω) ∩ H01 (Ω) ✈➔ ♥â✐ r✐➯♥❣ ❧➔ tr➯♥ C0∞ (Ω) ❚❛ ❝â − µ = η + ξ ✭✷✳✶✷✵✮ ▼➦t ❦❤→❝✱ t❤❡♦ ❝æ♥❣ t❤ù❝ tr✉♥❣ ❜➻♥❤ ✭❚❤❡♦ ❬✹❪✲❍➺ q✉↔ ✷✳✶✮✱ t❛ ❝â (β (yελ ) − vελ ) = θλ (pελ − (v ∗ − vελ )) ελ tr♦♥❣ ✤â θλ ελ ∈ ∂β (zλ ) , yελ ≤ zλ ≤ yελ − ελ (pελ − v ∗ + vελ ) (β (yελ ) − vελ ) ❧➔ ❜à ❝❤➦♥ ✤➲✉ tr➯♥ Ω\Eδ t❛ ❝â t❤➸ ❣✐↔ sû r➡♥❣ θελ → θ ✈➔ ❤➛✉ ❤➳t tr♦♥❣ Ω, ②➳✉✯ tr♦♥❣ L∞ (Ω\Eδ ) (β (yελ ) − vελ ) → θp ελ ②➳✉✯ tr♦♥❣ L∞ (Ω\Eδ ) ✺✷ ❉♦ θλ ✈➔ ✭✷✳✶✷✶✮ ✭✷✳✶✷✷✮ ❚ø ✭✷✳✶✷✶✮✱ t❛ ❝â β (zελ (x) w (x)) dx; ∀W ∈ L1 (ω\Eδ ) θελ (x) w (x) dx ≤ Ω\Eδ Ω\Eδ ❚ø ✤➙② s✉② r❛ ✭β ❧➔ ✤↕♦ ❤➔♠ t ữợ (z (x) w (x)) dx θελ (x) w (x) dx ≤ lim sup ελ →0 Ω\Eδ Ω\Eδ β (y ∗ (x) w (x)) dx ≤ Ω\Eδ ❉♦ ✤â θ (x) w ≤ β (y ∗ (x) w) ; ∀w ∈ R, ❤➛✉ ❤➳t x ∈ Ω\Eδ , ✈➔ ✈➻ t❤➳ ♥➯♥ ❞♦ ✤â θ (x) ∈ ∂β (y∗ (x)) ; ❤➛✉ ❤➳t x ∈ Ω\Eδ ✳ ❚❛ ✤➣ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ η ∈ L∞ (Ω\Eδ ) ✈➔ θη (x) ∈ ∂β (y∗ (x)) p (x) ; ❤➛✉ ❤➳t x ∈ Ω\Eδ ❚❤❡♦✱ ✭✷✳✶✷✵✮✱ t❛ t❤➜② r➡♥❣ ❤↕♥ ❝❤➳ µ tr➯♥ Ω\Eδ t❤✉ë❝ L∞ (Ω\Eδ ) ✈➔ µa (x) = µ (x) ∈ −∂β (y ∗ (x)) p (x) − ξ (x) t x \E tý ỵ t❛ ❦➳t ❧✉➟♥ −µa (x) ∈ ∂β (y ∗ (x)) p (x) + ξ (x) ❤➛✉ ❤➳t x ∈ Ω, ♥❤÷ ②➯✉ ❝➛✉✳ ❇➙② ❣✐í ❣✐↔ sû ≤ N ỵ ú H () ⊂ C Ω ✈➔ yε ❜à ❝❤➦♥ ✤➲✉ tr➯♥ Ω ▼➦t ❦❤→❝✱ tø ✭✷✳✶✵✾✮✱ t❛ ❝â A (εpε ) = −εξε − β (yε ) + vε ✈➔ ❞♦ {β (yε) − vε} ❜à ❝❤➦♥ tr♦♥❣ L2 (Ω) t❛ s✉② r❛ {εpε}ε>0 ❜à ❝❤➦♥ tr♦♥❣ H (Ω) ✈➔ tr♦♥❣ C Ω ✳ ❈✉è✐ ❝ò♥❣✱ t❛ ❝â yε − εpε + ε (v ∗ − vε ) ∈ β −1 (vε ) , y ∗ ∈ β −1 (v ∗ ) ✺✸ ❇➡♥❣ ❝→❝❤ trø ✈➔ ♥❤➙♥ ✈ỵ✐ (v∗ − vε) t❛ ✤÷đ❝ ε |v ∗ − vε | ≤ |yε | + ε |pε | ❤➛✉ ❤➳t tr♦♥❣ Ω ✈➔ ❞♦ β ❧➔ ▲✐♣s❝❤✐t③ ✤à❛ ♣❤÷ì♥❣✱ ♥➯♥ ✤✐➲✉ ♥➔② ❝❤♦ t❛ |Apε | ≤ |ξε | + L (|pε | + |v ∗ − vε |) ; ❤➛✉ ❤➳t x ∈ Ω ❉♦ ✤â✱ { Apε } ❧➔ ❜à ❝❤➦♥ tr♦♥❣ L2 (Ω) ✈➔ t❛ ❦➳t ❧✉➟♥ p ∈ H01 (Ω) ∩ H (Ω)✱ η ∈ L2 (Ω) ✈➔ Ap = (Ap∗ )a ♥❤÷ ②➯✉ ❝➛✉✳ ●✐↔ sû ✤✐➲✉ ❦✐➺♥ ✭✷✳✶✵✹✮ ✤ó♥❣✳ ❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ qε = 1ε (β (yε) − vε) ❧➔ ❝♦♠♣❛❝t ②➳✉ tr♦♥❣ L1 (Ω) ❚❛ ❝â qε = β (yε − ελ (pε − θε )) (pε − θε ) dλ ❤➛✉ ❤➳t tr♦♥❣ Ω, tr♦♥❣ ✤â β ❧➔ ✤↕♦ ❤➔♠ ❝õ❛ β ✈➔ θε = (v∗ − vε) ❚ø ✤✐➲✉ ❦✐➺♥ ✭✷✳✶✵✹✮ t❛ ❝â ≤ qε (x) ≤C ((pε (x) − θε (x))) × |β (yε (x) − ελ (pε (x) − θε (x)))| dλ + |yε (x)| + ε (pε (x) − θε (x)) ❤➛✉ ❤➳t tr♦♥❣ {x; pε (x) < θε (x)}✱ ≤ −qε (x) ≤C ((pε (x) − θε (x))) × |β (yε (x) − ελ (pε (x) − θε (x)))| dλ + |yε (x)| + ε |pε (x) − θε (x)| ❤➛✉ ❤➳t tr♦♥❣ {x; pε (x) < θε (x)}✳ ▼➦t ❦❤→❝✱ t❛ ❝â ✭❞♦ β ✤ì♥ ✤✐➺✉✮ β (yε − ελ (pε − θε )) ≤ β (yε ) ❉♦ ✤â β (yε − ελ (pε − θε )) ≤ β (yε − ε (pε − θε )) ✈ỵ✐ pε ≥ θε ✈ỵ✐ pε ≤ θε qε (x) ≤C |pε (x) − θε (x)| |β (yε (x))| + |β (yε (x) − ε (pε (x) − θε (x)))| + |yε (x)| + ε |pε (x) − θε (x)| ✺✹ ❤➛✉ ❤➳t x ∈ Ω ✭✷✳✶✷✸✮ ữ ự trữợ { p} ❝♦♠♣❛❝t tr♦♥❣ L2 (Ω) , ♥➯♥ yε → y ∗ tr♦♥❣ L2 (Ω) , θε → tr♦♥❣ L2 (Ω) , β (yε − ε (pε − θε )) → v ∗ tr♦♥❣ L2 (Ω) , ✈➔ {β (yε)} ❜à ❝❤➦♥ tr♦♥❣ L2 (Ω) , t❤❡♦ t✐➯✉ ❝❤✉➞♥ ❉✉♥❢♦r❞✲ P❡tt✐s ✈➔ ✭✷✳✶✷✸✮ { qε } ❧➔ ❝♦♠♣❛❝t ②➳✉ tr♦♥❣ L1 (Ω) ❉♦ ✤â { Apε } ❧➔ ❝♦♠♣❛❝t ②➳✉ tr♦♥❣ L1 (Ω) , s✉② r❛ ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝ơ♥❣ ❝â t❤➸ ♥❤➟♥ ✤÷đ❝ ❦➳t q✉↔ ✤â ♥➳✉ sû ❞ư♥❣ ❝→❝❤ t✐➳♣ ❝➟♥ ♠ỉ t↔ tr♦♥❣ ▼➺♥❤ ✤➲ ✷✳✺✳ ❈ö t❤➸✱ ①➜♣ ①➾ ❜➔✐ t♦→♥ ✭✷✳✾✹✮✱ ✭✷✳✾✹✮ ❜ð✐ ❤å ❝→❝ ❜➔✐ t♦→♥ ✤✐➲✉ ❦❤✐➸♥ tè✐ ÷✉ s❛✉✿ ❈ü❝ t✐➸✉ ❤♦→ gε (y) + h (u) + 12 |u − u∗|2U ✈ỵ✐ ♠å✐ ✭✷✳✶✷✹✮ (y, u) ∈ H01 (Ω) H () ì U, ợ r tr Ω, ✭✷✳✶✷✺✮ H01 (Ω) ∩ H (Ω) ✈➔ β ε ∈ C ∞ (R) ①→❝ Ay + β ε (y) = Bu + f tr♦♥❣ ✤â Ay = A0y ✈ỵ✐ D (A) = ✤à♥❤ ❜ð✐ +∞ ε βε r − ε2 θ − βε −ε2 θ β (r) = ρ (θ) dθ + βε (0) , r ∈ R −∞ Ð ✤➙② βε (r) tr♦♥❣ R✱ tù❝ R, = ε−1 r − (1 + εβ)−1 r ; r ∈ R ✈➔ ρ ❧➔ C0∞ ✭✷✳✶✷✻✮ ❝❤➾♥❤ ❤♦→ ρ ∈ C ∞ (R) , ρ (r) = 0, |r| > 1, ρ (r) = ρ (−r) , ρ (t) dt = r ữ ỵ st ✤ì♥ ✤✐➺✉ t➠♥❣✱ ✈➔ |β ε (r) − βε (r)| ≤ 2ε; ∀r ∈ R ❈❤ó♥❣ t❛ ✤❛♥❣ ð t➻♥❤ ❤✉è♥❣ ✤➣ ♠ỉ t↔ tr♦♥❣ ❬✹❪✲▼ư❝ ✷✳✷✱ tr♦♥❣ ✤â ϕε (y) = j ε (y) dx; ∀y ∈ L2 (Ω) , ∇j ε = β ε Ω ✺✺ ✭✷✳✶✷✼✮ • ❚r♦♥❣ ♣❤➛♥ t✐➳♣ t❤❡♦✱ t❛ ✤➦t β ε = (β ε) ●✐↔ sû (yε, uε) ❧➔ ❝➦♣ tè✐ ÷✉ ❝õ❛ ❜➔✐ t♦→♥ ✭✷✳✶✷✹✮✳ ❑❤✐ ✤â t❛ ❝â uε → u∗ ♠↕♥❤ tr♦♥❣ U, yε → y ∗ ♠↕♥❤ tr♦♥❣ H01 (Ω) , ②➳✉ tr♦♥❣ H (Ω) , ✭✷✳✶✷✽✮ βε (yε ) → β (y ∗ ) ②➳✉ tr♦♥❣ L2 (Ω) ❚❤➟t ✈➟②✱ tø ❜➜t ✤➥♥❣ t❤ù❝ g ε (yε ) + h (uε ) + ∗ |uε − u∗ |2U ≤ g ε y u + h (u∗ ) ✭✷✳✶✷✾✮ ✭yu ❧➔ ♥❣❤✐➺♠ ❝õ❛ ✭✷✳✶✷✺✮✱ t❛ t❤➜② {uε} ❧➔ ❜à ❝❤➦♥ tr♦♥❣ U ✳ ❉♦ ✤â t❤❡♦ ♠ët ❞➣② ❝♦♥✱ εn → t❛ ❝â uεn → u ②➳✉ tr♦♥❣ U ▼➦t ❦❤→❝✱ t❛ ❝â t❤➸ ✈✐➳t ✭✷✳✶✷✺✮ ♥❤÷ s❛✉ Ayε + βε (yε ) = Buε + f + βε (yε ) − β ε (yε ) ✈➔ t❤❡♦ ▼➺♥❤ ✤➲ ✷✳✷✱ yε → y ♠↕♥❤ tr♦♥❣ H (Ω) ✈➔ ②➳✉ tr♦♥❣ H (Ω) tr♦♥❣ ✤â y ❧➔ ♥❣❤✐➺♠ ❝õ❛ ✭✷✳✾✺✮✱ tr♦♥❣ ✤â u = u ✣✐➲✉ ♥➔② ❝❤♦ t❛ n ∗ g (y) + h (u) + lim sup |uε − u∗ |2U ≤ g y + h (u∗ ) = inf(P ) εn →0 ❉♦ ✤â uε → u∗ ♠↕♥❤ tr♦♥❣ U ✈➔ y∗ = y✱ u = u n tũ ỵ t õ ỡ ỳ ❝â p ∈ H01 (Ω) ∩ H (Ω) s❛♦ ❝❤♦ n − Ayε − βε (yε ) pε = ∇g ε (yε ) tr♦♥❣ Ω, ✭✷✳✶✸✶✮ B ∗ pε ∈ ∂h (uε ) + uε − u∗ ❇➙② ❣✐í✱ ♥❤➙♥ ✭✷✳✶✸✵✮ ✈ỵ✐ pε, ✈➔ ❦➼ ❤✐➺✉ pε, t❛ ữủ p H01 () (y ) p dx ≤ C; + Ω ✺✻ ✭✷✳✶✸✵✮ ∀ε > õ t ởt ỵ ❧➔ ε t❛ ❝â pε → p ♠↕♥❤ tr♦♥❣ L2 (Ω) ✱ ②➳✉ tr♦♥❣ H01 (Ω)✱ ∇gε (yε) → ξ ②➳✉ tr♦♥❣ L2 (Ω)✱ ✈➔ t❤❡♦ ❞➣② ❝♦♥ s✉② rë♥❣ {ελ}✱ • β ελ (yελ ) pελ → η ②➳✉✯ tr♦♥❣ (L∞ (Ω))∗ ❱➻ t❤➳✱ −Ap − η = ξ ∈ ∂g (y)∗ tr♦♥❣ Ω ❇➙② ❣✐í✱ t❤❡♦ ✣à♥❤ ỵ r ợ ộ > tỗ t t ❝♦♥ ✤♦ ✤÷đ❝✳ Eδ ❝õ❛ Ω s❛♦ ❝❤♦ m (Eδ ) ≤ δξ ✈➔ y ∗ , p ∈ L∞ (Ω\Eδ ) ✈➔ yε (x) → y ∗ (x) , pε (x) → p (x) ✤➲✉ tr➯♥ Ω\Eδ • ❚ø ✤â {β ε (yε)} ❧➔ ❜à ❝❤➦♥ tr♦♥❣ L∞ (Ω\Eδ ) t❛ ❝â t❤➸ ❣✐↔ sû r➠♥❣✱ t❤❡♦ ♠ët ❞➣② ❝♦♥ • β ε (yε ) → fδ ②➳✉✯ tr♦♥❣ L∞ (Ω\Eδ ) ❑❤✐ ✤â✱ t❤❡♦ ❇ê ✤➸ ✷✳✻ s❛✉ ✤➙② t❛ s✉② r❛ fδ (x) ∈ ∂β (y∗ (x)) ❤➛✉ ❤➳t x ∈ Ω\Eδ , ✈➔ ❞♦ ✤â ηa (x) = fδ (x) p (x) ∈ ∂β (y ∗ (x)) p (x) , ❤➛✉ ❤➳t x ∈ \E P ố ỵ ữủ s r ữ ự tr { (y )} ❧➔ ❜à ❝❤➦♥ tr♦♥❣ L∞ (Ω) ♥➳✉ ≤ N ≤ ✭✈➻ β ε (yε ) ≤ |β (yε )| ≤ C ∈ Ω ✮ ✈➔ ❧➔ ❝♦♠♣❛❝t ②➳✉ tr♦♥❣ L1 (Ω) ♥➳✉ β t❤♦↔ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ✭✷✳✶✵✹✮✳ ❨➳✉ tè ❝❤➼♥❤ ✤➸ ❝❤✉②➸♥ q✉❛ ❣✐ỵ✐ ❤↕♥ tr♦♥❣ ✤✐➲✉ ❦✐➺♥ ❝➛♥ tè✐ ÷✉ ①➜♣ ①➾ ✤÷đ❝ ①→❝ ✤à♥❤ ữợ X ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❝♦♠♣❛❝t ✤à❛ ♣❤÷ì♥❣ ✈➔ ❝❤♦ v ❧➔ ✤ë ✤♦ ❞÷ì♥❣ tr➯♥ X s❛♦ ❝❤♦ v(X) < ∞ ●✐↔ sû yε ∈ L1 (X, v) s❛♦ ❝❤♦ yε → y ♠↕♥❤ tr♦♥❣ L1 (X, v) ✈➔ • ε • ε β (yε ) → f0 ②➳✉ tr♦♥❣ L1 (X, v) ❑❤✐ ✤â f0 (x) ∈ ∂β (y (x)) ✺✼ ❤➛✉ ❤➳t x ∈ X ✷✳✷✳✹ ✣✐➲✉ ❦❤✐➸♥ tè✐ ÷✉ ❜➔✐ t♦→♥ ✈➟t ❝↔♥ ❚❛ s➩ ♥❣❤✐➯♥ ❝ù✉ ❜➔✐ t♦→♥ s❛✉✿ ❈ü❝ t✐➸✉ ❤♦→ g (y) + h (u) , ∀ (y, u) ∈ H01 (Ω) ∩ H (Ω) × U, y ∈ K ✭✷✳✶✸✷✮ ❱ỵ✐ r➔♥❣ ❜✉ë❝ a (y, y − z) ≤ (f + Bu, y − z) , ∀z ∈ K ✭✷✳✶✸✸✮ tr♦♥❣ ✤â a : H01 (Ω) × H01 (Ω) → R ✤à♥❤ ♥❣❤➽❛ ❜ð✐ ✭✷✳✷✮ ✈➔ ❑ ❧➔ t➟♣ ❤ñ♣ ỗ ữ t trữợ tữỡ ữỡ ợ t t (A0 y f − Bu) (y − ψ) = ❤➛✉ ❤➳t tr♦♥❣ Ω ✭✷✳✶✸✹✮ A0 y − f − Bu ≥ 0, y ≥ ψ ❤➛✉ ❤➳t tr♦♥❣ Ω y = tr♦♥❣ ∂Ω Ð ✤➙②✱ B ∈ L U, L2 (Ω) , f ∈ L2 (Ω) ✈➔ g : L2 (Ω) → R; h : U → R ●✐↔ sû ✭②✯✱ ✉✯✮ ❧➔ ❝➦♣ tè✐ ÷✉ ❝õ❛ ❜➔✐ t♦→♥ ✭✷✳✶✸✷✮✳ ❑❤✐ õ tỗ t p H01 () ợ Ap (L∞ (Ω))∗ ✈➔ ξ ∈ L2 (Ω) s❛♦ ❝❤♦ ξ ∈ ∂g (y∗) ✈➔ (Ap)a + ξ = ❤➛✉ ❤➳t tr♦♥❣ [x, y ∗ (x) > ψ (x)] ✭✷✳✶✸✺✮ p (Ay ∗ − Bu∗ − f ) = ❤➛✉ ❤➳t tr♦♥❣ Ω ✭✷✳✶✸✻✮ a (p, χ (y ∗ − ψ)) + (ξ, (y ∗ − ψ) χ) = 0; ∀χ ∈ C Ω ✭✷✳✶✸✼✮ B ∗ p ∈ ∂h (u∗ ) ✭✷✳✶✸✽✮ a (p, p) + (ξ, p) ≤ ✭✷✳✶✸✾✮ ◆➳✉ ≤ N ≤ õ ữỡ tr ữợ ỵ (Ap + ξ) (y ∗ − ψ) = ✺✽ tr ỵ t tỷ A A0 ✈ỵ✐ t➯♥ ♠✐➲♥ D (A) ∈ H01 (Ω) ∩ H (Ω) ✈➔ (Ap)a ❧✐➯♥ tö❝ ❤♦➔♥ t♦➔♥ ❝õ❛ ❆✯♣✳ ◆➳✉ N ≤ 3, ❦❤✐ ✤â y ∗ ∈ H (Ω) ⊂ C Ω ✈➔ Ap (y ∗ − ψ) rã r➔♥❣ ✤÷đ❝ ①→❝ ✤à♥❤ ❧➔ ♣❤➛♥ tû ❝õ❛ (L∞ (Ω))∗ ✤➦❝ ❜✐➺t✱ ✭✷✳✶✹✵✮ ❜❛♦ ❤➔♠ Ap∗ = tr♦♥❣ [x, y∗ (x) > ψ (x)] ❇✐➸✉ t❤ù❝ ✭✷✳✶✸✺✮✱ ✭✷✳✶✸✻✮ ✤↕✐ ❞✐➺♥ ❝❤♦ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ t ự ỵ t t♦→♥ ♣❤↕t ❈ü❝ t✐➸✉ ❤♦→ ✭✷✳✶✹✶✮ g ε (y) + h (u) + |u − u∗ |2U ✈ỵ✐ r➔♥❣ ❜✉ë❝ Ay + β ε (y) = f + Bu ✭✷✳✶✹✷✮ tr♦♥❣ ✤â ε β (r) = −ε = −ε −1 −1 ∞ r − ε2 θ − −∞ ∞ r−ε θ − ε2 θ− ρ (θ) dθ − − −ε θ θp (θ) dθ ρ (θ) dθ + ε ✭✷✳✶✹✸✮ ❚r♦♥❣ ♣❤➛♥ ❦❤→❝✱ β ε ✤÷đ❝ ①→❝ ✤à♥❤ t❤❡♦ ❝ỉ♥❣ t❤ù❝ ✭✷✳✶✷✻✮✱ tr♦♥❣ ✤â β ỗ t ứ t ✭✷✳✶✹✶✮ ❝â ➼t ♥❤➜t ♠ët ❝➦♣ tè✐ ÷✉ ε−2 r ( uε , yε ) ∈ U × H01 (Ω) ∩ H (Ω) ❚ø ❝❤ù♥❣ ♠✐♥❤ ❝õ❛ ✣à♥❤ ỵ t t ữ s❛✉ uε → u∗ ♠↕♥❤ tr♦♥❣ U, yε → y ∗ ♠↕♥❤ tr♦♥❣ H01 (Ω) , ②➳✉ tr♦♥❣ H (Ω) βε (yε − ψ) → Bu∗ − Ay ∗ + f ②➳✉ tr♦♥❣ L2 (Ω) ◆❣♦➔✐ r❛✱ t❛ ❝â t➼♥❤ tè✐ ÷✉ ❝õ❛ ✭✷✳✶✹✶✮ ✭①❡♠ ✭✷✳✶✸✵✮✱ ✭✷✳✶✸✶✮✮ − Apε − β ε (yε − ψ) pε = ∇g ε (yε ) B ∗ pε ∈ ∂h (uε ) + uε − u∗ ✺✾ tr♦♥❣ Ω ✭✷✳✶✹✹✮ ✭✷✳✶✹✺✮ ❑❤✐ ✤â ữỡ tr t ợ p, s õ ợ p, t ữủ ữợ ữủ • pε H01 (Ω) + β ε (yε − ψ) pε dx ≤ C ∇g ε (yε ) Ω L2 (Ω) ≤C ✭✷✳✶✹✻✮ ❉♦ ✤â✱ ❝â ❞➣② ❝♦♥✱ ❦➼ ❤✐➺✉ ε, s❛♦ ❝❤♦ pε → p ♠↕♥❤ tr♦♥❣ H01 (Ω) , ②➳✉ tr♦♥❣ L2 (Ω) , ∇gε (yε ) → ξ ∈ ∂g (y ∗ ) ②➳✉ tr♦♥❣ L2 (Ω) ●✐↔ sû ε → tr♦♥❣ ✭✷✳✶✹✺✮✱ t❛ ✤÷đ❝ B ∗ p ∈ ∂h (u∗ ) ❇➙② ❣✐í✱ ❣✐↔ sû ξε : Ω → R ✈➔ ηε : Ω → R ❧➔ ❤➔♠ ✤â ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ♥➳✉ |yε (x) − ψ (x)| > ε2 , ξε (x) = ♥➳✉ |yε (x) − ψ (x)| ≤ ε2 , ηε (x) = ❱➻ ♥➳✉ ♥➳✉ • ε β (r) = −ε |yε (x) − ψ (x)| > −ε2 , |yε (x) − ψ (x)| ≤ −ε2 ∞ −1 β (θ) dθ; ε−2 r ❦❤✐ ✤â ✭✷✳✶✹✸✮ trð t❤➔♥❤ ∀r ∈ R ✭✷✳✶✹✼✮ • ε (yε (x) − ψ (x)) β (yε (x) − ψ (x)) pε (x) − pε (x) β ε (yε (x) − ψ (x)) = ε |pε (x)| θρ (θ) dθ ≤ ε |pε (x)| ε−2 (yε (x)−ψ(x)) ❤➛✉ ❤➳t x ∈ Ω ✭✷✳✶✹✽✮ ▼➦t ❦❤→❝✱ t❛ ❝â ε −1 yε − ε2 θ − ψ ρ (θ) dθ pε (x) β (yε − ψ) = ε pε ξε ε−2 (yε (x)−ψ(x)) + ε−1 pε (yε − ψ) + εpε ✻✵ θp (θ) dθ ❤➛✉ ❤➳t tr♦♥❣ Ω ❈✉è✐ ❝ò♥❣ t❛ ❝â • pε β ε (yε − ψ) ≤ ε pε β ε (yε − ψ) ε−1 |yε − ψ| ξε + ε−1 |yε − ψ| ηε + 2ε |pε | ❤➛✉ ❤➳t x ∈ Ω ✭✷✳✶✹✾✮ ú ỵ r (y ) = ε−1 (yε − ψ) ηε + Cεηε tr♦♥❣ t➟♣ ❤ñ♣ ❝♦♥ ❜à ❝❤➦♥ ❝õ❛ L2 (Ω)tr♦♥❣ ❦❤✐ ①→❝ ✤à♥❤ ξε t❛ t❤➜② ❤➛✉ ❤➳t x ∈ Ω ❧➔ ❜à ❝❤➦♥ tr♦♥❣ L1 (Ω) , ♥❤÷ ✭✷✳✶✹✼✮ ✈➔ ✭✷✳✶✹✾✮ ε−1 |yε (x) − ψ (x)| ξε (x) ≤ ε • ❚ø ✤â pε β ε (yε − ψ) ❝❤♦ ❞➣② ❝♦♥ εn → pεn (x) β εn (yεn (x) − ψ (x)) → ❤➛✉ ❤➳t x ∈ Ω, ✭✷✳✶✺✵✮ tr♦♥❣ ✤â pεn β εn (yεn − ψ) → p (f + Bu∗ − Ay ∗ ) ②➳✉ tr♦♥❣ L1 () ũ ợ ỵ r t ❝â ❤➛✉ ❤➳t tr♦♥❣ Ω, p (f + Bu∗ − Ay ∗ ) = ❞♦ ✤â pεn β εn (yεn − ψ) → p (f + Bu∗ − Ay ∗ ) ♠↕♥❤ tr♦♥❣ L1 (Ω) ✭✷✳✶✺✶✮ ❑❤✐ ✤â ợ t t (yn ) n (yεn − ψ) pεn → ❚❤➟t ✈➟② β ε (yε − ψ) + ε−1 (yε − ψ)−1 ✭✷✳✶✺✷✮ trð t❤➔♥❤ + • εn ♠↕♥❤ tr♦♥❣ L1 (Ω) ≤ Cε, (yεn − ψ) β (yεn − ψ) pεn → ✻✶ ✭✷✳✶✺✷✮ ❦❤✐ ✤â ✭✷✳✶✹✼✮✱ ✭✷✳✶✺✶✮ ✈➔ ♠↕♥❤ tr♦♥❣ L1 (Ω) ❱➻ (yε n − ψ)+ ∈ H01 (Ω) , →♣ ❞ư♥❣ ❝ỉ♥❣ t❤ù❝ ●r❡❡♥✬s tr♦♥❣ ✭✷✳✶✹✹✮ a pεn , (yεn − ψ)+ χ + ∇g εn (yεn ) , (yεn − ψ)+ χ → ∀χ ∈ C Ω ❚ø ✤â pε → p ②➳✉ tr♦♥❣ H01 (Ω) ✈➔ (yε − ψ)+ → y∗ − ψ ♠↕♥❤ tr♦♥❣ H (Ω) t❛ ✤÷đ❝ ✭✷✳✶✸✼✮✳ ❚ø ❤➺ q✉↔ ❜➜t ✤➥♥❣ t❤ù❝ ✭✷✳✶✸✾✮ t❛ ❝â ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✹✹✮ ✈➻ t❛ ❝â n n (Apε + ∇gε (yε ) , pε ) ∀ε > ❇➙② ❣✐í✱ ❝❤å♥ ❞➣② ❝♦♥ t✐➳♣ t❤❡♦ ♥➳✉ ❝➛♥✱ t❛ ❣✐↔ sû r➡♥❣ ❤➛✉ ❤➳t x ∈ Ω yεn (x) → y ∗ (x) t ợ ữợ ữủ ∈ (L∞ (Ω))∗ ✈➔ ❞➣② ❝♦♥ ✤÷đ❝ tê♥❣ q✉→t ❤♦→ ελ ❝õ❛ εn, s❛♦ ❝❤♦ • ②➳✉ tr♦♥❣ β ελ (yελ − ψ) pελ → µ (L∞ (Ω))∗ ✣✐➲✉ ♥➔② ❝❤ù♥❣ tä r➡♥❣ Ap ❝â ♣❤➛♥ tû ♠ð rë♥❣ tr♦♥❣ (L∞ (Ω))∗ t❛ ❝â −A∗ p − µ = ξ ∈ ∂g (y ∗ ) ❇➙② ❣✐í✱ ✈ỵ✐ ỵ r ộ > õ E t➟♣ ❤đ♣ ❝♦♥ ✤♦ ✤÷đ❝ ❝õ❛ Ω s❛♦ ❝❤♦ m (Eδ ) ≤ δ, y∗ − ψ ❧➔ ❜à ❝❤➦♥ tr \E = ỗ t tr yεn − ψ → y ∗ − ψ ❑❤✐ ✤â tø ✭✷✳✶✺✷✮✱ ♥➯♥ µ (y∗ − ψ) = tr♦♥❣ Ωδ ♥❣❤➽❛ ❧➔✱ (y ∗ − ψ) µa ϕdx + µs ((y ∗ − ψ) ϕ) = ∀ϕ ∈ L∞ (Ω) , sup ϕ ⊂ Ωδ Ωδ ▼➦t ❦❤→❝✱ ❝â Ωk ❞➣② t➠♥❣ s❛♦ ❝❤♦ m ❞♦ ✤â (y ∗ − ψ) µa ϕdx = 0; Ω\Ωk ≤ k −1 ✈➔ µs = tr➯♥L∞ Ωk ∀ϕ ∈ L∞ (Ω) , sup ϕ ⊂ Ωδ ∩ Ωk Ωδ ∩Ωk ✻✷ ❉♦ ✤â✱ (y ∗ − ψ) µa = (y ∗ − ψ) µa = ❦❤✐ ✤â ❤➛✉ ❤➳t tr♦♥❣ Ωδ tr♦♥❣ − (Ap)a = ξ ∈ ∂g (y ∗ ) ◆➳✉ ≤N ≤ t❤➻ H (Ω) ⊂ C Ω yε (x) → y ∗ (x) ❱➻ ψ ∈ H (Ω) ⊂ C Ω , ✈➔ ❣✐↔ sû δ → t❛ s✉② r❛ [y ∗ > ψ] ✈➔ ✤➲✉ tr➯♥ Ω ♥➯♥ ✭✷✳✶✺✷✮ trð t❤➔♥❤ (y∗ − ψ) µ = ♥❣❤➽❛ ❧➔✱ (y ∗ − ψ) (Ap + ξ) = ỵ ữủ ự t ✈➠♥ ♥➔② t➻♠ ❤✐➸✉ ✈➲ ❝→❝ t♦→♥ tû ♣❤✐ t✉②➳♥ ỡ ỹ sỹ tỗ t ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❡❧❧✐♣t✐❝✱ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ❝❤♦ ❜➔✐ t♦→♥ ✤✐➲✉ ❦❤✐➸♥ tè✐ ÷✉ ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❡❧❧✐♣t✐❝✱ ù♥❣ ❞ö♥❣ ✈➔♦ ❜➔✐ t♦→♥ ✤✐➲✉ ❦❤✐➸♥ tè✐ ÷✉ ♣❤÷ì♥❣ tr➻♥❤ ❡❧❧✐♣t✐❝ ♥û❛ t✉②➳♥ t➼♥❤ ✈➔ ❜➔✐ t♦→♥ ✈➟t ❝↔♥✳ ✻✹ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ ❍✳ ❇r❡③✐s✱ ❋✉♥❝t✐♦♥❛❧ ❆♥❛❧②s✐s ❙♦❜♦❧❡✈ s♣❛❝❡s ❛♥❞ P❛rt✐❛❧ ❉✐❢❢❡r❡♥✲ t✐❛❧ ❊q✉❛t✐♦♥s✱ ❙♣r✐♥❣❡r✱ ✷✵✶✶✳ ❬✷❪ ◆❣✉②➵♥ ❍ú✉ ❉÷✱ ✣✐➲✉ ❦❤✐➸♥ tè✐ ÷✉ ❤➺ t➜t ✤à♥❤ ✈➔ ♥❣➝✉ ♥❤✐➯♥✱✱ ◆❳❇ ✣↕✐ ❤å❝ ◗✉è❝ r ự ỵ t ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✤↕♦ ❤➔♠ r✐➯♥❣✱ ✱ ◆❳❇ ✣↕✐ ❤å❝ ◗✉è❝ ❣✐❛ ❍➔ ◆ë✐✱ ✷✵✵✺✳ ❬✹❪ ❱✳ ❇❛r❜✉✱ ◆♦♥❧✐♥❡❛r ❞✐❢❢❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ♦❢ ♠♦♥♦t♦♥❡ t②♣❡s ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s✱ ❙♣r✐♥❣❡r✱ ✷✵✶✵✳ ✻✺