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(3.3) -vz(x,y,O)=g(x,y,v(x,y,O)), (x,Y)EIR2, trongdogthoacacdi~ukit%n(G1),(G2). CactiOOcha't(51),(52)d11c;1ccl,1th€ l<;tinh11sau (s:) vE C2(IR~)nC(IR~}Vz E C(IR~} 1 0)Jim sup Iv(x,y,z~=0, R-HO (s;) 8v 8v 8v (ii)Hm sup I x-(x,y,z)+y-(x,y'Z)+z-(x,y,Z~=O. R-HOOx'+y'+z'=R',z>Oax 8y ()z 1 16 Khi d6tac6dinhly sau Dinh ly 3.1 : Gid sa nghifm v caa bili loan (3.2), (3.3) wli g:IR2X[0,+00)--+[0,+00)la hamlient1!Cthodcactinhchat(s~),(S;).Khid6v la nghifm caaphu(/flgtrlnh richphanphi tuye'nsau (3.4) v(x,y,z)=~ If g(~,l1,v(~,l1,O))d~dl1,V(x,y,z)EIR~. 21tJR2~(x - ~Y+(y -l1Y +Z2 Ta clinggiasadinggiatribienu(x,y)=v(x,y,o)cuanghit;mv cuabai roan(3.2),(3.3)thai di€u kit;n (s;) Tichphan If g(~,l\V(~,11,0))2d~dl1t6nt~iV(x,y)E IR2. IR2~(x-~) +(y-11) Khi d6,ta dungdinhly hQitl;lbi cMn, choz --+0+trongphuongtr'inhtichphan (3.4),nhovao(S;), tathudu<;fc (3.5) v(x,y,O)=~If g(~,l1,V(~,l1,O))d~dl1,V(x,Y)EIR2. 21tIR2~(X-~Y+(y-l1Y Khid6,phuongtr'inhtichphan(3.5)du<;fcvi€t l~irheaanhamu(x,y)=v(x,y,O) nhusau (3.6) u(x,y) =A[g(~,TJ,u(~,TJ))Kx,y) =If g(~,l1,U(~,l1))d~dl1,V(x,Y)EIR2, IR2~(x-~Y +(y-l1Y trongd6A lamQtroantatuy€ntinhxacdinhbdicongthuc (3.7) A[G(~,~)Kx,y)~~If J G(~,~) ~d~,\i(x,Y)EIR'.21tIR2 (x - ~Y+(y-l1Y Nhu v~yphuongtr'inhtichphan(3.6)du<;fcthi€t l~ptttbai roanNeumann phituy€n (3.2),(3.3). D~chungminhdug phuongtr'inhtichphan(3.6)khongconghit;mduong lientl;lc,trudch€t tadn mQtb6d€ sauday 17 B6d~3.1.VaimQi(x,y)eIR2tac6: (i) Ntu 0<ex-y ~1, A[(.J~2+Y}2J(I +.J~2+Y}2tJX' y)= +00. Ntu ex- y >1, A[(.J~2+Y}2J(I +.J~2+Y}2t JX' y) h()it¥ va (ii) A[(.J~2+fl2J(I+.J~2+fl2tJx,y) > 1 - 2"'+1(ex-1-1)(1 +~x2+ y2r-,-l (iii) Ntu ex- y=2, A [(.J~2 +fl2 J(I +.J~2 +Y}2t](x, y):> ln(l ~~x'+y' )2<1 X2+y2 ChUngminhb6 d~3.1 (i) 0<ex- y~1: Sud\mgb~td~ngthuc (3.8) 1 > 1 ~(x-~Y +(y-Y}Y - .Jx2+y2+~~2+Y}2 \f~, Y},x, Y E JR. ta du<;1c (3.9)A[(.J~2+Y}2J(I+.J~2+Y}2tJx,y) 1 21t +00 rY+ldr =- fd<pf J21t 0 o(I+r)«(r+ X2+y2) 18 +00 ry+ldr = f(l+rt(r+fx2+y2) +00 ry+ldr > f =+00 - I(I+rt(r+~x2+y2) , ,rY r I. , +foo dr VI . ( )~ - khI r -++00va - =+00.(I+rt r+~x2+y2 r"-Y Ir"-Y (ii) a-y>l: Ta ki€m tra I~i A[(~~2+'I12J(1+~~2+'I12t](X,y)hQi tl,ln6u a-y>1. a)Xet t~i(x,y)=(0,0) :Ta co (3.10) A[(~~2+'112J(1+~~2+'112tJo,O) - +00rY +OO(I+rY - 1 - f( )"dr< f( )"dr- <+00.0 I+r 0 I+r a-y-I V~y,richphan (3.11) A[(~~2+'112J(I +~~2+'112tJo,O) hQitl,lkhia-y >1. b)Xet t~i(x,y):;t:(0,0): ChQnR>3~X2+y2 >0. Ta viet I~i A[(~~2+'I12J(I+~~2+'I12tJX,y)thanh t6nghairichphan: (3.12) A[(~~2+'112J(1+~~2+'112tJx,y) 19 - J (l)( ) J (2)( )= R x,y + R x,y. U)Danhgia Taco (3.13) x If d~dl1 ~(x-~f+(Y-'1f';R~(X-~Y +(y-l1Y x If d~dl1 ~~2+'12';R~~2 +112 2n R fd<pfdr 0 0 (jj) Danhgia 20 ChtiY dug (3.14)t~,Yl):~(X-~Y+(Y-YlY ~R}c t~,Yl):~~2+Yl2~R-~X2 +y2J (3.15)~(x-~Y +(y-YlY ~1~~2+Yl2_~X2+y21, yoimQi (~,Yl),(x,Y)EIR2. Taco = +J rY. rdr . R-~x2+y'(1+rt Ir-~x2 +y21 Do R >3~X2+y2 >0, taco Ir-~x2+y21=r-~x2+y2 ~R-2~X2+y2 >Jx2+l >0, yoimQi r~R-~x2+/. Dod6,tichphan +1 (r' t 'I rdr I Mih,Ivdi "-y>1.R-~x2+y'1+r r- ~x2+y2 V~yrichphan (3.17)J~)(x,y)hQit\,lyoi a-y>1. T6nghQp1~i(3.11),(3.12),(3.13)ya(3.17)tathuduQc (3.18)V(X,Y)E IR2, A[(~~2+Yl2n1+~~2+Yl2t'Jx,y) hQit\,lyoi a-y >1. Honnua,yoi a - y>1,taco (3.19) IX +00 rY rdr AUk +~'1(1+k +~'}}x,y)~ 1(1 +r)"\r+.Jx'+y') 21 :2: +J rY. rdr . ~x'+y'(I+rt(r+~x2+i) Tli ba'td~ngthucsail (3.20) r 1 r+~x2+y2 :2:2' Vr:2:-Jx2+i, tathuduQctli (3.19)dng " ] 1 +00 rYdr (3.21) A [(~~2+YJ2J(I+~~2+YJ2J(x'Y)~2 f (l+r)"~x2+y2 1 +00 ~- J ( r ) Ci 2 1-Ci 1+~x2+y2 1+r r dr +00 1 J r1-Cidr>-- 20+1 ~ 1+yx-+T 1 = 2Ci+1(ex- 1-1)(1 + ~X2+ y2r-1-1 . (iii) a - y=2, taco (3.22)A[(~~2+YJ2Hl+~~2+YJ2tJx,y) +00 rY rdr = I(I+rt2' r+p +i +OO ( r ) Y+2 dr ~!l+r 'r(r+~x2+y2r 22 Ta sadl:mgba'tding thuc r 1 -~-, \ir~l, 1+r 2 (3.23) tasuyra (3.24)A[(~~2+112Hl+~~2+112t'}x,y) ( 1 ) Y+2+00 dr ( 1 ) Y+2 1 +00 ( 1 1 J~"2 rr(r+~x2+y2)="2 ~X2+y2r ~- r+~x2+y2dr - 1 In ( r )] +00 _In(I+~x2+y2). - 2a~X2+y2 r+~x2+y2 1 2a~x2+i Dfnh ly 3.2 : Gid sa dingg thodcacgid thuye't(Gl), (G2)viYidi~uki?n 0<a -y::;2. Khi dophl1(!flgtrlnhrichphdnphi tuye'n(3.1)khongconghi?m dl1(!flglien t{lc. Chungminhdfnhly 3.2: B~ngphuongphilpphanchung,ta gia sar~ngphuongtrinhtichphanphi tuye'n (3.1)conghi~mlien tl,lCduongu =u(x,y). Gia sarhg t6nt~i(xo,Yo)E IR2sao chou(xo,Yo)>O.Dou lientl,lC,khido,t6nt~iro>0saocho: (3.25)u(x,y»!u(xo,yo)=mo,2 \i(x, Y)E Bra(xo,Yo)= {(x,y): (x - xO)2+(y - YO)2<r5}. Tasuyratu(G2),(3.6),(3.25)vatinhdondi~ucualoantaA, r~ng (3.26) u(x,y) =A[g(~,11,u(~,11))Kx,y) ~A[ M(~~2+112)Yua(~,11)}X,y) ~ M(maY' If (~~2+112J 2 d~d11, 2n ~(X- ~Y+(y-11) \i(X,Y)E IR2. 23 sadl,mgbfftdingthucsau (3.27)~(x-~Y+(Y-11Y~~X2+y2+~~2+112 ~(1+~X2+y2Xl+~~2+112) ~(1+~X2+y2Xl+~x~+y~+~(~-xoY +(11- JoY ) ~(1 +~x2+y2~1+~x6 +Y6 +r~), \i (x,y) E IR 2, \i (~,Y])E B'0(xo,Yo)' ta thudu'Qc (328) M(mo)" If (~~2+Y]2J d~dY] . 2n Bo(XO'Yo)~(x-~Y+(y-Y]Y M(moY' Z 2n If(~~2+Y]2)Yd~dY] (1+~x2 + y2Xl+~x~+ y~+r02)Bo(XO,Yo) M(mo)" ~ (I +~x'+d12:~x'+y' + ,)'jdq>'k'd'0 0 ro 0 0 M(mo Y' rt2 1 =(y+2)(I+~x~+y~+r02r1+~x2+i . Ta suyratlt(3.26),(3.28)dug (3.29)u(x,y)z( ~m] r= u](x,y), \ix, Y E IR,1+ X2+y2 vdi M(m )"rY+2 m]=(y+2)(1+J x~:y~+r~)' Ta xetcactru'onghQpkhacnhaucuaa - y. Truong hQ'p1 : 0<a -y ~1. Ta thudu'Qctlt(G2),(3.6),(3.29)vatinhddndi<$ucuatoantaA r~ng (3.30) u(x,y) =A[g(~,Y],u(~,Y]))Kx,y) Z A[ M(~~2+Y]2)Yua(~,Y])Jx, y) 24 ~A[ M(~~2+112)yU~(~,l1)}X,y) =Mm~A[(~~2+112HI +~~2+112t }X' y) =+00. Dob6d63.1,(i).Dayladi6uvoIy. Truong hqp2 : I <ex- y<2. Ap d\lllgb6d63.1,(ii) tathudu'<;1ctu (Gz),(3.6)vaunhdondi~ucuatmintuA, ding (3.31) u(x,y) =A[g(~,11,u(~,l1))Kx,y) ~A[ M(~~2+112)yua(~,l1)JX' y) ~A[ M(~~2+112}U~(~'l1)}X,y) =Mm~A[(~~2+112HI+~~2 +112t}x,y) a 1Milll. ) a-,-I ;:::2(x+1(ex-1-1) (1+~x2 +y2 =m2(1+~x2+y2 rq2 =U2(X,y), trongdo M a Y -Iilll . q =ex- ., 2 (3.32)ill2 =2(x+1(ex-1-1) B~ngquyn<;lptagiasur~ng (3.33)u(X,Y)~Uk-l(x,y)=mk-JI+~x2+y2jqk-l, 'Ii(x,Y)EIR2. Ne'uexqk-lY>I, khi do,sud\lllgb6d63.1,(ii) tathudu'<;1ctu(Gz),(3.6),(3.33) r~ng (3.34) u(x,y) =A[g(~,11,u(~,l1))Kx,y) ~A[M(~~2+112}ua(~'l1)}X,y) 25 ~A[ M(~~2+1121U~-I(~,l1)JX' y) =Mm~-IA[(~~2+1121(1+~~2+112tqk_1JX' y) > Mmk-l . 1 - 2"'Qk-1+1(exQk-l- "'1-1) ( ~2 2) ("'Qk-1-"f-l) 1+ x +y =Uk(x,y)=mk(I+~x2+y2 rqk, trongdo cac day sf) {qk}'{mk}du'<;Icxac dinhb~ngGongthuGquyn<;tp (3.35) qk=aqk-I- Y-1, k =2,3,...; ql =1, Mmk-l k =2,3,..., (3.36)mk= 2"'Qk-1+1qk Tli (3.35), (3.36)ta thudu'<;Ic 1 (2 ) k-l k-l 1 M '"- - ex ex ex - mk-l (3.37) qk = - "'I , mk = +1' k = 2,3,... ex- 1 ex- 1 2"'%-1 qk Do 1<a - Y <2, ta co th~ch<;msf)tv nhien 1<0phl,lthuQcvao a, y saocho (3.38)In (a-2Y ::;; ko <Ina2-a-2ay. 2-a+y 2-a+y Voi sf)tvnhien1<0du'cjc hQn,taco: (3.39)0<aqk - y::;;1.0 sa dl,lngb6d~3.1,(i), tathudu'<;Ictli (G2),(3.6),(3.30)r~ng (3.40)u(x,y)=A[g(~,11,u(~,l1))Kx,y) ~A[M(~~2+112U"(~'11)JX,y) ~A[ M(~~2+1121U~o(~,l1)JX'y) =Mm~oA[(~~2+1121(1+~~2+112tqkOJx, y)=+00. Di~udomallthu~nvadinhIy 3.2du'cjchungminhchotru'ongh<;lp2. Tniong hqp3: a-y =2. 26 Vdi ex-y =2, apdl:mgb6d~3.1,(iii)tathudu'<;fctu(G2),(3.6)vatinhdondi~u cualoantaA dng (3.41) u(x,y)= A[g(~,YJ,u(~,YJ))Kx,y) 2A[ M(~~2+YJ2Ju"(~'YJ)Jx,y) 2 A[ M(~~2+YJ2J u~(~,YJ)Jx, y) =Mm~A[(~~2+YJ2HI +~~2+YJ2tJx, y) 2 Mm~In(1+~x2+y2). 2".~X2+y2 Ta suytu(3.41)r~ng ! C2 Inp2 [ 1+~X2+y2 J(3.42)U(X,y)2V2(X,y)= ~x2+y2 2' 0, X2+y2 2 I, X2+y2::;1, trongd6 ( ) a 1 a m] (3.43)P2=1; C2=-Mm]=M- . 2a 2 Giasadug (3.44) ! Ck-] InPk-l ( I+~x2+y2 J 2 2 U(X,Y)2Vk-JX,y)= ~x2+y2 2' x +y 21, 0, x2+y2::;I, trongd6Pk-],Ck-lla cach~ngs6du'ong. Sadl,mg iathie't(G2)va(3.6),(3.44)tac6 (3.45) u(x,y) =A[g(~,YJ,u(~,YJ))](x,y) 2 A[ M(~~2+YJ2J u,,(~,YJ)Jx, y) 2A[ M(~~2+YJ2J v~j~,YJ)Jx, y) 27 = M f/vI~2+112Y v~)~,11)d~d11 21tIR'~(x-~Y +(Y-11Y ~ M If (vI~2+11:yv~)~,~)d~d11 21t~'+'1'~I~(x-~) +(Y-11) a InaPk_I [l+~~:+'1'J >MCk-l If ~ )d~d~- 21t ~2+r12~1( 2+112~~2+112+~X2 +y2 +00 InaPk-l ( l+r ) ~McL, f ~ 2 dr1 r r+~x2+y2) . Ta xet tru'ongh<;ipX2+y2~1 taco InaPk-J ( I+r ) InaPk-J ( I+r )+00 2 +00 2 (3.46) f ( )dr~ f ~ )dr1 r r+~x2+l ~x2+y2r r+~x2+l l ap [ 1+~x2+l J + f oo dr>n k-l . - 2 ~x2+y2r(r+~X2+y2) =InaPk-l ( I +~x2 +l J . 1 . +J [ !- 1 ] dr 2 ~X2+l ~x2+y2 r r+~x2+y2 ] +00 1+~x2+y2 1 .In r 2 ~ln""'-{ 2 }~X2+y2 r+~x2+yp.i =lnaPk_l ( I+~X2+y2 ) . In2 . 2 ~X2+y2 Voi X2+l ~1tadung(3.45),(3.46) InaPk-l ( 1+r )+00 2 (3.47)u(x,y)~MC~-lf ~ ~ )dr1 rr+ x2+l 28 ~ MC~_IIn 2 .Inapk-l ( 1+~x2+y2 J . ~X2+l 2 Tu (3.45),(3.47)tathuduQc ! C ( 1+~X2+ 2 J k InPk y x2 +y2 >1 (3.48) u(X,Y)~Vk(X,y)= ~x2+y2 2' -, 0, x2 +y2 ~1, trongdoPk,Ck la cach~ngs6duclngxacdinhbdicongthucquyn;;tp (3.49) Pk =apk-l; Ck =MC~_,In2, k =3,4,... Tu (3.43),(3.49)taco (3.50)Pk =ak-2, k-2 C. ~(MID2):"[(MID2)o"C;;f' ~(MID 2):~Hrn,M"~:::)('D2"-J . Nhovao(3.50),tavie'tI;;ti(3.48)voi X2+/ ~1nhusau (3.51)u(X,Y)~Vk(X,y) -I [ 2a-1 [ J] ak-2 - (MIn2)~! Ma(a-I)(I 2) ~ 1 1+~x2+y2 - ~ ml n a-I n .X 2 +y2 2 2 ChQn(x,y)saocho 2a-l 1 ( 1 -J 2 2 J ~mIMa(a-l)(In2)a-lIn + X2+y >1 hay (3.52)Jx2+y2>-1+2CX) -'"~ --,- } =po. 1mIMa(a-I)(1n2)a-1 Khi dotaco (3.53)u(x,y)~ Iim Vk(X,y)=+ct),~x2+y2>Po.k~+oo 29 Di~unayvo1y.Dinh1y3.2duQcchungminhchotntC1nghQp3. T6 hQpcactruC1nghQp1-3 tasuyfa d.ngdinh1y3.2duQcchungminh. ._.