Luận văn Phương pháp phần tử hữu hạn cho bài toán Elliptic phi tuyến biên cong

PhucJngphap phdn ta hT1uhf.:lncho bili loan elliptic phi tuyin bien cong 6 ClnJdNG2 : s1jTONTAl vA DUY NHAT LaI GIAI , ? '" 1. CAC GIA THIET Voip>1d~tp'=~ p-l CHI)<pEC([O,I]),<p(x»0, '\Ix E [0,1], cp La d titngkhuctren[0,1]. (H2) G EV'. (H3) H E If(IJ). .(H4) Ml'M2 :QxIR~IR, g :QxIR~IR Lacac hamthoa di€u ki?n Caratheodory, nghia La: Vz EIR,cachamMj(.,. ,z),g(.,.,z)dodu(/ctrenQ, vawJihduhet(x,y) En cachamMj (x,y,.)vag(x,y,.) lient1:tcheoz, i=1,2. (HS) MI, M2 dcJndi?u tang theobien tha 3, tac La: (Mj(x,y,z)- Mj (x,y,z)Xz- z)>0 , '\Iz,Z E IR, z:;i:Z, a.e(x,y)E Q i =12, (H6)T6ntf.:libahlings5ducJngC1,C~,C2vahamhELP 1(0) saDcho (i) zMj(x,y,z)~C1IzIP -C~ '\Iz E IR, (ii) a.e(x,Y)EQ , i=I,2; IMi(X,y,Z~~C2(lh(x,y~+lzIP-l), '\IzEIR, a.e(x,Y)EQ, i=I,2. (H7) T6ntf.:lihlings5ducJngC3<%~ thoa: Ig(x,y,z~~C3(1+lzIP-l) , '\IzEIR, , C { IIVIIWI'p V trongdo 0=sup Ilvllv 'v E , a.e(x,Y)EO, v*o}. (H8)p >2 thoa: Phl1ctngphapphtintahauhr;mchobai toanellipticphi tuylnbiencong 7 ( 2 ] P' [ C3ColoIX' +IIGllv'+CoCIIHIIU'(fl) J P' 4C~IOI ( 11:1 ) p (p-l)- C -CCP +C -CCp< /3CCoP 130 130 1 trongdolo! =roes0 =f<p(x)dx , 0 C=sup~lvllc(n) : VEW1,P(0) , IIvllwl,P=I}, ( C t8ntr;lidophepnhungW1,p(0) c~ C(o) facompact) [; fahangst!trongdinhfyvetchl1ctng!, C =suP { v Ir : v E c1(Qt Ilvllw"(o)=+ (H9) VetimtJi a E (0, 7Tf3)cohai hangst!dl1ctnga vakasaGcho: (i) (ii) ka~gacotga,. (iii) g(x,y,Z):?ga , tlz E[-a, a],a.e.(x,y) E.Q; I g(x,y,Z])-g(x,y,zz)I ~ka I z] - zzl , tlz],zzE [-a, a],a.e.(x,y)E.Q. BMtoan(0.1)-(0.3)duQcduav€ b~litoanbie'nphannhusau: Hili toaD(f}: TImu E VsaGcho (2.1) (M{X, Y, :l~)+(M,( X,y,: l :)+(g(x,y, u)sin u, w) + =(G,w)+ fHwds. VWEV. r) , J::. , ,,',? 2. DJNH LY TON T~I VA DUY NHAT LaI GIAI Djnhly 2.1: GidsaM], Mz, g, G, H thoa(Hl)-(H7).Khi dobaitoan(P) co fiJi gidi . Hctnnaa,nlu themVaGcacgiLlthilt (H8)va(H9)thlfiJigidicua (P) faduynh[{t. Chungminh: Phuangphapphdntahfluhr;lnchobili roanellipticphi tuytnbiencong 8 Bjnh19duQchungminhquanhi~ubudc: Bu'oc 1..;.Xap xl Galerkin Vi V tach duQCnen t6n t~imQt"co sd" de'mduQc {wj J.- theo nghla:J-I,2,... . Wj E V, . V m, {wI, ,W mJ dQcl~ptuye'ntinh, . T~pcact6hQptuye'ntinhhii'uh~ncaeWjtrum~ttrongV . TatlmWigiaixa'pxi dudid~ng: (2.2) m Um(X'Y)= ICmjWj(X,y), j=1 trongd6cac Cm.thoah~phuongtrlnhphi tuye'nsailJ (2.3) (M{ x,y,0;: ), ~j)+(M2(x,y,a;; ). a;j) +(g(x,y,um)sinum'Wj)=(G,Wj)+ fHWjds f1 j =1,..,m. Trudehe'tachungminhh~(2.3)c6Wigiai. B~tVm1Akh6nggianhii'uh~nchi~usinhbdi Wj, j =1.m. Coi Pm: Vm~ Vmxacdjnhbdi (2.4) (2.5) m Pm(Um)= Ipmj (Um)Wj , j=1 Pm;(Um)= (MI(X,Y,; ). ~j) +(M,(x,y,a;; ). ~j) +(g(x,y,um)sinum'Wj)-(G,Wj)- fHWjds, j=l,..,m, f1 m Urn=2:Cm.Wj . j=1 J (2.6) Khid6(2.3)tuongduongvdi: Pm(Um)=O. Tac6th€ nghi~ml~ikh6ngkh6khanr~ng: (2.7) Pm: Vm~ Vmlientvc. B€ apdvngb6d~Brouwer(b6d~1.3,chuang1)tachIdn chungminht6nt~iPm>0 saDcho Phu:cmgphap ph6.nt11hau hc;mcho bili roan elliptic phi tuyin bien cong 9 (2.8) II urn Ilv =Prn => (Prn(Urn),Urn)v ~O.m m CM '1r~nglien V rntaIffyHchvohuangsail rn (2.9) (Urn'Vrn)v=Icrn.drn. m j=l J J vdi rn urn=ICrn.Wj j=l J rn vrn = IdrnWj' j=l J Chufinlien Vrnsinhbdi tichvohuangVrnduQck'1hi~u11.11v .m Taco (2.10) rn (Prn(Urn),Urn)v=IPrn. (Urn)Crn. m j=l J J ~(M{ x,y,~). ~m)+(M,(x,y,~). ~m) +(g(x, y,urn)sin urn' Urn)- (G, Urn)- fHurnds f1 Tli giathie't(H6)(i),taduQc: (Ml(x, Y. a;;; ). a;;; ) +(M,( x,y,~m). ~m) (2.11) ~c1!IUrnll~-2C;IOI. Tli giathie't(H7)suyfa: (2.12) !(g(x,y,urn)sinurn,urn)!:S;C3CoIQlh'llurnllv+c3c~IIUrnll~. sa dl).ngdinh1'1ve't(b6d€ 1.2,chuang1)tathuduQc: (2.13) fHurnds f] :s; II H IILP'(f1) II You rn IIU (f]) :s; ccoIIHIIU'(fl) Ilurn !Iv . Tli (2.10)-(2.13)vadoG E V' suyfa: (Prn(Urn),Urn)v ~(C1 -c3C~)llurnll~ -2C;IQ! (2.14) m ( h ~ )-\C3CoIQI p'+IIGllv' +ccoIIHIIU'(fl) IUrnliv Phu(Jng phapphdntahr1uh{lnchobili toanellipticphi tuye'nbiencong 10 =(CI -C3C~Xllumll~-~IIIUmllv -YI)' trongd6~I>0, YI >0 dU<;1cxacdinhbdi - (C3CoIQIX,+IIGllv'+CCoIIHllu'(r~J~I- IC - C CP)'\: I 3 0 - 2C~IQII YI- I(CI-C3C~)' (2.15) Chuydingsadt:mgba'td~ngthucHoldertac6: (2.16) ~lllUmllv~~(ellUmlivJ'+;.(~1r, trongd6 E>0 dU<;1CchQnsao rho (2.17) EP 1 P 2 8=(~thay Khid6tli(2.14),(2.16),(2.17)suyra: (Pm(UmhUm)v:<:(C1-C3qhIIUmll~ -~ ( ~ J P'-YI )m \2 p' P = ~(C1-C3C:{IIUmll~-(p-le:J -zr} (2.18) ChI1Y r~ngffiQichu§:ntrenVm d€u tu'ongduong,dod6t6nt~ihai h~ngs6duongClmva C2msao rho: (2.19) Clm II vllvm ~II v !Iv ~C2m II v IIVm Vv EVm. ~ 0 ? ~ 1 ,.ChQnPm > thoaPm=-P VOl Clm (2.20) p{ (p-lr:l r +2y,r Khi d6ne'uII Um Ilv =Pmthltli (2.18)-(2.20)tasuyram (Pm(Um),Um)v~O.m V~y(2.8)thoa,dod6apdvngb6d€ Brouwersuyfa (2.6)c6Wigi,HUrnthoa PhucmgphapphJ.ntithiluht;mchobili toanellipticphi tuye'nbiencong 11 (2.21) II urn II v ~Prn'm Buck2: Danhgiatiennghi~m TuPrn(urn)=0,voitinhloantu'ongtl!d~nd€n (2.18),tasuyfa: (2.22) Dodo (2.23) ,[IIUmll~-(p -le:1 r- 2YI) ~o. IIUmllv~p=((p -Ie:! r+2y!r Tugiathi€t (H6)(ii)va(2.23)suyfa: (2.24) Ml ( X'Y' a; J ' ~c2 lllhIIL'(n)+ ~Ip-I ) .LP (0) LP(0) Do (2.23)va (2.24) ta du'<;5c (2.25) IIMl ( X' y, Gum JII , ~c ,ax LP(0) C 1ahangs6dQcl~pvoi m. TuongtvvOiM2 taclingco: M 2 ( X,y, Gurn )11 . ~c .ax LP(0) Danhgiatu'ongtv,tugiathi€t (H7)va(2.23)tasuyra (2.26) (2.27) Ilg(X,y, urn)sin urnIILP'(O)~C , C 1ahangs6dQcl~pvoi m. ChliyrangphepnhlingV c-+LP (0) 1acompact,khi do tu (2.23),(2.25),(2.26)suyra t6nt<;liffiQtdaycancua {urn}v~nky hi~u1a{urn}saocho (2.28) Urn ) u trongW1,p(0) yu, (2.29) Urn ) u trong U (0) ffi<;lnh, (2.30) Urn ) u a.e (x,y) E 0, (2.31) MI(x,y,8un/Ox) ) Xl tron U' (0) eu tfiKH'T\JN!!IENI g y, THtT\!!EN-'----'-'-- , ('1"".....1'1 Phuangphapph6.ntah11uhr;lnchobili toanellipticphi tuyenbiencong 12 (2.32) M2(x,y,0un/8y) ~ X2 tfong L p'(D) y€u. M~tkhactITgii thi€t (H4)suyfa: (2.33) g(X,y,um)sin Um g(X,y,u)sinu a.e(x,y) E D. Apdl,mgb6de1.4,chu'dng1vdi N=2, q=p', Q=D, Gm=g(x,y,um)sinUm va G =g(x,y,u)sinu, tIT(2.27)va (2.33)suyfa (2.34) g(X,y,um)sin Um ~ g(X,y,U)sinU tfong L p'(D) y€u . Bu'oc3: QuagieJihr;ln QuagiOih?ntrongphu'dngtrlnh(2.3),sadt;mg(2.31),(2.32)va (2.34)tasuyfa u thoa phudngtrlnh: (2.35) \Xl ,0;:)+(X, , :)+(g(x.y.u)sinu. w) =(G,w)+ IHwds r1 , \jw E V. Nhuv~yd~chungminhulaWigiii baitmln(P)tachidn chUngminh Xl =Ml( X,Y,:J va X, ~M,(X,y,: J. . TIT(2.3)ta co /Ml ( X' Y, Gum J , Gum ) +/M2 ( X' y, Gum ) , Gum )(2.36) \ Ox Ox \ Oy Oy =-(g(X,y,um)sinum,Um)+(G,Um)+fHumds r. sadl;lng(2.28),(2.29),(2.34),(2.35)vaquagidih?ntrong(2.36)taco (2.37) g~[(M{x,y,0;; ). 0;; )+(M,( x,y,~). ~m)] =(XI':)+(X2':). TIT(2.28),(2.31)-(2.33)va(2.37)suyfa Phuangphapphdntithiluh(;mchobili toanellipticphi tuyfnbiencong 13 (2.38) ~l1![(M{ x,Y,a;: )-MJx,y,$I)' a;; -$1) +(M,( x,y,a;; J - M,(x, y,~,), a;; -~,) ] = (Xl- MJx,y,$lh: -$1)+( X, - M,(x,y,$,), :-$,), V~i'~2ELP(0) . Do giathie't(H5) tasoyfa: (2.39) (Xl -MI(X,y,$J,: -$I)+(X2 -M2(X,y,$2)': -$2 )?O V~i'~2E LP(0). Trang (2.39)chQn au ~I=--AWI , Ox au ~2=-' ay Taco (2.40) WI ELP(O), A>O, (Xl - M{X,y,: -AWl),w} 0 ChoA ---+ 0+, sad1;lngiathie't(H6)(ii) va dodinh1yhQit1;lbi ch~nLebesguetasoyra (2.41) Dodo (2.42) (Xl -MI(X'Y' :}WI)~O , Vw I E LP(0) . Xl =M{X,Y.:} Ly 1u~nWong W', tu (2.39) ta cling co X, =M,(X,y,:). (2.43) V~y s1;1'tant<;tiWi giai u cua bai roan (P) du'Qcchung minh. PhucJngphapphtintahi1uh{lnchobili roanellipticphi tuyenbiencong 14 Blioc 4 : ST;tduy nhdtlili gild Trudehe'tachll '1dingWigiiHcuabai toan(P) t6nt~ivabi ch~ntrongV (2.44) II u II v ::;P, trongdopduqcxacdinhtit(2.23).Tli giathie't(H8)va(2.15)taco: (2.45) II u Ilc(n)~CCoP~1t.3 GQiUl. Uz Iahainghic$mcua(P)thoa (2.46) II Uj IIc(n)::;CCoP::;; , i =1,2. KhidoUj - Uzthoa (2.47) (M.( x,y,~)-M.(x,y,~), a;) +(M 2(X, y, ~) - M 2 (X, y, ~J :) +(g(x,y,Uj)sinUj - g(x,y,Uz)sinUz, w) =0 ,Vw EV. ChQnW=Uj- Uztrong(2.46)taco (M{ x,y,~)-M{x,y,a;:), ~(Ul - U,)) +(M2(x,y,a;; ) - M2( x,y,a;: ).~(Uj - uJ) +(g(x, y, Uj )(sinUj - sin uJ, Uj - U2) +((g(x,y,Uj)-g(x,y,uJ)sinU2' Uj -U2) =0 (2.48) Chu'1dingtlicacgiathie't(H9)vatli (2.46)taco (g(x, y, Uj )(sinUj - sinU2),Uj - U2) +((g(x,y, Uj) - g(x,y, U2))sin uz, Uj - Uz) ~(ga casa - ka sina ~Iuj - Uzl12~0 , (2.49) trongdoa =cCop. . Tli (2.48)va (2.49),do tfnhdondic$utangng~tcuaMj vaMz ta Suyfa Uj - Uz =O.V~y Uj=Uz. Dinh1'1duqcchungminh.. Phucmgphapphcintahiluht;mchobai loanellipticphi tuylnbienGong 15 3. TRUONGH(1PRIENG Trangtru'ongh<;1priengvai (2.50) M1(x,y,z)=M2 (x,y,z)=Z , bairoan(0.1)-(0.3)ITathanh ~:.5!) - AU+g(x,y,u)sinu=G (x,y) E Q, (2.52) U I =0 fo ' au, =H. avlf( mliroanbie'nphantu'dnglingvai(2.50)-(2.51)la: Baitoan(P'): l1muEo/?={ VEH1(Q): vir. =O}saoChO (2.53) a(u,w)+(g(x,y,u)sinu,w)=(G,w)+ fHwds , \lWE o//". f] trongdoa ( . , .) Ladt;mgsongtuytntinhxacdinhbiJi (2.54) ( auaw auaw J a(uw)=If --+-- dxdy., n OxOx 0y0y Taclingchuydingo//"lakhonggianHilbertd5ivaitichvohu'anga( . , . ) vachua':nsinh baiUchvohu'angla (2.55) I vii =~a(v,V). M~tkhac,trongWhai chua':n1.11va11.IIHI(n)Iatu'dngtu'dngdod6 (2.56):I Co>0 I V 11 ~ II V IIHl(n)~Co! V II ' V V E W Tathanhl~pcaegiiithie'tsan: (HI') 0 , \Ix E [0,1], cpC2titngkhuctren[0,1]. (H2') G Eo//"'(W' Laddingducuaq/:;' (H3') H E L2(1] ). (H4') g :nxIR~IR Lahamthoadiiu ki?n Caratheodory. PhucJngphapphdn ta hl1uhr;mcho blli roan elliptic phi tuytn bien cong 16 (H5') T6nt{lihlings{fducJngC 3 <Yc ~ thoa: Ig(x,y,z~~C3(1+lzl), 'v'zEIR , a.e (X,Y)EO, , { II V IIHI trongdo Co =supI V II ' V E W, V *0 } (H6') VaimlJi a E (0, ffl3)cohaihlings{fducJnggavakasaGcho: (iv) (v) kaS"gacotga, (vi) g(x,y,z)zga , \iz E [-a, a], a.e.(x,y)ED, I g(X,y,Zl)-g(X,y,Z2)I s"ka I Zl- Z2 I , \ill, Z2E [-a, a], a.e.(x,y)ED. DinhIf 2.2: Cidsa cacgidthitt (Hl')-(H5') Ladung.Khi doblli roan(P') coLCtigidi. HcJnnl1a,ntu thaythigidthief(H2')biJigidthiit (H2")G EL2(D) thi bai roan(P') co lCtigidi u E H2(D) n P/. HcJnnr1a,niub5sungthemgidthief(H6')vll thaygidthitt(H2')biJigidthief(H2") saG cho ( I J ~ (H7') !q>(x)dx +IIGII+IIHIIL2(f\)dTlnho thiblliroan(r) coduynhdtmQtlCtigidi trongH2(D)n W. Chungminh: Tu'dngtvdinh192.1,dinh192.2duQchUngminhquanhit~ubuoe. Buoc 11Xapxl GaLerkin Giasa{wj }.- "cdsa"de'mduQCcuaq~Tatlmloigiaixa'pXl duoid~ng:J-l,2,... (2.57) m um(x,y)= ?:CmjWj(X,y),J=1 trongd6cac cm.thoahc$phuongtrlnhphi tuye'nsauJ Phl1cJngphdpphdn ta hT1uht;mcho bili roan elliptic phi tuytn bien cong 17 a(um' W j )+ (g(x, y, Urn)sin urn' Wj ) = (G, Wj ) + fH W j ds (2.58) fl j =1,..,m. Trudehe"ttachungminhhc$(2.58)coWigiiiibich~n. D~tWmla khonggianhOOh(}.nehi€u sinhbdi Wj,j =Lm. Coi (2.59) Pm :Wm~Wm m Pm(Um)= Ipm. (Um)Wj. ) J=I Pmj(um)=a(um,wJ+(g(x,y,um)sinum, Wj)-(G,Wj)- fHWjdS ~~ ~ j =1,..,m, m Um(X,y)=LCm.Wj(X,y). j=1 ) (2.61) Khido(2.58)tuangduangvdi: Pm(Um)=O. (2.62) Ta coth€ nghic$ml(}.ikhongkhokhanding: Pm : Wm~ Wmlien t1,le. D€ apdl,mgb6d€ Brouwer(b6d€ 1.3,chuang1)tachidn chungminht6nt~iPm>0 saoeho (2.63) II Urn Ilv =Pmm (Pm(Um),Um)vm :2:0,=> trongdoll.llvmlaehu§:nsinhbdinehvohuangsau: (2.64) vdi Taco (2.65) m (Um,Vm)v =LCm.dm.,m . 1 ) ) J= m Um=LCm.Wj' j=1 ) m vm=Ldm.Wj' j=1 ) m (pm(um),Urn)v =L Pm. (Um)cm. m j=1 ) ) =a(Um,Um)+(g(X,y,um)sinum' Um)-(G,Um)- fHumds . f( Tugiathie"t(H5')suyra: Phuongphapph6ntahiluhr;mchobai roanellipticphi tuytnbiencong 18 (2.66) I(g(x,y, urn)sin urn'Urn)1::;c3coiOIliI urn 11 +C3C~I Urn I~ sad\mgdinhly ve't(b6d~1.2,chudng1)tathuduQc: (2.67) fHurnds fl ::; II H IIL2(f)1YoUrnIIL2(f1) ::; IIHIIL2(f)IYoUrn IIL2(f) ::; ell H IIL2(f)1Urn IIHI(Q) ::; ecoIIHIIL2(fl)I Urn 11 Tli (2.66)-(2.67)vadoG E q:/'suyfa: (Prn(Urn),Urn)Vrn ;:::(1-C3C~) urn I~ (2.68) (~~ )-\C3CoIOI2+IIGt +ccoIIHIIL2(fl) I urn11 =[(l-c,c;)1Urn I] - (C,CoIOIy, +11G t +CCollH!!L'(r,))]IUrn 11. Do ffiQichu:lntrenq:/rnd~utudngdudng,suyracohaih~ngs6dudngClmvaCZmsao cho: (2.69) Clmll V IIWm~ I V 11~ Czmll V IIWm, V V E q:~. ~ 0 ? ~ 1ChQnPm> thoaPm=- P vdi Clrn (2.70) ~ ~ IC3CoiOI2+IIGllv'+CCoIIHIL2(fl) P= (1-C3C~) Khidone'uII Urn Ilvrn=Pmthltli (2.68)tasuyra (Prn(Urn),Urn) Wrn~O. V~y(2.63)thoa,dod6apd1.).ngb6d~Brouwersuyra (2.58)c61Oigiai Urnthoa (2.71) II urn II ::; Pm .Vrn Buck2: Danhgialiennghi?mvaquagiaih(;m Phl1CJngphap phdn ta hr1uhf,'mcho bili roan elliptic phi tuyin bien cong 19 Tli (2,61)va (2.6'6)ta s,\l'j1:a'. (2.72) I urn 11 S;P , vdiP >0 chobCii(2.70). Tli (2.72)va giathi~t(BS') ta guyra (2.73) Ilg(x, y, Urn)sinurnII s;c , C1fth~ngs6dQcl?pValm.. ChIiyriingphepnhIing~/ c~ L2 (D) 1acompact,khi d6 tii (2.72)va (2,73)suyra t6nt?i m9tdaycancila{um} yinkf hj~uIii {um}saDcho: Tli giathie't(H4')suyfa: (2.77) g(x,y,Urn)sinUrn ~ g(x,y,U) sin U a.e (x,y) E Q , Ap dt,mgb6d~1.4,chuang1Val N=2, q=2, Q=Q, Grn=g(x,y,urn)sinUrn va G =g(x,y,u)sinu, tli(2.73)va(2.77)suyfa (2.78) g(x,y,urn)sinUrn > g(x,y,u)sinu tfongL 2(Q)ye'u. Do(2.74)va(2.78),quagidih~ntrong(2.58)tasuyfadingulaWigiaicuabairoan(P'). S1,1't6nt~iWigiaiducJchungminh. Bay giGtathaygia thie't(H2') bCiigiathie't(H2''). Chuydng Wi giai u E q/"'cua(P') thoamanphuongtrlnhsailday: (2.79) ~U =g(x,y,u)sinu - G trong D'(n). Tli cacgiathie't(H2") ,(H5')va(2.79)tasuyfa (2.80) ~uE L2(Q). Dodo (2.81) U E H2(Q)nw' . Giasli'(H2") thoadungthaycho(H2')vathemvaGcaegiathi~t(H6')va(HT). Khi do, clingtli (H2") (H5')va(2.79)tasuyfa (2.74) Urn > U trongHI (Q) ye'u, (2.75) Urn > U tfongL2(Q) mnh, (2.76) Urn > u a.e(x,y)E Q. PhucJng phap phtln t11hr1uh{Jncho bai loan elliptic phi tuye'nbien cong 20 (2.82) II~ull~C3~01~+lIull)+IIGII TaeMyr~ng: (2.83)TrongH2(Q)haichufinII V IIH2(n)va ~I v I~+II ~v112latu'ongdu'ong. (2.84)PhepnhungH2(Q)c-) C(O) lientt;lC(n=2). Dod6tu(2.83)va(2.84)tasuyfa: (2.85) 3Co >0 : Ilvllc(n)~Co~v 11+11~vll), \tv E H2(0). M~tkhac,vdigiathie't(H2") thayrho(H2')tac6thedanhgiaWigi,Hu trongo//"tu'ong t1!nhu'trong(2.72)nhu'sau: (2.86) vdi (2.87) IUrn 11 ~p, p= Co 2(C3IOI~+IIGII+CIIHIIL2(fIJ. 1- C3Co TIT(2.82),(2.85)-(2.87)tasuyra (2.88) (2.89) II u Ilc(n)~a a = 1-~oC' ((I+Co)(C3101y, +II G 11)+CoC(1+C3Co~1HIIL'lr)3 0 1 Chuyr~ngvdigiilthie't(HT) tac61°1=f<p(x)dx, II G II ' II H IIL2(f)dunhosaorho 0 I (2.90) a ~rc/3. Tasechungminhr~ngbairoan(P')c6Wigiaiduynha'trongH\Q)no/~ TMt v~y,giilsa u,v E H2(Q)no//"lahaiWigiaicua(P') thoa (2.91) II u Ilc(n)~a ~; , "v Ilc(n)~a~; . Khid6u- v thoa (2.92)a(u- v,u - v)+(g(x,y,u)sinu - g(x,y,v)sinv ,u- v)=0 hay PhU(Jflgphap ph8.ntzlhilu hg.ncho bili roan elliptic phi tuytn bien cong 21 (2.93) I u-v I~+((g(x,y,u)-g(x,y,v))sinu,u-v) +(g(x,y,vXsinu-sinv),u-v) =0 Sad\lnggiathi€t (H6')tadanhgiahais6h~ngthlihaivathlibacuav€ trai(2.93)nhtt sail: (2.94) ((g(x,y,u)- g(x,y,v))sinu,u- v)+(g(x,y,vXsinu - sinv),u - v) ~(gacosa- kasina~lu- vl12~0 T6 h<;:fp(2.93)va (2.94)tathud11<;:fcI u- v I~ ~0 hayu=v. E>inh11'd11<;:fCchungminh..