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£' '/' ~l.({(ht Chu'dng '0 ,~, Yrl;/& (tJao ,//' c/~{t;JIc/& 36 200/ Cf9c '" (i7;' ,,//ItNt;f CY/':; Ft? , , GIAI PHUdNG TRINH LAPLACE ~, ,., , ~ TRONG MIEN NGOAI CUA QUA CAU , ' I Ma DAlJ: Xel b,li l()(1ngia lri bien Dirichlel cua phlWng ldnh Laplace d6i vdi mot mi6n D (lIdl1lien va bi ch~n): LlU(X)= 0, \iXEW\i5 HeX) lit') r 1:1ham s6lien =rex), LlXEaD t~lc tren aD -DE chC(ng minh gl)n ma khong mat llnh l6ng quc1t,trong Chl(elng chung loj xel mi6n D la qua diu deJn vi B={XER" : x < I} vdi bien la a13={ I I x E R": x < I} khong gian RII vdi n~ I I -Bai loan gia lr! bien Dirichlet mi6n ngoai se c6 nhi€u nghi~m.Trong cac Sc1chgiao khoa,ngl(oi ta thl(Ong them tlnh chat CI}th~ cua u vung vo q(c lhl mdi xac lIinh dl(l}C bi~u thac cua u.Trong chuang ,chung toi l1l(a fa ml)t dang nghiQI11u1ng qm1t dll cIll(a 1'0tinh ch~fl cua 1I0 vo c~(c -N(ji dung cua dll(ling dlCl;5C tdnh bay d~t'a theo bai ban IS] II lYIQTs6 DJNH NGHIA vA DJNH LY Dilll, if' : Xel mi2n = 13 (0,1') \ {O} = {x E nn : < x < r}, vdi r> D Khi lIl) lc1nlc.lih~ng so c c6 cae tlnh chat sall : C E (0, I) Ml)i h2\1111i6uhoa c dlCOngu xac l1inh tren Q d€u th6a \ix, Y E 13 (o,~), Ixi = Iy\=> clI(y) < u (x) Clul thich: Hang sf) c khong ph\! thuoc ham di6u boa u Clllfllg mill" - Gl)i K IA m~t ceiu ban ldnh ~ (tac K =~ S) lhl K la t~p col11pacttrong O Theo l1inh 1)/Harnack thl tl1n t~\i h~ng s6 c E (0, 1) co tinh chat J;L(1 It 'jrt~t-(&to dt;c 37 200/ C U(Y) Jfiu;}IbJt- ,9'h{Wth- % < U(X) VtJi nH}ih~lIndi~ll hoa dl((jng II xac djnh tfen Q va v6i x, Y E K Cui t lllY Y thuoc (0, 1) va u 13ham di~u hoa dlf0ng tfen n - H~l1n s(1 x -1- u (tx) cung 13 ham di~u hoa duong tren n, d6 eung tho a bat dang lh(ic tren, t(ic la CU(ly) < u(tx) i M6i x E S se tl(ong (ing nhat mot ph~n to' tx E ~ S, d6 eu (y) < u (x) - Ta thay r '\Ix, y E - S , tr '\Ix, Y E - S , ~ c6 lh~ c6 gia lrj bat k5' khoang (o,~) ne'u t dtiQC chqn thich ht.ip (0, I) Do d6 '\Ix, Y E B (o,~), I x I = I y I=>eu (y) < u (x) 2) Villh If': Gi~i sli II : Rli -1- Ria ham di~u hoa tren B\{O}={XERII : 0< Ham sf) A[ullren Vdi x Y I x < I} I B\{O} ducjc d1nh nghla nhl( sall: thuQc B\{O}, A[uJ(x) la gia tr1 trung blnh eua ham u tren m~t diu b,1n Idnh lxi, tuc la Alul(x):= twile 111-l fu(y)dS(y) mix! lyl=lxl AluJ(x) := ~ fu~xl~}lS(~) 1~I=l trung lit) w la di~n rich cua m~t du don vi Khi l16 AluJ co bi~ll thue nhu sau : Tncc'lnghejp n = 2: l Yiui /1 'Prl~t' ~o ~~c 38 200/ Alu!(x)=b(-Inlxl)+c, /1j;?J/Zn ,9'~tV/t~ 'r:i VXE B\ {O} trung d6 b, c IA h~ng so - TnCongh(Jpn > : Alul(x)=b IxI2-11+C,VXEB\{O} 11, IAhhng so c CIll£llg lIlillh Xet ham so f : (0,1) ~ R dlCU= (do u la ham di~u boa) Suy f =-~ (V'lI)(~)dS(~) ~) r()S + f (\7u)(~)dS(~) ~ =0 1'1 S f Do d() , ~,s I~ (Vu)(~)dS(~):::: f ~S l~ I (Vu)(~)dS(~) Tl'( dJng lhere li~n tren, ta suy fa 'Vr E (0, I), f~ r rS (Vu)(~)ds(~) = h~ng sO' (h~ng sO'nay c1u~:!c Qi la kl) g - 1'1:(bi~u th(fc clh 1"(r) (j tren, ta SHYra fer) = k1 rl-n, 'Vr E (0,1) Do d6 k [In r + k2 neu n:::: k Ir2-11+ k) 1'(r) ::: neu n> J Ma Alulex) = ~f(lxl), SHYra Alul (x) c6 bi~u thac nhtI' phat bi~u (rang 0) c1jnh ly 11.2 (p ,J u(h, '/:'(0 ;/(/;/t c.iJ/' l-5ao Ol9C f 40 200/ - ~ './(p';jIMb ('::5,- o;/~ !3hCVlth- f/ tt 3) Djllh Ii : Gic1slt ham so th~(eu co cae tinh eha't salt: i) Lila ham oi~u boa tren B\{O}={XE RI1:0ug lren B(D,~) am G(.>iL == inf {w(x): x E B\{O}} l'{)n lai day {ad lwng B \{O} saD cho w (ad -7L k -7 Day [a"J BallI trung t~p compact 00 B nen t6n t~1iday {bd hOi l~1v~ b E D Tn('1ug IH.fpbEB\{O}: Ilam di~u boa w d'.ll clfc li6u x == b, lheo nguyen Iy c~tcti6u lhl w la ham hang lren B\{O}, w c6 gia ld khong am trang 13(o,~) nen suy L ~ O Tn('jng IH.fp bEGB : w(b)=D nen L = D Tn(Ojugh(}pb == : w khC>ng lren B(o,f) am nen L ~ O Ta 11IC>u L ~ 0, d6 w khong am lren B\{O} d) V~y la Cl) lI(X) Ihide - - (1) cAlul(x) ~ 0, '\Ix E B \{O} 2: (ChCrng minh HeX) - Alul(x) ~ 0, '\Ix E B \ {O}) Cluj \ lll: If! clay s6 dlfC}C c1jnh nghla bai lu == c 1,==c+lu(l-c) IIlI == c + llll._1(I - c) (2) Tl( l" ==c E (0, 1) la SHYHI lJ > va tl < c + (l - c) ==1, tltc t[ E (0, I) Bang ljUY IH,'P, la c6 till E (0, I) vdi ll1l)i Il1 EN > 0, d6 day {tm}tang, d6ng thai clay hi ch~n biji I Hen hoi tl,lv~ gidi IH,ln ta gqi la ma Ta U) tlll - tlll-J == C (1 - till-I) Cho 111 )-00, ll( (5.2) SHYra I ==c + t (l - c) ;; "'/) fUll ,~jtf(/( tIll ',,/' ,)1oc 42 : !{,i(J/ -)0 (3) I m -)0 00 Xel day ham {wm} yoi Will= U -tm - /(y; ~yg1t ?-ZUltlt 'Yci =1 l v ~y '(Jew Ta lh,l'y Wm - cA\wm] =u -tm = II - A[ul ' Alul cA [u -lmAlu]! tmAIu I - cA Ill! + rImAIIII =u-lc+tm(l-c)IAlunl =U-tlllt-1A\Llml = Wm+l n'rcla WII1+1 =wn-cAlwnl - Thcl) kel qLl21 d bt(dc I thl ham di~Ll hoa Wo khong am tren B\{ ()} (I) Ap dl,lI1gkc't LlU21I) Jai vdi ham W()thl Ol((,iC ( tinh chflt khong am clla ham WI tren13\{O:, Bang Lluy n"Lp, ta suy ham Wm kh6ng am tren B\{O} vdi mQi so' nguycn dll\!ng 111 , tLtCla W11l(X) u(x) - tmAlul(x) ~ 0, \imE N ,\ixE B\{O} = Clio Il) -} 00 thl tm -)0 (uo (3)), u(x)-AllIj(X)~O, sLoe (4) \ix E 13\{0} IhiO'c : (Chang minh u Giil suy = AluJ) It'\n ti;li Xu E B\ {O} san cho u(xo) > Alu](xo) Do u lien t~IC[(.IiXonen t6n li;limOt Ian c~n V clla Xosao cho HeX)> A[u](xo) , \iXEV, M~l kllac, theo ket qua (4) d bt(dc ta co lI(X)~ Alul(xo) , \ixEoB(O,l xol) SHY gia trj trung blnh cua u tren m~t dtu 8B(0,l xol) iOn h(in A[lIl(xo), llrc la AIul(xo) > AI ul(xo) Dietl n~IYvo 19 Do lit) khong c6 Xol1aO thuQc B\{O} ma co trnh chflt U(Xo)> Alul(xo), ket h(}p ydi tlnh cha't (4) ta dt«}c HeX) = Alul(x), \ix E B\{O} I L' Lw( , 1" It (// Fltlto l:;Y~ O{tO (7i9c 43 20{)/ Jf:f~VJlb;to L9'hCl/lth- %; Ihifie : (ke'llu~n) Alul c() bi~u lhlrc nhl( lrong dinh 19 1I.2(d1lWng 4), nen u Lung y~y Tn((Jng - Ta Ct) u(x) hl}P Il co: co: 2: be-In x I) + c, \lXE B\{O} I Khi Ix I-~ I lhlu (x) -) nen suy C Khi x E 13 (0, V~y u(x) - =o r) lhlu (x) ~ 0, suy fa b ~ = b (-In I x I) ydi b ~ Tnf(ing hl.ip n > 2: Tau')u(x)co:blxI2-n+c, \Ix E 8\{0} Khi I x 1-) I lhl u (x) -) nen SHYfa c Llu lk) u(x) =b ( I co: - b, X 12-11-I) Khi x E 13 (0, r) lhlu (x) ~ 0, SHY b ~ O [J 4/ Di"" /y : GiJ sLY lulll I la Llaycae ham di~u boa lfen l~p m6 Dc U'\day hQi It.!d~u vC:h~lmso u lfen m6i l~p B (a, r) c D Khi lit')u la ham aieu boa lren D Clll?llg milllt Cui a IllY y IhuQc D T6n 1 1) L1U(x)= 0, \:Ix E RII\ B HeX) ::: rex), \:Ix E 3B d6 r lien l~IC lren bien 3B Khi d6,mqi nghi~m lren nll\ B nilL(sau x ::: I X 11 u(x) w ~ l uEC\R"\B )nC(R'\B) +- [ Ixr ] clla bAi loan d~u c6 d x~O trung B thl f au w(x) >I.J -211: ' (ta gqi gidi h',111 1£1 ) L' l(~)dS(C;;) -Sau day ta se dllYng minh w bang tren B\{O}, Xct ham g,lx)=w(x)+E(-lnl xl) IHlIl1gt;ui~u boa tren B\{O} Khi X~O lhl gix) ~oo, suy g,,(x»O nell x (hi gan O l-)~t m=inf'{ g,,(x) :xEB\{O}} (m c6 thi la -.00) T6n t~liday{adlrong B\{O}sao cho gt;(ak)~m k~oo Day {ad Ham t~p compaGt B nen t6n tO.Neli bE aB thl m=O dn go;{h)=O Neu h=O thl III;?0 g,,(x»O x gaB O.Do d6 ta luon c6 III ;:::O,suy g,,(x) khong am tren B\{O} vdi mQi E >0, d6 w(x) khtHlg [1mtren B\{OI, Trong biiu lluk cua gix) LaxcL ham di~lI boa -w thay VIw,ly lu~11lu'dng H,rla c6 ket qua -w(x) khong am trenB\{O Lt(rc fa w(x) khol1g llLrdngLren B\{OI, Ta ua ch(rng Illinh w(x) VITakhong am VITakhong dlrOng tren B\{O},l1cn C()the suy w(x) bang Lren B\{O} Dn (It>va bi6u th((c clia u LrenU,z\B la u(x) = w(x*)+h(x) = hex) ') =~ 211: j au l - lxl- - ~ f(C;;)uS(C;;) x-c;; I -Ham s6 u c6 biill th(rc d Ln3n th~t slf la nghi~m clla bai loan nell u Ihl1a de linh chftL (i),(ii),(iii).Ta Lhfty 1Ithda (i)va (ii),cho x~oo Lhl u(x) ~~ f f(~)dS(~) 211: an (j.) ( ",' '/' oLU-{?-/t Frz,/t c.~) 55 C)(tu en t?c !to{N Jt:;~0, d6 w(x) khang am lfen B\{O}, Trung hit3u Iinte cLia g~(x) ta xet ham di€u boa -w thay VI w,ly lu~n ll(ling llf Ll c() kc't L/u.i -w(x) khong am IrenB\{ },t(rc Iii w(x) khong dl(Ong Ire n B \ ( () I, 'I'a dfJ ch((ng minh w(x) vua khong am vua kh6ng dl(0ng tren B\{0 !,ncn c() the SHY w(x) bang tren B\{O}, T6m l'.li, nghi~11lclla bai loan Iren Iii dllY nhat va c6 bit3u thl'i'c tTen n2\ B la u(x) =w(x*)+h(x) = hex) =~ 2rc lxl2-1 fIx -( DB f(~)ds(~) ":> 5) Vi d{l: Xcl hai [min Dirichkt d6i vdi l1li€n ngoai clla dla Iron dl1n vi B trang n2, ~ - = 0, \Ix E n.-\ B = rex), \Ix E 8B ,lHl( x) ,u(x) u(x) ~ -111X I I kill x~ CX) ( /J ,La~i/t ," frl/t i /J ()(to ,.ij';; ,;1(9(; I' - 6.cy/,./ 'jI-a'pc-'ll :7h£Mth j/ /;; I 57 5!(){i/ U() r lien l~IClren bien aB va nghit%m UE C2 (R2\B) n C (U2\B) IHy clurng l11inh nghi~111 CUi.!bai loan la ') u(x) 1xl ] = -211: J-2f(~)dS(~) ~ 3B x I I Cluj thieh :Vi cl~l3 (Iu(x) I bi ch~n Ix I kha 16n )la lru'ong h(.jp d~c bi~l CUi.! i cll.1 (I u(x) Ico th6 lien 00 x.~ oo).Chung toi v~n tdnh bay vi v cl~\3 vll11116n phan bit%lr5 tnc()ng hc;iPu bi ch~n va khong bi ch~n CluIng lIlillh Gd Sl~btli locin co nhgit%1111a 1I Thel> dinh Iy IV 2-chlcung , nghit%m l\ c6 dO,N6u bE aB thl m=O gt;(b)=0,N6u b=O Ihl m 20 gt:(x»O x gfin O,Do d6 ta luon c6 111 O,suy gE(X) khong am Lren B\{O} vdi mqi E >0, d6 w(x) khong am Lren B\{O}, Twng Il(dng It,r la Lren Ii \ {° I hi6u LhC(cclla g,,(x) la xet h~lIn di€u boa -w thay VI w,ly lu~n kCI qua -w(x) khong am trenI3\{O},tC(cIii w(x) khong dl(Ong Cl) Ta da cIlll'ng minh w(x) vua khong am vua khong dlWng tren B\{O},ncn c() Ih6 SHY w(x) bang lren B\{O} Tl)l111',li,nghi~m dla bili Loan tren Iii nha't va c6 bitSu tllll'Clren n?\ B Iii u(x) =w(x*)+h(x) = hex) ? =~ 2IT f 'X'- -~ r(QdS(~) x-~ aB I I 6) \Ii d{l: , XCI hili loan Dirichlet l!()i vdi mi€n ngoiii clla dJa Lron ddn vi B n- ")- ,L\u(x) = 0, '\Ix E R-\ B (i) ,u(x) = rex), '\Ix E aB (ii) u(x) _ II ~ L:;t)( "' ' kl11 x~ CD ( III ) III X Lrung r lien ll,lCtren bien aB , L c6 thtS bang +CDhay -CD, Vtl nghi0m UE C2 (R2\ B) n C (Ie\B), Hay chCrng minh ke'lquLl saIl : Khi L hun tH~nIh1nghi~ m Ii}0 clIo w lllan Iuan dL((Jnghay Illan Iuan am tren ! mi~n 13(0,1').M:;tt klulc, hiim s6 di~u hoa w tri~t Lieu tren bien ClB Theo dinh 19 Il.3-chlcdng ,ham W c6 bi€ll thue tren B\{O} Hi w(x)=k(-Inlxl) '\IxeB\{O} Ll1c {~O i;t~(; !!()()/ V~y L hang A:f;t;YIJ/t !JZ,Wtli% +lfJ hay -0) lhi hai lOLin v() nghi~lll,con L hecn h! (0 ff(~)dS(~) DB w(x) -+ -UJ neu L < ~ ff(~)dS(~) DB Nhl!' Ih6 ,[(in tO cho w lu6n Juan dlfcjng hay lu6n lu6n am tren mien B(O,r), M~t klHlc, h~\ln s6 dieu hoa w tri~t lieu tren bien DB Thco d!nh Iy I1.3-dn((jng ,ham w c6 bi€u thlic lren B\{O} la ?-u II- w(x)=k( x -I), Luc LIco biEu thu'c tren If\B u(x)= ?-n x II- la w(x*)+h(x) =Ix12-11.k ~x *12-11=k(I-lxI2-U) Cho x~co \ixEB\{O} 1)+h(X) +h(x) lhl u(x) ~k+~ ff(~)ds(~) 2n DB (huu h