Luận văn Nghiệm một số bài toán uốn tấm nhiều lớp

ChlidDg1 Nghi~mye'ucuabai toaDI Trongchudngnay,bai loanI du<jcphatbiSuduaid~ngbie"nphanb~ng cachduavaokhonggianhamthichh<jp.Sv t6nt~iva duynhfftnghi~rn cuabai loanbie"nphan(nghi~rnye"ucuabai loanI) du<jchungrninh.Cu6i clingla ke"tquavS tinhtrdncuanghi~rn. 1.1 Phat biiu bie'nphan Duavaokhonggianham { I 2 8v 8v A }V = v I v E HonH ,ax=8y=0trenr . TrenV tatrangbi tichvahuang (18) r 82u82v r 82u 82v r 82u82v r (u,v)v= in ax28x2dx+in 8xay8x8ydx+in 8y28y2dx+in uvdx(19) vairnQiu,v E V. Tichvahuangnaysinhrachuftn Ilvllv= J(v, v)v (20) vai v E V. Lu'uy, doV C H2nentrongV tacochuftncarnsinhtuchuftn trohgH2, xacdinhbdi 1 ( 82 2 82 2 82 2 )Ilvll1-2 = n 8x~ + 8x;y + 8Y~ +I \l vl2+Ivl2 dx. Ch. 1Nghi?mytu cilahili toanI 17 M~nhd'e3. TrangkhanggianV, chudn(20)va chuilntheaH2 ia tl1(jflg dl1(jflg. Changminh.Hi~nnhienta co Ilvll~<Ilvll~2' M~tkhac,apd\!ngba'td~ngthuc(14),ta co (21) ( 02 2 02 2 ) II\7v1112<2C 0 ~ + 0 ~ . <2Cllvll~, . x L2 y L2 suyra ( 02 2 02 2 ) 02 2 Ilvll~2 < (1+2C) 0 ~ + 0 ~ + 0; +IIvll12x L2 y L2 X Y L2 < (1+2C)llvll~. (22) Tli cacba'td~ngthuc(21),(22)tanh~nduqcdi~uphiiichungminh. 0 M~nhd'e4. V ia khanggianHilberttachdl1f/Cvdi tichvahl1dng(19). Changminh.Ta chungminh V Ia kh6nggian con dongtrongH2 nHJ. Cho {vn}Ia day trong V sao cho Vn ---+V trong H2 nHJ- Ta chung minhdingv E V. Vi c;;E HI vac;; =0I r nell,theodinh19vSt,t6nt£;liffiQth~ngs6 C > 0 (ph\!thuQc[2)saocho ov oxIIP(r) ovn- ov <C ovn- OV ox ox L2(r)- ox ox"Hl ( 02vn 02v 2 02Vn< C --- +- OX2 OX2 L2 oxoy 2 112OV oxoy liP 2 ) OVn OV ---+0; + II ox - ox L2 Ch. 1 Nghi?mytu Gilabili loan I 19 Tim w E V thoa a(w,v) = .Infvdx (27) voi mQiv E V. 1.2 811fon t~iva duynha'tnghi~m Dinh Iy 1. V6'if E L2 thlhili loanI conghi?mytu duynh({tw E V. Changminh.Tli congthuc(26),voi mQiw,v E V, ta c6 fpw 02v 02w 02v la(w,v)1 < Dnllox2. £2 Ox2 £2+2D12 8xoy £2 oxoy11£2 02W 02v + D22118y2 £2 oy211£2 ( ) 1/2 02w 2 02w 2 02w 2 <M-+ +- ox2 £2 oxoy £2 oy2 £2 ( 02v 2 02v 2 02v 2 ) 1/2 X - + + - Ox2 £2 oxoy £2 oy2 £2 < Mllwllvllvllv, trongd6 M =max{Dn,2D12,D22}.Do d6, d~ngsongtuytn tinha(-,.) lien t\lCtrenV. Ch. 1Nghi?myeuGuahili loanI 20 Clingtu congthuc(26),voi ffiQiv E V, taco a2v 2 a2v 2 a2v 112 a(v,v) = Dn II a2 +2D12aa +D22 a 2x £2 x Y £2 Y II£2 ( a2v 2 a2v 2 a2v 2 )>m~+ +2'ax £2 axay £2 ay £2 trongdo m =ffiin{D11,2D12,D22}. Tu bfftd~ngthuc(13),tad€ dangsuyra ( 2 2 a2 2 a2v 2 ) av v 2 2 + + 2 . Ilvll£2<2C ax2 £2 axay £2 ay £2 Nhu'v~y, 2 2 2 ) a2 2 a2v a v 2 v + - . Ilvll~< (2C +1)( ox2 L' + oxoy L' fJy2 £2 (28) Voi 0: =m/(2C2+1)thl a(v,v) > o:llvll~ voi ffiQiv E V. f)i~unaychungto d~ngsongtuye'ntinha(.,.) Ia khangtu. Do f E L2 va 1fvdx <IlfllL'IIVIIL' <IlfllL'llvllv voi ffiQiv E v, Dend~ngtuye'ntinh (f, .), dinhbdi (f,v)=.k fvdx Ch. 1Nghi~mytucuabili toanI 21 vdi mQiu E V, lien tl}ctrenV. Vdi cactinhchffttren,nhaDinh19Lax - Milgram,tacodi~uphai chungminh. D 1.3 Tinh trdn cuanghi~mye'u Gia sli'w la nghit:;mcuabaitoaD1.Trongml}cnay,chungt6ikhaosattinh troncuanghit:;mye'umaslft6ntc;tivaduynhfftnghit:;mdiiduQcchungminh trongml}ctrudc.Ke'tquachinhduQcphatbiSutrongdinh19dudiday Djnh Iy 2. Ntu f E L2 thl w E H4. Changminh.Dlfa trenkhai nit:;mtl sai phan,chungminhcuadinh192 duQctie'nhanhHinluQtchocactruanghQpD =R2,D=R~vaDlat~p mdbfftky. A. Tnlifnghf/PD =R2 Lffyh E R2. Khi do,D+h =D. Trongphuongtrlnh(27)cuabaitoaD I, Iffyu =D-hDhWE V. Ap dl}ngb6d~1,phuongtrlnhnh~nduQccothS vie'tgQnlc;tila VT =VP. Taco VT = Dn r 82DhW82DhWdx+2D12r 82DhW82DhWdx }D 8x2 8x2 }D 8x8y 8x8y + D22r 82Dhw82Dhw }D 8y2 8y2 dx ( 82Dh W 2 82Dh W 2 82Dh W 2 )>m 2 + + 2 '8x £2 8x8y £2 8y £2 (29) trongdo m =min{Dn, 2D12,D22}. Ch. 1 Nghi?mytu Gilabid loan I 22 DS danhgia VP, tadungbfftd~ngthae(14) VP < IIfll£2I1D_hDhw\\£2 < Ilfll£211\7Dhwll£2 Tu (29),(30)ta sur ra a2Dhw 2 82Dhw 2 82Dhw 2 2C llfl1 2 a 2 + a a + a 2 < 2 £2.X £2 X Y £2 X £2 - m Do f E £2ehotru'de,Den a2D 2 a2D 2 a2D 2 hW hW hW <a 2 + aa + a2 - Const.,x £2 x Y £2 Y £2 sur ra a2W a2w a2w Dh~ ,Dh ,Dh- a 2ax £2 axay £2 y 11£2 bi eh~n.Theom~nhdS 1,WE H3. Baygio,thayu =D-h(D-k(Dk(DhW)))E V trongphu'dngtrlnh(27). Cupg ap d\lngb6 dS 1, phu'dngtrlnhnh~ndu'<;1eco thS vie'tgQnlc;tila < V2CllfllL' ( /J2w 2 + /J2DhW 2 ) 1/2ax £2 ay2 £2 < V2Cllfllu ( fj2 w2 + ff2 Dhw 2 + ff2 DhW 2 ) 1/28x £2 axay ay2 (30)£2 £2 Ch. 1 Nghi?mye'ucuabili toanI 23 VT' =VP'. Taco VT' = Dll r 82Dk(DhW)82Dk(DhW) in 8x2 8X2 dx +2D12r 82Dk(DhW)82Dk(DhW)dx in 8x8y 8x8y . D 1 82Dk(DhW)82Dk(DhW)d+ 22 8 2 8 2 Xn y Y > ( 82Dk(DhW) 2 + 82Dk(DhW) 112 m 8x2 £2 8x8y liP 82Dk(DhW) 2 )+II 8y2 £2' (31) va VP' < 11f11£21ID_h(D_k(Dk(Dhw)))11£2 < IlfllvllV'D_k(Dk(Dhw))11£2 < V2llfllv [ 82D~(~W) 2 + 82D..(~w) ,,2x V 8x8y 11£2 2 ] 1/2 82Dk(DhW) . (32) +II 8y2 £2 Tli (31), (32), ta sur fa 82Dk(DhW) 2 82Dk(DhW) 2 82Dk(DhW) 2 ~ llfI12. 8 2 + 8 8 + 8 2 <2 £2,X £2 X Y £2 Y £2 m Ch. 1Nghi?mytucilahili loanI 24 nghlaIa 82Dk(DhW) 8x2 II£2 ' 82Dk(DhW) 8x8y 11£2' 82Dk(DhW) 8y2 II£2 bi ch~n.Theam~nhd~1,WE H4([2). B. TntiJnghtJp[2=R~={(x,y)E R2,Y>O} ChQnh =(h,O)E R2,taco[2+h=[2.Lffyu=D-h(DhW) E V, thay u vaa(27),danhgia tu'dngtl]'nhu'tru'ongh<jp[2=R2 ta du'<jcke'tqua 83w 83w 83w 8x3' 8x8y2' 8x28y E L2. Baygio,voi h = (h,O),k = (k,O), Iffyu =D_h(D_k(DhW))) E V, danhgia tu'dngtv nhu'tru'ongh<jp[2=R2 ta ding co 84W 84w 84w 2 -, 2 2' 8 3 EL.8X4 8x 8y x 8y Nh " d~ kK h" J:: h" . h 84w 83w L2 v '" coo bK u'VC;ly,e et t uc, ta canc ling mIll 8y4' 8y3 E . 01U E c at ky, tIT(27),saumQts6bie'nd6i taco J 84w 1 1 84W J 84w D22 _8 4udx = fudx - Du 8 4udx - 2D12 8 28 2udx., 0 y 0 0 x OX y Ch. 1Nghifmye'ucaahili toanI 25 Do d6 .k84w ( 84w 84w ) _ 4udx < C _8 4 + 8 282 + IIIIIL2 IIullL2n Y X £2 X Y L2 < C/llull£2. (33) f)~t ( 84w Tu =in 8y4udx. R5rang,T Ia anhx~tuye'ntinhlientt;letuC~(S1)vaoR (do(33)).Theo djnh 19Riez, ta tlm dlt<;1cr E L2 sao cho r =~y~'nghlaJa ~:~E L2, f){j' ki{j'm Ira ~:~E L2, ta vie't (27) dUdi dl).ng { ( 8290 D128290 ) { I Dl1 in 8x2+ Dl18y2 udx = inI udx VuE V, (34) { ( 8W8S D128w 8S ) { 1 in 8x8x +Dl1 8y8y dx = in 90sdx"IsE Ho, (35) trongd6 82w D1282w E £2,- -+ 8 290 - 8x2 Dl1 Y I I - I ( D12 D ) 84w L2- + - - 22 - E . Dl1 8y4 Ch. 1Nghi?myeuGilahili loanI 26 DO"ivoi bai loanbie'nphan(35),sii'd1;lngke'tquav6 tinhchinhquy cualoi giai ye'u(xemdinh19IX.25, [4], IT. 325),ta du'Qcw E HI. Con voi bai loan(34),ne'udu'Qcvie'thilidu'oid<;tng D 1 ( 8908U DI28908U ) d - 1 f ' d \.../ V.11 --+--- U x- u x vuE. D 8x 8x Dn 8y 8y D 83W Thl (dinh 19IX.25) go E HI, suy ra -8 3 E £2.. Y Tom l<;ti,W E H4. C. TntiJnghf/Pflingquat:D fa t(ipmdbfftTry Cl. Danhgia hentrang(danhgia trent(ipQ cc D) cO"dinht~pmaQ cc D, la'ymQt~pmaQ' saochoQ cc Q'cc D. X6t hamch~tc1;lt~E Cgo(R2),0 <~<1 saocho { ~-1 ~=O trenQ trenR2- Q' GQiv lamQthamba'tkythuQclopV(Q')la kh6nggianV voiDdu'Qcthay baiQ' . Lucdo~vE V(Q')lahamnoirQngcuav rabellngoaiQ'vab~ng 0 bellngoaiQ'. B~ngcacph6ptinhdongiantaco ( 82(~w)82vdx = { { 82w82(~v)+28~ [ 8w82v- 82W8v ]}Q' 8X2 8x2 }Q' 8x2 8x2 8x 8x 8x2 8x28x 82 ( 82~ ) 82w82~ + [8x2 w8x2 - 8x28x2]v}dx (36) Ch. 1Nghi~mytu cuahili roanI 27 r 82(~w)82vdx - r { 82w82(~v)+8~ [ 8w 82v - 82w8v ] } dx JQ' 8x8y 8x8y JQ' 8x8y8x8y 8x 8y8x8y 8x8y8y r { 8~ [ 8w 82v 82w8v ]+JQ' 8y 8x 8x8y - 8x8y8x 82 82~ 82w 82~ }+ [8x8y(w8x8y)- 8x8y8x8y]v dx(37) r 82(~w)82vdx = r {82W82(~v)+28~[8w82v- 82W8v]JQ' 8y2 8y2 JQ' 8y2 8y2 8y 8y 8y2 8y28y 82 82~ 82w82~ + [8y2(w8y2)- 8y28y2]v}dx (38) Bay gid, ta giai b~Liloan(27)trongt~pQt. Ta ki~mchungduQcding~w thoab~Liloanbitn phansau r " r [ 8w8~82v 8w 8 ( 8~8V ) ]a(~w,v) = JQ,f vdx+2DnJQ, 8x8x8x2+8x8x 8x8x dx 2D r [ ( 8W8~ 8w8~ ) 82v 8w 8 ( 8~8V )+ 12JQ, 8y8x +8x8y 8x8y+ 8y8x 8x8y +8w~( 8~8V ) ] dx 8x8y 8y8x r [ 8w8~82v 8w 8 ( 8~8V ) ]+2D22JQ, 8y8y8y2+ 8y8y 8y8y dx, (39) Ch. 1Nghi~mytu cilahili roanI 28 voi ffiQiv E V (Q'),trongd6I" E £2(Q')va " fj2 fJ2~ fJ2wfJ2~ I = I~+Dn [fJx2(wfJx2)- fJx2fJx2] fJ2 fJ2~ fJ2w fJ2~ +2D12 [ (W ) - ]fJxfJy fJx8y fJxfJyfJxfJy fJ2 fJ2~ fJ2wfJ2~ ]+D22[fJy2(wfJy2) - fJy28y2 . ? d daytadasad\lngcacc6ngth1i'c(36)- (38). Voi V1 =Dh(~W)E V(Q')vaIhlkhabe.Lffyv =D-hV1thayv vao (39).Dungb6 dS 1,phu'dngtrlnhnh~ndu'Qcvie'tgQnl(;lila VT =VP. DanhgiaVP. Do~E cgo(R2)va~bi ch~nDen IVP! < c {llrIUID-hV1L,+11&~~wt,(II~:;t, +II:~~~IU +118~wt(II:;~IIL'+11~:2t,)} < c {1If"IIL'IIVV111L'+Ilv~:IIL'(II~:;IlL'+II:~~~IU +llv~;lljll:;~~IIL'+11~:2t,)} 1/2 ( fJ2V1 11 2 II fJ2V1 11 2 Il fJ2V1 11 2 )< C1 IIV'v111i2+II fJx2 £2 + 2 fJxfJy L2+ fJy2 L2 ( 3 fJ2v 2 82v 2 3 fJ2v 2 ) 1/2 < C1 IIv111i2+211 fJx2111L2+21IfJxfJ~IIL2 +211 fJy2111£2 < 2C1I1v11Iv(Q/)' Ch. 1Nghi?mye'ucuabili toanI 29 vdi ( 8 2 8 2 ) 1/2 C1 = Ilf"lli2+211~8: 11£2+211~8; 11£2 . Ba'td~ngthuc cu6i cung thu du'Qcnha (14). Vi f" E £2,W E H2 DenCO h~ngsO'C2 >0 saocho v p < C21IDh(~w)IIV(Q')' Danhgia VT. Do (28) (40) VT > m (11 82Dh(~w) 11 2 + 11 82Dh(~W) 11 2 + 11 82Dh(~w) 11 2 )8x2 £2 8x8y £2 8y2 £2 > C31IDh(~w)II~(QI)' (41) Tli (40),(41)va d~ngthucVT = VP , ta suyra IIDh(~w)II~(QI)<CIIDh(~w)llv(Q')' nghlala IIDh(~w)llv(Q/)bi ch~n,do Q cc Q' Den IIDh(~w)IIV(Q)< IIDh(~w)llv(Q') hay IIDh(~w)llv(Q)bi ch~n. Ke'tquala li D 82(~w) II li D 82(~w) II li D 82(~w) IIh 8x2 £2(Q)' h 8x8y £2(Q)' h 8y2 £2(Q) bi ch~n.Ap d\lngb6d~1taco~wE H3(Q) .Vanhu'v~yw E H3(Q). Chungminhtu'dngtvnhu'trenvdiv =D_k(D_hVl),VI=Dh(Dk(~w)) taclingchungminhdu'Qc 11D.(Dk8~~~)t"11D.(Dk8;~~:)t2'IIDh(Dk82~~~)t2 Ch. 1Nghi?mytucuabili loanI 30 bi ch~n.va nhu'v~ytheob6 d€ 1 tading du'QcWE H4(Q). C2. Ddnhgid iJ fanc(incuabien Tru'oche'ttaxettru'onghQpdongianSl Ia lllla quacffudonvi Sl =B(0,1)nR~ D~tQ =B(O,1/2)nR~ tachQnhamch~tC\lt~E C;:O(R2),o<~<1 sao cho { ~=1 ~=O trenB(O,1/2) trenR2- B(O,1) Khi do~- 1trenQ va ~tri~tlieu tc;tigffnbiencuaSl. Vi W=odQctheo du'ongth~ngy=° va~=° gffnbiencuaSlnen~wE V(Sl). Voi VI E V(Sl),h >okhabe,Iffyh = (h,O),thayv =D-hVlva~w vao(27),dungb6 d€ 1 taco a(Dt.(~w),VI) = 1f" D-hVldx 2 (D 1 [ aWa~a2D-hVl aw a ( a~aD-hVl )] d+ 11 -- +-- - x n axax ax2 8x8x 8x 8x D 1{ 8w [ 8~82D_hVl 8 ( 8~8D_hVl )]+ 12 - - +- -n 8x ay 8x8y 8y 8y 8x 8w [ a~82D_hVl 8 ( 8~8D_hVl )] }d+-- +-- x8y ax 8x8y 8x 8x 8y +D22{ [ 8w8~82D-hVl +8w~(8~ 8D_hVl) ] dX )(42)in 8y8y 8y2 8y8y 8y 8y Danhgiatu'ongt\1'nhu'trongC1taco bfftd~ngthlic : IIDh(~w)IIV(Q)<C Ch. 1Nghi?mytu cuabili roanI 31 Voi vI E V(~), h,k > 0 khanho,d~th = (h,O),k= (k,O)thayV = D_k(D_hVI) sau do thay VI = Dh(Dk(~w))ta cling chungminh du'Qc IIDhDk(~w)llv(Q)bi ch~n. Chungminhtu'dngtl!nhu'trongtru'onghQpB. (tru'onghQpR~)tacling co du'Qcke'tquaw E H4(Q). Baygiotaxettru'onghQp~t6ngquatcobieno~ trdnvathuOclopC2. La'yba'tky (xo,Yo)E o~. Do b6 d~phanho~chddnvi, tant~imOtHinc~n U cua(xo,Yo)vamOtsonganhtuhlnhtrollddnvi Q ={(x,y) : x2+y2 <I} leDU H : Q ---+ U, J =H-I : U ---+ Q saocho H E C2(Q), J E C2(U) va H(Q+)=Un~, H(Qo)=Unf. Ky hi~u Qo= {(x,y)E Q,y =a},Q+= {(x,y)E Q,y >a}. Tavie't(YI,Y2)= J(x, y),(x,y)=H(yl,Y2). ChQns >0khabesaochonii'ahlnhtrollU' =B(O,s)n{y> O}chua trongQ+. D~tQ' = B(O,~)n {y>a}. La'yhamch~tc\lt~E Cr:(R2),0 < ~<1 saocho ~-g trenB(O,~) trenR2- B(O,s) Khi do~- 1trenQ' va~tri~tlieu t~igffnbiencuaU'. Cu6i clingtadinh nghla w'=w 0 H, W'= (~w)0 H, v'=v0 H, 9 =iff 0 H. Ta cow'E V(U'). , VI w =0 dQctheodu'ongthiingy = ° va~= 0 gffnbienU' Den ~wE V(U'). Ky hi~u .._OYiOYj b.._OYiOYj .._OYiOYj aI] - ox oX' I] - ox oy' cI] - oY oY . Ch.1Nghifmye'ucaahili roanI 32 Vdi ffiQiv' E V(U'), thay~wvao(27)bi6nd6inhu'trong(42)tadu'Qc VT =VP, trongdo 2 { 1 a2w' a2v' VT = L Dll aijakZa a a IJHldx . .k Z=l U' Yi Yj Yk YzZ,J, , 1 a2w' a2v' +2D12 bijbkza a a a IJHIdxU' Yi Yj Yk Yz 1 a2w' a2v' }+D22 CijCkZa a a a IJHIdxU' Yi Yj Yk Yz (43) VP = r gv~I J H I dx Ju' 2 l aw'( a~ a2v' a a~av')+2Dll L aijakZaaa a +a(aa) IJHldx. .kZ=l U' Yi Yj Yk Yz Yj Yk YzZ,J,, 2 l aw'( a~ 82v' a a~av')+2D12L 2bjibkZa 8 a +a(aa) IJHIdx. .kZ=l U' Yi Yj Yk Yz Yj Yk YzZ,J,, 2 l aw'(a~82v' a a~av')+2D22L CijCkZaa a a +a(aa) IJHIdx(44). .kZ=l U' Yi Yi Yk Yz Yj Yk YzZ,J,, Vdi v~E V(U'), h >0 khabe, Iffyh =(h,0),v' = D_hv~thayvao(43)- Ch. 1Nghi?mye'ucuahili toanI 33 (44),dungb6 dS 1, ta du'<;1c 2 { 1 ( 82W' ) 82v~ VT = L Dll ,Db aijakl8.a. 8 8 IJHldx . .k l=l U Yz YJ Yk YlZ,J, , 1 ( 82W' ) 82v' +2D12 Db bijbkl8 8 8 81 IJHIdxU' Yi Yj Yk Yl 1 ( 82W' ) 82v~ }+D22 Db CijCkl8 8 8 8 IJHI dxU' Yi Yj Yk Yl - VT1+ VT2, trongd6 2 { 1 82DbW' 82v~ VT1 = L Dll aijakl 8 8 8 8 IJHIdx . .kl=l U' Yi Yj Yk YlZ,J, , J 82DbW' 82v~ + 2D12 bijbkl 8 8 8 8 IJHIdxU' Yi Yj Yk Yl 1 82DbW' 82v~ }+D22 CijCkl8 8 8 8 IJHIdxU' Yi Yj Yk Yl Ch. 1 Nghifm ye'ucua hili loan1 34 2 { 1 a2w' a2v~ VT2 = L Du Dh(aijakl)a .a .a a IJHI dx . .kl-1 U' Yz YJ Yk YlZ,J, ,- 1 a2w' a2v' +2D12 Dh(bijbkl)a a a a1 IJHIdxU' Yi Yj Yk Yl 1 a2w' a2v~ }+D22 Dh(CijCkl)a a a a IJHIdxU' Yi Yj Yk Yl VP = {9D-hV~IJHI dx }u' ~ {2D l aw'[ a~a2D_hv~ a ( a~aD-hv~)] IJ Id+ L..,; u aijakl- +- - H ). .kl=l u' aYi aYj aYkaYl aYj aYk aYlZ,J,, +2D;2 { 2bjibklaw' ( a~a2D_hv~+~( a~aD_hv~)) IJHI dx}u' aYi aYj aYkOYl aYj aYk aYl l aw'( a~a2D_hv~ a a~aD_hv~) I }+2D22 CijCkl-a _a 0 a +-a (-a 0 ) IJH dx .U' Yi Yi Yk Yl Yj Yk Yl ChQn v~=DhW' E V(U'). Danh gia VT1. D~t 2 a2Dhw' 2 a2Dhw' 2 a2Dhw' h = L aij a .a . ' l2= L bija .a . ' l3= L Cija .a . '. . 1 YzYJ .. 1 Yz YJ . . 1 Yz YJz,J= z,J= z,J= Ch. 1 Nghi?mye'uGilahili roanI 35 taco d~ngthuc [ EPDw' EPDw' ] 8;? 8Y18Y2 - (J T ) -1 [ h l2 ] J -l 82DhW' 82DhW' - H l l H 8Yl8Y2 8y~ 2 ! dohams6H E C2 nent6nt~iC1> 0 saocho 82DhW' 82DhW' 8y? 8Yl8Y2 82Dhw' 82Dhw' 8Yl8Y2 8y~ <C111h l2 l2 lj suy fa ~ II 82DhW' I1 2 <c1(li+2l~+l~).~ 8Yi8y. £2. . 1 )Z,)= (j dayta dungchuffncuamatf~nA =(aij) : IIAII =(L:~j=1a7j}I/2.Cu6i cungta co danhgia sau IVT11 > C2 { (li +2l~+l~)IJHIdx Jul > C3 (t II&DhW'11 2 ) .z,)=1 8yz8Yj £2 Danhgia VT2 Do caehamDh(aijakl),Dh(bijbkl~,Dh(CijCkl)bi ch~nnen { 2 82w, 2 82' IVT21 <C~Ju' ij; 8yJ)Yj k~l8Yk~~IIIJHIdx Ch. 1Nghi?mytu cilabili todnI 36 Dungbfitd~ngthucSchwarztadu'cjc ( 2 a2w' )( 2 II a2v~ II )IVT21 < C~ ij; IIOYiOYJ L' i~l OYiOYj L' ( ) 1/2 ( ) 1/2 2 a2w' 2 2 a2v' 2 < 8C~ij; IlaYiayJ2 i~,IIaYia~J2 BanhgiaVP NhA "aYi OYi a~ . 1 21, h- h' kh? " 1" A A U'~nxet -a ' -a ' -a ' 'l =, a n ling am a VI len tl;tctrenx Y Yi do do chungbi ch~n.Vi the": !V?! < c~ ( r 19D_hv;ldx+t r a~w' a2v; dX )}U' i,k,I=1}u' aYi aYkaYI 1/2 2 ) 1/2 2 aD W' 2 a2v' 2 < C?(1lglll'+811 8:, t,) ("D-hv;"l,+'~1118Yi8~jt, 1/2 2 ) 1/2 2 aw 2 &d 2 < C? (11911h+811~aYit2) (11~V;lliO+ij; IlaYia~J2 1/2 2 ) 1/2 2 a2w' 2 a2v~ 2 < c;("9111,+ij; II OYiOy)lL')(j; II OYiOy)lL' Bfitd~ngthuccu6icungcuadanhgiatrenco du'cjcdo (14) Ch. 1Nghi~mytu Gilahili loanI 37 TITbfftd~ngthuc /VT11< IVT21+ IV?I taduQc { ( ) 1~ 2 82D W' 2 2 82w' 2 ij; II aYi~Yj IlL' <C ij; II OyiaYj IlL' ( ) 1/2 } 2 82W' 2 2 82v' 2 1/2 + IIglll2+ij; IIOYiOYjt, X {ij; IIOYiO~jIIJ (45) thayv~=DhW' vao(45)taduQc ( 1/2 ( 1/2 2 82D W' 2 2 82w' 2 ,j;118Yi~Yj IlL') < c{ ,j;118Yi8yjt,) 1/2 } 2 82w' 2 +(119111,+ij; II oyJ}YjIlL') tUclillla;~~:l"i,j =1,2 bj chi).n.Vi Q' cc U' nen 82DW' 82Dw' II h II < II h II .. 1 2, 2,) = , . 8Yi8Yj £2(Q')- 8Yi8Yi £2 ~ 11 82DhW' l1 bi ch~n. suyfa 8Yi8Yi £2(Q') B ,:!d~ 1 h 82W' H 1(Q ' ) .. 1 2 0 e c ota 8 8 E ,2,)= , .Yi Yi Ch. 1Nghi~myeucilabili toanI 38 Voi vI E V(U'), Vt,l > 0 kha nho. Lffy h = (h,0),k = (k,O),V = D_k(D_hVI) sau doIffy VI = Dh(DkW) thayvao(27)chungminhtudngt1;1' nhutfongC.2 d 1, 82Dh(DkW') II .. 1 2 b " h - T ' b,A! dll:: 1 ta U<;lc 8Yi8Yi ' 1" J =, ! c ~n. U 0 e ta suy fa 84W' 84W' 84W' 7]4' 8 38 '8 28 2E L2(Q'),YI YI Y2 YI Y2 Vi tfen Q', W'=W' nen 84W' 84w' 84'- w 8Yf ' 8yr8Y2'8yi8y§ E L2(Q') suy fa 84W 84w 84w 8x4' 8x38y' 8x28y2E L2(H(Q')) Cu5i clIng ta din kitJ'rn Ira ~:~ t6nt(li vii thuQcvila L2(UnO). Vi U' c Q+ dodoH(U') c H(Q+)=un sl,dungcongthucd6ibiSntadU<;lc 1 82w82v 1 ( 82w82v 82w 82v )D22 --dx = fv - Dn-- - 2DI2 dxH(Q')8y28y2 H(Q') 8x28x2 8x8y8x8y Ch. 1Nghi?mytu cuahili todnI 39 biSn d6i tich phan ta du'Qc 1 82w82v 1 ( 84w 84w )D22 _8 2 8 2dx = f - Dn-8 4 - 2D128 28 2 vdxH(Q') Y Y H(Q') X X Y < M (lIfllL2+II ~x~IlL' +118~;y211L')IlvllL' hay (82w82v M [ II B4w ll II 84W II ]JH(Q') 8y2 8y2dx <D22 Ilfll£2 + 8X4 £2 + 8x28y2 £2 IIvll£2 (46) B~t Tv = ( 82w82v JH(Q') 8y2 8y2dx Tv tuySntinhva lienWc(do((46)» dodo theodinh19biSudi~nRiez co duynhfft9 E L2(H(Q')) saocho 1 82w82v 1Tv = --dx = TvdxH(Q')8y28y2 H(Q') 84W hay T = 8y4 E L2(H(Q')) Tu'dngtt!nhu'tfongtru'onghQpR~ ta suyfa w E H4(D} D