Luận văn Sự không tồn tại lời giải dương của một số bài toán neumann phi tuyến trong nửa không gian trên

s~ ~ tk- teJi&t (jidtd~ ~ m#48-' i$'citt~~pMt~~~~~~ CHUONG 3 " ;:: ... . <) sTj KHONG TON T~I Lal GIAI DUONG ? ... , , CUA BAI TOAN VOl N =3 Chungtaxetbai loan(1.1),(1.2)Cl;1thSvoi n=3nhusau: (3.1) l::1u=O,(x,y,z)ER~={(x,y,Z)ER3,z>o}, (3.2) -uz(x,y,O)=g(x,y,u(x,y,O)), (x,y)E R2. voi g : R2x[0,+00)-+[0,+00)thoa di~uki~n: (G) g la ham lien tl;1C, (G2)T6n t~ihaih~ngsa M >0, a 2::0saa cho: g(x,y,u) 2::M ua, Vx,y E R, VU 2::O. Cac Hnhchftt(s,),(S2)du<;jccl;1thSl~inhusau: (S;) U E C2(R~)(1C(R~), uzEC(R~), (8;)(i) lim sup I u(x,y,z)I =0, R-+toox2+i+z2=R2,z>O (ii) . au au au hm sup I x-(x,y,z) +y-(x,y,z) +z-(x,y,z) 1=O. R-HOO x2+i+z2=R2,z>O ax 0' az Khi dotacodinh19sau: Dinh It 2: Ne'u[iJi gidi u ciLa hai loan (3.1), (3.2) vdi g : R2 x [0,+00) -+[0,+00)la ham lien tl;lc thoa cac tinh chat (s;),(8;),khido u la liJi gidiciLaphLtdngtrlnhrichphanphi tuytnsau: 12 S 'f"~ tk t~ t~ 9idtd~ ~ ffl# u' g>eUt~~ftMt~~~~~~" (3.3) u(x,y,z)=~ H g(e;,77,U(e;,ry,0))de;dry,V(X,y,z)ERl. 21rR2~(X-e;)2 +(y_ry)2 +Z2 Ta cling gia sa r~nggia tri bien u(x,y,O)cua Wi giai u cua bai loan (3.1), (3.2) thoadi~uki~n: (S;) Tichphan H g(e;,ry,u(e;,ry,O))de;dry tan tc;ziV(x,y) E R2. R2~(x- e;)2+(y - ry)2 Ta phat bi~uk€t quachinhtrongph~nnaynhu'Jau: Dinh Iv 3: Gid S~(rling g thoacae gid thilt (Gj),(G2) v6'i 0<a ~2. Khi do belitoan (3.1),(3.2)khongcolCii gidi du:ang thoa (s;),(s;),(s;). ChungminhdinhIy 3: B~ngphu'dngphapphanchung,giasar~ngbai loan (3."1),(3.2)co Wi giai du'dngu=u(x,y,z)thoa(Sn,(S;),(S;). Dung dinh 19"hQi t1,lbi ch?n, cho z ~ 0+ trongphu'dngtrlnh . tichphan(3.3),nhoVaG(s;) , tathuduQc: (3.4) u(x,y,O) =~ If g(e;,ry,u(e;,ry,O))de;dry, V(x,y) E R2. 21rR"2~(X-e;)2+(y_ry)2 Ta d~t:u(x,y,O)==u(x,y). Khi do,tavi€t 1~i(3.4)nhu' sail: (3.5) u(x,y)=A[g(e;,ry,u(e;,ry))](x,y) =~ II g(e;,ry,u(e;,ry))2 d~d1], 'v'(x,y)E R2,- 27rR2~(x - e;)2+(y -ry) 13 s~~ tk t<#tla~ d~ ~ ffl# d 1$'eUt~~ftMt~~~~~~ trongdoA la mQttoantli'tuye'ntlnhxacdinhb~ngcongthuc: (3.6) A[v(~,17)](X,'y)=2~If v(~,:)d~d17?' V(x,y)E R2.--J...!(x -J: ) .../..(1 -77) - IC v \ ':> '\J DS chungminhdinhly 3, tachicgnchungminhr~ng phudngtrlnhtichphan(3.5)khongcoWigiaidudnglientlJc. Trudehe'taegnmQts6ba'td~ngthuedanhgiasauday: B6 d~4. V6'imQi(x,y)ER2taco: (i) A[(1+~~2 +17 2 )-a ](x,y) =+00 , niu 0<a ~1, 1 (ii) Al(! +~e+~2F"](x,y) "2(a -1)( 1+~x2+~2)0-' ' niu a >1, (iii) A[(1+~~2+172)-2](X,Y)21n(1+~x2+y2) , niu a=2. 4 ~x2+y2 niu x2+y2 2 1 Chungminhb6d~4: (i)'0<a ~1: Su dlJngba'td~ngthuesailday (3.7) 1 1> ~(x-~)2+(y-17)2- ~x2+y2 +V~2+172 Vx,Y,~,17E R, va saudod6ibiC'ns6quat9adQeve,tathuduQe 14 S'f"~ t~ t~ to:i9-idid~ ~ m#<14' ~4it~ ~~t~~tUkt~~~ (3.8) A[(1+~;2 +172)-a](X,y) 2~ Sf d;d77 . 21l:R2 (1 +~;2 +772)a ( ~;2 +172+~x2 +y2 ) +oo f rdr2 - +00 . 0 (1+r) a ( r +~x2+y2 ) (ii) 1<a <2: Tu'dngtt!nhu'(3.8),taco (3.9) +00 rdr A[(1+Je+'l' )-"](x,y) " I (1+r)"(r+Jx'+y2) +00 2 f rdr ~x2+i ( 1+r ) a ( r +~X 2 + y2 ) Tli bfftd~ngthucSail (3.10) r 1 r +~x2+y2 2"2' I 2 2 \lr 2 '\jX +y , ta thudu'Qctli'(3.9) ding (3.11) +00 dr 1 f - A [(1 +~I;' +~2)-a](x,y) "2 Jx'+i ( 1+r)" - 2(a -1)( l'I--~X2+)/ )a-I . (iii) a =2: Tu'dngtlfnhu'(3.8),taco vdi x2+y221 (3.12) +00 rdr A[(1+~1;2+rt' )-2](x,y) " I (1+r),<r+~x2+y2) 15 Sf!' ~ tiH t~ &i tJidi d~ ed4~ 48: i$'.ut~~pMt~~~~~~ +00 r rdr> - - t(1 +r) 2( r +~X2+y2) . . Sadl;lngbfftd~ngthuc (3.13) r 1->- (1+r)2-4r' Vr ~1, tasuyfa (3.14) 1+00 dr A [(I +~l;2+~2 )-2](x, y) ;"4 ! r( r +P +y2 ) 1 +00 1 1 )drj (-- ~ 2~ 2 2 r r + x2+y4 x +y I = +00 1 r = xIn( ) 4~x2+y2 r +~x2+y2 - In(l+~x2+y2 ) - 4~x2+y2 B6d~4du'ychungminh. Baygio,dStie'pWcchungminh,tagiltsading t6nt~i (xo,Yo)ER2saocho u(xo,Yo)>O.Do u lien tl;lc,khi d6 t6nt~i ro>0 saocho (3.15) 1 u(x,y)>-u(xo,Yo) ==mo,2 V(x,y)E Bro(xo,Yo)={(x,y):(x-xO)2 +(y- YO)2<ri}. Ta suytu (G2),(3.5),(3.15)va Hnhddndi~ucuatminta A , , . rang (3.16) u(x,y)=A[g(~,7],u(~,7]))](x,y)~A[M ua(~,7])](x,y) 16 Srf"~ t~ t~ tiJ:i~ d~ ~ m#48-' i$'citt~ ~ pMt&Pf~~ nem~ ~ ~ =M H ua(~,ry)d~dry 2ff R2~(X - ~)2+(y - ry)2 >M(mo)a H d~dry , \;I(x,y)EOR2. - 2ff BI!)(XO'Yo)~(x-~)2 +(y_ry)2 sa d\lngbfftd£ngthucsailday (3 1'7\ IC _' ):,2. (y '7 ,2 / ~x2 . -:-i. ~ ..2. I) -v x-,=,) T -) :::,\j x- + Y T \j <;-- + 'I s (1+~x2 +y 2 )( 1+~~2 + 772) s(1+~X2+y2)(1+~x5+Y5 +~(~-xO)2+(ry_YO)2) - S (1+~x2 +y2 )(1+~x5+Y5 +ro2) , \;Ix,yE R, \;I(~,ry)E Bro(xo,Yo). ta thu du<;5c (3.18) M(mo)a H d~d77 2ff BI!)(XO'Yo)~(x-~)2+(y-77)2 M(mo)a ;;:: 2ff ffd~dry ~ 2 2 ~ 2 ., ')(1+ x +y )(1+ Xo+Yo+rO)B'1j(xO,YO) M(mo)a--- > 2ff ff r2 - (1+~x2+y2)(1+~x5+Y5 +ro2) 0 \7 Stf"~7 ti.~t~ tG:t~ d~ c.d4ffl# 44' g'c:Ut~~pMt~~~~9i4H~ M(mo)a 2 1fro 1= 21f X . (1+~x~+y~+ro2) (1+~X2+y2) Ta suytli' (3.16),(3.18)ding (3.19) m) u(x,y) > ~ ;: U)(x,y)1+ x2 +y2 M a 2 O' mOrO VIm) =2(1+~x5+Y5 +rt) . Vx,Y E R , Ta xetcaetrttongh<Jpkhacnhaucua a. Tru<fngh(jp 1: 0<a ~1. Ta thudtt<Jctli'(G2),(3.5),(3.19)va tinhddndi~ucua loan ii'iA, r~ng (3.20) U(x,y)=A[g(~,1],u(~,1]))](x,y)~A[M ua(~,1])](x,y) ~A[M ui (~,7])](X,y) =M miA[(1+~~2+1]2)-a](x,y) =+00, dob6d~4,(i). Day la di~uVO19. Truong h(lp 2: 1<a -::2. Ap dt,mgb6d~4, (ii), tathudtf<JCtli'(G2),(3.5),va ttnhddndi~ucuatmintll'A , r~ng (3.21) u(x,y)=A[g(~,1],u(~,1]))](x,y)~A[M ua(~,1])](x,y) 18 S¥ ~ t~ t~ tia tJidi d~ ~ m# 44' g'.ut~~ft4tt~~~~~~ ~ A[M uf (~,1J)](X,Y) =M mfA[ (1+)~2 +1J2)-a ](X,y) Mma > I - 2(a-1)(1+~X2+y2 )a-I . (2 2 -'12 =m2(1+-Vx +y ) =U2(X,y), trongdo (3.22) a Mml , m2=2(a -1) q2=a -1 . Bangguyn~ptagiltsarang (3.23) u(x,y)~uk-l(x,y)=mk-I(1+Jx2+y2 )-Qk-I, Vx,YER. N€u aqk-I>1, khi do,sadl;lngb6d64, (ii), tathudu'Qc tu (G2),(3.5),(3.23)ding (3.24) u(x,y)=A[g(~,1J,u(~,1J))](x,y)~A[M ua(~,1J)](x,y) ~A[M Uk-l(~,1J)](x,y) =Mmk-IA[(1+-!f+1J2 )-aQk-1](x,y) > M mk-l - 2(aqk-1-1)(1+~x2+y2 )aqk-I-I (2 2 -qk =mk(1+-Vx +Y ) =Uk(X'Y)' 19 s~~ tik t~ tia 9Ut d~ uU m#44-' g'4i t~ ~ {14tt~ ~ ~ ~ 9i4# ~ trong do cacdays6 {qd,{mk}duQcxacdinhbC5icongthucqui n~p: (3.25) qk =aqk-I -] , k=2,3,...; ql =1, (3.26) mk=M mk-I 2qk ' k =2,3,... Tli (3.25),(3.26)ta thu du'<;ic (3.27) qk=I-(2-a)ak-1a-I' a Mmk-I , mk= 2qk Vk =2,3,.... Do 1<a <2, tacoth€ chQnduQcs6tlj nhien ko,phV thuQcvao a, saocho: (328) -In(2 - a) . In(2- a). ~ko<1-Ina Ina . Vdi s6tlfnhien koduQcchQn,taco (3.29) O<aqko~1. Sa dvngb6d€ 4, (i), tathuduQctu (G2),(3.5),(3.24), (3.29), r~ng (3.30) u(x,y)=A[g(;,7],u(;,7]))](x,y)~A[M ua(;,7])](x,y) ~A[Mu~ (~,1J)](x,y) =M 111roA[ (1+~C;2+7]2faqko ](x,y) =+00. Dinh ly 3duQcchung'minhchotruonghQp2. 20 S'f"~ tk- t'!i &i 9idi d~ cda.ffl# M.' Z$'eitto<i#~ p4tt~ Wu«J~ ~ 9ia4 tUH, Truong hc;ip3: a =2. Voi a =2,apdvngb6d~4,(iii),tathuduQctu (G2)' (3.5),vatinhdondi~ucuatoanta A , r~ng (3.31) u(x,y)=A[g(~,17,1l(~,1]»](x,y)~A[M u2(~,1])](x,y) ~A[M uf (~,'7)](x,y) =MmfA[(1+~~2+'72)-2](x,y) ~M mt lne1+~x2+y2 ) 4 ~H2. 2 ..\, -r-y Ta suyratu (3.31)r~ng (3.32) ( » V2(X'Y) 2 J "2 2 1 U x,y - ( 1+~X2 +y) , X2 +y ~ , C2 In( 2 Jx'+y' x'+y',;;J.0, = trongd6 (3.33) P2=1, C ' 1M. 22 =- ml 4 Giasar~ng: (3.34) U(X,y)~vk-I(x,y) ( ~ J "k_' C 1+ X2+y2 k-I In( ), X2+/ ~1, ~X2+y2 2 0 , X2+y2 S 1. = 21 s¥~ t~ t~ t~ ~ d~ c:-d4m# d g>.utMH~p4tt~~~~~~ trongdo Pk-I,Ck-1ladieh~ngs6duong. Sad\,mgiathie't(G2)va (3.5),(3.34),taco: (3.35) u(x,y)=A[g(~,17,u(~,17»](X,Y);:::A[M u2(~,77)](x,y) ;:::A[MvLI(~,17)](x,y) 2 . =M H vk-I(~,17)d~d17 2" R2~(x- ~)2+(y -17)2 ;:::M If 2" .;2+r/2~1 vi-I (~,17)d~d17 ~(x - ~)2+(y - 77)2 [ ~ 2 2 J 2Pk_l In(l+ ~2+17) d~d17 (~2+172)~(x-~)2+(y-17)2 >Mcf-I Sf - 2" .;2+1J2 1 McLI If> 2 2- " .;2+1/~I ( ~ 2 2 J 2Pk_1 In(l+ ~2+17) d~d17 (~2+772)(~~2+772+~x2+y2 ) . +<X.J ( In(1+r ) 2Pk-1 ;:::MC2 f 2) dr k-I I r(r +~x2+y2) Ta xettruonghQpx2+y2;:::1,taco (3.36) ( 1+r ) 2f1k-1 ( 1+r ) 2f1k-1 +CI)In( --2- ) dr +0.0 In(2-- ) dr f > f 1 r(r+~x2+y2) - ~x2+/ r(r+~x2+y2) 22 S 'f' ~ t~ t~ &t ~ dU?1~ ~ ffl# #' . goat~ ~ ftM t~ WuuJI1da~ ~ tUn J 2Pk-1 +00 dr 1+~x2 +y2 . 2 2 ;'[lll( 2 ) N~y"(r+P +y ) J 2Pk-1 ~ 2 2 11+ x +y - =(In( 2 ). ,J;2+ y' +00 J (!- 1 ~x2+i r r +~x2+y2 )dr 1+00 J 2Pk_1 r 1+~x2+y2 1 xin . 2 2 =(In( 2 ) ~X2 +y2 r+p +y I)x'+/ ( } 2Pk-l 1+ ~ x 2 + y 2 ) In2 = I~ . 2 ~x2+y2 Voi x2+y2 ~1,ta suy tli (3.35), (3.36) ding (3.37) ( 1 1+r ) 2Pk-l u(x,y) ~MC2 +J oo n(2) dr k-( . 1 r(r +~X2+y2) 2 ( ~2 2 J 2Pk-1 >MCk-I In2 1 1+ x +y -~2 2 n( 2 ).'x +y Tli (3.35),(3.37), tathudu'Qc (3.38) u(x,y) ~Vk(x,y) [ ~ J Pk C 1+ x2 + 2 k In( y), X2+y2~1, ~X2 +y2 2 0, x2+/s1. = trongd6 Pk,Cklacachhngsf)du'dngxacdinhbdicaccong thlicquin~p. 23 s«"~ tk t4i tlTt..,...:<..dUXdLL ~ I1e8t4-6.'. '.' ~ -_-or, . g'eUt~~put~~m:a~~~ (3.39) - . 2Pk - 2Pk-b Ck =MCk-lln2, k =3,4,.... Tli (3.33),(3.39)taco (3.40) Pk =2k-2 , C' - 1 M 2k-l-l ( 1 ~l 2) 2k-1 k - - -m\-v1l1L.ln2 2 Nhd vao(3.40)tavi~tl~i(3.38)vdi x2+y2~1nhu'sau . (3.41) u(x,y) ~Vk(x,y) - 1 [ 1 2 2 2'-2 M~x2+y21n2 4M m;ln21n(I+~X +y' J2 ) . 1 1+Jx2 +y2 ChQnx,y saocho 4.M2mfln21n( 2 ) >1 hay (3.42) I 2 2 4vx +y >-1 +2exp( 2 2 ) ==Po. M m(ln2 Khi dotaco (3.43) u(x,y)~ Jim vk(x,y)=+co, k -HOO / 2 2 '\j x +y >Po. Di~unayvo 1;'.Dinh1;'3 du'Qchungminhchotru'dng hQp3. T6 hQpcactru'dnghQp1-3chungtathftyr~ngdinh1;'3 du'Qchungminh. Chu thich 1. Ke'tquacuadjnh1;'3m~nh onke'tquatrongl8]Ruy, Long,Blnh.Th~tv~y,tuonglingvoiclingphuongtrlnh(3.1), 24 s~ ~ t~ t¥ tfa9idtd~ ~ m?t~' ~citto<iH-~;It4t t~ WuetJ~ ~ ~ ~. (3.2), cacgia thitt saildayda:dungtrong [8]khongdin thitt d day . (°3) g(x,y,u)khonggidmd61Wli bientJutba,tttcla, (g(x,y,u)-g(x,y,v))(u-v)~O, \/x,YER, \/u,v~o. (04) Tichphan ff g(x,y,O)dxdy R2 1+~x2+y2 tbntgivadU(Jng. Chti thich 2. Chu yrhng voi 0<a::;2, ham g(x,y,u)=.uakhong giJi quyttduQctrongl8]VIkhongthoagiathitt (°4),trongkhid6 chungWiaa:giaiduQcvi d\lnaytranglu~nvan. Chti thich3. Truong hQp a =2, caclac gia Bunkin,Galaktionov, Kirichenko, Kurdyumov,Samarsky[1] c6 cho mQtdanhgia tuongtlf nhu (3.38)nhungphlict~pbon,mad d6VkduQccho dlfoid~ngcuamQtchu6iham. Chti thich4. Ke"tquacuadinhly 3 khongcondungvoi a >2. Ta xet phanvi d\l sail day vdi a =3 va g(x,y,u)=u3.Ta c6 g thoa cacgiathitt (°1),(°2). Khi d6hamsf) 1 u(x,y,z)= ~x2+y2+(z+1)2 la WigiJi dl((1ngcuabailoan(3.1),(3.2)vathoa(sn,(s;),(s;). 25