Luận văn Nghiệm một số bài toán uốn tấm nhiều lớp

NGHIỆM MỘT SỐ BÀI TOÁN UỐN TẤM NHIỀU LỚP KIỀU TRÍ THỊNH Trang nhan đề Mục lục Bảng ký hiệu Mở đầu Chương1: Nghiệm yếu của bài toán I. Chương2: Nghiệm yếu của bài toàn II. Kết luận Phụ lục Tài liệu tham khảo

pdf23 trang | Chia sẻ: maiphuongtl | Lượt xem: 1968 | Lượt tải: 0download
Bạn đang xem trước 20 trang tài liệu Luận văn Nghiệm một số bài toán uốn tấm nhiều lớp, để xem tài liệu hoàn chỉnh bạn click vào nút DOWNLOAD ở trên
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

Các file đính kèm theo tài liệu này:

  • pdf4.pdf
  • pdf0.pdf
  • pdf1.pdf
  • pdf2.pdf
  • pdf3.pdf
  • pdf5.pdf
  • pdf6.pdf
  • pdf7.pdf
  • pdf8.pdf