Luận văn Sự không tồn tại nghiệm dương của một số phương trình tích phân phi tuyến liên kết với bài toán neumann trong nửa không gian trên

Lugnvantotnghi?p Trang13 CHU<1NG3 st; KHONG TON T~I NGHI:tM DU<1NG CUA PHU<1NGTRINH TICH PHAN PHI TUYEN VOl N =2 ChungWi xetslfkh6ngt6nt~inghi<$mdu'dngcuaphu'dngtrinhtichphan phituy€nsailday(tu'dnglingvoi N =2.) (3.1) - ~ If g«!;,7],u«!;,7])) d<!;d7], V(x,y) E IR2, u(x,y) - 21ffR2 ~(x- <!;)2+(y - 7])2 voi R: IR2 x [0,+00)--+[0,+00)thoacac di~uki<$n: (G1)g la ham lien t1;lC, (G2) T6n t~ihaih~ngsf{ M >O,a2 0saocho: g(x,y,v)2Mva, Vx,YEIR,Vv20. ChungWi xetbai toan(1.1),(1.2)c1;lth~voi N =2 nhu'sau: (3.2) (3.3) L'lv=0, (x,y,z) E IR: ={(x,y,z) E IR3, z> O}, -vo(x,y,O)=g(x,y,v(x,y,O)),(x,y)E IR2, trongdo g thoacacdi~uki<$n(G1),(G2). Cac tinhchfit (81),(82)du'QC1;lth~l~inhu'sau: (8;) V E C\IR:) () CUR:), VZ E C(IR:), (i) lim sup Iv(x,y,z}1=0, 11~+OOx2+/+z2=R2,z>0 (8; ) (ii) lim S . up I xOv(x,y,z)+y Ov(x,y,z)+ZOv(x,y,z)1=0. II-HOO x2+/+z2=1I2,DO ax cy az Khidotacodinhly sail: Dinh ly 3.1: Ntu nghi~m v cua hai loan (3.2), (3.3) vcJi g : IR2 x [0,+00)--+[0,+00)la hamlientT;lCthoacactinhchat (8;), (8;). Khi d6 v fa nghi~mcuaphuangtrinhtichphanphi tuytnsau: LU{Jnwintotnghifp Trang14 (3.4) v(x,y,z)=~ If g(e;,7J,v(e;,7J,O)) de;d7J,V(x,y,z)EIR~. 2Jr R2~(x-e;)2 +(Y_7J)2 +Z2 Ta clinggiil sd'dinggiatq bien u(x,y)=v(x,y,O)cuanghic%mv cuab~liroan (3.2),(3.3)thoa di~ukic%n: (S;) Tichphan If g(e;,7J'2v(e;,7J,0))2 de;d7J t6nt1;li V(x,Y)EIR2. ,,' ~(x-e;) +(Y-7J) Khi do, dungdinh 19hQi tl;lbi ch~n,cho z -+0+trongphuongtrinhrich phan (3.3),nhovao (S;), tathudu<;1c: u(x,y,O)=~ If g(e;,7J,u(e;,7J,O))de;d7J,V(x,Y)EIR2. 2Jr R2~(x-e;)2 +(Y_7J)2 Khi do,phuongtrinhrichphan(3.5)du<;1cvi€t l1;lirheaffnham u(x,y)=v(x,y,O) (3.5) nhusail: (3.6) u(x,y) =A[g(e;,7J,u(e;,7J))](x,y) ==~ If g(e;,7J,u(e;,7J))2 de;d7J, V(x,y) E IR2. 2Jr R2~(x-e;)2 +(Y-7J) trongdoA Ia mQtroantd'tuy€ntinhxacdinhbAngcongthuc: (3.6) A[G(e;'7J)](x,y)=~If G(e;,7J)de;d7J , V(x,y)EIR2. 2Jr R2~(x-e;)2 +(Y_7J)2 Ta chi cgn chungminh rAngphuongtrinh rich phan (3.6) khong co nghic%l11 du'onglien tl;lC.Trudch€t tacgnmQtb6 d~sailday: B,:J d~ 3 1 tT" . ( ) R 2 , 0 e . . v(Jl mQl x,y E ta co: (i) A[(1+Je;2 +7J2ra](x,y) =+00,ntu O<a::;;l, (ii) A[ (1+JI;' +~2 ra ](x,y) " ~ ' ntu a> 1,2(a -1)( 1+ x2+y2 t-1 ~ 1n(1+JX2 +y2 ) A' (iii)A[(1+ e;2+7J2)-2](x,y)2: ~ ,neua=2.4 x2+y2 Chung minh. (i) 0<a::;;1: Sd'dl;lngbfftd!ng thucsailday Lu(mvantotnghi~p Trang15 (3.7) 1 1> ~(x-.;)2 +(y_7])2 - ~x2+y2 +~.;2+7]2 1 1 ~ x ~ ' Vx,y,.;,7]E JR,1+~X2 +y2 1+ .;2+7]2 vasaildodBibie'nsf)quatQadQeve,tathuducJe (3.8) A[(l+~.;2+7]2ra](x,y)=~ H(l+~.;2 +7]2rad.;d7] 2n:R, ~(x-';Y+(Y-7]Y ~~ fJ d.;d7] 2n:R2(1+~.;2+7]2)a(~.;2+7]2+~x2+y2) +00 rdr ~ f -+00. 0 (1+r) a ( r +~x2+y2 ) (ii) a>1:Tuongtvnhu(3.8),taco (3.9) +<x:J rdr A[(l+~l;'+~' )-a](x,y);" f (1+r)"(r+Jx2+y2) > +I rdr - J~2+y'(l+rt( r+~x2 +y2 ) . Tli batd£ngthuesail (3.10) r 1>- r +~x2+y2 - 2' I 2 2 Vr~vx +y , tathudu'cJetU(3.9)ding A[(l+~.;2+7]2)-a](x,y)~! 1 ~ a2 u-i( l+r) -yx-+r (3.11) 1 - 2(a -1)( 1+~x2+y2 t-I' (iii) a =2:Tu'ongtvnhu(3.8),taco (3.12) +00 rdr A[(l+~I;'+1J' F'](x,y) " f (l+r)'(r+~x'+y') Lugnvantotnghi~p Trang16 >] rdr - 1(1+r)2(r+~x2+y2)' Sa dl;lngba'td~ngthuc (3.13) r 1->- (1+r)2 - 4r' Vr ~1, tasoyfa (3.14) 1+00 dr A[(l+~~' +ry'F'](x,y) ;"4 fr(r+.,J;2+y2 ) +00 = 1 f(~- 1 )dr 4~x2+y2 I r r +~x2+y2 +00 1 r = xln( ) 4~x2+y2 r +~x2+y2 _In(1+Jx2 +y2) - 4 ~x2+y2 B6 d~3.1du'<;5cchangminh. Ta phatbi€u ketquachinhtrongph~nnaynhusau: Djnh Iy 3.2:Gidsading g thoacacgiGthitt (Gi),(G2)vai 0<as;2. Khi do phuangtrlnhtichphan(3.1)khongconghi~mduanglientlf,c. Chung minh. B~ngphuongphapphan chang,gia sa r~ngphuongtrlnh tich phan (3.1) co nghi~mduonglien tl;lc u=u(x,y). Gia sa r~ngt6n t<;li (xo,Yo)EIR2saocho u(xo,Yo)>0.Do u lientl;lc,khidot6nt0 saocho (3.15) 1 u(x,y)>-u(xo,Yo)==mo,2 Vex,y) E Brn(xo,Yo)={(x,y) :(x - xo)2+(y - YO)2<r02}. Tasuytu (G2),(3.5),(3.15)vatinhdondi~ucualoanta A, r~ng (3.16) u(x,y)=A[g(~,1],u(~,1]))](x,y)~A[M ua(~,1])](x,y) LucJ-nvantotnghifp Trang17 =M If ua(;,17)d;d17 ~M(mot If d;d17 , 27rR2~(x-;)2 +(Y-17)2 27r B'1J(XO,yo)~(X-;)2+(Y-17)2 V(x,y) E JR2. SU'dl;lngba'td£ngthucsauday (3.17) ~(x- ;)2 +(y -17)2s;~x2+y2 +~;2 +172 S;(l+~X2 +y2 )(1+~;2 +172) S;(1+~x2+y2)(l+~X5 +Y5 +~(;-xO)2 +(17-YO)2) S;(1+ ~x2 +y2 )(1 +J xg + yg + f02), Vx,y E JR, V(;,17) E Bro(xo,Yo), ta thudu'<;1c: (3.18) M(mo)a If d;d17 27r BI'[)(XO'Yo)~(x-;)2+(y-17)2 M(mo)a > 27r Ifd; d17 - ~ 2 2 ~ 2 2 2)(1+ x +y )(1+ Xo+Yo +fO BI'[)(xo,Yo) M(mo)a > 27r 7r~2 - (1+~x2+y2)(1+~X5+Y5 +fi ) 0 M(mot 2 7rfo 1 > 27r x - (1+~x5+Y5 +f02) (1+~X2+y2)' Ta suytu (3.16),(3.18)r~ng (3.19) m) u(x,y)~ ~ ==u\(x,y)1+ x2 +y2 Vx,y E JR, voi M a 2mOfO ml= ~ 2 2 2'2(1+ Xo +Yo +rO) Ta xetcaetntongh<;1pkhacnhaucua a. Tru'ongh(jp 1: 0<a S;1. Lu~nwin totnghifp Trang18 TathuduQctu (Gz),(3.5),(3.19)vatinhddndi~ucuatoanta A, r~ng (3.20) u(x,y) =A[g(q,7],u(q,7]))](x,y);:::A[M ua(q,7])](x,y) ;:::A[M uf (q,7])](x,y) =Mm~A[(I+Jqz+7]z ra](X,y) =+00, dobe)d~3.1,(i). Dayla di~uvo19. Truong hqp 2: 1<a <2. Ap dl,lngbe)d~3.1(ii), tathuduQctu (Gz),(3.5),va tinhddndi~ucuatoanta A, r~ng (3.21) u(x,y)=A[g(q,7],u(q,7]))](x,y);:::A[Mua(q,7])](x,y) ;:::A[M uf (q,7])](x,y) =M mf A[ (1+~qZ+7]z)-a](x,y) Mma> 1 - 2(a-I)(I+~xZ+yZ )a-l f z z-n =mz(1+vx +y ) ==uz(x,y), trongdo (3.22) M a mz= ml 2(a-I)' qz=a-l. Bangquyn<;tpagiasar~ng (3.23) u(x,y);:::Uk-l(x,y) ==mk-l (1 +Jxz +y2 rQk-l, V(x,y) E IR2. Ne'u aqk-l>1,khi do, sa d\lngbe)d~3.1,(ii), ta thuduQctu (Gz), (3.5),(3.23) ding (3.24) u(X,y) =A [g(q,7],u(q,7]»)](x,y);:::A[M ua(q,7])](x,y) ;:::A[M uk-l (q,7])](x,y) =Mmk-IA[(I+~qZ +7]z)-aQk-l](x,y) Lu(mvantotnghi~p Trang19 > M mk-I - 2(aqk-l -1)( 1+~x2 +y2 )aqk-J-l f 2 2 -qk =mk(I+-yx +y ) =Uk(x,y), trongdocacdaysO'{qd,{mddU<;1cxacdinhb(ji c6ngthucquin(;lp: (3.25) qk=aqk-I-I,k=2,3,...; q]=I, (3.26) M m:-1 k =2,3,...- , mk- 2qk Tu (3.25),(3.26)tathudU<;1c (3.27) (2- ) k-] M mk-I _1- a a ,mk= , Vk=2,3,.... qk - a -I 2qk Do 1<a <2, taco th~chQndU<;1csO'tv nhienko,ph\!thuQcVaGa, saDcho: (3.28) - In(2- a) 5:,ko<1- In(2- a) .Ina Ina Voi sO'tv nhien kodu<;1cchQn,taco (3.29) O<aqk 5:,1.0 Sad\!ngb6d~3.1,(i),tathudu<;1ctu (G2),(3.5),(3.24),(3.29),ding (3.30) U(x,y)=A[g(q,17,u(q,17))](x,y)~A[M ua(q,17)](x,y) ~A[M u~(q,17)](x,y) =M m:oA[(1+r+ 172raqko](x,y)=+00. Btnh 193.2 dU<;1cchung minh cho truongh<;1p2. Tru'ong hqp 3: a =2. Voi a =2, apdl;1ngb6d~3.1,(iii), tathudU<;1ctu (G2),(3.5),va t1nhdondit%u cuatoan ta A, ding (3.31) u(x,y) = A [g(q, 17,u(q,17))](x,y) ~A[M u2(q,17)](x,y) Lu{mvantotnghi~p Trang20 ~A[Muf(~,7])](x,y) =MmfA[(1+~~2+7]2)-2](x,y) ~M mf In(1+Jx2 +y2 ) 4~x2+y2 . TaSuyfatu(3.31)ding (3.32) trangdo (3.33) Giasading: (3.34) u(x,y)~v2(x,y) = C2 ~x2+y2 [ ~ ] P2 1+ x2 + 2 In( 2 y) , x2 +y2 ~1, x2 +y2 ~1.0, _ 1 C 1 2P2 -, 2 =-Mm].4 u(x,y) ~vk-l(x,y) = Ck-l 0y2 ( ~ 2 2 J Pk-l In(1+ x2 +y) , x2 +y2 ~1, x2 +y2 ~1.0, trangdo Pk-l'Ck-l la caeh~ngs6duong. SadlJnggiathie't(G2)va (3.5),(3.34),taco: (3.35) u(x,y)=A[g(~,7],u(~,7]))](x,y)~A[M u2(~,7])](x,y) ~A[M VLl (~,lJ)](x,y) 2 = M If Vk-l (~,7])d~d7] 2ff R2~(x- ~)2+(y - 7])2 ~ M If VLl (~,7])d~d7] 2ff ~2+7J2;::1~(x-~)2 +(y_7])2 Lwjn vantotnghi~p Trang21 >McLI If- 2 2 7r ;;2+1];:::1 ( ~ 2 2 J 2Pk_1 lln( 1 +{;2 +1]) d{; d1] (~2+172)~(x-~)2+(Y-17)2 >McLI If - 27r ;;2+1]2;:::1 ( ~ 2 2 ] 2Pk-I lln(l+ ';2 +'1) d.;d'l (~2+172)(~~2+172+~x2 +y2 ) +00 ( In(1+r ) 2Pk_1 ~MC2 f 2) dr k-I I r(r +~X 2 2 . +y ) Ta xettru'ongh<;1px2 +y2~1, taco (3.36) ( 1+r ) 2Pk-1 ( 1+r ) 2Pk-1 +a:J In(2 ) dr > +a:J In(2 ) dr f ~ 2 2 - f ~ 2 21 r(r + x +y) ~x2+i r(r + x +y ) ( J 2Pk_1 ' 1+~x2+y2 +a:Jf dr ~ In() 2 2 2 ~x2+ir(r+p +y ) - [ 1+~x2+y2 ) J 2PH 1 1 (! - 1 )dr - In( 2 ~x2+y2 ~x2+i r r+~x2 +y2 1+a:J ) 2Pk_1 r2 2 1 1 1+ x +y xn 2 2 =[lll( ~2 ) ~x2+y2 r+p +y l~x2+Y' [ 1+~x2+y2 )J 2Pk-I ln2 = In( . 2 ~x2+y2 Vdi x2+y2~1,tasuytu (3.35),(3.36)r~ng ( l+r ) 2Pk-l 2 +a:J In( 2 ) dr u(x,y) ~MCk-1 f ~1 r(r + x2 +y2 ) (3.37) Lu(jnvantotnghi~p Trang22 ( ~ J 2Pk_1 >McLl In2 1 1+ x2+y2 - ~ n( ).X2+y2 2 Tli (3.35),(3.37),tathuduqc (3.38) u(x,y) ~Vk(x,y) 0, ( ~ 2 2 J Pk 1+ x +y 2 2 In( 2 ), x +y ~1, x2 +y2 ~1. = ck ~x2+y2 trongdo Pk,Ck lacach~ngsf)dudngxacdinhboicaccongthucquin(;lp. (3.39) Pk =2Pk-P Ck =McLlln2, k=3,4,.... Tli (3.33),(3.39)taco k-2 1 2k-I-1.1 ~ 2k-1 Pk =2 , Ck=-M (-m]vln2) .In2 2 Nho VaG(3.40)tavie'tl(;li(3.38)voi x2+y2 ~1nhu'sau (3.40) (3.41) u(x,y) ~vk(x,y) = 1 [ 1 M~x' +y' In2 4M'ml'ln21n(I+~X'+y' ) 2k_2 2 ) . 1 1+~x2 +y2 ChQnx,y saocho 4M2m]21n21n( 2 ) >1 hay (3.42) ~X2+y2 >-1 +2exp( 2 4212) ==Po'M m] n Khi dotaco (3.43) u(x,y) ~ lim Vk(X,y)=+00,~X2+y2 >Po'k-++oo Bi~unayvo 19.Binh 193.2 duqcchungminhchotru'onghqp3. T6hqpcactHronghqp1-3chungthftyr~ngdinh193.2duqcchungminh.- LwJn wintotnghi~p Trang23 Chu thich3.1.Ke'tquacuadinhIy 3 m<;lnhhonke'tquatrong[8]Ruy,Long, Binh.Th~tv~y,tu'ongli'ngvoi cungphuongtrlnh(3.1),(3.2),cacgiathie'tsail daydil dungtrong[8]kh6ngcfinthie't(j day (G3) g(x,y,u) kh6nggiamd6i voi bie'nthli'ba,i.e., (g(x,y,u)-g(x,y,v))(u-v)~O, Vx,YEIR, Vu,v~O. (G4) Tich phan H g(x,y,O)dxdy R2 1+~x2+y2 t6nt<;liva duong. Chu thich3.2.Chuydingvoi 0<a ~2, ham g(x,y,u)=ua kh6nggiai quye't du'<;5ctrong[8]vi kh6ngthoagia thie't(G4),trongkhi do chungt6i dil giai du<;5c vi d1,lnaytronglu~nvan. Chu thich 3.3.Truongh<;5pa =2,cactacgiaBunkin,Galaktionov,Kirichenko, Kurdyumov,Samarsky[I] cochomQtdanhgiatuongtt;1'nhu(3.38)nhungphli'c t<;lphon,ma(jdo vk du<;5cchoduoid<;lngcuamQtchu6iham. Chu thich3.4.Ke'tquacuadinhIy 3.2kh6ngcondungvoi a >2.Ta xetphan vi d1,lsaildayvoi a =3 va g(x,y,u)=U3.Tacogthoacacgiathie't(Gi),(G2)va ham s6 u(x,y) ~~ 1 Iii nghi~mdlCdngcuo phlCdngtrinh tich phan (3.1).X2 +y2 +1