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LU{J.nvan tot nghi~p Trang 24 CHUONG 4 st; KHONG TON T~I NGHIEM DUONG ? "'" , CUA PHUONG TRINH TICH PHAN Val (J = N -1, N > 2 Trang phgn nay chung ta xet sv kh6ng t6n t~i nghit%mdu'dngcua phu'dng trlnh tich phan phi tuye'n sau day (4.1) U ( x ) =b f g(y,u(y)) d y "dx E IRN N N-l' , IRN Iy - xl trang do bN = 2((N-l)lUN+ltl voi lUN+1la dit%ntich cua m~t c~u ddn vi trong IRN+I, N > 2 va g: IRN xIR+ ~ IR la ham lien t\!Ccho tru'oc thoa di~u kit%n: T6n t~i cae hftng s6 a,fJ ~ 0, M > 0 sao cho (4.2) g(x,u) ~ MlxlP ua, "dxE IRN, "du~ 0, va mQt sf) di~u kit%nph\! sau do. Phudng trlnh tich phan (4.1) duQc thanh l~p tu bai loan Neumann phi tuye'n sau dayvoiN=n-l>2: TIm mQt ham v Ia nghit%mcua bai loan Neumann (4.3) (4.4) ~v=O, xEIR: ={(xl,xn):xl EIRn-l,xn >O}, - vxn(Xl ,0) = g(XI, V(XI ,0)), Xl E IRn-l, thoa cae tinh cha't: (8]) VEC2(IR:)nC(IR:), vxn EC(IR:), lim ( SUP I vex) I + R. sup ov (x) J = 0, k HOO Ixl=R,xn>O Ixl=R,xn>O fun (82) d day g: IRn-1x[0,+00)~ [0,+00)cho tru'oc thoa cac di~u kit%nsau: (G]) (G2) g la ham lien t\!e, 3a~0,3M>0: g(xl,v)~Mva, "dv~O, "dxl EIRn-l. va mQt sf) di~u kit%nph\! se d~t sau. Lu(jn van tot nghi~p Trang25 Khi do, n€u g 1a ham lien t\lCva nghi~m v bai loan (4.3), (4.4) co cac tinh cha'"t(SI)' (S2)' thi v 1anghi~mcua phudngtrinhtichphan sau day I - 2 f g(l,vel ,0))dl I n (4.5) vex ,xn) - 2 (n-2)/2 ' VeX ,xn) E IRp (n-2)OJn Rn-I (1 I I I 2 ) Y -x' +Xn trang do OJn1a di~n tich cua m~t c~u ddn vi trong IRn. Day 1a k€t qua trong ph~n thi€t l~p phudng trinh tich phan (chudng 2, dinh 1y 2.1), trang do co stf thay d6i cac ky hi~u trang cach vi€t bang cach thay (a/,an) va (xl,xn) 1~n1u'<!tbdix=(xl,xn) va Y=(/,Yn)' Ta cling gia sa rang gia tri bien V(XI,0) cua nghi~m v cua bai loan (4.3), (4.4) thoa tinh cha'"t: (s3) Tich phan f g(/, v(/ ,0))d/ /Rn-I I yl - xl In-2 t<3n t~i, VXI E IRn-l. Gia sa rang bai loan (4.3), (4.4) co nghi~m dudng v= V(XI,xn) thoa cac di~u ki~n (SI)- (S3)' Dung dinh 1'9hQi t\l bi ch~n Lebesgue, cho Xn ~ 0+ trang phu'dngtrlnh tich phan (4.5), nho vao (S3)'ta thu duQc: v(xl,0)= 2 f g(l, vel ,0))_~l , vxl EIRn-l. (n - 2)OJn /Rn-I Il - Xl In (4.6) Ta vi€t l~i phudng trinh tich phan (4.6) bang cach thay l~i cac ky hi~u n-1 = N, Xl = x, l =Y, V(XI ,0)= U(XI), i.e., (4.7) u(x) = 2 f g(y,u(y») dy (N -l)OJ ' I I N-I' '\Ix E IRN. N+l IR' y-x Khi do, ta phat bi~u k€t qua chinh trang ph~n nay nhu sau: Djnh ly 4.1. Ntu g thoa cae gia thitt (GJ, (Gz) vdi N > 2 va 0 ~ a ~ N~l' Khi do, phl1ang trinh tick phdn (4.7) khong c6.nghi~m lien t~c dl1ang. Lu(in win tot nghifp Trang 26 Ch6 thich 4.1, K€t qua nay m~nh hdn k€t qua tfong [2], [8]. Th~t v~y, vOi CY= N -1, d cling phu'dngtrlnh rich phan (4.7), cae gia thi€t sau day dii sa dt,mg trong cae bai baa [2], [8] ma trong ehu'dngnay khong e~n d€n: (G3) g(x,u) la ham khong giam d6i vdi bi€n u, i.e., (g(x,u)-g(x,v))(u-v)~O VxEIRN, Vu~O, Vv~O. (G4) Tich phan J g (1,0; ~-I t6n t~iva du'dng. 1/1' ( 1+ x ) Tru'de h€t ta e~n mQt sO'ba't d&ngthue sau day: B6 d~ 4.1. Vai mQi q ~ 0,X E IRN, fa dijt: (4.8) A[q](x):=A[(1+lylrq](x)= J(1+lylr:_,dy. lRN Iy - x I Khi an (4.9) A[q](x) = +00, ne'u q:::; 1, (4.10) A[q](x) hQifl;l va A[q](x)~ OJNN-I 111 -I' ne'u q>1. (q-I)2 (1+ x)q Chung minh b6 d~ 4.1. a) Gia sa q :::;1. Chti Y d€n ba't d&ng thue tam giae (4.11 ) Iy - xl :::;Iyl+ Ix! vdi mQi x, y E IRN , ta suy fa tu eong thue (4.8) ding A[q](x) = J (1+lyl )-:-~y [RN Iy - x I > J (1 + Iyl rq d = + J oo (1+rrq d J d:S - 1III N1Y II Nlr r' 1/' ( Y + x ) - 0 ( r + x ) - lyl=r (4.12) trong d6 J dSr la rich phan m~t tren m~t e~u, tam 0, ban kinh r trong IRN. Iyl=r Tich phan n~y ehinh la dit%nrich eua m~t tren m~t e~u Iyl=r, tue la: (4.13) J N-l dSr = r OJN' Iyl=r LucJnvan tot nghifp Trang27 Do do, ta suy tu (4.12), (4.13) ding (4.14) +00 N-} dr J A[q](x)~wN I( r:'xl)N 1(1+r)q =wN q' +00 N-I d Tich philo Jq = f rll N-I r philo ky khi q ~ 1 va hQi t1;1khi q > 1. 0 (r+ x) (1+r)q Do do, rich philo (4.15) A[q](x) philo ky khi q ~ 1. a) Gia sa q > 1. i) Xet t~i x = 0, ta co (4.16) - f (1+ Iylrq dy - + f oo(1+ rrq rl-Ndr =w + f oo~ . A[q](O)- I I N-I -wN N-I N (I+r)q m~ y 0 r, 0 / / A +00 dr A' , Do do, hch P han f hOI tu VI q > 1. 0 (1+r)q . . V~y, rich philo (4.17) A [q](0) hQi t1;1khi q > 1. ii) Xet t~i x =F0, chQn R > 31xJ> O. Ta vie't l~i A[q](x) thanh t6ng hai tich philo A[q](x)= f (1+IYI)~q_~y+ f (1+IYI)~q_~y =J~I>CX)+J~2)(X). IY-Xl$/?Iy - xl Jy-xl"/? Iy - xl (4.18) U)Banhgia J~I)(X)= f (1+lylrqdy I N 1 . IY-Xl$/? Y - xl - Ta co: (4.19) J (l) () = f (1+lylrqdy< (I II) -q f ~ Ii X N-I - sup + Y N-I IY-XI$R Iy - xl ly-xl:>R ly-xl:SRIy - xl d R N-Id = sup (1+ !ylrq f :-1 = sup (1 + !ylrq wN rN-/ IY-XI$R Izl:SRIzi ly-xl:SR 0 r = sup (1 + Iylrq wNR < +00. ly-xl:SR Lugn wln tot nghi~p Trang 28 OJ) Danhgia J~2)(X)= f (1+lyl)-qdy I N I . ly-4~1I Y - xl - Ta co: (4.20) (21 = f (1+lylrqdy < f (1+lylrqdy < f (1+lylrqdy JII (x) NI - NI - NI ly-xl~R Iy-xl - lyl~R-lxl Iy-xl - IYI~R-Ixillyl-Ixil - +00 (1 ) -'1 N-I d +00 N-I d f +r r r f r r =OJN N 1 =OJN N I - . II-Ixl Ir-Ixll - R-Ixllr-Ixll - (1+r)q Chu y rang, do R>3Ixl>O,ta colr-lxll=r-lxl:=::R-2Ixl>lxl>O, voi mQi r:=::R-Ixl. +00 N-] d D d ' ' h h A f r r h A' ~. 1 0 0, tIc p an N I 'I Q1 tl,l VOl q> . R-Ixl I r -Ixll - (1+ r) V~y, tich phan (4.21 ) J~2)(x) hQi W khi q > 1. T6 h<;5pl(;li(4.17), (4.18), (4.19) va (4.21) ta thu du<;5c (4.22) \Ix E JRN, A[q](x) hQi tl,lkhi q > 1. Hdn nua, voi q > 1, ta vie"t (4.23) +00 N-l d +00 N-I d J = f r r :=:: f r r q o(r+lxl)N-I(1+r)q Ixl(r+lxl)N-I(1+r)q +00 rN-Idr 1 +00 dr :=::J( r+r )N-I(1+r)q =2N-I J(1+r)q = 1 1 \Ix E JRN (q-l)2N-l (1+lxl)q-l . Do do b6 d~ 4.1 du<;5cchung minh Chung minh dinh ly 4.1. Bang cach thay ham g(x,u) bdi gI(x,u) = bNg(x,u) va hang s6 M trong (4.2) thay bdi bNM, ta co th~ gia sa rang bN= 1 ma khong lam m!t tinh t6ng quat. LucJnvan tot nghifp Trang 29 (4.24) trongdo (4.25) Ta vie't phuong trlnh tich phan (4.7) voi bN = 1 theo d~ng u(x) = Tu(x) = A[g(y,u(y))](x), \/x EIRN, A [w(y)](x) = J w(y) d~-I' X E IRN. iii' I y - x I Ta chung mint b~ng phan chung. Gia su u Ia nghi~m lien t\lCva duong cua (4.24). Khi do t6n t~i XoEIRN sao cho u(xo)> o. VI u lien t\lc nen t6n t~i ro > 0 sao cho: u(x»~u(xo)=L \/xEIRN, Ix-xol:::;;ro. 2 Ta suy tu gia thie't (G2),(4.24)-(4.26) r~ng (4.26) (4.27) u(x) = A[g(y,u(y))](x) ~ MA[ua(y)](x) 2::MLa J dy N-l' \/x E IRN. Iy-xol:s:ro I y - x I Su d\lng ba't d~ng thuc sau (4.28) I y - x I :::;;Iyl + Ixl :::;;(1 + Ixl)(1 + Iyl) =(1+ Ixl)(1+ Iyl- Xo + xo) :::;;(1 + Ixl)( 1+ jxoI+ Iy - Xo I ) :::;;(1+lxl)(1+lxol+ro)' \/x,YEIRN, Iy-xo I:::;;ro' ta suy tu (4.27), (4.28) dng (4.29) u(x) 2:: MLa J ~ N-l Iy-xol:s:ro I y - x I Ta vie't l~i (4.30) trong do > MLa 1 -(1+lxol+ro)N-lx(1+lx l )N-l J dy Iy-xol:s:ro = MLa 1 OJ N X NrO (l+lxol+ro)N-l (1+lxl)N-l N ' \/xEIRN. u(x) 2::u1(x) = m](1 + Ixlrq), \/x E IRN, Lugn win tot nghifp Trang30 (4.31 ) a N M L ())NrO ql = N -1, m] = N(1+lxo!+ro)N-I' Sa dl;lngffiQtl~n nii'a d&ng thuc (4.24), ta sur tITghl thi~t (G2), (4.27) r[tng (4.32) u(x) 2 MA[ua (y)](x) 2 M4[u~ (y)](x) =Mm~A[(1 + Iylraq, ](x) \::IxE IRN. Bay gid ta xet cac tru'dng hQpkhac nhau cua gia tti a. 1 O::;a::; N-1 Ta sur ra tU (4.9), (4.32) voi q = a ql = a(N -1)::; 1, dng Truong hQ'p1: (4.33) u(x) = +00 \::IxE IRN. D6 la di~u vo 19. Truong hdp 2: ~ < a <~. . N-1 N-1 Sa dl;lng (4.10) voi q = a q] = a(N -1) > 1,ta sur ra tIT(4.32) r[tng: (4.34) u(x) 2 Mm~A[(1+ Iylraq,](x) = Mm~A[a ql](x) ()) 2 Mmla N N-I(1+lx!)I-aq" \::IxEIRN. (aql -1)2 hay (4.35) U(X)2u2(x)=m2(1+lxlrq2, \::IxEIRN, trong d6 (4.36) q2 =aq ] -1 m - M()) N ma , 2 - I 2N-l . q2 Giasa dng (4.37) u(x) 2 Uk-I (X) =mk-I(1+!X!rqk-l, \::IXEIRN. N€u aqk-I > 1, khi d6 ta dung ba"td&ngthuc (4.10) voi q =aqk-I > 1, ta thu du'Qc tITgia thi€t (G2), (4.24), (4.37), r[tng (4.38) u(X) 2 M4[ua (y)](x) 2 M m:_]A[(1 + Iylraqk-' ](x) Luc7nvan tot nghi~p Trang 31 = M m:-lA[a qk-I ](x) 2 M ma ()) k-l N (aqk-I -1)2N-l (1+IXI)I-aqk-1 2mk(I+lxlrqk =Uk(X), '\IxEIRN, trong d6 cac dtiy {qk},{mk} duQCxac d~nh bdi cac cong thuc qui n~p sau: (4.39) a M())N mk-I k = 2,3,., 1 m = N I ' qk=aqH-' k 2 qk Tli (4.31), (4.39) ta thu duQc (4.40) { N - k, ntu a = 1, qk = k I I-a k-I A'" 1 N (N-l)a - - , neu -<a<-, a=t:l, I-a N-l N-l Ta suy tli (G2),(4.10) va (4.24) ding (4.41) U(x) 2 Mm: A [(1+ Iylraqk ](x), '\Ix E IRN. Nhu v~y ta chI cftn chQn ffiQts6 t1,1'nhien k saGcho: (4.42) 0 < aqk ::;1. Do (4.40), ta chQn ffiQts6 t1,1'nhien k nhusau: i) N€u a=1, tachQn k=N-1,khid6: aqk =a(N-k)=a(N-N+l)=a=1, ii) N€u ~<a<~ va a=t:1, tachQnk thoa ko :=;;k<ko+l, N-l N-l voi k() = 21n[N -(N -1)a]. 1na N Tni<tng htjp 3: a = N -1 . Ta vi€t l~i (4.20) (4.43) u(x) 2 M A[ua(y)](x) 2 Mm~A[(1+ Iylraql](x) =Mm~A[(1+lylrN](x), '\IxEIRN, M~Hkhac,voiffiQi xEIRN, IxI21,tac6. (4.44) A[(1+lylrN](x)= f (1+IYI~~IN dy RN Iy - xl Lugn van tot nghifp Trang 32 > f (1+lylrN d > + f '" (1+rrN d I dS - IIII NIY- II Nlr r Ii \ ( y + x ) - 0 ( r + x ) - lyl=r +"'(1+rrNrN-I 1\I+rrNrN-I = OJv f II dr ~ OJN f II dr . 0 (r + x )N-I I (r + x )N-I Ixl rN-Idr ~OJN [(1+r)N(r+lxj)N-I. Chu yr~ng voi mQi r sao cho 1~ r ~ Ix!ta co (4.45) ( ) N r 1, 1 1 1+ r ~ 2N va r + Ixl~ 21xJ. V~y, ta co ta (4.45) dug Ixl rN-Idr 1 1 Ixl dr !(1+ r)N ( r + Ixl)N-I ~ 2N ( 21xl)N-2 !r( r + Ixl) (4.46) 1 1 1+ Ixl N = 4N-I x Ixl N-I x In( 2)' "Ix E IR , Ix!~ 1. Ta (4.43), (4.44), (4.46) ta suy ra ding (4.47) 0, Ixl~1, u(x) ~ V2(x) = ~ ~ ( In 1+ Ixl ) PZ, Ixl ~ 1, IxIN-I 2 voi (4.48) PZ = 1, Cz = MOJNm~ 4N-I Gia su r~ng (4.49) 0, Ixl~1, u(x) ~ vk-l (x) = ~ Ck-l ( In 1+ ixi J Pk-l, Ixl ~ 1, IxlN-l 2 trong do Pk-l>Ck-lla cae h~ng s6 dtiong. Su d\lng gia thie't (G2)va (4.49), ta suy ra dug (4.50) u(x) ~ M A[ua(y)](x) Lwjn van tot nghi~p Trang 33 ~ M A[v:-1(y)](x) =M J V:-J~~I dy RN Iy - xl > M J v:-I(y) d > M J v:-I(y) d - I?' (lyl+lxl)N-1 Y - lyl~1(lyl+lxl)N-1 Y +W V:-I (y) dSr = M Jdr J (r + Ixl)N I I Iyl=r ) a Pk-I I+r ( In(- ) +w 2 dr = M OJNC:-1J r(r + Ixl)N I 1 Ta xet tru'ong hcJp Ixl~ I, ta co (4.51) ( 1+ r ) a Pk-l ( 1+ r ) a Pk-I +00 In( -) +00 In( -) J 2 J 2 dr~ dr I r(r+lxl)N-1 Ixl r(r+lxl)N-l ( 1+ Ixl J a Pk-I +00 dr ~ In(-) J ( I I) N-l 2 Ixl r r + x [ II J a Pk-I +00 d ;, In(l: x). I~r(r +:)N-I - 1 ( - 1+ x aPk-I (N -1)2N-Ilxt-1 In-fl) . Tli (4.50), (4.51), ta suy ra r~ng 0, Ixl~ 1, U(X)~Vk(X)=~ Ck ( 1+lxl ) Pk II - In- , x:2:1, IxlN-l 2 (4.52) trong do Pk>Ckla cae h~ng s6 du'dngxac dinh b~ng cae cong thu qui n(;lpnhu' sau: (4.53) Pk =apk-I' C MOJ C a k = N k I (N -1)2N-I' k = 3,4, Ta tinh fa cDng thuc hiSn cua Pk>Ck nho vao (4.48), (4.53), nhu'sau